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General Relativity, Black Holes, And Cosmology

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General Relativity, Black Holes, and Cosmology Andrew J. S. Hamilton
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Page 1: General Relativity, Black Holes, And Cosmology

General Relativity, Black Holes, and Cosmology

Andrew J. S. Hamilton

Page 2: General Relativity, Black Holes, And Cosmology
Page 3: General Relativity, Black Holes, And Cosmology

Contents

Preface page 1

Notation 6

PART ONE SPECIAL RELATIVITY 9

Concept Questions 11

What’s important in Special Relativity 13

1 Special Relativity 14

1.1 The postulates of special relativity 14

1.2 The paradox of the constancy of the speed of light 16

1.3 Paradoxes and simultaneity 17

1.4 Time dilation 18

1.5 Lorentz transformation 20

1.6 Paradoxes: Time dilation, Lorentz contraction, and the Twin paradox 22

1.7 The spacetime wheel 23

1.8 Scalar spacetime distance 26

1.9 4-vectors 27

1.10 Energy-momentum 4-vector 28

1.11 Photon energy-momentum 30

1.12 Abstract 4-vectors 31

1.13 What things look like at relativistic speeds 32

1.14 How to programme Lorentz transformations on a computer 33

PART TWO COORDINATE APPROACH TO GENERAL RELATIVITY 35

Concept Questions 37

What’s important? 39

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iv Contents

2 Fundamentals of General Relativity 40

2.1 The postulates of General Relativity 40

2.2 Existence of locally inertial frames 41

2.3 Metric 41

2.4 Basis gµ of tangent vectors 42

2.5 4-vectors and tensors 43

2.6 Covariant derivatives 45

2.7 Coordinate 4-velocity 51

2.8 Geodesic equation 51

2.9 Coordinate 4-momentum 52

2.10 Affine parameter 52

2.11 Affine distance 53

2.12 Riemann curvature tensor 53

2.13 Symmetries of the Riemann tensor 54

2.14 Ricci tensor, Ricci scalar 55

2.15 Einstein tensor 55

2.16 Bianchi identities 55

2.17 Covariant conservation of the Einstein tensor 56

2.18 Einstein equations 56

2.19 Summary of the path from metric to the energy-momentum tensor 57

2.20 Energy-momentum tensor of an ideal fluid 57

2.21 Newtonian limit 58

3 ∗More on the coordinate approach 59

3.1 Weyl tensor 59

3.2 Evolution equations for the Weyl tensor 59

3.3 Geodesic deviation 61

3.4 Commutator of the covariant derivative revisited 62

4 ∗Action principle 65

4.1 Principle of least action for point particles 66

4.2 Action for a test particle 67

4.3 Action for a charged test particle in an electromagnetic field 68

4.4 Generalized momentum 69

4.5 Hamiltonian 69

4.6 Derivatives of the action 70

PART THREE IDEAL BLACK HOLES 71

Concept Questions 73

What’s important? 75

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Contents v

5 Observational Evidence for Black Holes 76

6 Ideal Black Holes 78

6.1 Definition of a black hole 78

6.2 Ideal black hole 78

6.3 No-hair theorem 79

7 Schwarzschild Black Hole 80

7.1 Schwarzschild metric 80

7.2 Birkhoff’s theorem 81

7.3 Stationary, static 81

7.4 Spherically symmetric 82

7.5 Horizon 83

7.6 Proper time 84

7.7 Redshift 84

7.8 Proper distance 85

7.9 “Schwarzschild singularity” 85

7.10 Embedding diagram 85

7.11 Energy-momentum tensor 86

7.12 Weyl tensor 86

7.13 Gullstrand-Painleve coordinates 86

7.14 Eddington-Finkelstein coordinates 87

7.15 Kruskal-Szekeres coordinates 88

7.16 Penrose diagrams 89

7.17 Schwarzschild white hole, wormhole 90

7.18 Collapse to a black hole 91

7.19 Killing vectors 92

7.20 Time translation symmetry 92

7.21 Spherical symmetry 92

7.22 Killing equation 93

8 Reissner-Nordstrom Black Hole 95

8.1 Reissner-Nordstrom metric 95

8.2 Energy-momentum tensor 96

8.3 Weyl tensor 96

8.4 Horizons 97

8.5 Gullstrand-Painleve metric 97

8.6 Complete Reissner-Nordstrom geometry 98

8.7 Antiverse: Reissner-Nordstrom geometry with negative mass 100

8.8 Ingoing, outgoing 100

8.9 Mass inflation instability 101

8.10 Inevitability of mass inflation 103

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vi Contents

8.11 The black hole particle accelerator 104

8.12 The X point 104

8.13 Extremal Reissner-Nordstrom geometry 105

8.14 Reissner-Nordstrom geometry with charge exceeding mass 106

8.15 Reissner-Nordstrom geometry with imaginary charge 106

9 Kerr-Newman Black Hole 109

9.1 Boyer-Lindquist metric 109

9.2 Oblate spheroidal coordinates 110

9.3 Time and rotation symmetries 110

9.4 Ring singularity 111

9.5 Horizons 111

9.6 Angular velocity of the horizon 113

9.7 Ergospheres 113

9.8 Antiverse 114

9.9 Closed timelike curves 114

9.10 Energy-momentum tensor 116

9.11 Weyl tensor 116

9.12 Electromagnetic field 116

9.13 Doran coordinates 117

9.14 Extremal Kerr-Newman geometry 117

9.15 Trajectories of test particles in the Kerr-Newman geometry 118

9.16 Penrose process 122

9.17 Constant latitude trajectories in the Kerr-Newman geometry 123

9.18 Principal null congruence 123

9.19 Circular orbits in the Kerr-Newman geometry 124

PART FOUR HOMOGENEOUS, ISOTROPIC COSMOLOGY 133

Concept Questions 135

What’s important? 137

10 Homogeneous, Isotropic Cosmology 138

10.1 Observational basis 138

10.2 Cosmological Principle 139

10.3 Friedmann-Robertson-Walker metric 140

10.4 Spatial part of the FRW metric: informal approach 140

10.5 Comoving coordinates 142

10.6 Spatial part of the FRW metric: more formal approach 143

10.7 FRW metric 144

10.8 Einstein equations for FRW metric 144

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Contents vii

10.9 Newtonian “derivation” of Friedmann equations 145

10.10 Hubble parameter 146

10.11 Critical density 147

10.12 Omega 147

10.13 Redshifting 148

10.14 Types of mass-energy 148

10.15 Evolution of the cosmic scale factor 149

10.16 Conformal time 151

10.17 Looking back along the lightcone 151

10.18 Horizon 152

PART FIVE TETRAD APPROACH TO GENERAL RELATIVITY 157

Concept Questions 159

What’s important? 161

11 The tetrad formalism 162

11.1 Tetrad 162

11.2 Vierbein 162

11.3 The metric encodes the vierbein 163

11.4 Tetrad transformations 164

11.5 Tetrad Tensor 165

11.6 Raising and lowering indices 165

11.7 Gauge transformations 165

11.8 Directed derivatives 166

11.9 Tetrad covariant derivative 166

11.10 Relation between tetrad and coordinate connections 168

11.11 Torsion tensor 168

11.12 No-torsion condition 168

11.13 Antisymmetry of the connection coefficients 169

11.14 Connection coefficients in terms of the vierbein 169

11.15 Riemann curvature tensor 170

11.16 Ricci, Einstein, Bianchi 171

11.17 Electromagnetism 171

12 ∗More on the tetrad formalism 174

12.1 Spinor tetrad formalism 174

12.2 Newman-Penrose tetrad formalism 177

12.3 Electromagnetic field tensor 179

12.4 Weyl tensor 183

12.5 Petrov classification of the Weyl tensor 186

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12.6 Raychaudhuri equations and the Sachs optical scalars 187

12.7 Focussing theorem 189

13 ∗The 3+1 (ADM) formalism 191

13.1 ADM tetrad 192

13.2 Traditional ADM approach 193

13.3 Spatial tetrad vectors and tensors 194

13.4 ADM connections, gravity, and extrinsic curvature 194

13.5 ADM Riemann, Ricci, and Einstein tensors 195

13.6 ADM action 196

13.7 ADM equations of motion 199

13.8 Constraints and energy-momentum conservation 200

14 ∗The geometric algebra 201

14.1 Products of vectors 202

14.2 Geometric product 203

14.3 Reverse 204

14.4 The pseudoscalar and the Hodge dual 205

14.5 Reflection 206

14.6 Rotation 207

14.7 A rotor is a spin- 12 object 209

14.8 A multivector rotation is an active rotation 210

14.9 2D rotations and complex numbers 210

14.10 Quaternions 212

14.11 3D rotations and quaternions 213

14.12 Pauli matrices 215

14.13 Pauli spinors 216

14.14 Pauli spinors as scaled 3D rotors, or quaternions 218

14.15 Spacetime algebra 219

14.16 Complex quaternions 221

14.17 Lorentz transformations and complex quaternions 223

14.18 Spatial Inversion (P ) and Time Inversion (T ) 224

14.19 Electromagnetic field bivector 225

14.20 How to implement Lorentz transformations on a computer 225

14.21 Dirac matrices 229

14.22 Dirac spinors 231

14.23 Dirac spinors as complex quaternions 232

14.24 Non-null Dirac spinor — particle and antiparticle 235

14.25 Null Dirac Spinor 236

14.26 Chiral decomposition of a Dirac spinor 237

14.27 Dirac equation 238

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Contents ix

14.28 Antiparticles are negative mass particles moving backwards in time 239

14.29 Dirac equation with electromagnetism 240

14.30 CPT 240

14.31 Charge conjugation C 241

14.32 Parity reversal P 242

14.33 Time reversal T 243

14.34 Majorana spinor 243

14.35 Covariant derivatives revisited 243

14.36 General relativistic Dirac equation 244

14.37 3D Vectors as rank-2 spinors 244

PART SIX BLACK HOLE INTERIORS 247

Concept Questions 249

What’s important? 250

15 Black hole waterfalls 251

15.1 Tetrads move through coordinates 251

15.2 Gullstrand-Painleve waterfall 252

15.3 Boyer-Lindquist tetrad 258

15.4 Doran waterfall 259

16 General spherically symmetric spacetime 265

16.1 Spherical spacetime 265

16.2 Spherical electromagnetic field 276

16.3 General relativistic stellar structure 277

16.4 Self-similar spherically symmetric spacetime 278

17 The interiors of spherical black holes 290

17.1 The mechanism of mass inflation 290

17.2 The far future? 293

17.3 Self-similar models of the interior structure of black holes 295

17.4 Instability at outer horizon? 310

PART SEVEN GENERAL RELATIVISTIC PERTURBATION THEORY 311

Concept Questions 313

What’s important? 315

18 Perturbations and gauge transformations 316

18.1 Notation for perturbations 316

18.2 Vierbein perturbation 316

18.3 Gauge transformations 317

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18.4 Tetrad metric assumed constant 317

18.5 Perturbed coordinate metric 317

18.6 Tetrad gauge transformations 318

18.7 Coordinate gauge transformations 319

18.8 Coordinate gauge transformation of a coordinate scalar 319

18.9 Coordinate gauge transformation of a coordinate vector or tensor 320

18.10 Coordinate gauge transformation of a tetrad vector 320

18.11 Coordinate gauge transformation of the vierbein 321

18.12 Coordinate gauge transformation of the metric 321

18.13 Lie derivative 322

19 Scalar, vector, tensor decomposition 323

19.1 Decomposition of a vector in flat 3D space 323

19.2 Fourier version of the decomposition of a vector in flat 3D space 324

19.3 Decomposition of a tensor in flat 3D space 325

20 Flat space background 326

20.1 Classification of vierbein perturbations 326

20.2 Metric, tetrad connections, and Einstein and Weyl tensors 328

20.3 Spinor components of the Einstein tensor 330

20.4 Too many Einstein equations? 331

20.5 Action at a distance? 332

20.6 Comparison to electromagnetism 333

20.7 Harmonic gauge 337

20.8 What is the gravitational field? 338

20.9 Newtonian (Copernican) gauge 338

20.10 Synchronous gauge 339

20.11 Newtonian potential 341

20.12 Dragging of inertial frames 342

20.13 Quadrupole pressure 343

20.14 Gravitational waves 344

20.15 Energy-momentum carried by gravitational waves 347

PART EIGHT COSMOLOGICAL PERTURBATIONS 349

Concept Questions 351

21 An overview of cosmological perturbations 352

22 ∗Cosmological perturbations in a flat Friedmann-Robertson-Walker background 357

22.1 Unperturbed line-element 357

22.2 Comoving Fourier modes 358

22.3 Classification of vierbein perturbations 358

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22.4 Metric, tetrad connections, and Einstein tensor 360

22.5 ADM gauge choices 362

22.6 Conformal Newtonian gauge 362

22.7 Synchronous gauge 363

23 Cosmological perturbations: a simplest set of assumptions 364

23.1 Perturbed FRW line-element 364

23.2 Energy-momenta of ideal fluids 364

23.3 Diffusive damping 367

23.4 Equations for the simplest set of assumptions 368

23.5 Unperturbed background 370

23.6 Generic behaviour of non-baryonic cold dark matter 371

23.7 Generic behaviour of radiation 372

23.8 Regimes 373

23.9 Superhorizon scales 373

23.10 Radiation-dominated, adiabatic initial conditions 376

23.11 Radiation-dominated, isocurvature initial conditions 379

23.12 Subhorizon scales 380

23.13 Matter-dominated 381

23.14 Recombination 382

23.15 Post-recombination 383

23.16 Matter with dark energy 384

23.17 Matter with dark energy and curvature 385

24 ∗Cosmological perturbations: a more careful treatment of photons and baryons 387

24.1 Lorentz-invariant spatial and momentum volume elements 388

24.2 Occupation numbers 388

24.3 Occupation numbers in thermodynamic equilibrium 389

24.4 Boltzmann equation 390

24.5 Non-baryonic cold dark matter 392

24.6 The left hand side of the Boltzmann equation for photons 394

24.7 Spherical harmonics of the photon distribution 395

24.8 Energy-momentum tensor for photons 396

24.9 Collisions 396

24.10 Electron-photon scattering 398

24.11 The photon collision term for electron-photon scattering 399

24.12 Boltzmann equation for photons 402

24.13 Diffusive (Silk) damping 403

24.14 Baryons 403

24.15 Viscous baryon drag damping 405

24.16 Photon-baryon wave equation 405

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xii Contents

24.17 Damping of photon-baryon sound waves 407

24.18 Ionization and recombination 409

24.19 Neutrinos 409

24.20 Summary of equations 409

24.21 Legendre polynomials 410

25 Fluctuations in the Cosmic Microwave Background 411

25.1 Primordial power spectrum 411

25.2 Normalization of the power spectrum 412

25.3 CMB power spectrum 412

25.4 Matter power spectrum 413

25.5 Radiative transfer of CMB photons 413

25.6 Integrals over spherical Bessel functions 415

25.7 Large-scale CMB fluctuations 417

25.8 Monopole, dipole, and quadrupole contributions to Cℓ 418

25.9 Integrated Sachs-Wolfe (ISW) effect 418

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Preface

Illusory preface

As of writing (May 2010), this book is incomplete. If you happen to discover this draft on the internet, you

are welcome to it (and especially welcome to send me helpful advice and criticism). This book has been

written during two semesters of teaching graduate general relativity at the University of Colorado, Boulder,

during Spring 2008 and 2010. I hope to complete the book the next time I teach the course, which could

possibly be Spring 2012.

Meanwhile, I am vividly aware of the book’s shortcomings. The book is incomplete in many parts, and

needs pruning in others. If the early chapters read more like notes than a book, that is true; I was some

way into writing before I realised that a book was taking shape. Especially, the book is missing many

planned figures. Many of the anticipated figures can be found at three websites: “Special Relativity”

(http://casa.colorado.edu/~ajsh/sr/sr.shtml), “Falling into a Black Hole” (http://casa.colorado.

edu/~ajsh/schw.shtml), and “Inside Black Holes” (http://jila.colorado.edu/~ajsh/insidebh/index.

html).

Although the book is incomplete, I have tried hard to keep mathematical errors from creeping in. If you

find an error — especially in a minus sign or a factor — please let me know.

True preface

This book is driven by one overriding question: “What do students want?”

A fundamental premise of this book is that, in the field of general relativity, the number one thing, by

far, that students want to learn about is black holes. And the second thing they want to learn about is the

cosmic microwave background.

This book is born in part out of frustration with the relentlessly left-brained character of so many texts

on general relativity. I’m probably one of those left-brained characters myself. However, my experience in

general relativistic visualization has convinced me that those interested in black holes from a mathematical

perspective are vastly outnumbered by those fascinated by black holes for other reasons. Among those

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2 Preface

reasons are that black holes are, like dinosaurs, awesomely powerful, and supremely mysterious. I worry that

general relativity is (as I hear from students) often taught as if it were little more than tensor calculus. I

fear that the abstract approach repels and culls our right-brained students until only the most left-leaning

of our students remain to trasmit left-brained relativity to the next generation.

Although I was fascinated by general relativity already as a graduate student, my active involvement with

relativity was stimulated by students who insisted that I teach it. From the beginning it seemed obvious

that the way to teach relativity was through visualization. Thus, through teaching, I began to do general

relativistic visualizations of black holes. Initially the visualizations were simple animations, which I put

together into a website “Falling into a Black Hole” in 1997 and 1998. The visualizations touched a chord

with the outside world. In 2001/2 I had the privelege of spending a year’s sabbatical with the Denver

Museum of Nature and Science, where I began developing the Black Hole Flight Simulator (BHFS). That

sabbatical eventually led to a large-format immersive digital dome show “Black Holes: The Other Side of

Infinity,” produced at the DMNS and directed by Tom Lucas. Premiering in 2006, that dome show has

been distributed to some 40 digital domes worldwide. Since that time visualizations with the BHFS have

appeared in several TV documentaries and in a number of exhibits. The experience of working with non-

science professionals has been, and continues to be, intensely enjoyable, and has sensitized me to the insidious

cultural chasms that divide us in an increasingly specialized society.

These experiences have left a prominent dent in my thinking. For example, I think that the highlight of

special relativity is the question of what you see and experience when you pass through a scene at near the

speed of light. Yet most texts scarcely mention the subject, if at all. Similarly, I think that the highlight

of black holes is what (general relativity predicts) you see and experience when you fall into a black hole.

Again, most texts hardly address the issue. Texts often do mention Penrose-Hawking singularity theorems.

Yet few texts mention the all-important mass inflation instability discovered by E. Poisson & W. Israel

(1990). The inflationary instability probably plays the central role in determining the interior structure of

astronomically realistic black holes, and in particular in cutting off the wormhole and white hole connections

to other universes that exist in the ideal Kerr geometry of a rotating black hole. Even E. Poisson (1994) “A

Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics” mentions the inflationary instability only

in the problems at the end of the last chapter. An important goal of this book is to redress this hole in the

teaching of general relativity.

The second focus of this book, after black holes, is the Cosmic Microwave Background (CMB). The CMB

offers a profound window on the genesis of our Universe. Observations of fluctuations in the CMB are in

astonishing agreement with the predictions of general relativistic perturbation theory coupled with some

well-understood physics and some less well-understood but neverthess successful ideas about inflation. For

this book, the goal I set myself was to attempt the simplest possible treatment of CMB fluctuations that

would yield a result that could be compared to observation. This is not an easy goal, since calculation of

CMB fluctuations presents many technical challenges. I applaud recent texts such as M. P. Hobson, G.

P. Efstathiou, & A. N. Lasenby (2006) “General Relativity: An Introduction for Physicists,” and T. Pad-

manabhan (2010) “Gravitation: Foundations and Frontiers,” which include chapters on the CMB power

spectrum. General texts like Hobson et al., Padmanabhan, and the present book by no means replace spe-

cialized books on the Cosmology and the CMB, such as S. Dodelson (2002) “Modern Cosmology,” R. Durrer

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Preface 3

(2008) “The Cosmic Microwave Background,” or D. H. Lyth & A. R. Liddle (2009) “The Primordial Density

Perturbation.” However, for many students a course on general relativity may be the only opportunity they

get to learn about the CMB, and I think that a modern course on general relativity should include a basic

introduction to the CMB.

Notwithstanding its intended focus on applications rather than mathematics, this is not an easy book.

It is a serious graduate-level text. R. M. Wald (2006 “Teaching the mathematics of general relativity”,

Am. J. Phys. 74, 471–477 http://arxiv.org/abs/gr-qc/0511073) describes the challenges of teaching the

necessary mathematics in a course on general relativity. This book will not make climbing the mathematical

mountain of general relativity any easier. But the intention is that this book will help you get a clear view

from the top, and not abandon you in fog. While the book does not shirk mathematics, I have tried hard to

make the logic and derivations as clear and tight as possible.

So much for the overall goals of this book. What about the strategy to achieve those goals? Firstly, this

book is intended as a book from which a student can learn, and a lecturer can teach. It is not intended as

a reference book. In a learning/teaching book, one must choose carefully not only what to include, but also

what not to include, because the latter distracts and dilutes.

The grand strategy is to go through general relativity in two passes. In the first pass, the aim is to

run through the foundations of general relativity, and to get to ideal black holes as quickly as possible.

In the second pass, the book essentially starts all over again, using a tetrad-based approach rather than a

coordinate-based approach. The tetrad approach provides the basis for the subsequent treatment of non-ideal

black holes, and of the cosmic microwave background.

The emphasis of (the second half of) this book on tetrads is unusual for a textbook, but consistent with

its right-brained emphasis. If you want to see what’s happening in a spacetime, then you need to look

at it with respect to the frame of an observer, which means working in a locally inertial (orthonormal)

frame. The problem with coordinate frames is that they prescribe that the axes of the spacetime are the

tangent vectors to the coordinates. These tangent vectors are skewed, not orthonormal. Looking at things

in a coordinate frame is like looking at a scene with eyes crossed. Even with geometries as simple as the

Friedmann-Robertson-Walker geometry of homogeneous, isotropic cosmology, it is necessary to play games

to see plainly what its energy-momentum is (for FRW, raise one of the indices on the energy-momentum

tensor — but that trick fails in more complicated spacetimes). Tetrads obviate the need to waste time

attempting to conceptualize the distinction between vectors and covectors (one-forms). My own suspicion

is that the locally flat structure of general relativity may be more fundamental than its globally geometric

character, which could be an emergent phenomenon.

A virtue of the two-pass approach is that the student gets to revisit the fundamentals of general relativity

from two similar but not identical perspectives. This reaffirmation of fundamentals is especially important

given the fast pace and stripped-down coverage.

The course that I teach to senior physics undergraduates and beginning graduate students at the University

of Colorado covers the following 8 topics, each topic taking about 2 weeks during the 16-week semester:

1. Pass 1.

a. Chapter 1: Special relativity.

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4 Preface

b. Chapter 2: Coordinate approach to general relativity.

c. Chapters 5–9: Ideal black holes, namely Schwarzschild, Reissner-Nordstrom, and Kerr-Newman.

d. Chapter 10: Homogeneous, isotropic cosmology.

2. Pass 2.

a. Chapter 11: Tetrad approach to general relativity.

b. Chapters 15–17: Black hole interiors.

c. Chapters 18–20: General relativistic perturbation theory.

d. Chapters 21, 23, and 25: Cosmological perturbations.

The first of the eight topics is special relativity. Special relativity is an essential precursor to general

relativity, since a fundamental postulate of general relativity is the Principle of Equivalence, which asserts

that at any point there exist frames, called locally inertial, or free-fall, with respect to which special relativity

operates locally. The strategy of Chapter 1 on Special Relativity is first to confront the paradox of the

constancy of the speed of light, and from there to proceed rapidly to the highlight of special relativity, the

question of what you see and experience when you pass through a scene at near the speed of light. I choose

not to pause to discuss electromagnetism, actions, or other important topics in special relativity, since that

would get in the way of the driving goal, to head to a black hole at the fastest possible pace1.

The second of the eight topics is what I call the coordinate approach to general relativity. This is a lightning

introduction to the fundamental ingredients of general relativity, from the metric through to the Einstein

tensor, using the traditional coordinate-based approach, where components of tensors are expressed relative

to a basis of coordinate tangent vectors. To make the material more accessible, and to lay the groundwork

for tetrads, the book builds on concepts of vectors familiar from high school, and avoids unncecessary

mathematical distractions, such as emphasizing the distinction between vectors and 1-forms. Typically,

texts go through this material at a more leisurely pace, taking time to convey challenging conceptual issues.

Here however I choose not to linger, for two reasons. The first is the obvious one: the goal is to get to black

holes post-haste. The second reason is that, as mentioned earlier in this preface, looking at tensors in a

coordinate basis is like looking at the world with eyes crossed. As my mother used to say, “If you do that

and the wind changes, you’ll be stuck like that forever.”

The third topic is ideal black holes, and here the pace slows. An ideal black hole is one that is stationary

(time translation invariant), and empty outside its singularity, except for the contribution of a static electric

field. In the 4 dimensions of the spacetime we live in, ideal black holes come in just a few varieties: the

Schwarzschild geometry for a spherical, uncharged black hole, the Reissner-Nordstrom for a spherical, charged

black hole, and the Kerr-Newman geometry for a rotating, charged black hole.

The fourth topic is homogeneous, isotropic cosmology, the Friedmann-Robertson-Walker (FRW) geometry.

The FRW geometry forms the essential background spacetime for the cosmological perturbation theory to

be encountered later.

The book now enters the second pass. Tetrads — systems of locally inertial (or other) frames attached to

1 A strategy that might fail for students like myself. I learned special relativity from L. D. Landau & E. M. Lifshitz’sincomparable “The Classical Theory of Fields.” I recall vividly the extraordinary delight in discovering a text that, incontrast to those dreadful books that conveyed the idea that electromagnetism was something to do with resistors andcapacitors, put relativity, Maxwell’s equations, and actions up front.

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Preface 5

each point of spacetime — are well known to, and widely used by, general relativists. The tetrad approach

to general relatiivity is more complicated than the coordinate approach in that it requires an additional

superstructure. However, the advantage of being able to see straight, because you are working in an or-

thonormal frame, outweighs the disadvantage of the additional overhead. While the coordinate approach is

adequate for simple spacetimes — ideal black holes, and the FRW geometry — its defects are a barrier to

understanding more complicated spacetimes. Most texts do not cover tetrads, or cover them as an aside. In

this book, tetrads are developed systematically, in one self-contained chapter.

The sixth topic is black hole interiors. Cool, but needs more work.

The seventh topic is general relativistic perturbation theory, some understanding of which is prerequisite

for dealing with cosmological perturbations and the CMB. The approach starts — in a thankfully short

chapter — with one of the most difficult aspects of general relativistic perturbation theory, namely the

problem of coordinate and tetrad gauge ambiguities. This might seem a peculiar starting point. A more

typical starting point is to vary the metric, pick a gauge, and lo there are waves. However, I think that it

is important to show how, at least in flat or FRW background spacetimes, the real physical perturbations

emerge naturally from the formalism, without having to pick a gauge. In flat spacetime, the formalism

picks out one particular gauge, the Newtonian gauge (though I think it should be called the Copernican

gauge, because in the solar system it would pick out a Sun-centered almost-Cartesian frame), in which the

perturbations retained are precisely the physical perturbations and no others. The Newtonian/Copernican

gauge provides the natural arena for elucidating general relativistic phenomena such as the dragging of

inertial frames, and gravitational waves.

The eighth and final topic is cosmological perturbation theory, emphasizing the calculation of fluctuations

in the CMB. This was one of the most challenging parts of the book to write, because of the shear volume of

physics that goes into the calculation. I did my best to condense the core of the calculation into “a simplest

set of assumptions,” Chapter 23. However, if you want to calculate a CMB power spectrum that you can

actually compare to observations, then you’ll have to go beyond the “simple” chapter. Subsequent chapters

will help you do that.

Beyond the eight topics described above, there are several chapters, all starred2, that contain relevant,

but lower priority, material. The material contains some fun stuff, my favourite being “How to implement

Lorentz transformations on a computer,” §14.20.

2 The idea of starred chapters comes from Steven L. Weinberg’s classic 1972 text “Gravitation and Cosmology,” from which Ilearned general relativity. Curiously, I found his starred chapters often more interesting than the unstarred ones.

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Notation

Except where actual units are needed, units are such that the speed of light is one, c = 1, and Newton’s

gravitational constant is one, G = 1.

The metric signature is −+++.

Greek (brown) letters α, β, ..., denote dummy 4D coordinate indices. Latin (black) letters a, b, ..., denote

dummy 4D tetrad indices. Mid-alphabet Latin letters i, j, ... denote 3D indices, either coordinate (brown)

or tetrad (black). To avoid distraction, colouring is applied only to coordinate indices, not to the coordinates

themselves.

Specific (non-dummy) components of a vector are labelled by the corresponding coordinate (brown) or

tetrad (black) direction, for example Aµ = At, Ax, Ay, Az or Am = At, Ax, Ay, Az. Allowing the same

label to denote either a coordinate or a tetrad index risks ambiguity, but it should apparent from the context

what is meant. Some texts distinguish coordinate and tetrad indices for example by a caret on the latter,

but this produces notational overload.

Boldface denotes abstract vectors, in either 3D or 4D. In 4D, A = Aµgµ = Amγγm, where gµ denote

coordinate tangent axes, and γγm denote tetrad axes.

Repeated paired dummy indices are summed over, the implicit summation convention. In special and

general relativity, one index of a pair must be up (contravariant), while the other must be down (covariant).

If the space being considered is Euclidean, then both indices may be down.

∂/∂xµ denotes coordinate partial derivatives, which commute. ∂m denotes tetrad directed derivatives,

which do not commute. Dµ and Dm denote respectively coordinate-frame and tetrad-frame covariant deriva-

tives.

Choice of metric signature

There is a tendency, by no means unanimous, for general relativists to prefer the −+++ metric signature,

while particle physicists prefer +−−−.

For someone like me who does general relativistic visualization, there is no contest: the choice has to be

−+++, so that signs remain consistent between 3D spatial vectors and 4D spacetime vectors. For example,

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Notation 7

the 3D industry knows well that quaternions provide the most efficient and powerful way to implement

spatial rotations. As shown in Chapter 14, complex quaternions provide the best way to implement Lorentz

transformations, with the subgroup of real quaternions continuing to provide spatial rotations. Compatibility

requires −+++. Actually, OpenGL and other graphics languages put spatial coordinates in the first three

indices, leaving time to occupy the fourth index; but in these notes I stick to the physics convention of

putting time in the zeroth index.

In practical calculations it is convenient to be able to switch transparently between boldface and index

notation in both 3D and 4D contexts. This is where the +−−− signature poses greater potential for

misinterpretation in 3D. For example, with this signature, what is the sign of the 3D scalar product

a · b ? (0.1)

Is it a · b =∑3

i=1 aibi or a · b =

∑3i=1 a

ibi? To be consistent with common 3D usage, it must be the

latter. With the +−−− signature, it must be that a · b = −aibi, where the repeated indices signify implicit

summation over spatial indices. So you have to remember to introduce a minus sign in switching between

boldface and index notation.

As another example, what is the sign of the 3D vector product

a× b ? (0.2)

Is it a×b =∑3

jk=1 εijkajbk or a×b =

∑3jk=1 ε

ijka

jbk or a×b =∑3

jk=1 εijkajbk? Well, if you want to switch

transparently between boldface and index notation, and you decide that you want boldface consistently to

signify a vector with a raised index, then maybe you’d choose the middle option. To be consistent with

standard 3D convention for the sign of the vector product, maybe you’d choose εijk to have positive sign for

ijk an even permutation of xyz.

Finally, what is the sign of the 3D spatial gradient operator

∇ ≡ ∂

∂x? (0.3)

Is it ∇ = ∂/∂xi or ∇ = ∂/∂xi? Convention dictates the former, in which case it must be that some boldface

3D vectors must signify a vector with a raised index, and others a vector with a lowered index. Oh dear.

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Page 21: General Relativity, Black Holes, And Cosmology

PART ONE

SPECIAL RELATIVITY

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Concept Questions

1. What does c = universal constant mean? What is speed? What is distance? What is time?

2. c+ c = c. How can that be possible?

3. The first postulate of special relativity asserts that spacetime forms a 4-dimensional continuum. The fourth

postulate of special relativity asserts that spacetime has no absolute existence. Isn’t that a contradiction?

4. The principle of special relativity says that there is no absolute spacetime, no absolute frame of reference

with respect to which position and velocity are defined. Yet does not the cosmic microwave background

define such a frame of reference?

5. How can two people moving relative to each other at near c both think each other’s clock runs slow?

6. How can two people moving relative to each other at near c both think the other is Lorentz-contracted?

7. All paradoxes in special relativity have the same solution. In one word, what is that solution?

8. All conceptual paradoxes in special relativity can be understood by drawing what kind of diagram?

9. Your twin takes a trip to α Cen at near c, then returns to Earth at near c. Meeting your twin, you see

that the twin has aged less than you. But from your twin’s perspective, it was you that receded at near

c, then returned at near c, so your twin thinks you aged less. Is it true?

10. Blobs in the jet of the galaxy M87 have been tracked by the Hubble Space Telescope to be moving at

about 6c. Does this violate special relativity?

11. If you watch an object move at near c, does it actually appear Lorentz-contracted? Explain.

12. You speed towards the center of our Galaxy, the Milky Way, at near c. Does the center appear to you

closer or farther away?

13. You go on a trip to the center of the Milky Way, 30,000 lightyears distant, at near c. How long does the

trip take you?

14. You surf a light ray from a distant quasar to Earth. How much time does the trip take, from your

perspective?

15. If light is a wave, what is waving?

16. As you surf the light ray, how fast does it appear to vibrate?

17. How does the phase of a light ray vary along the light ray? Draw surfaces of constant phase on a spacetime

diagram.

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12 Concept Questions

18. You see a distant galaxy at a redshift of z = 1. If you could see a clock on the galaxy, how fast would the

clock appear to tick? Could this be tested observationally?

19. You take a trip to α Cen at near c, then instantaneously accelerate to return at near c. If you are looking

through a telescope at a clock on the Earth while you instantaneously accelerate, what do you see happen

to the clock?

20. In what sense is time an imaginary spatial dimension?

21. In what sense is a Lorentz boost a rotation by an imaginary angle?

22. You know what it means for an object to be rotating at constant angular velocity. What does it mean for

an object to be boosting at a constant rate?

23. A wheel is spinning so that its rim is moving at near c. The rim is Lorentz-contracted, but the spokes are

not. How can that be?

24. You watch a wheel rotate at near the speed of light. The spokes appear bent. How can that be?

25. Does a sunbeam appear straight or bent when you pass by it at near the speed of light?

26. Energy and momentum are unified in special relativity. Explain.

27. In what sense is mass equivalent to energy in special relativity? In what sense is mass different from

energy?

28. Why is the Minkowski metric unchanged by a Lorentz transformation?

29. What is the best way to program Lorentz transformations on a computer?

Page 25: General Relativity, Black Holes, And Cosmology

What’s important in Special Relativity

See

http://casa.colorado.edu/~ajsh/sr/

1. Postulates of special relativity.

2. Understanding conceptually the unification of space and time implied by special relativity.

a. Spacetime diagrams.

b. Simultaneity.

c. Understanding the paradoxes of relativity — time dilation, Lorentz contraction, the twin paradox.

3. The mathematics of spacetime transformations

a. Lorentz transformations.

b. Invariant spacetime distance.

c. Minkowski metric.

d. 4-vectors.

e. Energy-momentum 4-vector. E = mc2.

f. The energy-momentum 4-vector of massless particles, such as photons.

4. What things look like at relativistic speeds.

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1

Special Relativity

1.1 The postulates of special relativity

The theory of special relativity can be derived formally from a small number of postulates:

1. Space and time form a 4-dimensional continuum:

2. The existence of locally inertial frames;

3. The speed of light is constant;

4. The principle of special relativity.

The first two postulates are assertions about the structure of spacetime, while the last two postulates form

the heart of special relativity. Most books mention just the last two postulates, but I think it is important

to know that special (and general) relativity simply postulate the 4-dimensional character of spacetime, and

that special relativity postulates moreover that spacetime is flat.

1. Space and time form a 4-dimensional continuum. The correct mathematical word for continuum

is manifold. A 4-dimensional manifold is defined mathematically to be a topological space that is locally

homeomorphic to Euclidean 4-space R4.

The postulate that spacetime forms a 4-dimensional continuum is a generalization of the classical Galilean

concept that space and time form separate 3 and 1 dimensional continua. The postulate of a 4-dimensional

spacetime continuum is retained in general relativity.

Physicists widely believe that this postulate must ultimately break down, that space and time are quantized

over extremely small intervals of space and time, the Planck length√

G~/c3 ≈ 10−35 m, and the Planck time√

G~/c5 ≈ 10−43 s, where G is Newton’s gravitational constant, ~ ≡ h/(2π) is Planck’s constant divided by

2π, and c is the speed of light.

2. The existence of globally inertial frames. Statement: “There exist global spacetime frames with

respect to which unaccelerated objects move in straight lines at constant velocity.”

A spacetime frame is a system of coordinates for labelling space and time. Four coordinates are needed,

because spacetime is 4-dimensional. A frame in which unaccelerated objects move in straight lines at constant

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1.1 The postulates of special relativity 15

velocity is called an inertial frame. One can easily think of non-inertial frames: a rotating frame, an

accelerating frame, or simply a frame with some bizarre Dahlian labelling of coordinates.

A globally inertial frame is an inertial frame that covers all of space and time. The postulate that

globally inertial frames exist is carried over from classical mechanics (Newton’s first law of motion).

Notice the subtle shift from the Newtonian perspective. The postulate is not that particles move in straight

lines, but rather that there exist spacetime frames with respect to which particles move in straight lines.

Implicit in the assumption of the existence of globally inertial frames is the assumption that the geometry

of spacetime is flat, the geometry of Euclid, where parallel lines remain parallel to infinity. In general

relativity, this postulate is replaced by the weaker postulate that local (not global) inertial frames exist. A

locally inertial frame is one which is inertial in a “small neighbourhood” of a spacetime point. In general

relativity, spacetime can be curved.

3. The speed of light is constant. Statement: “The speed of light c is a universal constant, the same in

any inertial frame.”

This postulate is the nub of special relativity. The immediate challenge of this chapter, §1.2, is to confront

its paradoxical implications, and to resolve them.

Measuring speed requires being able to measure intervals of both space and time: speed is distance travelled

divided by time elapsed. Inertial frames constitute a special class of spacetime coordinate systems; it is with

respect to distance and time intervals in these special frames that the speed of light is asserted to be constant.

In general relativity, arbitrarily weird coordinate systems are allowed, and light need move neither in

straight lines nor at constant velocity with respect to bizarre coordinates (why should it, if the labelling

of space and time is totally arbitrary?). However, general relativity asserts the existence of locally inertial

frames, and the speed of light is a universal constant in those frames.

In 1983, the General Conference on Weights and Measures officially defined the speed of light to be

c ≡ 299,792,458 m s−1, (1.1)

and the meter, instead of being a primary measure, became a secondary quantity, defined in terms of the

second and the speed of light.

4. The principle of special relativity. Statement: “The laws of physics are the same in any inertial

frame, regardless of position or velocity.”

Physically, this means that there is no absolute spacetime, no absolute frame of reference with respect to

which position and velocity are defined. Only relative positions and velocities between objects are meaningful.

It is to be noted that the principle of special relativity does not imply the constancy of the speed of light,

although the postulates are consistent with each other. Moreover the constancy of the speed of light does

not imply the Principle of Special Relativity, although for Einstein the former appears to have been the

inspiration for the latter.

An example of the application of the principle of special relativity is the construction of the energy-

momentum 4-vector of a particle, which should have the same form in any inertial frame (§1.10).

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16 Special Relativity

1.2 The paradox of the constancy of the speed of light

The postulate that the speed of light is the same in any inertial frame leads immediately to a paradox.

Resolution of this paradox compels a revolution in which space and time are united from separate 3 and

1-dimensional continua into a single 4-dimensional continuum.

Here, Figure ??, is Vermilion. She emits a flash of light. Vermilion thinks that the light moves outward

at the same speed in all directions. So Vermilion thinks that she is at the centre of the expanding sphere of

light.

But here also, Figure ??, is Cerulean, moving away from Vermilion, at about 12 the speed of light. Vermilion

thinks that she is at the centre of the expanding sphere of light, as before. But, says special relativity,

Cerulean also thinks that the light moves outward at the same speed in all directions from him. So Cerulean

should be at the centre of the expanding light sphere too. But he’s not, is he. Paradox!

Concept question 1.1 Would the light have expanded differently if Cerulean had emitted the light?

1.2.1 Challenge

Can you figure out Einstein’s solution to the paradox? Somehow you have to arrange that both Vermilion

and Cerulean regard themselves as being in the centre of the expanding sphere of light.

1.2.2 Spacetime diagram

A spacetime diagram suggests a way of thinking which leads to the solution of the paradox of the constancy

of the speed of light. Indeed, spacetime diagrams provide the way to resolve all conceptual paradoxes in

special relativity, so it is thoroughly worthwhile to understand them.

A spacetime diagram, Figure ??, is a diagram in which the vertical axis represents time, while the

horizontal axis represents space. Really there are three dimensions of space, which can be thought of as

filling additional horizontal dimensions. But for simplicity a spacetime diagram usually shows just one

spatial dimension.

In a spacetime diagram, the units of space and time are chosen so that light goes one unit of distance

in one unit of time, i.e. the units are such that the speed of light is one, c = 1. Thus light always moves

upward at 45 from vertical in a spacetime diagram. Each point in 4-dimensional spacetime is called an

event. Light signals converging to or expanding from an event follow a 3-dimensional hypersurface called

the lightcone. Light converging on to an event in on the past lightcone, while light emerging from an

event is on the future lightcone.

Here is a spacetime diagram of Vermilion emitting a flash of light, and Cerulean moving relative to

Vermilion at about 12 the speed of light. This is a spacetime diagram version of the situation illustrated

in Figure ??. The lines along which Vermilion and Cerulean move through spacetime are called their

worldlines.

Consider again the challenge problem. The problem is to arrange that both Vermilion and Cerulean are

at the centre of the lightcone, from their own points of view.

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1.3 Paradoxes and simultaneity 17

Here’s a clue. Cerulean’s concept of space and time may not be the same as Vermilion’s.

1.2.3 Centre of the lightcone

Einstein’s solution to the paradox is that Cerulean’s spacetime is skewed compared to Vermilion’s, as illus-

trated by Figure ??. The thing to notice in the diagram is that Cerulean is in the centre of the lightcone,

according to the way Cerulean perceives space and time. Vermilion remains at the centre of the lightcone

according to the way Vermilion perceives space and time. In the diagram Vermilion and her space are drawn

at one “tick” of her clock past the point of emission, and likewise Cerulean and his space are drawn at one

“tick” of his identical clock past the point of emission. Of course, from Cerulean’s point of view his spacetime

is quite normal, and it’s Vermilion’s spacetime that is skewed.

In special relativity, the transformation between the spacetime frames of two inertial observers is called a

Lorentz transformation. In general, a Lorentz transformation consists of a spatial rotation about some

spatial axis, combined with a Lorentz boost by some velocity in some direction.

Only space along the direction of motion gets skewed with time. Distances perpendicular to the direction

of motion remain unchanged. Why must this be so? Consider two hoops which have the same size when at

rest relative to each other. Now set the hoops moving towards each other. Which hoop passes inside the

other? Neither! For suppose Vermilion thinks Cerulean’s hoop passed inside hers; by symmetry, Cerulean

must think Vermilion’s hoop passed inside his; but both cannot be true; the only possibility is that the hoops

remain the same size in directions perpendicular to the direction of motion.

Cottoned on? Then you have understood the crux of special relativity, and you can now go away and

figure out all the mathematics of Lorentz transformations. Just like Einstein. The mathematical problem is:

what is the relation between the spacetime coordinates t, x, y, z and t′, x′, y′, z′ of a spacetime interval,

a 4-vector, in Vermilion’s versus Cerulean’s frames, if Cerulean is moving relative to Vermilion at velocity v

in, say, the x direction? The solution follows from requiring

1. that both observers consider themselves to be at the centre of the lightcone, and

2. that distances perpendicular to the direction of motion remain unchanged,

as illustrated by Figure ??. (An alternative version of the second condition is that a Lorentz transformation

at velocity v followed by a Lorentz transformation at velocity −v should yield the unit transformation.)

Note that the postulate of the existence of globally inertial frames implies that Lorentz transformations

are linear, that straight lines (4-vectors) in one inertial spacetime frame transform into straight lines in other

inertial frames.

You will solve this problem in the next section but two, §1.5. As a prelude, the next two sections, §§1.3

and 1.4 discuss simultaneity and time dilation.

1.3 Paradoxes and simultaneity

Most (all?) of the apparent paradoxes of special relativity arise because observers moving at different veloc-

ities relative to each other have different notions of simultaneity.

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18 Special Relativity

1.3.1 Operational definition of simultaneity

How can simultaneity, the notion of events ocurring at the same time at different places, be defined opera-

tionally?

One way is illustrated in Figure ??. Vermilion surrounds herself with a set of mirrors, equidistant from

Vermilion. She sends out a flash of light, which reflects off the mirrors back to Vermilion. How does Vermilion

know that the mirrors are all the same distance from her? Because the relected flash returns from the mirrors

to Vermilion all at the same instant.

Vermilion asserts that the light flash must have hit all the mirrors simultaneously. Vermilion also asserts

that the instant when the light hit the mirrors must have been the instant, as registered by her wristwatch,

precisely half way between the moment she emitted the flash and the moment she received it back again. If it

takes, say, 2 seconds between flash and receipt, then Vermilion concludes that the mirrors are 1 lightsecond

away from her.

Figure ?? shows a spacetime diagram of Vermilion’s mirror experiment above. According to Vermilion, the

light hits the mirrors everywhere at the same instant, and the spatial hyperplane passing through these events

is a hypersurface of simultaneity. More generally, from Vermilion’s perspective, each horizontal hyperplane

in the spacetime diagram is a hypersurface of simultaneity.

Cerulean defines surfaces of simultaneity using the same operational setup: he encompasses himself with

mirrors, arranging them so that a flash of light returns from them to him all at the same instant. But

whereas Cerulean concludes that his mirrors are all equidistant from him and that the light bounces off

them all at the same instant, Vermilion thinks otherwise. From Vermilion’s point of view, the light bounces

off Cerulean’s mirrors at different times and moreover at different distances from Cerulean. Only so can the

speed of light be constant, as Vermilion sees it, and yet the light return to Cerulean all at the same instant.

Of course from Cerulean’s point of view all is fine: he thinks his mirrors are equidistant from him, and

that the light bounces off them all at the same instant. The inevitable conclusion is that Cerulean must

measure space and time along axes that are skewed relative to Vermilion’s. Events that happen at the same

time according to Cerulean happen at different times according to Vermilion; and vice versa. Cerulean’s

hypersurfaces of simultaneity are not the same as Vermilion’s.

From Cerulean’s point of view, Cerulean remains always at the centre of the lightcone. Thus for Cerulean,

as for Vermilion, the speed of light is constant, the same in all directions.

1.4 Time dilation

Vermilion and Cerulean construct identical clocks, consisting of a light beam which bounces off a mirror.

Tick, the light beam hits the mirror, tock, the beam returns to its owner. As long as Vermilion and Cerulean

remain at rest relative to each other, both agree that each other’s clock tick-tocks at the same rate as their

own.

But now suppose Cerulean goes off at velocity v relative to Vermilion, in a direction perpendicular to the

direction of the mirror. A far as Cerulean is concerned, his clock tick-tocks at the same rate as before, a tick

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1.4 Time dilation 19

at the mirror, a tock on return. But from Vermilion’s point of view, although the distance between Cerulean

and his mirror at any instant remains the same as before, the light has further to go. And since the speed

of light is constant, Vermilion thinks it takes longer for Cerulean’s clock to tick-tock than her own. Thus

Vermilion thinks Cerulean’s clock runs slow relative to her own.

1.4.1 Lorentz gamma factor

How much slower does Cerulean’s clock run, from Vermilion’s point of view? In special relativity the factor

is called the Lorentz gamma factor γ, introduced by the Dutch physicist Hendrik A. Lorentz in 1904, one

year before Einstein proposed his theory of special relativity. Let us see how the Lorentz gamma factor is

related to Cerulean’s velocity v.

In units where the speed of light is one, c = 1, Vermilion’s mirror is one tick away from her, and from her

point of view the vertical distance between Cerulean and his mirror is the same, one tick. But Vermilion

thinks that the distance travelled by the light beam between Cerulean and his mirror is γ ticks. Cerulean is

moving at speed v, so Vermilion thinks he moves a distance of γv ticks during the γ ticks of time taken by

the light to travel from Cerulean to his mirror. Thus, from Vermilion’s point of view, the vertical line from

Cerulean to his mirror, Cerulean’s light beam, and Cerulean’s path form a triangle with sides 1, γ, and γv,

as illustrated. Pythogoras’ theorem implies that

12 + (γv)2 = γ2 . (1.2)

From this it follows that the Lorentz gamma factor γ is related to Cerulean’s velocity v by

γ =1√

1− v2, (1.3)

which is Lorentz’s famous formula.

1.4.2 Paradox

Vermilion thinks Cerulean’s clock runs slow, by the Lorentz factor γ. But of course from Cerulean’s perspec-

tive it is Vermilion who is moving, and Vermilion whose clock runs slow. How can both think the other’s

clock runs slow? Paradox!

The resolution of the paradox, as usual in special relativity, involves simultaneity, and as usual it helps to

draw a spacetime diagram, such as the one in Figure ??.

While Vermilion thinks events happen simultaneously along horizontal planes in this diagram, Cerulean

thinks events occur simultaneously along skewed planes. Thus Vermilion thinks her clock ticks when Cerulean

is at the point before Cerulean’s clock ticks. Conversely, Cerulean thinks his clock ticks when Vermilion is

at the point before Vermilion’s clock ticks.

Where do the two light beams in Vermilion’s and Cerulean’s clocks go in this spacetime diagram? Figure ??

shows a 3D spacetime diagram.

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20 Special Relativity

Concept question 1.2 Figure ?? shows a picture of a 3D cube. Is one edge shorter than the other?

Projected on to the page, it appears so, but in reality all the edges have equal length. In what ways is this

situation similar or disimilar to time dilation in 4D relativity?

1.5 Lorentz transformation

A Lorentz transformation is a rotation of space and time. Lorentz transformations form a 6-dimensional

group, with 3 dimensions from spatial rotations, and 3 dimensions from Lorentz boosts.

If you wish to understand special relativity mathematically, then it is essential for you to go through the

exercise of deriving the form of Lorentz transformations for yourself. Indeed, this problem is the challenge

problem posed in §1.2, recast as a mathematical exercise. For simplicity, it is enough to consider the case of

a Lorentz boost by velocity v along the x-axis.

You can derive the form of a Lorentz transformation either pictorially (geometrically), or algebraically.

Ideally you should do both.

Exercise 1.3 Pictorial derivation of the Lorentz transformation. Construct, with ruler and com-

pass, a spacetime diagram that looks like the one in Figure ??. You should recognize that the square

represents the paths of lightrays that Vermilion uses to define a hypersurface of simultaneity, while the

rectangle represents the same thing for Cerulean. Notice that Cerulean’s worldline and line of simultaneity

are diagonals along his light rectangle, so the angles between those lines and the lightcone are equal. Notice

also that the areas of the square and the rectangle are the same, which expresses the fact that the area is

multiplied by the determinant of the Lorentz transformation matrix, which must be one (why?). Use your

geometric construction to derive the mathematical form of the Lorentz transformation.

Exercise 1.4 3D model of the Lorentz transformation. Make a 3D spacetime diagram of the Lorentz

transformation, with not only an x-dimension, as in the previous problem, but also a y-dimension. Resist the

temptation to use a 3D computer modelling program. Believe me, you will learn much more from hands-on

model-making. Make the lightcone from flexible paperboard, the spatial hypersurface of simultaneity from

stiff paperboard, and the worldline from wooden dowel.

Exercise 1.5 Mathematical derivation of the Lorentz transformation. Relative to person A (Ver-

milion, unprimed frame), person B (Cerulean, primed frame) moves at velocity v along the x-axis. Derive

the form of the Lorentz transformation between the coordinates (t, x, y, z) of a 4-vector in A’s frame and the

corresponding coordinates (t′, x′, y′, z′) in B’s frame from the assumptions:

1. that the transformation is linear;

2. that the spatial coordinates in the directions orthogonal to the direction of motion are unchanged;

3. that the speed of light c is the same for both A and B, so that x = t in A’s frame transforms to x′ = t′ in

B’s frame, and likewise x = −t in A’s frame transforms to x′ = −t′ in B’s frame;

4. the definition of speed; if B is moving at speed v relative to A, then x = vt in A’s frame transforms to

x′ = 0 in B’s frame;

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1.5 Lorentz transformation 21

5. spatial isotropy; specifically, show that if A thinks B is moving at velocity v, then B must think that A

is moving at velocity −v, and symmetry (spatial isotropy) between these two situations then fixes the

Lorentz γ factor.

Your logic should be precise, and explained in clear, concise English.

You should find that the Lorentz transformation for a Lorentz boost by velocity v along the x-axis is

t′ = γt− γvxx′ = − γvt+ γx

y′ = y

z′ = z

,

t = γt′ + γvx′

x = γvt′ + γx′

y = y′

z = z′

. (1.4)

The transformation can be written elegantly in matrix notation:

t′

x′

y′

z′

=

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

t

x

y

z

, (1.5)

with inverse

t

x

y

z

=

γ γv 0 0

γv γ 0 0

0 0 1 0

0 0 0 1

t′

x′

y′

z′

. (1.6)

A Lorentz transformation at velocity v followed by a Lorentz transformation at velocity v in the opposite

direction, i.e. at velocity −v, yields the unit transformation, as it should:

γ γv 0 0

γv γ 0 0

0 0 1 0

0 0 0 1

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (1.7)

The determinant of the Lorentz transformation is one, as it should be:

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

= γ2(1− v2) = 1 . (1.8)

Indeed, requiring that the determinant be one provides another derivation of the formula (1.3) for the Lorentz

gamma factor.

Concept question 1.6 Why must the determinant of a Lorentz transformation be one?

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22 Special Relativity

1.6 Paradoxes: Time dilation, Lorentz contraction, and the Twin paradox

There are several classic paradoxes in special relativity. Two of them have already been met above: the

paradox of the constancy of the speed of light in §1.2, and the paradox of time dilation in §1.4. This section

collects three famous paradoxes: time dilation (reiterating §1.4), Lorentz contraction, and the Twin paradox.

If you wish to understand special relativity conceptually, then you should work through all these paradoxes

yourself. As remarked in §1.3, most (all?) paradoxes in special relativity arise because different observers

have different notions of simultaneity, and most (all?) paradoxes can be solved using spacetime diagrams.

The Twin paradox is particularly helpful because it illustrates several different facets of special relativity,

not only time dilation, but also how light travel time modifies what an observer actually sees.

1.6.1 Time dilation

If a timelike interval t, r corresponds to motion at velocity v, then r = vt. The proper time along the

interval is

τ =√

t2 − r2 = t√

1− v2 =t

γ. (1.9)

This is Lorentz time dilation: the proper time interval τ experienced by a moving person is a factor γ less

than the time interval t according to an onlooker.

Exercise 1.7 On a spacetime diagram, show how two observers moving relative to each other can both

consider the other’s clock to run slow compared to their own.

1.6.2 Fitzgerald-Lorentz contraction

Consider a rocket of proper length l, so that in the rocket’s own rest frame (primed) the back and front ends

of the rocket move through time t′ with coordinates

t′, x′ = t′, 0 and t′, l . (1.10)

From the perspective of an observer who sees the rocket move at velocity v in the x-direction, the worldlines

of the back and front ends of the rocket are at

t, x = γt′, γvt′ and γt′ + γvl, γvt′ + γl . (1.11)

However, the observer measures the length of the rocket simultaneously in their own frame, not the rocket

frame. Solving for γt′ = t at the back and γt′ + γvl = t at the front gives

t, x = t, vt and

t, vt+l

γ

(1.12)

which says that the observer measures the front end of the rocket to be a distance l/γ ahead of the back

end. This is Lorentz contraction: an object of proper length l is measured by a moving person to be shorter

by a factor γ.

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1.7 The spacetime wheel 23

Exercise 1.8 On a spacetime diagram, show how two observers moving relative to each other can both

consider the other to be contracted along the direction of motion.

1.6.3 Twin paradox

See Exercise 1.11 at the end of the chapter.

1.7 The spacetime wheel

1.7.1 3D wheel

Figure ?? shows an ordinary 3D wheel. As the wheel rotates, a point on the wheel describes an invariant

circle. The coordinates x, y of a point on the wheel relative to its centre change, but the distance r between

the point and the centre remains constant

r2 = x2 + y2 = constant . (1.13)

More generally, the coordinates x, y, z of the interval between any two points in 3-dimensional space (a

vector) change when the coordinate system is rotated in 3 dimensions, but the separation r of the two points

remains constant

r2 = x2 + y2 + z2 = constant . (1.14)

1.7.2 4D spacetime wheel

Figure ?? shows a 4D spacetime wheel. The diagram here is a spacetime diagram, with time t vertical and

space x horizontal. A rotation between time t and space x is a Lorentz boost in the x-direction. As the

spacetime wheel boosts, a point on the wheel describes an invariant hyperbola. The spacetime coordinates

t, x of a point on the wheel relative to its centre change, but the spacetime separation s between the point

and the centre remains constant

s2 = − t2 + x2 = constant . (1.15)

More generally, the coordinates t, x, y, z of the interval between any two events in 4-dimensional spacetime

(a 4-vector) change when the coordinate system is boosted or rotated, but the spacetime separation s of the

two events remains constant

s2 = − t2 + x2 + y2 + z2 = constant . (1.16)

1.7.3 Lorentz boost as a rotation by an imaginary angle

The − sign instead of a + sign in front of the t2 in the spacetime separation formula (1.16) means that time

t can often be treated mathematically as if it were an imaginary spatial dimension. That is, t = iw where

i ≡√−1 and w is a “fourth spatial coordinate”.

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24 Special Relativity

A Lorentz boost by a velocity v can likewise be treated as a rotation by an imaginary angle. Consider a

normal spatial rotation in which a primed frame is rotated in the wx-plane clockwise by an angle a about

the origin, relative to the unprimed frame. The relation between the coordinates w′, x′ and w, x of a

point in the two frames is(

w′

x′

)

=

(

cos a − sina

sin a cos a

)(

w

x

)

. (1.17)

Now set t = iw and α = ia with t and α both real. In other words, take the spatial coordinate w to be

imaginary, and the rotation angle a likewise to be imaginary. Then the rotation formula above becomes(

t′

x′

)

=

(

coshα − sinhα

− sinhα coshα

)(

t

x

)

(1.18)

This agrees with the usual Lorentz transformation formula (??) if the boost velocity v and boost angle α

are related by

v = tanhα , (1.19)

so that

γ = coshα , γv = sinhα . (1.20)

This provides a convenient way to add velocities in special relativity: the boost angles simply add (for boosts

along the same direction), just as spatial rotation angles add (for rotations about the same axis). Thus a

boost by velocity v1 = tanhα1 followed by a boost by velocity v2 = tanhα2 in the same direction gives a

net velocity boost of v = tanhα where

α = α1 + α2 . (1.21)

The equivalent formula for the velocities themselves is

v =v1 + v21 + v1v2

, (1.22)

the special relativistic velocity addition formula.

1.7.4 Trip across the Universe at constant acceleration

Suppose that you took a trip across the Universe in a spaceship, accelerating all the time at one Earth

gravity g. How far would you travel in how much time?

The spacetime wheel offers a cute way to solve this problem, since the rotating spacetime wheel can

be regarded as representing spacetime frames undergoing constant acceleration. Specifically, points on the

right quadrant of the rotating spacetime wheel represent worldlines of persons who accelerate with constant

acceleration in their own frame.

If the units of space and time are chosen so that the speed of light and the gravitational acceleration are

both one, c = g = 1, then the proper time experienced by the accelerating person is the boost angle α, and

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1.7 The spacetime wheel 25

Table 1.1 Trip across the Universe.

Time elapsed Time elapsedon spaceship on Earth Distance travelled To

in years in years in lightyears

α sinhα coshα− 1

0 0 0 Earth (starting point)1 1.175 .54312 3.627 2.762

2.337 5.127 4.223 Proxima Cen3.962 26.3 25.3 Vega6.60 368 367 Pleiades10.9 2.7 × 104 2.7 × 104 Centre of Milky Way15.4 2.44 × 106 2.44 × 106 Andromeda galaxy18.4 4.9 × 107 4.9 × 107 Virgo cluster19.2 1.1 × 108 1.1 × 108 Coma cluster25.3 5 × 1010 5 × 1010 Edge of observable Universe

the time and space coordinates of the accelerating person, relative to a person who remains at rest, are those

of a point on the spacetime wheel, namely

t, x = sinhα, coshα . (1.23)

In the case where the acceleration is one Earth gravity, g = 9.80665 m s−2, the unit of time is

c

g=

299,792,458 m s−1

9.80665 m s−2= 0.97 yr , (1.24)

just short of one year. For simplicity, Table 1.1, which tabulates some milestones along the way, takes the

unit of time to be exactly one year, which would be the case if one were accelerating at 0.97 g = 9.5 m s−2.

After a slow start, you cover ground at an ever increasing rate, crossing 50 billion lightyears, the distance

to the edge of the currently observable Universe, in just over 25 years of your own time.

Does this mean you go faster than the speed of light? No. From the point of view of a person at rest

on Earth, you never go faster than the speed of light. From your own point of view, distances along your

direction of motion are Lorentz-contracted, so distances that are vast from Earth’s point of view appear

much shorter to you. Fast as the Universe rushes by, it never goes faster than the speed of light.

This rosy picture of being able to flit around the Universe has drawbacks. Firstly, it would take a huge

amount of energy to keep you accelerating at g. Secondly, you would use up a huge amount of Earth time

travelling around at relativistic speeds. If you took a trip to the edge of the Universe, then by the time

you got back not only would all your friends and relations be dead, but the Earth would probably be gone,

swallowed by the Sun in its red giant phase, the Sun would have exhausted its fuel and shrivelled into a

cold white dwarf star, and the Solar System, having orbited the Galaxy a thousand times, would be lost

somewhere in its milky ways.

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26 Special Relativity

Technical point. The Universe is expanding, so the distance to the edge of the currently observable Universe

is increasing. Thus it would actually take longer than indicated in the table to reach the edge of the currently

observable Universe. Moreover if the Universe is accelerating, as recent evidence from the Hubble diagram

of Type Ia Supernovae suggests, then you will never be able to reach the edge of the currently observable

Universe, however fast you go.

1.8 Scalar spacetime distance

One of the most fundamental features of a Lorentz transformation is that its leaves invariant a certain

distance. the scalar spacetime distance, between any two events in spacetime. The scalar spacetime distance

∆s between two events separated by ∆t,∆x,∆y,∆z is given by

∆s2 = −∆t2 + ∆r2

= −∆t2 + ∆x2 + ∆y2 + ∆z2 . (1.25)

A quantity such as ∆s2 that remains unchanged under any Lorentz transformation is called a scalar. It is

left to you in exercise 1.9 to show explicitly that ∆s2 is unchanged under Lorentz transformations. Lorentz

transformations can be defined as linear spacetime transformations that leave ∆s2 invariant.

The single scalar spacetime squared interval ∆s2 replaces the two scalar quantities

time interval ∆t

distance interval ∆r =√

∆x2 + ∆y2 + ∆z2(1.26)

of classical Galilean spacetime.

Exercise 1.9 Invariant spacetime interval. Show that the squared spacetime interval ∆s2 defined by

equation (1.25) is unchanged by a Lorentz transformation, that is, show that

−∆t′2 + ∆x′2 + ∆y′2 + ∆z′2 = −∆t2 + ∆x2 + ∆y2 + ∆z2 . (1.27)

You may assume without proof the familiar results that the 3D scalar product ∆r2 = ∆x2 + ∆y2 + ∆z2 is

unchanged by a spatial rotation, so it suffices to consider a Lorentz boost, say in the x direction.

1.8.1 Proper time, proper distance

The scalar spacetime squared interval ∆s2 has a physical meaning.

If an interval ∆t,∆r is timelike, ∆t > ∆r, then the square root of minus the spacetime interval squared

is the proper time ∆τ along it

∆τ =√

−∆s2 =√

∆t2 −∆r2 . (1.28)

This is the time experienced by an observer moving along that interval.

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1.9 4-vectors 27

If an interval ∆t,∆r is spacelike, ∆t < ∆r, then the spacetime interval equals the proper distance

∆l along it

∆l =√

∆s2 =√

∆r2 −∆t2 . (1.29)

This is the distance between two events measured by an observer for whom those events are simultaneous.

Concept question 1.10 Justify the assertions (1.28) and (1.29).

1.8.2 Timelike, lightlike, spacelike

A spacetime interval ∆xm is called

timelike if ∆s2 < 0 ,

null or lightlike if ∆s2 = 0 ,

spacelike if ∆s2 > 0 .

(1.30)

1.8.3 Minkowski metric

The scalar spacetime squared interval ∆s2 associated with an interval ∆xm = ∆t,∆r = ∆t,∆x,∆y,∆zcan be written

∆s2 = ∆xm∆xm

= ηmn∆xm∆xn (1.31)

where ηmn is the Minkowski metric

ηmn ≡

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (1.32)

Equation (1.31) uses the implicit summation convention, according to which paired indices are explicitly

summed over. Invariably, one of a pair of repeated indices is raised, the other lowered.

1.9 4-vectors

A 4-vector in special relativity is a quantity am = at, ax, ay, az that transforms under Lorentz transfor-

mations like an interval xm = t, x, y, z of spacetime

a′m = Lmna

n (1.33)

where Lmn denotes a Lorentz transformation.

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28 Special Relativity

1.9.1 Index notation

In special and general relativity it is convenient to introduce two versions of the same 4-vector quantity, one

with raised indices, called the contravariant components of the 4-vector,

am ≡ at, ax, ay, az , (1.34)

and one with lowered indices called the covariant components of the 4-vector,

am ≡ −at, ax, ay, az (1.35)

(the naming is crazy, and you do not need to remember it).

The indices run over m = t, x, y, z, or sometimes m = 0, 1, 2, 3.

Why introduce raised and lowered indices? Because

amam ≡

m

amam = ata

t + axax + aya

y + azaz

= − (at)2 + (ax)2 + (ay)2 + (az)2 (1.36)

is a Lorentz scalar.

1.10 Energy-momentum 4-vector

Symmetry argument:

Symmetry Conservation law

Time translation Energy

Space translation Momentum

suggests

energy = time component

momentum = space component

of 4-vector. (1.37)

The Principle of Special Relativity requires that the equation of energy-momentum conservation

energy

momentum= constant (1.38)

should take the same form in any inertial frame. The equation should be Lorentz covariant, that is, the

equation should transform like a Lorentz 4-vector.

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1.10 Energy-momentum 4-vector 29

1.10.1 Construction of the energy-momentum 4-vector

Require:

1. it’s a 4-vector

2. goes over to the Newtonian limit as v → 0.

Newtonian limit:

Momentum p is mass m times velocity v

p = mv = mdr

dt. (1.39)

4D version:

Need to do two things to Newtonian momentum:

• replace r by a 4-vector xm = t, r• replace dt by a scalar — the only obvious choice is the proper time interval τ .

Result:

pk = mdxk

= m

dt

dτ,dr

= m γ, γv (1.40)

which are special relativistic versions of energy E and momentum p

pk = E,p = mγ,mγv . (1.41)

1.10.2 Special relativistic energy

E = mγ (units c = 1) (1.42)

or, restoring standard units

E = mc2γ . (1.43)

Taylor expand γ for small velocity v:

γ =1

1− v2/c2= 1 +

1

2

v2

c2+ ... (1.44)

so

E = mc2(

1 +1

2

v2

c2+ ...

)

= mc2 +1

2mv2 + ... . (1.45)

The first term, mc2, is the rest-mass energy. The second term, 12mv

2, is the non-relativistic kinetic energy.

Higher-order terms give relativistic corrections to the kinetic energy.

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30 Special Relativity

1.10.3 Rest mass is a scalar

The scalar quantity constructed from the energy-momentum 4-vector pk = E,p is

pkpk = −E2 + p2

= −m2(γ2 − γ2v2)

= −m2 (1.46)

minus the square of the rest mass.

1.11 Photon energy-momentum

Photons have zero rest mass

m = 0 . (1.47)

Thus

pkpk = −E2 + p2 = −m2 = 0 (1.48)

whence

p ≡ |p| = E . (1.49)

Hence

pk = E,p= E1,n= hν1,n (1.50)

where ν is the photon frequency.

The photon velocity is n, a unit vector. The photon speed is one (the speed of light).

1.11.1 Lorentz transformation of photon energy-momentum 4-vector

Follows the usual rules for 4-vectors.

In the case that the Lorentz transformation is a Lorentz boost along the x-axis, the transformation is

p′t

p′x

p′y

p′z

=

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

pt

px

py

pz

=

γ(pt − vpx)

γ(px − vpt)

py

pz

. (1.51)

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1.12 Abstract 4-vectors 31

Equivalently

hν′

1

n′x

n′y

n′z

=

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

1

nx

ny

nz

= hν

γ(1− nxv)

γ(nx − v)ny

nz

.

These mathematical relations imply the rules of 4-dimensional perspective, §1.13.1.

1.11.2 Redshift

Astronomers define the redshift z of a photon by

z ≡ λobs − λemit

λemit. (1.52)

In relativity, it is often more convenient to use the redshift factor 1 + z

1 + z ≡ λobs

λemit=νemit

νobs. (1.53)

1.11.3 Special relativistic Doppler shift

If the emitter frame (primed) is moving with velocity v in the x-direction relative to the observer frame

(unprimed) then

hνemit = hνobsγ(1− nxv) (1.54)

so

1 + z =νemit

νobs

= γ(1− nxv)

= γ(1− n.v) . (1.55)

This is the general formula for the special relativistic Doppler shift.

1.12 Abstract 4-vectors

A = Amγγm (1.56)

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32 Special Relativity

1.13 What things look like at relativistic speeds

1.13.1 The rules of 4-dimensional perspective

The diagram below illustrates the rules of 4-dimensional perspective, also called “special relativistic beam-

ing,” which describe how a scene appears when you move through it at near light speed.

1

1γv

γ

On the left, you are at rest relative to the scene. Imagine painting the scene on a celestial sphere around

you. The arrows represent the directions of light rays (photons) from the scene on the celestial sphere to

you at the center.

On the right, you are moving to the right through the scene, at some fraction of the speed of light. The

celestial sphere is stretched along the direction of your motion into a celestial ellipsoid. You, the observer,

are not at the center of the ellipsoid, but rather at one of its foci (the left one, if you are moving to the

right). The scene appears relativistically aberrated, which is to say concentrated ahead of you, and expanded

behind you.

The lengths of the arrows are proportional to the energies, or frequencies, of the photons that you see.

When you are moving through the scene at near light speed, the arrows ahead of you, in your direction

of motion, are longer than at rest, so you see the photons blue-shifted, increased in energy, increased in

frequency. Conversely, the arrows behind you are shorter than at rest, so you see the photons red-shifted,

decreased in energy, decreased in frequency. Since photons are good clocks, the change in photon frequency

also tells you how fast or slow clocks attached to the scene appear to you to run.

Numbers? On the right, you are moving through the scene at v = 0.6 c. The celestial ellipsoid is stretched

along the direction of your motion by the Lorentz gamma factor, which here is γ = 1/√

1− 0.62 = 1.25. The

focus of the celestial ellipsoid, where you the observer are, is displaced from center by γv = 1.25×0.6 = 0.75.

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1.14 How to programme Lorentz transformations on a computer 33

1.14 How to programme Lorentz transformations on a computer

3D gaming programmers are familiar with the fact that the best way to program spatial rotations on a

computer is with quaternions. Compared to standard rotation matrices, quaternions offer increased speed

and require less storage, and their algebraic properties simplify interpolation and splining.

Section 1.7 showed that a Lorentz boost is mathematically equivalent to a rotation by an imaginary

angle. Thus suggests that Lorentz transformations might be treated as complexified spatial rotations, which

proves to be true. Indeed, the best way to program Lorentz transformation on a computer is with complex

quaternions, as will be demonstrated in Chapter 14.

Exercises

Exercise 1.11 Twin paradox. Your twin leaves you on Earth and travels to the spacestation Alpha, ℓ

= 3 lyr away, at a good fraction of the speed of light, then immediately returns to Earth at the same speed.

The accompanying spacetime diagram shows the corresponding worldlines of both you and your twin. Aside

from part (a) and the first part of (b), I want you to derive your answers mathematically, using logic and

Lorentz transformations. However, the diagram is accurately drawn, and you should be able to check your

answers by measuring.

1. Label the worldlines of you and your twin. Draw the worldline of a light signal which travels from you on

Earth, hits Alpha just when your twin arrives, and immediately returns to Earth. Draw the twin’s “now”

when just arriving at Alpha, and the twin’s “now” just departing from Alpha (in the first case the twin is

moving toward Alpha, while in the second case the twin is moving back toward Earth).

2. From the diagram, measure the twin’s speed v relative to you, in units where the speed of light is unity,

c = 1. Deduce the Lorentz gamma factor γ, and the redshift factor 1 + z = [(1 + v)/(1 − v)]1/2, in the

cases (i) where the twin is receding, and (ii) where the twin is approaching.

3. Choose the spacetime origin to be the event where the twin leaves Earth. Argue that the position 4-vector

of the twin on arrival at Alpha is

t, x, y, z = ℓ/v, ℓ, 0, 0 . (1.57)

Lorentz transform this 4-vector to determine the position 4-vector of the twin on arrival at Alpha, in the

twin’s frame. Express your answer first in terms of ℓ, v, and γ, and then in (light)years. State in words

what this position 4-vector means.

4. How much do you and your twin age respectively during the round trip to Alpha and back? What is the

ratio of these ages? Express your answers first in terms of ℓ, v, and γ, and then in years.

5. What is the distance between the Earth and Alpha from the twin’s point of view? What is the ratio of

this distance to the distance between Earth and Alpha from your point of view? Explain how your arrived

at your result. Express your answer first in terms of ℓ, v, and γ, and then in lightyears.

6. You watch your twin through a telescope. How much time do you see (through the telescope) elapse

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34 Special Relativity

on your twin’s wristwatch between launch and arrival on Alpha? How much time passes on your own

wristwatch during this time? What is the ratio of these two times? Express your answers first in terms of

ℓ, v, and γ, and then in years.

7. On arrival at Alpha, your twin looks back through a telescope at your wristwatch. How much time does

your twin see (through the telescope) has elapsed since launch on your watch? How much time has elapsed

on the twin’s own wristwatch during this time? What is the ratio of these two times? Express your answers

first in terms of ℓ, v, and γ, and then in years.

8. You continue to watch your twin through a telescope. How much time elapses on your twin’s wristwatch,

as seen by you through the telescope, during the twin’s journey back from Alpha to Earth? How much

time passes on your own watch as you watch (through the telescope) the twin journey back from Alpha

to Earth? What is the ratio of these two times? Express your answers first in terms of ℓ, v, and γ, and

then in years.

9. During the journey back from Alpha to Earth, your twin likewise continues to look through a telescope

at the time registered on your watch. How much time passes on your wristwatch, as seen by your twin

through the telescope, during the journey back? How much time passes on the twin’s wristwatch from the

twin’s point of view during the journey back? What is the ratio of these two times? Express your answers

first in terms of ℓ, v, and γ, and then in years.

Exercise 1.12 Lines intersecting at right angles. Prove that if two lines appear to intersect at right-

angles projected on the sky in one frame, then they appear to intersect at right-angles in another frame

Lorentz-transformed with respect to the first.

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PART TWO

COORDINATE APPROACH TO GENERAL RELATIVITY

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Concept Questions

1. What assumption of general relativity makes it possible to introduce a coordinate system?

2. Is the speed of light a universal constant in general relativity? If so, in what sense?

3. What does “locally inertial” mean? How local is local?

4. Why is spacetime locally inertial?

5. What assumption of general relativity makes it possible to introduce clocks and rulers?

6. Consider two observers at the same point and with the same instantaneous velocity, but one is accelerating

and the other is in free-fall. What is the relation between the proper time or proper distance along an

infinitesimal interval measured by the two observers? What assumption of general relativity implies this?

7. Does the (Strong) Principle of Equivalence imply that two unequal masses will fall at the same rate in a

gravitational field? Explain.

8. In what respects is the Strong Principle of Equivalence (gravity is equivalent to acceleration) stronger

than the Weak Principle of Equivalence (gravitating mass equals inertial mass)?

9. Standing on the surface of the Earth, you hold an object of negative mass in your hand, and drop it.

According to the Principle of Equivalence, does the negative mass fall up or down?

10. Same as the previous question, but what does Newtonian gravity predict?

11. You have a box of negative mass particles, and you remove energy from it. Do the particles move faster

or slower? Does the entropy of the box increase or decrease? Does the pressure exerted by the particles

on the walls of the box increase or decrease?

12. You shine two light beams along identical directions in a gravitational field. The two light beams are

identical in every way except that they have two different frequencies. Does the Equivalence Principle

imply that the interference pattern produced by each of the beams individually is the same?

13. What is a “straight line”, according to the Principle of Equivalence?

14. If all objects move on straight lines, how is it that when, standing on the surface of the Earth, you throw

two objects in the same direction but with different velocities, they follow two different trajectories?

15. In relativity, what is the generalization of the “shortest distance between two points”?

16. What kinds of general coordinate transformations are allowed in general relativity?

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38 Concept Questions

17. In general relativity, what is a scalar? A 4-vector? A tensor? Which of the following is a scalar/vector/

tensor/none-of-the-above? (a) a set of coordinates xµ; (b) a coordinate interval dxµ; (c) proper time

τ?

18. What does general covariance mean?

19. What does parallel transport mean?

20. Why is it important to define covariant derivatives that behave like tensors?

21. Is covariant differentiation a derivation? That is, is covariant differentiation a linear operation, and does

it obey the Leibniz rule for the derivative of a product?

22. What is the covariant derivative of the metric tensor? Explain.

23. What does a connection coefficient Γκµν mean physically? Is it a tensor? Why, or why not?

24. An astronaut is in free-fall in orbit around the Earth. Can the astronaut detect that there is a gravitational

field?

25. Can a gravitational field exist in flat space?

26. How can you tell whether a given metric is equivalent to the Minkowski metric of flat space?

27. How many degrees of freedom does the metric have? How many of these degrees of freedom can be removed

by arbitrary transformations of the spacetime coordinates, and therefore how many physical degrees of

freedom are there in spacetime?

28. If you insist that the spacetime is spherical, how many physical degrees of freedom are there in the

spacetime?

29. If you insist that the spacetime is spatially homogeneous and isotropic (the cosmological principle), how

many physical degrees of freedom are there in the spacetime?

30. In general relativity, you are free to prescribe any spacetime (any metric) you like, including metrics with

wormholes and metrics that connect the future to the past so as to violate causality. True or false?

31. If it is true that in general relativity you can prescribe any metric you like, then why aren’t you bumping

into wormholes and causality violations all the time?

32. How much mass does it take to curve space significantly (significantly meaning by of order unity)?

33. What is the relation between the energy-momentum 4-vector of a particle and the energy-momentum

tensor?

34. It is straightforward to go from a prescribed metric to the energy-momentum tensor. True or false?

35. It is straightforward to go from a prescribed energy-momentum tensor to the metric. True or false?

36. Does the Principle of Equivalence imply Einstein’s equations?

37. What do Einstein’s equations mean physically?

38. What does the Riemann curvature tensor Rκλµν mean physically? Is it a tensor?

39. The Riemann tensor splits into compressive (Ricci) and tidal (Weyl) parts. What do these parts mean,

physically?

40. Einstein’s equations imply conservation of energy-momentum, but what does that mean?

41. Do Einstein’s equations describe gravitational waves?

42. Do photons (massless particles) gravitate?

43. How do different forms of mass-energy gravitate?

44. How does negative mass gravitate?

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What’s important?

This part of the notes adopts the traditional coordinate-based approach to general relativity. The approach

is neither the most insightful nor the most powerful, but it is the fastest route to connecting the metric to

the energy-momentum content of spacetime.

1. Postulates of general relativity. How do the various postulates imply the mathematical structure of general

relativity?

2. The road from spacetime curvature to energy-momentum:

metric gµν

→ connection coefficients Γκµν

→ Riemann curvature tensor Rκλµν

→ Ricci tensor Rκµ and scalar R

→ Einstein tensor Gκµ = Rκµ − 12gκµR

→ energy-momentum tensor Tκµ

3. 4-velocity and 4-momentum. Geodesic equation.

4. Bianchi identities guarantee conservation of energy-momentum.

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2

Fundamentals of General Relativity

2.1 The postulates of General Relativity

General relativity follows from three postulates:

1. Spacetime is a 4-dimensional manifold;

2. The (Strong) Principle of Equivalence;

3. Einstein’s Equations.

2.1.1 Spacetime is a 4-dimensional manifold

A 4-dimensional manifold is defined mathematically to be a topological space that is locally homeomorphic

to Euclidean 4-space R4.

This postulate implies that it is possible to set up a coordinate system (possibly in patches)

xµ ≡ x0, x1, x2, x3 (2.1)

such that each point of (the patch of) spacetime has a unique coordinate.

Andrew’s convention:

Greek (brown) dummy indices label curved spacetime coordinates.

Latin (black) dummy indices label locally inertial (more generally, tetrad) coordinates.

2.1.2 (Strong) Principle of Equivalence (PE)

“The laws of physics in a gravitating frame are equivalent to those in an accelerating frame”.

The Weak Principle of Equivalence is “Gravitating mass = inertial mass”.

PE ⇒ spacetime is locally inertial (see §2.2).

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2.2 Existence of locally inertial frames 41

2.1.3 Einstein’s equations

Einstein’s equations comprise a 4× 4 symmetric matrix of equations

Gµν = 8πGTµν . (2.2)

Here G is the Newtonian gravitational constant, Gµν is the Einstein tensor, and Tµν is the energy-

momentum tensor.

Physically, Einstein’s equations signify

(compressive part of) curvature = energy-momentum content . (2.3)

Einstein’s equations generalize Poisson’s equation

∇2Φ = 4πGρ (2.4)

where Φ is the Newtonian gravitational potential, and ρ the mass-energy density. Poisson’s equation is the

time-time component of Einstein’s equations in the limit of a weak gravitational field and slowly moving

matter.

2.2 Existence of locally inertial frames

The Principle of Equivalence implies that at each point of spacetime it is possible to choose a locally inertial,

or free-fall, frame, such that the laws of special relativity apply within an infinitesimal neighbourhood of

that point. By this is meant that at each point of spacetime it is possible to choose coordinates such that

(a) the metric at that point is Minkowski, and (b) the first derivatives of the metric are all zero.

It is built into the Principle of Equivalence that general relativity is, like special relativity, a metric

theory. Notably, the proper times and distances measured by an accelerating observer are the same as

those measured by a freely-falling observer at the same point and with the same instantaneous velocity.

2.3 Metric

The metric is the essential mathematical object that converts an infinitesimal coordinate interval

dxµ ≡ dx0, dx1, dx2, dx3 (2.5)

to a proper measurement of an interval of time or space.

Postulate (1) of general relativity means that it is possible to choose coordinates

xµ ≡ x0, x1, x2, x3 (2.6)

covering (a patch of) spacetime.

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42 Fundamentals of General Relativity

Postulate (2) of general relativity implies that at each point of spacetime it is possible to choose locally

inertial coordinates

ξm ≡ ξ0, ξ1, ξ2, ξ3 (2.7)

such that the metric is Minkowski,

ds2 = ηmn dξmdξn , (2.8)

in an infinitesimal neighborhood of the point. The spacetime distance squared ds2 is a scalar, a quantity

that is unchanged by the choice of coordinates.

Since

dξm =∂ξm

∂xµdxµ (2.9)

it follows that

ds2 = ηmn∂ξm

∂xµ

∂ξn

∂xνdxµdxν (2.10)

so the scalar spacetime distance squared is

ds2 = gµν dxµdxν (2.11)

where gµν is the metric, a 4× 4 symmetric matrix

gµν = ηmn∂ξm

∂xµ

∂ξn

∂xν. (2.12)

2.4 Basis gµ of tangent vectors

You are familiar with the idea that in ordinary 3D Euclidean geometry it is often convenient to treat vectors

in an abstract coordinate-independent formalism. Thus for example a 3-vector is commonly written as an

abstract quantity r. The coordinates of the vector r may be x, y, z in some particular coordinate system,

but one recognizes that the vector r has a meaning, a magnitude and a direction, that is independent of the

coordinate system adopted. In an arbitrary Cartesian coordinate system, the Euclidean 3-vector r can be

expressed

r =∑

i

xi xi = x x+ y y + z z (2.13)

where xi ≡ x, y, z are unit vectors along each of the coordinate axes.

The same kind of abstract notation is useful in general relativity. Define gµ

gµ ≡ g0, g1, g2, g3 (2.14)

to be the basis of axes tangent to the coordinates xµ. Each axis gµ is a 4D vector object, with both magnitude

and direction in spacetime. (Some texts represent the tangent vectors gµ with the notation ∂µ, but this

notation is not used here, to avoid the potential confusion between ∂µ as a derivative and ∂µ as a vector.)

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2.5 4-vectors and tensors 43

An interval dxµ of spacetime can be expressed in coordinate-independent fashion as the abstract vector

interval dx

dx ≡ gµ dxµ = g0 dx

0 + g1 dx1 + g2 dx

2 + g3 dx3 . (2.15)

The scalar length squared of the abstract vector interval dx is

ds2 = dx · dx = gµ · gν dxµdxν (2.16)

whence

gµν = gµ · gν (2.17)

the metric is the (4D) scalar product of tangent vectors.

The tangent vectors gµ form a basis for a 4D tangent space that has three important mathematical

properties. First, the tangent space is a vector space, that is, it has the properties of linearity that define

a vector space. Second, the tangent space has an inner (or scalar) product, defined by the metric (2.17).

Third, vectors in the tangent space can be differentiated with respect to coordinates, as will be elucidated

in §2.6.3.

2.5 4-vectors and tensors

2.5.1 Contravariant coordinate 4-vector

Under a general coordinate transformation

xµ → x′µ (2.18)

a coordinate interval dxµ transforms as

dx′µ =∂x′µ

∂xνdxν . (2.19)

In general relativity, a coordinate 4-vector is defined to be a quantity Aµ = A0, A1, A2, A3 that trans-

forms under a coordinate transformation (2.18) like a coordinate interval

A′µ =∂x′µ

∂xνAν . (2.20)

2.5.2 Abstract 4-vector

A 4-vector may be written in coordinate-independent fashion as

A = gµAµ . (2.21)

The quantity A is an abstract 4-vector. Although A is a 4-vector, it is by construction unchanged by a

coordinate transformation, and is therefore a coordinate scalar. See §2.5.6 for commentary on the distinction

between abstract and coordinate vectors.

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44 Fundamentals of General Relativity

2.5.3 Lowering and raising indices

Define gµν to be the inverse metric, satisfying

gλµ gµν = δν

λ =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (2.22)

The metric gµν and its inverse gµν provide the means of lowering and raising coordinate indices. The

components of a coordinate 4-vector Aµ with raised index are called its contravariant components, while

those Aµ with lowered indices are called its covariant components,

Aµ = gµν Aν , (2.23)

Aµ = gµν Aν . (2.24)

2.5.4 Covariant coordinate 4-vector

Under a general coordinate transformation (2.18), the covariant components Aµ of a coordinate 4-vector

transform as

A′µ =

∂xν

∂x′µAν . (2.25)

You can check that the transformation law (2.25) for the covariant components Aµ is consisistent with the

transformation law (2.20) for the contravariant components Aµ.

You can check that the tangent vectors gµ transform as a covariant coordinate 4-vector.

2.5.5 Scalar product

If Aµ and Bµ are coordinate 4-vectors, then their scalar product is

AµBµ = AµBµ = gµνA

µBν . (2.26)

This is a coordinate scalar, a quantity that remains invariant under general coordinate transformations.

In abstract vector formalism, the scalar product of two 4-vectors A = gµAµ and B = gµB

µ is

A ·B = gµ · gν AµBν = gµνA

µBν . (2.27)

2.5.6 Comment on vector naming and notation

Different texts follow different conventions for naming and notating vectors and tensors.

In this book I follow the convention of calling both Aµ (with a dummy index µ) and A ≡ Aµgµ vectors.

Although Aµ and A are both vectors, they are mathematically different objects.

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2.6 Covariant derivatives 45

If the index on a vector indicates a specific coordinate, then the indexed vector is the component of the

vector; for example A0 (or At) is the x0 (or time t) component of the coordinate 4-vector Aµ.

In this book, the different species of vector are distinguished by an adjective:

1. A coordinate vector Aµ, identified by Greek (brown) indices µ, is one that changes in a prescribed way

under coordinate transformations. A coordinate transformation is one that changes the coordinates of the

spacetime without actually changing the spacetime or whatever lies in it.

2. An abstract vector A, identified by boldface, is the thing itself, and is unchanged by the choice of

coordinates. Since the abstract vector is unchanged by a coordinate transformation, it is a coordinate

scalar.

All the types of vector have the properties of linearity (additivity, multiplication by scalars) that identify

them mathematically as belonging to vector spaces. The important distinction between the types of vector

is how they behave under transformations.

In referring to both Aµ and A as vectors, I am following the standard physics practice of mentally regarding

Aµ and A as equivalent objects. You are familiar with the advantages of treating a vector in 3D Euclidean

space either as an abstract vector A, or as a coordinate vector Ai. Depending on the problem, sometimes

the abstract notation A is more convenient, and sometimes the coordinate notation Ai is more convenient.

Sometimes it’s convenient to switch between the two in the middle of a calculation. Likewise in general

relativity it is convenient to have the flexibility to work in either coordinate or abstract notation, whatever

suits the problem of the moment.

2.5.7 Coordinate tensor

In general, a coordinate tensor Aκλ...µν... is an object that transforms under general coordinate transforma-

tions (2.18) as

A′κλ...µν... =

∂x′κ

∂xα

∂x′λ

∂xβ...∂xγ

∂x′µ∂xδ

∂x′ν... Aαβ...

γδ... . (2.28)

You can check that the metric tensor gµν and its inverse gµν are indeed coordinate tensors, transforming

like (2.28).

The rank of a tensor is the number of indices. A scalar is a tensor of rank 0. A 4-vector is a tensor of

rank 1.

2.6 Covariant derivatives

2.6.1 Derivative of a coordinate scalar

Suppose that Φ is a coordinate scalar. Then the coordinate derivative of Φ is a coordinate 4-vector

∂Φ

∂xµis a coordinate tensor (2.29)

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46 Fundamentals of General Relativity

transforming like equation (2.25).

As a shorthand, the ordinary partial derivative is often denoted in the literature with a comma

∂Φ

∂xµ= Φ,µ . (2.30)

For the most part this book does not use the comma notation.

2.6.2 Derivative of a coordinate 4-vector

The ordinary partial derivative of a covariant coordinate 4-vector Aµ is not a tensor

∂Aµ

∂xνis not a coordinate tensor (2.31)

because it does not transform like a coordinate tensor.

However, the 4-vector A = gµAµ, being by construction invariant under coordinate transformations, is a

coordinate scalar, and its partial derivative is a coordinate 4-vector

∂A

∂xν=∂gµA

µ

∂xν

= gµ∂Aµ

∂xν+∂gµ

∂xνAµ is a coordinate tensor . (2.32)

The last line of equation (2.32) assumes that it is legitimate to differentiate the tangent vectors gµ, but

what does this mean? The partial derivatives of basis vectors gµ are defined in the usual way by

∂gµ

∂xν≡ lim

δxν→0

gµ(x0, ..., xν+δxν , ..., x3)− gµ(x0, ..., xν , ..., x3)

δxν. (2.33)

This definition relies on being able to compare the vectors gµ(x) at some point x with the vectors gµ(x+δx)

at another point x+δx a small distance away. The comparison between two vectors a small distance apart

is made possible by the existence of locally inertial frames. In a locally inertial frame, two vectors a small

distance apart can be compared by parallel-transporting one vector to the location of the other along

the small interval between them, that is, by transporting the vector without accelerating or precessing with

respect to the locally inertial frame. Thus gµ(x+δx) in the definition (2.33) should be interpreted as its

value parallel-transported from position x+δx to position x along the small interval δx between them.

2.6.3 Coordinate connection coefficients (Christoffel symbols)

The partial derivatives of the basis vectors gµ that appear on the right hand side of equation (2.32) define

the coordinate connection coefficients Γκµν , also known as Christoffel symbols,

∂gµ

∂xν≡ Γκ

µν gκ is not a coordinate tensor . (2.34)

The definition (2.34) shows that the connection coefficients express how each tangent vector gµ changes,

relative to parallel-transport, when shifted along an interval δxν .

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2.6 Covariant derivatives 47

2.6.4 Covariant derivative of a contravariant 4-vector

Expression (2.32) along with the definition (2.34) of the connection coefficients implies that

∂A

∂xν= gµ

∂Aµ

∂xν+ Γκ

µνgκAµ

= gκ

(

∂Aκ

∂xν+ Γκ

µνAµ

)

is a coordinate tensor . (2.35)

The expression in parentheses is a coordinate tensor, and defines the covariant derivative DνAκ of the

contravariant coordinate 4-vector Aκ

DνAκ ≡ ∂Aκ

∂xν+ Γκ

µνAµ is a coordinate tensor . (2.36)

As a shorthand, the covariant derivative is often denoted in the literature with a semi-colon

DνAκ = Aκ

;ν . (2.37)

For the most part this book does not use the semi-colon notation.

2.6.5 Covariant derivative of a covariant coordinate 4-vector

Similarly,

∂A

∂xν= gκDνAκ is a coordinate tensor (2.38)

where DνAκ is the covariant derivative of the covariant coordinate 4-vector Aκ

DνAκ ≡∂Aκ

∂xν− Γµ

κνAµ is a coordinate tensor . (2.39)

2.6.6 Covariant derivative of a coordinate tensor

In general, the covariant derivative of a coordinate tensor is

DαAκλ...µν... =

∂Aκλ...µν...

∂xα+ Γκ

βαAβλ...µν... + Γλ

βαAκβ...µν... + ...− Γβ

µαAκλ...βν... − Γβ

ναAκλ...µβ... − ... (2.40)

with a positive Γ term for each contravariant index, and a negative Γ term for each covariant index.

2.6.7 No-torsion condition

The existence of locally inertial frames requires that it must be possible to arrange not only that the tangent

axes gµ are orthonormal at a point, but also that they remain orthonormal to first order in a Taylor expansion

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48 Fundamentals of General Relativity

about the point. That is, it must be possible to choose the coordinates such that the tangent axes gµ are

orthonormal, and unchanged to linear order:

gµ · gν = ηµν , (2.41)

∂gµ

∂xν= 0 . (2.42)

In view of the definition (2.34) of the connection coefficients, the second condition (2.42) is equivalent to the

vanishing of all the connection coefficients:

Γκµν = 0 . (2.43)

Under a general coordinate transformation xµ → x′µ, the tangent axes transform as gµ → g′µ = ∂xν/∂x′µ gν .

The 4×4 matrix ∂xν/∂x′µ of partial derivatives provides 16 degrees of freedom in choosing the tangent axes

at a point. The 16 degrees of freedom are enough — more than enough — to accomplish the orthonormality

condition (2.41), which is a symmetric 4 × 4 matrix equation with 10 degrees of freedom. The additional

16− 10 = 6 degrees of freedom are Lorentz transformations, which rotate the tangent axes gµ, but leave the

metric ηµν unchanged.

Just as it is possible to reorient the tangent axes gµ at a point by adjusting the matrix ∂x′ν/∂xµ of first

partial derivatives of the coordinate transformation xµ → x′µ, so also it is possible to reorient the derivatives

∂gµ/∂xν of the tangent axes by adjusting the matrix ∂2x′ν/∂xλ∂xµ of second partial derivatives. The second

partial derivatives comprise a set of 4 symmetric 4×4 matrices, for a total of 4×10 = 40 degrees of freedom.

However, there are 4 × 4 × 4 = 64 connection coefficients Γκµν , all of which the condition (2.43) requires to

vanish. The matrix of second derivatives is thus 64 − 40 = 24 degrees of freedom short of being able to

make all the connections vanish. The resolution of the problem is that, as shown below, equation (2.51),

there are 24 combinations of the connections that form a tensor, the torsion tensor. If a tensor is zero in one

frame, then it is automatically zero in any other frame. Thus the requirement that all the connections vanish

requires that the torsion tensor vanish. This requires, from the expression (2.51) for the torsion tensor, the

no-torsion condition that the connection coefficients are symmetric in their last two indices

Γκµν = Γκ

νµ . (2.44)

It should be emphasized that the condition of vanishing torsion is an assumption of general relativity, not

a mathematical necessity. It has been shown in this section that torsion vanishes if and only if spacetime is

locally flat, meaning that at any point coordinates can be found such that conditions (2.41) are true. The

assumption of local flatness is central to the idea of the principle of equivalence. But it is an assumption,

not a consequence, of the theory.

Concept question 2.1 If torsion does not vanish, then there is no locally inertial frame. What does

parallel-transport mean in such a case?

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2.6 Covariant derivatives 49

2.6.8 Aside: torsion and the integrability of the position vector

From the definitions (2.34) of the connection coefficients, the no-torsion condition (2.44) is equivalent to

∂gµ

∂xν− ∂gν

∂xµ= 0 . (2.45)

According to Frobenius’ theorem, this condition (2.45) is precisely the condition for the system gµ to be

integrable, that is, there exists a position vector X whose partial derivatives are

gµ =∂X

∂xµ. (2.46)

Equivalently, the total differential of the position vector X is

dX = gµdxµ . (2.47)

The abstract vector interval dx ≡ gµ dxµ was defined by equation (2.15) as the coordinate-independent

version of a spacetime interval dxµ. The notation dx was merely symbolic: dx was not necessarily a total

differential of something. However, the no-torsion condition (2.47) implies that dx is in fact the total

differential dX of the position vector X

dx = dX . (2.48)

The no-torsion condition (2.44) is equivalent to the commutation of partial derivatives of X:

Γκµν gκ ≡

∂gµ

∂xν=

∂2X

∂xν∂xµ=∂gν

∂xµ≡ Γκ

νµ gκ . (2.49)

The physical meaning of torsion is discussed further in §3.4.

2.6.9 Torsion tensor

General relativity assumes no torsion, but it is possible to consider generalizations to theories with torsion.

The torsion tensor Sµκλ is defined by the commutator of the covariant derivative acting on a scalar Φ

[Dκ, Dλ] Φ = Sµκλ

∂Φ

∂xµis a coordinate tensor . (2.50)

Note that the covariant derivative of a scalar is just the ordinary derivative, DλΦ = ∂Φ/∂xλ. The expres-

sion (2.39) for the covariant derivatives shows that the torsion tensor is

Sµκλ = Γµ

κλ − Γµλκ is a coordinate tensor (2.51)

which is evidently antisymmetric in the indices κλ.

In Einstein-Cartan theory, the torsion tensor is related to the spin content of spacetime. Since this vanishes

in empty space, Einstein-Cartan theory is indistinguishable from general relativity in experiments carried

out in vacuum.

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50 Fundamentals of General Relativity

Exercise 2.2 Show that

DκAλ −DλAκ =∂Aλ

∂xκ− ∂Aκ

∂xλ+ Sµ

κλAµ . (2.52)

Conclude that, if torsion vanishes as general relativity assumes, Sµκλ = 0, then

DκAλ −DλAκ =∂Aλ

∂xκ− ∂Aκ

∂xλ. (2.53)

2.6.10 Connection coefficients in terms of the metric

The connection coefficients have been defined, equation (2.34), as derivatives of the tangent basis vectors gµ.

However, the connection coefficients can be expressed purely in terms of the (first derivatives of the) metric,

without reference to the individual basis vectors. The partial derivatives of the metric are

∂gλµ

∂xν=∂gλ · gµ

∂xν

= gλ ·∂gµ

∂xν+ gµ ·

∂gλ

∂xν

= gλ · gκ Γκµν + gµ · gκ Γκ

λν

= gλκ Γκµν + gµκ Γκ

λν

= Γλµν + Γµλν , (2.54)

which is a sum of two connection coefficients. Here Γλµν with all indices lowered is defined to be Γκµν with

the first index lowered by the metric,

Γλµν ≡ gλκΓκµν . (2.55)

Combining the metric derivatives in the following fashion yields an expression for a single connection:

∂gλµ

∂xν+∂gλν

∂xµ− ∂gµν

∂xλ= Γλµν + Γµλν + Γλνµ + Γνλµ − Γµνλ − Γνµλ

= 2 Γλµν − Sλµν − Sµνλ − Sνµλ

= 2 Γλµν , (2.56)

the last line of which follows from the no-torsion condition Sλµν = 0. Thus

Γλµν =1

2

(

∂gλµ

∂xν+∂gλν

∂xµ− ∂gµν

∂xλ

)

is not a coordinate tensor . (2.57)

This is the formula that allows connection coefficients to be calculated from the metric.

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2.7 Coordinate 4-velocity 51

2.6.11 Mathematical aside

General relativity is a metric theory. Many of the structures introduced above can be defined mathematically

without a metric. For example, it is possible to define the tangent space of vectors with basis gµ, and to

define a dual vector space with basis gµ such that gµ · gν = δνµ. Elements of the dual vector space are

commonly called one-forms. Similarly it is possible to define connections and covariant derivatives without

a metric. However, this book follows general relativity in assuming that spacetime has a metric.

2.7 Coordinate 4-velocity

Consider a particle following a worldline

xµ(τ) , (2.58)

where τ is the particle’s proper time. The proper time along any interval of the worldline is dτ ≡√−ds2.

Define the coordinate 4-velocity uµ by

uµ ≡ dxµ

dτis a coordinate 4-vector . (2.59)

The magnitude squared of the 4-velocity is constant

uµuµ = gµν

dxµ

dxν

dτ=ds2

dτ2= −1 . (2.60)

The negative sign arises from the choice of metric signature: with the signature −+++ adopted here, there

is a − sign between ds2 and dτ2. Equation (2.60) can be regarded as an integral of motion associated with

conservation of particle rest mass.

2.8 Geodesic equation

Let u ≡ gµuµ be the 4-velocity in coordinate-independent notation. The principle of equivalence implies

that the geodesic equation, the equation of motion of a freely-falling particle, is

du

dτ= 0 . (2.61)

Why? Because du/dτ = 0 in the particle’s own free-fall frame, and the equation is coordinate-independent.

In the particle’s own free-fall frame, the particle’s 4-velocity is uµ = 1, 0, 0, 0, and the particle’s locally

inertial axes gµ = g0, g1, g2, g3 are constant.

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52 Fundamentals of General Relativity

What does the equation of motion look like in coordinate notation? The acceleration is

du

dτ=dxν

∂u

∂xν

= uνgκDνuκ

= uνgκ

(

∂uκ

∂xν+ Γκ

µνuµ

)

= gκ

(

duκ

dτ+ Γκ

µνuµuν

)

. (2.62)

The geodesic equation is then

duκ

dτ+ Γκ

µνuµuν = 0 . (2.63)

Another way of writing the geodesic equation is

Duκ

Dτ= 0 , (2.64)

where D/Dτ is the covariant proper time derivative

D

Dτ≡ uνDν . (2.65)

2.9 Coordinate 4-momentum

The coordinate 4-momentum of a particle of rest mass m is defined to be

pµ ≡ muµ = mdxµ

dτis a coordinate 4-vector . (2.66)

The momentum squared is

pµpµ = m2uµu

µ = −m2 (2.67)

minus the square of the rest mass. Again, the minus sign arises from the choice −+++ of metric signature.

2.10 Affine parameter

For photons, the rest mass is zero, m = 0, but the 4-momentum pµ remains finite. Define the affine

parameter λ by

λ ≡ τ

mis a coordinate scalar (2.68)

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2.11 Affine distance 53

which remains finite in the limit m → 0. The affine parameter λ is unique up to an overall linear transfor-

mation (that is, αλ + β is also an affine parameter, for constant α and β), because of the freedom in the

choice of mass m and the zero point of proper time τ . In terms of the affine parameter, the 4-momentum is

pµ =dxµ

dλ. (2.69)

The geodesic equation is then in coordinate-independent notation

dp

dλ= 0 , (2.70)

or in component form

dpκ

dλ+ Γκ

µνpµpν = 0 , (2.71)

which works for massless as well as massive particles.

Another way of writing this is

Dpκ

Dλ= 0 , (2.72)

where D/Dλ is the covariant affine derivative

D

Dλ≡ pνDν . (2.73)

2.11 Affine distance

The affine parameter is also called the affine distance, because it provides a measure of distance along null

geodesics. When you look at a scene with your eyes, you are looking along null geodesics, and the natural

measure of distance to objects that you see is the affine distance. The freedom in the overall scaling of the

affine distance is fixed by setting it equal to the proper distance near the observer in the observer’s locally

inertial rest frame.

In special relativity, the affine distance coincides with the perceived (e.g. binocular) distance to objects.

2.12 Riemann curvature tensor

The Riemann curvature tensor Rκλµν is defined by the commutator of the covariant derivative acting

on a 4-vector

[Dκ, Dλ]Aµ = RκλµνAν is a coordinate tensor . (2.74)

The expression (2.74) assumes vanishing torsion; the more general expression with non-zero torsion is (3.20).

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54 Fundamentals of General Relativity

The expression (2.39) for the covariant derivative yields the following formula for the Riemann tensor in

terms of connection coefficients

Rκλµν =∂Γµνλ

∂xκ− ∂Γµνκ

∂xλ+ Γα

µλΓανκ − ΓαµκΓανλ is a coordinate tensor . (2.75)

This is the formula that allows the Riemann tensor to be calculated from the connection coefficients.

In flat (Minkowski) space, covariant derivatives reduce to partial derivatives, Dκ → ∂/∂xκ, and

[Dκ, Dλ]→[

∂xκ,∂

∂xλ

]

= 0 in flat space (2.76)

so that Rκλµν = 0 in flat space.

Comment: In quantum field theories (QED, QCD), the commutator of the gauge-covariant derivative is

taken to be the field. In conventional general relativity, by contrast, the metric is taken to be the fundamental

field, rather than the curvature. Another difference between quantum field theories and general relativity is

that the Lagrangian of quantum field theories is taken to be quadratic in the field, whereas the Lagrangian

of general relativity is taken to be linear in the curvature (specifically, the general relativity Lagrangian is

the Ricci scalar R).

2.13 Symmetries of the Riemann tensor

In a locally inertial frame, the connection coefficients all vanish, Γλµν = 0, but their partial derivatives,

which are proportional to second derivatives of the metric tensor, do not vanish. Thus in a locally inertial

frame the Riemann tensor is

Rκλµν =∂Γµνλ

∂xκ− ∂Γµνκ

∂xλ

=1

2

(

∂2gµν

∂xκ∂xλ+

∂2gµλ

∂xκ∂xν− ∂2gνλ

∂xκ∂xµ− ∂2gµν

∂xλ∂xκ− ∂2gµκ

∂xλ∂xν+

∂2gνκ

∂xλ∂xµ

)

=1

2

(

∂2gµλ

∂xκ∂xν− ∂2gνλ

∂xκ∂xµ− ∂2gµκ

∂xλ∂xν+

∂2gνκ

∂xλ∂xµ

)

. (2.77)

You can check that the bottom line of equation (2.77):

1. is antisymmetric in κ↔ λ,

2. is antisymmetric in µ↔ ν,

3. is symmetric in κλ↔ µν,

4. has the property that the sum of the cyclic permutations of the last three indices vanishes

Rκλµν +Rκνλµ +Rκµνλ = 0 . (2.78)

The first three of these four symmetries can be summarized by the shorthand notation

R([κλ][µν]) (2.79)

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2.14 Ricci tensor, Ricci scalar 55

in which [ ] denotes anti-symmetrization and ( ) symmetrization. These symmetries imply that the Riemann

tensor is a symmetric matrix of antisymmetric matrices. An antisymmetric matrix has 6 degrees of freedom.

A symmetric matrix of these things is a 6×6 symmetric matrix, which has 21 degrees of freedom. The final,

cyclic symmetry of the Riemann tensor, equation (2.78), removes 1 degree of freedom. Thus the Riemann

tensor has a net 20 degrees of freedom.

Although the above symmetries were derived in a locally inertial frame, the fact that the Riemann tensor

is a tensor means that the symmetries hold in any frame. If you prefer, you can add back the products of

connection coefficients in equation (2.75), and check that the claimed symmetries remain.

2.14 Ricci tensor, Ricci scalar

The Ricci tensor Rκµ and Ricci scalar R are the essentially unique contractions of the Riemann curvature

tensor. The Ricci tensor, the compressive part of the Riemann tensor, is

Rκµ ≡ gλνRκλµν is a coordinate tensor . (2.80)

The symmetries of the Riemann tensor imply that the Ricci tensor is symmetric

Rκµ = Rµκ (2.81)

and therefore has 10 independent components.

The Ricci scalar is

R ≡ gκµRκµ is a coordinate tensor (a scalar) . (2.82)

2.15 Einstein tensor

The Einstein tensor Gκµ is defined by

Gκµ ≡ Rκµ − 12 gκµR is a coordinate tensor . (2.83)

The symmetry of the Ricci and metric tensors imply that the Einstein tensor is likewise symmetric

Gκµ = Gµκ . (2.84)

The Einstein tensor has 10 independent components.

2.16 Bianchi identities

The Jacobi identity

[Dκ, [Dλ, Dµ]] + [Dλ, [Dµ, Dκ]] + [Dµ, [Dκ, Dλ]] = 0 (2.85)

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56 Fundamentals of General Relativity

implies the Bianchi identities

DκRλµνπ +DλRµκνπ +DµRκλνπ = 0 (2.86)

which can be written in shorthand

D[κRλµ]νπ = 0 . (2.87)

The Bianchi identities constitute a set of differential relations between the components of the Riemann

tensor, which are distinct from the algebraic symmetries of the Riemann tensor.

There are 20 independent Bianchi identities. If just the symmetries (2.79) of the Riemann tensor are

taken into account, then there are 24 identities; but the cyclic symmetry (2.78) eliminates 4, leaving 20

independent identities.

2.17 Covariant conservation of the Einstein tensor

The most important consequence of the Bianchi identities (2.87) is obtained from the double contraction

gκνgλπ (DκRλµνπ +DλRµκνπ +DµRκλνπ) = −DκRκµ −DλRλµ +DµR = 0 (2.88)

which implies that

DκGκµ = 0 . (2.89)

This equation is a primary motivation for the form of the Einstein equations, since it implies energy-

momentum conservation, equation (2.91).

2.18 Einstein equations

Einstein’s equations are

Gκµ = 8πGTκµ is a coordinate tensor equation . (2.90)

What motivates the form of Einstein’s equations?

1. The equation is generally covariant;

2. The Bianchi identities guarantee conservation of energy-momentum;

3. The Einstein tensor depends on the lowest (second) order derivatives of the metric tensor that do not

vanish in a locally inertial frame;

The covariant conservation of the Einstein tensor, equation (2.89), implies the conservation of energy-

momentum

DκTκµ = 0 . (2.91)

Einstein’s equations (2.90) constitute a complete set of gravitational equations, generalizing Poisson’s

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2.19 Summary of the path from metric to the energy-momentum tensor 57

equation of Newtonian gravity. However, Einstein’s equations by themselves do not constitute a closed set

of equations: in general, other equations, such as Maxwell’s equations of electromagnetism, and equations

describing the microphysics of the energy-momentum, must be adjoined to form a closed set.

2.19 Summary of the path from metric to the energy-momentum tensor

1. Start by defining the metric gµν .

2. Compute the connection coefficients Γλµν from equation (2.57).

3. Compute the Riemann tensor Rκλµν from equation (2.75).

4. Compute the Ricci tensor Rκµ from equation (2.80), the Ricci scalar R from equation (2.82), and the

Einstein tensor Gκµ from equation (2.83).

5. The Einstein equations (2.90) then imply the energy-momentum tensor Tκµ.

The path from metric to energy-momentum tensor is straightforward to program on a computer, but the

results are typically messy and complicated, even for fairly simple spacetimes. Inverting the path to recover

the metric from a given energy-momentum content is typically highly non-trivial, the subject of a huge

literature.

The great majority of metrics gµν yield an energy-momentum tensor Tκµ that cannot be achieved with

normal matter.

2.20 Energy-momentum tensor of an ideal fluid

The simplest non-trival energy-momentum tensor is that of an ideal fluid. In this case T µν is isotropic in

the locally inertial rest frame of the fluid, taking the form

T µν =

ρ 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

(2.92)

where

ρ is the proper mass-energy density ,

p is the proper pressure .(2.93)

The expression (2.92) is valid only in the locally inertial rest frame of the fluid. An expression that is valid

in any frame is

T µν = (ρ+ p)uµuν + p gµν , (2.94)

where uµ is the 4-velocity of the fluid. Equation (2.94) is valid because it is a tensor equation, and it is true

in the locally inertial rest frame, where uµ = 1, 0, 0, 0.

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58 Fundamentals of General Relativity

2.21 Newtonian limit

The Newtonian limit is obtained in the limit of a weak gravitational field and non-relativistic (pressureless)

matter. In Cartesian coordinates, the metric in the Newtonian limit is

ds2 = − (1 + 2Φ)dt2 + (1− 2Φ)(dx2 + dy2 + dz2) , (2.95)

in which

Φ(x, y, z) = Newtonian potential (2.96)

is a function only of the spatial coordinates x, y, z, not of time t.

For this metric, to first order in the potential Φ the only non-vanishing component of the Einstein tensor

is the time-time component

Gtt = 2∇2Φ , (2.97)

where ∇2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 is the usual 3D Laplacian operator. This component (2.97) of the

Einstein tensor plugged into Einstein’s equations (2.90) implies Poisson’s equation (2.4).

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3

∗More on the coordinate approach

3.1 Weyl tensor

The trace-free, or tidal, part of the Riemann curvature tensor defines the Weyl tensor Cκλµν

Cκλµν ≡ Rκλµν − 12 (gκµRλν − gκνRλµ + gλνRκµ − gλµRκν) + 1

6 (gκµgλν − gκνgλµ)R is a coordinate tensor .

(3.1)

The Weyl tensor has 10 independent components. These 10 components together with the 10 components

of the Ricci tensor account for the 20 components of the Riemann tensor. The Weyl tensor inherits the

symmetries (2.79) of the Riemann tensor

C([κλ][µν]) . (3.2)

Whereas the Einstein tensor Gκµ, necessarily vanishes in a region of spacetime where there is no energy-

momentum, Tκµ = 0, the Weyl tensor does not. The Weyl tensor expresses the presence of tidal gravitational

forces, and of gravitational waves.

3.2 Evolution equations for the Weyl tensor

This section is included because (a) the comparison to Maxwell’s equations is neat and insightful, (b) it helps

to account for the degrees of freedom of the gravitational field, (c) it shows how the Weyl tensor encodes

gravitational waves.

Contracted on one index, the Bianchi identities (2.86) are

D[κRλµ]νκ = 0 . (3.3)

There are 20 such independent contracted identities. Since this is the same as the number of independent

Bianchi identities, it follows that the contracted Bianchi identities (3.3) are equivalent to the full set of

Bianchi identities (2.87). If the Riemann tensor is separated into its trace (Ricci) and traceless (Weyl) parts,

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60 ∗More on the coordinate approach

equation (3.1), then the contracted Bianchi identities (3.3) become the Weyl evolution equations

DκCκλµν = Jλµν , (3.4)

where Jλµν is the Weyl current

Jλµν ≡ 12 (DµGλν −DνGλµ)− 1

6 (gλνDµG− gλµDνG) . (3.5)

The Weyl evolution equations (3.4) can be regarded as the gravitational analogue of Maxwell’s equations of

electromagnetism.

The Weyl current Jλµν is a vector of bivectors, which would suggest that it has 4 × 6 = 24 components,

but it loses 4 of those components because of the cyclic identity (2.78), which implies the cyclic symmetry

J[λµν] = 0 . (3.6)

Thus Jλµν has 20 independent components, in agreement with the above assertion that there are 20 inde-

pendent contracted Bianchi identities. Since the Weyl tensor is traceless, contracting the Weyl evolution

equations (3.4) on λν yields zero on the left hand side, so that the contracted Weyl current satisfies

Jλλµ = 0 . (3.7)

This doubly-contracted Bianchi identity, which is the same as equation (2.89), enforces conservation of

energy-momentum. Unlike the cyclic symmetries (3.6), which are automatically satisfied, equations (3.7)

constitute a non-trivial set of 4 conditions on the Einstein tensor. Besides the algebraic relations (3.6) and

(3.7), the Weyl current satisfies 6 differential identities comprising the conservation law

DλJλµν = 0 (3.8)

in view of equation (3.4) and the antisymmetry of Cκλµν with respect to the indices κλ. The Weyl current

conservation law (3.8) follows automatically from the form (3.5) of the Weyl current, coupled with energy-

momentum conservation (2.89), so does not impose any additional non-trivial conditions on the Riemann

tensor.

The 4 relations (3.7) and the 6 identities (3.8) account for 10 of the 20 contracted Bianchi identities (3.3).

The remaining 10 equations comprise Maxwell-like equations (3.4) for the evolution of the 10 components

of the Weyl tensor.

Whereas the Einstein equations relating the Einstein tensor to the energy-momentum tensor are postulated

equations of general relativity, the 10 evolution equations for the Weyl tensor, and the 4 equations enforcing

covariant conservation of the Einstein tensor, follow mathematically from the Bianchi identities, and do not

represent additional assumptions of the theory.

Exercise 3.1 Confirm the counting of degrees of freedom. ⋄

Exercise 3.2 From the Bianchi identities, show that the Riemann tensor satisfies the covariant wave

equation

Rκλµν = DκDµRλν −DκDνRλµ +DλDνRκµ −DλDµRκν , (3.9)

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3.3 Geodesic deviation 61

where is the D’Alembertian operator, the 4-dimensional wave operator

≡ DπDπ . (3.10)

Show that contracting equation (3.9) with gλν yields the identity Rκµ = Rκµ. Conclude that the wave

equation (3.9) is non-trivial only for the trace-free part of the Riemann tensor, the Weyl tensor Cκλµν . Show

that the wave equation for the Weyl tensor is

Cκλµν = (DκDµ − 12 gκµ )Rλν − (DκDν − 1

2 gκν )Rλµ

+ (DλDν − 12 gλν )Rκµ − (DλDµ − 1

2 gλµ )Rκν

+ 16 (gκµgλν − gκνgλµ)R . (3.11)

Conclude that in a vacuum, where Rκµ = 0,

Cκλµν = 0 . (3.12)

3.3 Geodesic deviation

This section on geodesic deviation is included not because the equation of geodesic deviation is crucial to

everyday calculations in general relativity, but rather for two reasons. First, the equation offers insight into

the physical meaning of the Riemann tensor. Second, the derivation of the equation offers a fine illustration

of the fact that in general relativity, whenever you take differences at infinitesimally separated points in

space or time, you should always take covariant differences.

Consider two objects that are free-falling along two infinitesimally separated geodesics. In flat space the

acceleration between the two objects would be zero, but in curved space the curvature induces a finite

acceleration between the two objects. This is how an observer can measure curvature, at least in principle:

set up an ensemble of objects initially at rest a small distance away from the observer in the observer’s

locally inertial frame, and watch how the objects begin to move. The equation (3.18) that describes this

acceleration between objects an infinitesimal distance apart is called the equation of geodesic deviation.

The covariant difference in the velocities of two objects an infinitesimal distance δxµ apart is

Dδxµ

Dτ= δuµ . (3.13)

In general relativity, the ordinary difference between vectors at two points a small interval apart is not

a physically meaningful thing, because the frames of reference at the two points are different. The only

physically meaningful difference is the covariant difference, which is the difference in the two vectors parallel-

transported across the gap between them. It is only this covariant difference that is independent of the frame

of reference. On the left hand side of equation (3.13), the proper time derivative must be the covariant proper

time derivative, D/Dτ = uλDλ. On the right hand side of equation (3.13), the difference in the 4-velocity

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62 ∗More on the coordinate approach

at two points δxκ apart must be the covariant difference δ = δxκDκ. Thus equation (3.13) means explicitly

the covariant equation

uλDλδxµ = δxκDκu

µ . (3.14)

To derive the equation of geodesic deviation, first vary the geodesic equation Duµ/Dτ = 0 (I’ve put the

index µ downstairs so that the final equation (3.18) looks cosmetically better, but of course since everything

is covariant the µ index could just as well be put upstairs everywhere):

0 = δDuµ

= δxκDκ

(

uλDλuµ

)

= δuλDλuµ + δxκuλDκDλuµ . (3.15)

On the second line, the covariant diffence δ between quantities a small distance δxκ apart has been set equal

to δxκDκ, while D/Dτ has been set equal to the covariant time derivative uλDλ along the geodesic. On the

last line, δxκDκuλ has been replaced by δuµ. Next, consider the covariant acceleration of the interval δxµ,

which is the covariant proper time derivative of the covariant velocity difference δuµ:

D2δxµ

Dτ2=Dδuµ

= uλDλ (δxκDκuµ)

= δuκDκuµ + δxκuλDλDκuµ . (3.16)

As in the previous equation (3.15), on the second line D/Dτ has been set equal to uλDλ, while δ has been

set equal to δxκDκ. On the last line, uλDλδxκ has been set equal to δuµ, equation (3.14). Subtracting (3.15)

from (3.16) gives

D2δxµ

Dτ2= δxκuλ[Dλ, Dκ]uµ , (3.17)

or equivalently

D2δxµ

Dτ2+Rκλµνδx

κuλuν = 0 , (3.18)

which is the desired equation of geodesic deviation.

3.4 Commutator of the covariant derivative revisited

The commutator of the covariant derivative is of fundamental importance because it defines what is meant

by the field in gauge theories. It was seen above that the commutator of the covariant derivative acting

on a scalar defined the torsion tensor, equation (2.50), which general relativity assumes vanishes, while the

commutator of the covariant derivative acting on a vector defined the Riemann tensor, equation (2.74). Does

the commutator of the covariant derivative acting on a general tensor introduce any other distinct tensor? No:

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3.4 Commutator of the covariant derivative revisited 63

the torsion and Riemann tensors completely define the action of the commutator of the covariant derivative

on any tensor, equation (3.22).

The general expression (3.22) for the commutator of the covariant derivative reveals the meaning of the

torsion and Riemann tensors. The torsion and Riemann tensors describe respectively the displacement and

the Lorentz transformation experienced by an object when parallel-transported around a curve. Displacement

and Lorentz transformations together constitute the Poincare group, the complete group of symmetries of

flat spacetime.

How can an object detect a displacement when parallel-transported around a curve? If you go around

a curve back to the same coordinate in spacetime where you began, won’t you necessarily be at the same

position? This is a question that goes to heart of the meaning of spacetime. To answer the question, you

have to consider how fundamental particles are able to detect position, orientation, and velocity. Classically,

particles may be structureless points, but quantum mechanically, particles possess frequency, wavelength,

spin, and (in the relativistic theory) boost, and presumably it is these properties that allow particles to

“measure” the properties of the spacetime in which they live. Specifically, a Dirac spinor (relativistic spin- 12

particle) has 8 degrees of freedom, of which 6 define a Lorentz transformation (a Lorentz rotor, a member of

the group of spin- 12 Lorentz transformations), and the remaining 2 comprise a complex number REALLY?

THE COMPLEX NUMBER IS WITH RESPECT TO THE PSEUDOSCALAR I ∼ e−ip·x whose phase

encodes the displacement. Thus a Dirac spinor could potentially detect a displacement through a change in

its phase when parallel-transported around a curve back to the same point in spacetime. General relativity,

which assumes that torsion vanishes, asserts that there is no such change of phase.

In the presence of torsion, the expression for the connection coefficients Γλµν is, from equation (2.57),

Γλµν =1

2

(

∂gλµ

∂xν+∂gλν

∂xµ− ∂gµν

∂xλ+ Sλµν + Sµνλ + Sνµλ

)

. (3.19)

The first part 12 (gλµ,ν + gλν,µ − gµν,λ) of this expression is called the Christoffel symbol of the first kind [the

same thing with the first index raised, 12g

κλ (gλµ,ν + gλν,µ − gµν,λ), is called the Christoffel symbol of the

second kind], while the second part 12 (Sλµν + Sµνλ + Sνµλ) is called the contortion (not contorsion!) tensor.

There’s no need to remember the crazy jargon, but in case you meet it, that’s what it means.

If torsion does not vanish, then the commutator of the covariant derivative acting on a contravariant

4-vector is

[Dκ, Dλ]Aµ = SνκλDνAµ +RκλµνA

ν is a coordinate tensor (3.20)

where the Riemann tensor Rκλµν is given in terms of the connection coefficients by the same formula (2.75)

as before, but the connection coefficients Γλµν themselves are given by (3.19). The Riemann tensor is still

antisymmetric in each of κ↔ λ and µ↔ ν, but with torsion it is no longer symmetric in κλ↔ µν. In other

words, the symmetries of the Riemann tensor with torsion are

R[κλ][µν] . (3.21)

As a matrix of antisymmetric tensors, the Riemann tensor with torsion has 6 × 6 = 36 degrees of freedom.

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64 ∗More on the coordinate approach

Because the Riemann tensor Rκλµν is no longer symmetric in κλ↔ µν, the Ricci tensor Rκµ ≡ gλνRκλµν is

no longer symmetric, and likewise the Einstein tensor Gκµ ≡ Rκµ− 12Rgκµ is no longer symmetric. Evidently

the antisymmetric part of the Einstein tensor depends on torsion.

Acting on a general tensor, the commutator of the covariant derivative is

[Dκ, Dλ]Aπρ...µν... = Sα

κλDαAπρ...µν... +Rκλµ

αAπρ...αν... +Rκλν

αAπρ...µα... −Rκλα

πAαρ...µν... −Rκλα

ρAπα...µν... . (3.22)

In more abstract notation, the commutator of the covariant derivative is the operator

[Dκ, Dλ] = Sκλ ·D + Rκλ (3.23)

where Sκλ ≡ gµSµκλ and D ≡ gµDµ, and the Riemann curvature operator Rκλ is an operator whose action

on any tensor is specified by equation (3.22). The action of the operator Rκλ is analogous to that of the

covariant derivative (2.40): there’s a positive R term for each covariant index, and a negative R term for each

contravariant index. The action of Rκλ on a scalar is zero, which reflects the fact that a scalar is unchanged

by a Lorentz transformation.

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4

∗Action principle

Hamilton’s principle of least action postulates that any dynamical system is characterized by a scalar action

S, which has the property that when the system evolves from one specified state to another, the path by

which it gets between the two states is such as to minimize the action. The action need not be a global

minimum, just a local minimum with respect to small variations in the path between fixed initial and final

states.

That nature appears to respect the principle of least action is of the profoundest significance.

x0

x1

λ

Figure 4.1 The action principle considers various paths through spacetime between fixed initial and finalconditions, and chooses that path that minimizes the action.

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66 ∗Action principle

4.1 Principle of least action for point particles

The path of a point particle through spacetime is specified by its coordinates xµ(λ) as a function of some

arbitrary parameter λ. In non-relativistic mechanics it is usual to take the parameter λ to be the time t,

and the path of a particle through space is then specified by three spatial coordinates xi(t). In relativity

however it is more natural to treat the time and space coordinates on an equal footing, and regard the path

of a particle as being specified by four spacetime coordinates xµ(λ) as a function of an arbitrary parameter

λ. The parameter λ is simply a continuous parameter that labels points along the path, and has no physical

significance (for example, it is not necessarily an affine parameter).

The path of a system of N point particles through spacetime is specified by 4N coordinates xµ(λ). The

action principle postulates that, for a system of N point particles, the action S is an integral of a Lagrangian

L(xµ, dxµ/dλ) which is a function of the 4N coordinates xµ(λ) together with the 4N velocities dxµ/dλ with

respect to the arbitrary parameter λ. The action from an initial state at λi to a final state at λf is thus

S =

∫ λf

λi

L

(

xµ,dxµ

)

dλ . (4.1)

The principle of least action demands that the actual path taken by the system between given initial and

final coordinates xµi and xµ

f is such as to minimize the action. Thus the variation δS of the action must be

zero under any change δxµ in the path, subject to the constraint that the coordinates at the endpoints are

fixed, δxµi = 0 and δxµ

f = 0,

δS =

∫ λf

λi

(

∂L

∂xµδxµ +

∂L

∂(dxµ/dλ)δ(dxµ/dλ)

)

dλ = 0 . (4.2)

The change in the velocity along the path is just the velocity of the change, δ(dxµ/dλ) = d(δxµ)/dλ.

Integrating the second term in the integrand of equation (4.2) by parts yields

δS =

[

∂L

∂(dxµ/dλ)δxµ

]λf

λi

+

∫ λf

λi

(

∂L

∂xµ− d

∂L

∂(dxµ/dλ)

)

δxµ dλ = 0 . (4.3)

The surface term in equation (4.3) vanishes, since by hypothesis the coordinates are held fixed at the end

points, so δxµ = 0 at the end points. Therefore the integral in equation (4.3) must vanish. Indeed least

action requires the integral to vanish for all possible variations δxµ in the path. The only way this can

happen is that the integrand must be identically zero. The result is the Euler-Lagrange equations of

motion

d

∂L

∂(dxµ/dλ)− ∂L

∂xµ= 0 . (4.4)

It might seem that the Euler-Lagrange equations (4.4) are inadequately specified, since they depend on

some arbitrary unknown parameter λ. But in fact the Euler-Lagrange equations are the same regardless of

the choice of λ. An example of the irrelevance of λ will be seen in the next section, §4.2. Since λ can be

chosen arbitrarily, it is usual to choose it in some convenient fashion. For a massive particle, λ can be taken

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4.2 Action for a test particle 67

equal to the proper time τ of the particle. For a massless particle, whose proper time never progresses, λ

can be taken equal to an affine parameter.

4.2 Action for a test particle

According to the principle of equivalence, a test particle in a gravitating system moves along a geodesic, a

straight line relative to local free-falling frames. A geodesic is the shortest distance between two points. In

relativity this translates, for a massive particle, into the longest proper time between two points. The proper

time along any path is dτ =√−ds2 =

−gµνdxµdxν . Thus the action Sm of a test particle of rest mass m

in a gravitating system is

Sm = −m∫ λf

λi

dτ = −m∫ λf

λi

−gµνdxµ

dxν

dλdλ . (4.5)

The factor of rest massm brings the action, which has units of angular momentum, to standard normalization.

The overall minus sign comes from the fact that the action is a minimum whereas the proper time is a

maximum along the path. The action principle requires that the Lagrangian be written as a function

of the coordinates xµ and velocities dxν/dλ, and it is seen that the integrand in the last expression of

equation (4.5) has the desired form, the metric gµν being considered a given function of the coordinates.

Thus the Lagrangian Lm of a test particle of mass m is

Lm = −m√

−gµνdxµ

dxν

dλ. (4.6)

The partial derivatives that go in the Euler-Lagrange equations (4.4) are then

∂Lm

∂(dxκ/dλ)= −m

−gκνdxν

dλ√

−gπρ(dxπ/dλ)(dxρ/dλ), (4.7a)

∂Lm

∂xκ= −m

−1

2

∂gµν

dxκ

dxµ

dxν

dλ√

−gπρ(dxπ/dλ)(dxρ/dλ). (4.7b)

The denominators in the expressions (4.7) for the partial derivatives of the Lagrangian are√

−gπρ(dxπ/dλ)(dxρ/dλ) = dτ/dλ. It was not legitimate to make this substitution before taking the partial

derivatives, since the Euler-Lagrange equations require that the Lagrangian be expressed in terms of xµ and

dxµ/dλ, but it is fine to make the substitution now that the partial derivatives have been obtained. The

partial derivatives (4.7) thus simplify to

∂Lm

∂(dxκ/dλ)= mgκν

dxν

dτ= muκ , (4.8)

∂Lm

∂xκ=

1

2m∂gµν

dxκ

dxµ

dxν

dτ= mΓµνκu

µuν dτ

dλ, (4.9)

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68 ∗Action principle

in which uκ ≡ dxκ/dτ is the usual 4-velocity, and the derivative of the metric has replaced by connections

in accordance with equation (2.54). The resulting Euler-Lagrange equations of motion (4.4) are

dmuκ

dλ= mΓµνκu

µuν dτ

dλ. (4.10)

As remarked in §4.1, the choice of the arbitrary parameter λ has no effect on the equations of motion.

With a factor of mdτ/dλ cancelled, and with the derivative converted to a covariant derivative by (2.39),

equation (4.10) becomes

Duκ

Dτ= Sµνκu

µuν , (4.11)

where Sµνκ is the torsion, equation (2.51). If torsion vanishes, as general relativity assumes, then the result

is the usual equation of geodesic motion

Duκ

Dτ= 0 . (4.12)

The fact that motion is geodesic only if torsion vanishes is to be expected, since, as argued in §2.6.7, space

is locally inertial only if torsion vanishes.

4.3 Action for a charged test particle in an electromagnetic field

Aim is to reproduce the Lorentz force law.

S = Sm + Sq (4.13)

where

Sq = q

∫ λf

λi

Aµ dxµ = q

∫ λf

λi

Aµdxµ

dλdλ . (4.14)

The Lagrangian is therefore

Lq = qAµdxµ

dλ. (4.15)

Partial derivatives are

∂Lq

∂(dxκ/dλ)= qAκ ,

∂Lq

∂xκ= q

∂Aµ

∂xκ

dxµ

dλ= q

∂Aµ

∂xκuµ dτ

dλ. (4.16a)

Applied to the Lagrangian L = Lm + Lq, the Euler-Lagrange equations (4.4) are

d

dλ(muκ + qAκ) =

(

mΓµνκuµuν + q

∂Aµ

∂xκuµ

)

dλ. (4.17)

If torsion vanishes, as general relativity assumes, then the result is the Lorentz force law for a test particle

of mass m and charge q moving in a prescribed gravitational and electromagnetic field

Dmuκ

Dτ= qFµκu

µ . (4.18)

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4.4 Generalized momentum 69

Exercise 4.1 Show that if torsion does not vanish, then the Lorentz force law becomes

Dmuκ

Dτ= [qFµκ + Sνµκ (muν + qAν)]uµ . (4.19)

[Hint: Recall the relations (2.51) and (2.52).] ⋄

4.4 Generalized momentum

The generalized momentum

πκ ≡∂L

∂(dxκ/dλ). (4.20)

The generalized momentum πκ of a test particle coincides with the ordinary momentum pκ:

πκ = pκ = muκ . (4.21)

The generalized momentum of a test particle of charge q in an electromagnetic field of potential Aκ

πκ = pκ + qAκ . (4.22)

4.5 Hamiltonian

Work with coordinates and generalized momenta instead of coordinates and velocities. Define Hamiltonian

H by

H ≡ πµdxµ

dλ− L (4.23)

S =

∫(

πµdxµ

dλ−H

)

dλ (4.24)

δS = [πµδxµ] +

−(

dπµ

dλ+∂H

∂xµ

)

δxµ +

(

dxµ

dλ− ∂H

∂πµ

)

δπµ

dλ (4.25)

dπµ

dλ= − ∂H

∂xµ,

dxµ

dλ=∂H

∂πµ. (4.26)

What is this useful for?

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70 ∗Action principle

4.6 Derivatives of the action

Besides being a scalar whose minimum value between fixed endpoints defines the path between those points,

the action can also be treated as a function of its endpoints along the actual path. Along the actual path,

the equations of motion are satisfied, so the integral in the variation (4.3) of the action vanishes identically.

The surface term in the variation (4.3) then implies that δS = πµδxµ, so that the partial derivatives of the

action with respect to the coordinates are equal to the generalized momenta,

∂S

∂xµ= πµ . (4.27)

This is the basis of the Hamilton-Jacobi method.

Page 83: General Relativity, Black Holes, And Cosmology

PART THREE

IDEAL BLACK HOLES

Page 84: General Relativity, Black Holes, And Cosmology
Page 85: General Relativity, Black Holes, And Cosmology

Concept Questions

1. What evidence do astronomers currently accept as indicating the presence of a black hole in a system?

2. Why can astronomers measure the masses of supermassive black holes only in relatively nearby galaxies?

3. To what extent (with what accuracy) are real black holes in our Universe described by the no-hair theorem?

4. Does the no-hair theorem apply inside a black hole?

5. Black holes lose their hair on a light-crossing time. How long is a light-crossing time for a typical stellar-

sized or supermassive astronomical black hole?

6. Relativists say that the metric is gµν , but they also say that the metric is ds2 = gµν dxµdxν . How can

both statements be correct?

7. The Schwarzschild geometry is said to describe the geometry of spacetime outside the surface of the Sun

or Earth. But the Schwarzschild geometry supposedly describes non-rotating masses, whereas the Sun

and Earth are rotating. If the Sun or Earth collapsed to a black hole conserving their mass M and angular

momentum L, roughly what would the spin a/M = L/M2 of the black hole be relative to the maximal

spin a/M = 1 of a Kerr black hole?

8. What happens at the horizon of a black hole?

9. As cold matter becomes denser, it goes through the stages of being solid/liquid like a planet, then electron

degenerate like a white dwarf, then neutron degenerate like a neutron star, then finally it collapses to a

black hole. Why could there not be a denser state of matter, denser than a neutron star, that brings a

star to rest inside its horizon?

10. How can an observer determine whether they are “at rest” in the Schwarzschild geometry?

11. An observer outside the horizon of a black hole never sees anything pass through the horizon, even to the

end of the Universe. Does the black hole then ever actually collapse, if no one ever sees it do so?

12. If nothing can ever get out of a black hole, how does its gravity get out?

13. Why did Einstein believe that black holes could not exist in nature?

14. In what sense is a rotating black hole “stationary”, but not “static”?

15. What is a white hole? Do they exist?

16. Could the expanding Universe be a white hole?

17. Could the Universe be the interior of a black hole?

18. You know the Schwarzschild metric for a black hole. What is the corresponding metric for a white hole?

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74 Concept Questions

19. What is the best kind of black hole to fall into if you want to avoid being tidally torn apart?

20. Why do astronomers often assume that the inner edge of an accretion disk around a black hole occurs at

the innermost stable orbit?

21. A collapsing star of uniform density has the geometry of a collapsing Friedmann-Robertson-Walker cos-

mology. If a spatially flat FRW cosmology corresponds to a star that starts from zero velocity at infinity,

then to what do open or closed FRW cosmologies correspond?

22. Is the singularity of a Reissner-Nordstrom black hole gravitationally attractive or repulsive?

23. If you are a charged particle, which dominates near the singularity of the Reissner-Nordstrom geometry,

the electrical attraction/repulsion or the gravitational attraction/repulsion?

24. Is a white hole gravitationally attractive or repulsive?

25. What happens if you fall into a white hole?

26. Which way does time go in Parallel Universes in the Reissner-Nordstrom geometry?

27. What does it mean that geodesics inside a black hole can have negative energy?

28. Can geodesics have negative energy outside a black hole? How about inside the ergosphere?

29. Physically, what causes mass inflation?

30. Is mass inflation likely to occur inside real astronomical black holes?

31. What happens at the X point, where the ingoing and outgoing inner horizons of the Reissner-Nordstrom

geometry intersect?

32. Can a particle like an electron or proton, whose charge far exceeds its mass (in geometric units), be

modeled as Reissner-Nordstrom black hole?

33. Does it makes sense that a person might be at rest in the Kerr-Newman geometry? How would the

Boyer-Linquist coordinates of such a person vary along their worldline?

34. In identifying M as the mass and a the angular momentum per unit mass of the black hole in the Boyer-

Linquist metric, why is it sufficient to consider the behaviour of the metric at r →∞?

35. Does space move faster than light inside the ergosphere?

36. If space moves faster than light inside the ergosphere, why is the outer boundary of the ergosphere not a

horizon?

37. Do closed timelike curves make sense?

38. What does Carter’s fourth integral of motion Q signify physically?

39. What is special about a principal null congruence?

40. Evaluated in the locally inertial frame of a principal null congruence, the spin-0 component of the Weyl

scalar of the Kerr geometry is C = −M/(r− ia cosθ)3, which looks like the Weyl scalar C = −M/r3 of the

Schwarzschild geometry but with radius r replaced by the complex radius r− ia cos θ. Is there something

deep here? Can the Kerr geometry be constructed from the Schwarzschild geometry by complexifying the

radial coordinate r?

Page 87: General Relativity, Black Holes, And Cosmology

What’s important?

1. Astronomical evidence suggests that stellar-sized and supermassive black holes exist ubiquitously in nature.

2. The no-hair theorem, and when and why it applies.

3. The physical picture of black holes as regions of spacetime where space is falling faster than light.

4. A physical understanding of how the metric of a black hole relates to its physical properties.

5. Penrose (conformal) diagrams. In particular, the Penrose diagrams of the various kinds of vacuum black

hole: Schwarzschild, Reissner-Nordstrom, Kerr-Newman.

6. What really happens inside black holes. Collapse of a star. Mass inflation instability.

Page 88: General Relativity, Black Holes, And Cosmology

5

Observational Evidence for Black Holes

It is beyond the scope of this course to discuss the observational evidence for black holes in any detail.

However, it is useful to summarize a few facts.

1. Observational evidence supports the idea that black holes occur ubiquitously in nature. They are not

observed directly, but reveal themselves through their effects on their surroundings. Two kinds of black

hole are observed: stellar-sized black holes in x-ray binary systems, mostly in our own Milky Way galaxy,

and supermassive black holes in Active Galactic Nuclei (AGN) found at the centers of our own and other

galaxies.

2. The primary evidence that astronomers accept as indicating the presence of a black hole is a lot of mass

compacted into a tiny space.

a. In an x-ray binary system, if the mass of the compact object exceeds 3 M⊙, the maximum theoretical

mass of a neutron star, then the object is considered to be a black hole. Many hundreds of x-ray binary

systems are known in our Milky Way galaxy, but only 10s of these have measured masses, and in about

20 the measured mass indicates a black hole.

b. Several tens of thousands of AGN have been cataloged, identified either in the radio, optical, or x-rays.

But only in nearby galaxies can the mass of a supermassive black hole be measured directly. This is

because it is only in nearby galaxies that the velocities of gas or stars can be measured sufficiently close

to the nuclear center to distinguish a regime where the velocity becomes constant, so that the mass

can be attribute to an unresolved central point as opposed to a continuous distribution of stars. The

masses of about 40 supermassive black holes have been measured in this way. The masses range from

the 4 × 106 M⊙ mass of the black hole at the center of the Milky Way to the 3 × 109 M⊙ mass of the

black hole at the center of the M87 galaxy at the center of the Virgo cluster at the center of the Local

Supercluster of galaxies.

3. Secondary evidences for the presence of a black hole are:

a. high luminosity;

b. non-stellar spectrum, extending from radio to gamma-rays;

c. rapid variability.

d. relativistic jets.

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Observational Evidence for Black Holes 77

Jets in AGN are often one-sided, and a few that are bright enough to be resolved at high angular resolution

show superluminal motion. Both evidences indicate that jets are commonly relativistic, moving at close

to the speed of light. There are a few cases of jets in x-ray binary systems.

4. Stellar-sized black holes are thought to be created in supernovae as the result of the core-collapse of

stars more massive than about 25 M⊙ (this number depends in part on uncertain computer simulations).

Supermassive black holes are probably created initially in the same way, but they then grow by accretion of

gas funnelled to the center of the galaxy. The growth rates inferred from AGN luminosities are consistent

with this picture.

5. Long gamma-ray bursts (lasting more than about 2 seconds) are associated observationally with super-

novae. It is thought that in such bursts we are seeing the formation of a black hole. As the black hole gulps

down the huge quantity of material needed to make it, it regurgitates a relativistic jet that punches through

the envelope of the star. If the jet happens to be pointed in our direction, then we see it relativistically

beamed as a gamma-ray burst.

6. Astronomical black holes present the only realistic prospect for testing general relativity in the strong field

regime, since such fields cannot be reproduced in the laboratory. At the present time the observational

tests of general relativity from astronomical black holes are at best tentative. One test is the redshifting

of 7 keV iron lines in a small number of AGN, notably MCG-6-30-15, which can be interpreted as being

emitted by matter falling on to a rotating (Kerr) black hole.

7. At present, no gravitational waves have been definitely detected from anything. In the future, gravitational

wave astronomy should eventually detect the merger of two black holes. If the waveforms of merging black

holes are consistent with the predictions of general relativity, it will provide a far more stringent test of

strong field general relativity than has been possible to date.

8. Although gravitational waves have yet to be detected directly, their existence has been inferred from the

gradual speeding up of the orbit of the Hulse-Taylor binary, which consists of two neutron stars, one of

which, PSR1913+16, is a pulsar. The parameters of the orbit have been measured with exquisite precision,

and the rate of orbital speed-up is in good agreement with the energy loss by quadrupole gravitational

wave emission predicted by general relativity.

Page 90: General Relativity, Black Holes, And Cosmology

6

Ideal Black Holes

6.1 Definition of a black hole

What is a black hole? Doubtless you have heard the standard definition many times: It is a region whose

gravity is so strong that not even light can escape.

But why can light not escape from a black hole? A standard answer, which John Michell (1784, Phil.

Trans. Roy. Soc. London 74, 35) would have found familiar, is that the escape velocity exceeds the speed of

light. But that answer brings to mind a Newtonian picture of light going up, turning around, and coming

back down, that is altogether different from what general relativity actually predicts.

A better definition of a black hole is that it is a

region where space is falling faster than light.

Inside the horizon, light emitted outwards is carried inward by the faster-than-light inflow of space, like a

fish trying but failing to swim up a waterfall.

The definition may seem jarring. If space has no substance, how can it fall faster than light? It means

that inside the horizon any locally inertial frame is compelled to fall to smaller radius as its proper time goes

by. This fundamental fact is true regardless of the choice of coordinates.

A similar concept of space moving arises in cosmology. Astronomers observe that the Universe is expand-

ing. Cosmologists find it convenient to conceptualize the expansion by saying that space itself is expanding.

For example, the picture that space expands makes it more straightforward, both conceptually and math-

ematically, to deal with regions of spacetime beyond the horizon, the surface of infinite redshift, of an

observer.

6.2 Ideal black hole

The simplest kind of black hole, an ideal black hole, is one that is stationary, electrovac outside its singularity,

and extends to asymptotically flat empty space at infinity. Electrovac means that the energy-momentum

tensor Tµν is zero except for the contribution from a stationary electromagnetic field.

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6.3 No-hair theorem 79

The next several chapters deal with ideal black holes. The importance of ideal black holes stems from

the no-hair theorem, discussed in the next section. The no-hair theorem has the consequence that, except

during their initial collapse, or during a merger, real astronomical black holes are accurately described as

ideal outside their horizons.

6.3 No-hair theorem

I will state and justify the no-hair theorem, but I will not prove it mathematically, since the proof is technical.

The no-hair theorem states that a stationary black hole in asymptotically flat space is characterized by

just three quantities:

1. Mass M ;

2. Electric charge Q;

3. Spin, usually parameterized by the angular momentum a per unit mass.

The mechanism by which a black hole loses its hair is gravitational radiation. When initially formed,

whether from the collapse of a massive star or from the merger of two black holes, a black hole will form

a complicated, oscillating region of spacetime. But over the course of several light crossing times, the

oscillations lose energy by gravitational radiation, and damp out, leaving a stationary black hole.

Real astronomical black holes are not isolated, and continue to accrete (cosmic microwave background

photons, if nothing else). However, the timescale (a light crossing time) for oscillations to damp out by

gravitational radiation is usually far shorter than the timescale for accretion, so in practice real black holes

are extremely well described by no-hair solutions almost all of their lives.

The physical reason that the no-hair theorem applies is that space is falling faster than light inside the

horizon. Consequently, unlike a star, no energy can bubble up from below to replace the energy lost by

gravitational radiation, so that the black hole tends to the lowest energy state characterized by conserved

quantities.

As a corollary, the no-hair theorem does not apply from the inner horizon of a black hole inward, because

there space ceases to fall superluminally.

If there exist other absolutely conserved quantities, such as magnetic charge (magnetic monopoles), or

various supersymmetric charges in theories where supersymmetry is not broken, then the black hole will also

be characterized by those quantities.

Black holes are expected not to conserve quantities such as baryon or lepton number that are thought not

to be absolutely conserved, even though they appear to be conserved in low energy physics.

Other stationary solutions exist that describe black holes in spacetimes that are not asymptotically flat,

such as spacetimes with a cosmological constant, or with a uniform electromagnetic field.

It is legitimate to think of the process of reaching a stationary state as analogous to reaching a condition

of thermodynamic equilibrium, in which a macroscopic system is described by a small number of parameters

associated with the conserved quantities of the system.

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7

Schwarzschild Black Hole

The Schwarzschild geometry was discovered by Karl Schwarzschild in late 1915 at essentially the same time

that Einstein was arriving at his final version of the General Theory of Relativity.

7.1 Schwarzschild metric

The Schwarzschild metric is, in a polar coordinate system t, r, θ, φ, and in geometric units c = G = 1,

ds2 = −(

1− 2M

r

)

dt2 +

(

1− 2M

r

)−1

dr2 + r2do2 , (7.1)

where do2 (this is the Landau & Lifshitz notation) is the metric of a unit 2-sphere

do2 = dθ2 + sin2θ dφ2 . (7.2)

The Schwarzschild geometry describes the simplest kind of black hole: a black hole with mass M , but no

electric charge, and no spin.

The geometry describes not only a black hole, but also any empty space surrounding a spherically sym-

metric mass. Thus the Schwarzschild geometry describes to a good approximation the spacetime outside the

surfaces of the Sun and the Earth.

Comparison with the spherically symmetric Newtonian metric

ds2 = − (1 + 2Φ)dt2 + (1− 2Φ)(dr2 + r2do2) (7.3)

with Newtonian potential

Φ(r) = −Mr

(7.4)

establishes that the M in the Schwarzschild metric is to be interpreted as the mass of the black hole.

The Schwarzschild geometry is asymptotically flat, because the metric tends to the Minkowski metric in

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7.2 Birkhoff’s theorem 81

polar coordinates at large radius

ds2 → − dt2 + dr2 + r2do2 as r →∞ . (7.5)

Exercise 7.1 The Schwarschild metric (7.1) does not have the same form as the spherically symmetric

Newtonian metric (7.3). By a suitable transformation of the radial coordinate r, bring the Schwarschild

metric (7.1) to the isotropic form

ds2 = −(

1−M/2R

1 +M/2R

)2

dt2 + (1 +M/2R)4 (dR2 +R2do2) . (7.6)

What is the relation between R and r? Hence conclude that the identification (7.4) is correct, and therefore

that M is indeed the mass of the black hole. Is the isotropic form (7.6) of the Schwarzschild metric valid

inside the horizon?

7.2 Birkhoff’s theorem

Birkhoff’s theorem states that the geometry of empty space surrounding a spherically symmetric matter

distribution is the Schwarzschild geometry. That is, if the metric is of the form

ds2 = A(t, r) dt2 +B(t, r) dt dr + C(t, r) dr2 +D(t, r) do2 , (7.7)

where the metric coefficients A, B, C, and D are allowed to be arbitrary functions of t and r, and if the

energy momentum tensor vanishes, Tµν = 0, outside some value of the circumferential radius r′ defined by

r′2 = D, then the geometry is necessarily Schwarzschild outside that radius.

This means that if a mass undergoes spherically symmetric pulsations, then those pulsations do not affect

the geometry of the surrounding spacetime. This reflects the fact that there are no spherically symmetric

gravitational waves.

7.3 Stationary, static

The Schwarzschild geometry is stationary. A spacetime is said to be stationary if and only if there exists

a timelike coordinate t such that the metric is independent of t. In other words, the spacetime possesses

time translation symmetry: the metric is unchanged by a time translation t → t + t0 where t0 is some

constant. Evidently the Schwarzschild metric (7.1) is independent of the timelike coordinate t, and is

therefore stationary, time translation symmetric.

The Schwarzschild geometry is also static. A spacetime is static if and only if the coordinates can be

chosen so that, in addition to being stationary with respect to a time coordinate t, the spatial coordinates

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82 Schwarzschild Black Hole

do not change along the direction of the tangent vector gt. This requires that the tangent vector gt be

orthogonal to all the spatial tangent vectors

gt · gµ = gtµ = 0 for µ 6= t . (7.8)

The Gullstrand-Painleve metric for the Schwarzschild geometry, discussed in section 7.13, is an example of a

metric that is stationary but not static (although the underlying spacetime, being Schwarzschild, is static).

The Gullstrand-Painleve metric is independent of the free-fall time tff , so is stationary, but observers who

follow the tangent vector gtff fall into the black hole, so the metric is not manifestly static.

The Schwarzschild time coordinate t is thus identified as a special one: it is the unique time coordinate

with respect to which the Schwarzschild geometry is manifestly static.

7.4 Spherically symmetric

The Schwarzschild geometry is also spherically symmetric. This is evident from the fact that the angular

part r2do2 of the metric is the metric of a 2-sphere of radius r. This can be see as follows. Consider the

metric of ordinary flat 3-dimensional Euclidean space in Cartesian coordinates x, y, z:

ds2 = dx2 + dy2 + dz2 . (7.9)

Convert to polar coordinates r, θ, φ, defined so that

x = r sin θ cosφ ,

y = r sin θ sinφ ,

z = r cos θ .

(7.10)

Substituting equations (7.10) into the Euclidean metric (7.9) gives

ds2 = dr2 + r2(dθ2 + sin2θ dφ2) . (7.11)

Restricting to a surface r = constant of constant radius then gives the metric of a 2-sphere of radius r

ds2 = r2(dθ2 + sin2θ dφ2) (7.12)

as claimed.

The radius r in Schwarzschild coordinates is the circumferential radius, defined such that the proper

circumference of the 2-sphere measured by observers at rest in Schwarschild coordinates is 2πr. This is a

coordinate-invariant definition of the meaning of r, which implies that r is a scalar.

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7.5 Horizon 83

7.5 Horizon

The horizon of the Schwarzschild geometry lies at the Schwarzschild radius r = rs

rs =2GM

c2. (7.13)

where units of c and G have been restored. Where does this come from? The Schwarzschild metric shows

that the scalar spacetime distance squared ds2 along an interval at rest in Schwarzschild coordinates, dr =

dθ = dφ = 0, is timelike, lightlike, or spacelike depending on whether the radius is greater than, equal to, or

less than r = 2M :

ds2 = −(

1− 2M

r

)

dt2

< 0 if r > 2M ,

= 0 if r = 2M ,

> 0 if r < 2M .

(7.14)

Since the worldline of a massive observer must be timelike, it follows that a massive observer can remain at

rest only outside the horizon, r > 2M . An object at rest at the horizon, r = 2M , follows a null geodesic,

which is to say it is a possible worldline of a massless particle, a photon. Inside the horizon, r < 2M , neither

massive nor massless objects can remain at rest.

A full treatment of what is going on requires solving the geodesic equation in the Schwarzschild geometry,

but the results may be anticipated already at this point. In effect, space is falling into the black hole. Outside

the horizon, space is falling less than the speed of light; at the horizon space is falling at the speed of light;

and inside the horizon, space is falling faster than light, carrying everything with it. This is why light cannot

escape from a black hole: inside the horizon, space falls inward faster than light, carrying light inward even if

that light is pointed radially outward. The statement that space is falling superluminally inside the horizon

of a black hole is a coordinate-invariant statement: massive or massless particles are carried inward whatever

their state of motion and whatever the coordinate system.

Whereas an interval of coordinate time t switches from timelike outside the horizon to spacelike inside the

horizon, an interval of coordinate radius r does the opposite: it switches from spacelike to timelike:

ds2 =

(

1− 2M

r

)−1

dr2

> 0 if r > 2M ,

= 0 if r = 2M ,

< 0 if r < 2M .

(7.15)

It appears then that the Schwarzschild time and radial coordinates swap roles inside the horizon. Inside the

horizon, the radial coordinate becomes timelike, meaning that it becomes a possible worldline of a massive

observer. That is, a trajectory at fixed t and decreasing r is a possible wordline. Again this reflects the fact

that space is falling faster than light inside the horizon. A person inside the horizon is inevitably compelled

as time goes by to move to smaller radial coordinate r.

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84 Schwarzschild Black Hole

7.6 Proper time

The proper time experienced by an observer at rest in Schwarzschild coordinates, dr = dθ = dφ = 0, is

dτ =√

−ds2 =

(

1− 2M

r

)1/2

dt . (7.16)

For an observer at rest at infinity, r →∞, the proper time is the same as the coordinate time,

dτ → dt as r →∞ . (7.17)

Among other things, this implies that the Schwarzschild time coordinate t is a scalar: not only is it the

unique coordinate with respect to which the metric is manifestly static, but it coincides with the proper time

of observers at rest at infinity. This coordinate-invariant definition of time t implies that it is a scalar.

At finite radii outside the horizon, r > 2M , the proper time dτ is less than the Schwarzchild time dt, so

the clocks of observers at rest run slower at smaller than at larger radii.

At the horizon, r = 2M , the proper time dτ of an observer at rest goes to zero,

dτ → 0 as r→ 2M . (7.18)

This reflects the fact that an object at rest at the horizon is following a null geodesic, and as such experiences

zero proper time.

7.7 Redshift

An observer at rest at infinity looking through a telesope at an emitter at rest at radius r sees the emitter

redshifted by a factor

1 + z ≡ λobs

λemit=νemit

νobs=

dτobs

dτemit=

(

1− 2M

r

)−1/2

. (7.19)

This is an example of the universally valid statement that photons are good clocks: the redshift factor is

given by the rate at which the emitter’s clock appears to tick relative to the observer’s own clock.

It should be emphasized that the redshift factor (7.19) is valid only for an observer and an emitter at rest

in the Schwarzschild geometry. If the observer and emitter are not at rest, then additional special relativistic

factors will fold into the redshift.

The redshift goes to infinity for an emitter at the horizon

1 + z →∞ as r → 2M . (7.20)

Here the redshift tends to infinity regardless of the motion of the observer or emitter. An observer watching

an emitter fall through the horizon will see the emitter appear to freeze at the horizon, becoming ever slower

and more redshifted. Physically, photons emitted vertically upward at the horizon by an emitter falling

through it remain at the horizon for ever, taking an infinite time to get out to the outside observer.

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7.8 Proper distance 85

7.8 Proper distance

The proper radial distance measure by observers at rest in Schwarzschild coordinates, dr = dθ = dφ = 0, is

dl =√ds2 =

(

1− 2M

r

)−1/2

dr . (7.21)

For an observer at rest at infinity, r → ∞, an interval of proper radial distance equals an interval of

circumferential radial distance, as you might expect for asymptotically flat space

dl→ dr as r →∞ . (7.22)

At the horizon, r = 2M , a proper radial interval dl measured by an observer at rest goes to infinity

dl→∞ as r→ 2M . (7.23)

7.9 “Schwarzschild singularity”

The apparent singularity in the Schwarzschild metric at the horizon r = 2M is not a real singularity, because

it can be removed by a change of coordinates, such as to Gullstrand-Painleve coordinates (7.26). Prior to

as late as the 1950s, people, including Einstein, thought that the “Schwarzschild singularity” at r = 2M

marked the physical boundary of the Schwarzschild spacetime. After all, an outside observer watching stuff

fall in never sees anything beyond that boundary.

Schwarzschild’s choice of coordinates was certainly a natural one. It was natural to search for static

solutions, and his time coordinate t is the only one with respect to which the metric is manifestly static.

The problem is that physically there can be no static observers inside the horizon: they must necessarily fall

inward as time passes. The fact that Schwarzschild’s coordinate system shows an apparent singularity at the

horizon reflects the fact that the assumption of a static spacetime necessarily breaks down at the horizon,

where space is falling at the speed of light.

Does stuff “actually” fall in, even though no outside observer ever sees it happen? Classically, the answer

is yes: when a black hole forms, it does actually collapse, and when an observer falls through the horizon,

they really do fall through the horizon. The reason that an outside observer sees everything freeze at the

horizon is simply a light travel time effect: it takes an infinite time for light to lift off the horizon and make

it to the outside world.

7.10 Embedding diagram

An embedding diagram is a visual aid to understanding geometry. It is a depiction of a lower dimensional

geometry in a higher dimension. A classic example is the illustration of the geometry of a 2-sphere embedded

in 3-dimensional space. The 2-sphere has a meaning independent of any embedding in 3 dimensions because

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86 Schwarzschild Black Hole

the geometry of the 2-sphere can be measured by 2-dimensional inhabitants of its surface without reference

to any encompassing 3-dimensional space. Nevertheless, the pictorial representation aids imagination.

Textbooks sometimes illustrate the Schwarzschild geometry with an embedding diagram that shows the

spatial geometry at a fixed instant of Schwarzschild time t. The diagram illustrates the stretching of proper

distances in the radial direction. I’ll let you figure out how to construct this embedding diagram.

It should be emphasized that the embedding diagram of the Schwarzshild geometry at fixed Schwarzschild

time t has a limited physical meaning. Fixing the time t means choosing a certain hypersurface through the

geometry. Other choices of hypersurface will yield different diagrams. For example, the Gullstrand-Painleve

metric is spatially flat at fixed free-fall time tff , so in that case the embedding diagram would simply illustrate

flat space, with no funny business at the horizon.

7.11 Energy-momentum tensor

The energy-momentum tensor of the Schwarzschild geometry is zero, by construction.

7.12 Weyl tensor

It turns out that the 10 components of the Weyl tensor, the tidal part of the Riemann tensor, can be decom-

posed in any locally inertial frame into 5 complex components of spin 0, ±1, and ±2. In the Schwarzschild

metric, all components vanish except the real spin 0 component. This component is a coordinate-invariant

scalar, the Weyl scalar C

C = −Mr3

. (7.24)

The Weyl scalar, which expresses the presence of tidal forces, goes to infinity at zero radius,

C →∞ as r → 0 , (7.25)

signalling the presence of a real singularity at zero radius.

7.13 Gullstrand-Painleve coordinates

The Gullstrand-Painleve metric is an alternative metric for the Schwarzschild geometry, discovered indepen-

dently by Allvar Gullstrand and Paul Painleve in (1921). When we have done tetrads, we will recognize

that the standard way in which metrics are written encodes not only metric but also a complete tetrad. The

Gullstrand-Painleve line-element (7.26) encodes a tetrad that represents locally inertial frames free-falling

radially into the black hole at the Newtonian escape velocity. Unlike Schwarzschild coordinates, there is no

singularity at the horizon in Gullstrand-Painleve coordinates. It is striking that the mathematics was known

long before physical understanding emerged.

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7.14 Eddington-Finkelstein coordinates 87

The Gullstrand-Painleve metric is

ds2 = − dt2ff + (dr − β dtff)2 + r2do2 . (7.26)

Here β is the Newtonian escape velocity (with a minus sign because space is falling inward)

β = −(

2M

r

)1/2

(7.27)

and tff is the proper time experienced by an object that free falls radially inward from zero velocity at infinity.

The free fall time tff is related to the Schwarzschild time coordinate t by

dtff = dt− β

1− β2dr , (7.28)

which integrates to

tff = t+ 2M

[

2( r

2M

)1/2

+ ln

(r/2M)1/2 − 1

(r/2M)1/2 + 1

]

. (7.29)

The time axis gtff in Gullstrand-Painleve coordinates is not orthogonal to the radial axis gr, but rather is

tilted along the radial axis, gtff · gr = gtffr = −β.

The proper time of a person at rest in Gullstrand-Painleve coordinates, dr = dθ = dφ = 0, is

dτ = dtff√

1− β2 . (7.30)

The horizon occurs where this proper time vanishes, which happens when the infall velocity β is the speed

of light

|β| = 1 . (7.31)

According to equation (7.27), this happens at r = 2M , which is the Schwarzschild radius, as it should be.

7.14 Eddington-Finkelstein coordinates

In Schwarzschild coordinates, radially infalling or outfalling light rays appear never to cross the horizon

of the Schwarzschild black hole. This feature of Schwarzschild coordinates contributed to the historical

misconception that black holes stopped at their horizons. In 1958, David Finkelstein carried out a trivial

transformation of the time coordinate which seeemed to show that infalling light rays could indeed pass

through the horizon. It turned out that Eddington had already discovered the transformation in 1924,

though at that time the physical implications were not grasped. Again, it is striking that the mathematics

was in place long before physical understanding.

In Schwarzschild coordinates, light rays that fall radially (dθ = dφ = 0) inward or outward follow null

geodesics

ds2 = −(

1− 2M

r

)

dt2 +

(

1− 2M

r

)−1

dr2 = 0 . (7.32)

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88 Schwarzschild Black Hole

Radial null geodesics thus follow

dr

dt= ±

(

1− 2M

r

)

(7.33)

in which the ± sign is + for outfalling, − for infalling rays. Equation (7.33) shows that dr/dt → 0 as

r → 2M , suggesting that null rays, whether infalling or outfalling, never cross the horizon. The solution to

equation (7.33) is

t = ± (r + 2M ln|r − 2M |) , (7.34)

which shows that Schwarzschild time t approaches ±∞ logarithmically as null rays approach the horizon.

Finkelstein defined his time coordinate tF by

tF ≡ t+ 2M ln |r − 2M | , (7.35)

which has the property that infalling null rays follow

tF + r = 0 . (7.36)

In other words, on a spacetime diagram in Finkelstein coordinates, radially infalling light rays move at 45,the same as in special relativistic spacetime diagrams.

7.15 Kruskal-Szekeres coordinates

Since Finkelstein transformed coordinates so that radially infalling light rays moved at 45 in a spacetime

diagram, it is natural to look for coordinates in which outfalling as well as infalling light rays are at 45.Kruskal and Szekeres independently provided such a transformation, in 1960.

Define the tortoise (or Regge-Wheeler 1959) coordinate r∗ by

r∗ ≡∫

dr

1− 2M/r= r + 2M ln |r − 2M | . (7.37)

Then radially infalling and outfalling null rays follow

r∗ + t = 0 infalling ,

r∗ − t = 0 outfalling .(7.38)

In a spacetime diagram in coordinates t and r∗, infalling and outfalling light rays are indeed at 45. Unfor-

tunately the metric in these coordinates is still singular at the horizon r = 2M :

ds2 =

(

1− 2M

r

)

(

− dt2 + dr∗2)

+ r2do2 . (7.39)

The singularity at the horizon can be eliminated by the following transformation into Kruskal-Szekeres

coordinates tK and rK :

rK + tK = e(r∗+t)/2 ,

rK − tK = ±e(r∗−t)/2 ,(7.40)

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7.16 Penrose diagrams 89

r = ∞

r =∞Hor

izon

Antihorizon Universe

Singularity (r = 0)

Black Hole

Tim

e

Space

Light

Light

Figure 7.1 Penrose diagram of the Schwarzschild geometry.

where the ± sign in the last equation is + outside the horizon, − inside the horizon. The Kruskal-Szekeres

metric is

ds2 = r−1e−r(

− dt2K + dr2K)

+ r2do2 , (7.41)

which is non-singular at the horizon. The Schwarzschild radial coordinate r, which appears in the factors

r−1e−r and r2 in the Kruskal metric, is to be understood as an implicit function of the Kruskal coordinates

tK and rK .

7.16 Penrose diagrams

Roger Penrose, as so often, had a novel take on the business of spacetime diagrams. Penrose conceived that

the primary purpose of a spacetime diagram should be to portray the causal structure of the spacetime, and

that the specific choice of coordinates was largely irrelevant. After all, general relativity allows arbitrary

choices of coordinates.

In addition to requiring that light rays be at 45, Penrose wanted to bring regions at infinity (in time or

space) to a finite position on the spacetime diagram, so that the entire spacetime could be seen at once. He

calls these thing conformal diagrams, but the rest of us commonly call them Penrose diagrams.

Penrose diagrams are bona-fide spacetime diagrams. For example, a coordinate transformation from

Kruskal to “Penrose” coordinates (the following transformation is not analytic, but Penrose does not care)

rP + tP =rK + tK

2 + |rK + tK |,

rP − tP =rK − tK

2− |rK − tK |,

(7.42)

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90 Schwarzschild Black Hole

brings spatial and temporal infinity to finite values of the coordinates, while keeping infalling and outfalling

light rays at 45 in the spacetime diagram. However, there are many such transformations, and Penrose

would be the last person to advocate any one of them in particular.

r = 0

r = ∞

r =∞

r =∞

r = ∞

Horiz

on

Universe

Black Hole

r = 0

Parallel Horizon

Antihorizon

Para

llel Ant

ihor

izon

Parallel Universe

White Hole

Figure 7.2 Penrose diagram of the complete, analytically extended Schwarzschild geometry.

7.17 Schwarzschild white hole, wormhole

The Kruskal-Szekeres spacetime diagram reveals a new feature that was not apparent in Schwarzschild or

Finkelstein coordinates. Dredged from the depths of t = −∞ appears a null line rK + tK = 0. The null line

is at radius r = 2M , but it does not correspond to the horizon that a person might fall into. The null line is

called the antihorizon. The horizon is sometimes called the true horizon, and the antihorizon the illusory

horizon. In a real black hole, only the true horizon is real. The antihorizon is replaced by an exponentially

dimming and redshifting image of the star that collapsed to form the black hole.

The Kruskal-Szekeres (= Schwarzschild) geometry is analytic, and there is a unique analytic continuation

of the geometry through the antihorizon. The analytic continuation is a time-reversed copy of the original

Schwarzschild geometry, glued at the antihorizon. Whereas the original Schwarzschild geometry showed an

asymptotically flat region and a black hole region separated by a horizon, the complete analytically extended

Schwarzschild geometry shows two asymptotically flat regions, together with a black hole and a white hole.

Relativists label the regions I, II, III, and IV, but I like to call them by name: “Universe”, “Black Hole”,

“Parallel Universe”, and “White Hole”.

The white hole is a time-reversed version of the black hole. Whereas space falls inward faster than light

inside the black hole, space falls outward faster than light inside the white hole. In the Gullstrand-Painleve

metric (7.26), the velocity β = ±(2M/r)1/2 is negative for the black hole, positive for the white hole.

The Kruskal or Penrose diagrams show that the universe and the parallel universe are connected, but

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7.18 Collapse to a black hole 91

only by spacelike lines. This spacelike connection is called the Einstein-Rosen bridge, and constitutes a

wormhole connecting the two universes. Because the connection is spacelike, it is impossible for a traveler

to pass through this wormhole.

Although two travelers, one from the universe and one from the parallel universe, cannot travel to each

other’s universe, they can meet, but only inside the black hole. Inside the black hole, they can talk to each

other, and they can see light from each other’s universe. Sadly, the enlightenment is only temporary, because

they are doomed soon to hit the central singularity.

It should be emphasized that the white hole and the wormhole in the Schwarzschild geometry are a

mathematical construction with as far as anyone knows no relevance to reality. Nevertheless it is intriguing

that such bizarre objects emerge already in the simplest general relativistic solution for a black hole.

7.18 Collapse to a black hole

Realistic collapse of a star to a black hole is not expected to produce a white hole or parallel universe.

The simplest model of a collapsing star is a spherical ball of uniform density and zero pressure which free

falls from zero velocity at infinity. In this simple model, the interior of the star is described by a collapsing

Friedmann-Robertson-Walker metric (the canonical cosmological metric), while the exterior is described by

the Schwarzschild solution. The assumption that the star collapses from zero velocity at infinity implies

that the FRW metric is spatially flat, the simplest case. To continue the geometry between Schwarzchild

and FRW metrics, it is neatest to use the Gullstrand-Painleve metric, with the Gullstrand-Painleve infall

velocity β at the edge of the star set equal to minus r times the Hubble parameter −rH ≡ −r d ln a/dt of

the collapsing FRW metric.

The simple model shows that the antihorizon of the complete Schwarzschild geometry is replaced by the

surface of the collapsing star, and that beyond the antihorizon is not a parallel universe and a white hole,

but merely the interior of the star (and the distant Universe glimpsed through the star’s interior).

Since light can escape from the collapsing star system as long as it is even slightly larger than its Schwarz-

schild radius, it is possible to take the view that the horizon comes instantaneously into being at the moment

the star collapses through its Schwarzschild radius. This definition of the horizon is called the apparent hori-

zon.

Hawking has advocated that a better definition of the horizon is to take it to be the boundary between

outgoing null rays that fall into the black hole versus those that go to infinity. In any evolving situation,

this definition of the horizon, which is called the absolute horizon, depends formally on what happens in the

infinite future, though in slowly evolving systems the absolute horizon can be located with some precision

without knowing the future. The absolute horizon of the collapsing star forms before the star has collapsed,

and grows to meet the apparent horizon as the star falls through its Schwarzchild radius.

In this simple model, the central singularity forms slightly before the star has collapsed to zero radius.

The formation of the singularity is marked by the fact that light rays emitted at zero radius cease to be able

to move outward. In other words, the singularity forms when space starts to fall into it faster than light.

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92 Schwarzschild Black Hole

7.19 Killing vectors

The Schwarzschild metric presents an opportunity to introduce the concept of Killing vectors (after Wil-

helm Killing, not because the vectors kill things, though the latter is true), which are associated with

symmetries of the spacetime.

7.20 Time translation symmetry

The time translation invariance of the Schwarzschild geometry is evident from the fact that the metric is

independent of the time coordinate t. Equivalently, the partial time derivative ∂/∂t of the Schwarzschild

metric is zero. The associated Killing vector ξµ is then defined by

ξµ ∂

∂xµ=

∂t(7.43)

so that in Schwarzschild coordinates t, r, θ, φ

ξµ = 1, 0, 0, 0 . (7.44)

In coordinate-independent notation, the Killing vector is

ξ = gµξµ = gt . (7.45)

This may seem like overkill – couldn’t we just say that the metric is independent of time t and be done

with it? The answer is that symmetries are not always evident from the metric, as will be seen in the next

section 7.21.

Because the Killing vector gt is the unique timelike Killing vector of the Schwarzschild geometry, it has

a definite meaning independent of the coordinate system. It follows that its scalar product with itself is a

coordinate-independent scalar

ξµξµ = gt · gt = gtt = −

(

1− 2M

r

)

. (7.46)

In curved spacetimes, it is hugely important to be able to identify scalars, which have a physical meaning

independent of the choice of coordinates.

7.21 Spherical symmetry

The rotational symmetry of the Schwarzschild metric about the azimuthal axis is evident from the fact that

the metric is independent of the azimuthal coordinate φ. The associated Killing vector is

gφ (7.47)

with components 0, 0, 0, 1 in Schwarzschild coordinates t, r, θ, φ.

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7.22 Killing equation 93

The Schwarzschild metric is fully spherically symmetric, not just azimuthally symmetric. Since the 3D

rotation group O(3) is 3-dimensional, it is to be expected that there are three Killing vectors. You may

recognize from quantum mechanics that ∂/∂φ is (modulo factors of i and ~) the z-component of the angular

momentum operator L = Lx, Ly, Lz in a coordinate system where the azimuthal axis is the z-axis. The 3

components of the angular momentum operator are given by:

iLx = y∂

∂z− z ∂

∂y= − sinφ

∂θ− cot θ cosφ

∂φ,

iLy = z∂

∂x− x ∂

∂z= cosφ

∂θ− cot θ sinφ

∂φ,

iLz = x∂

∂y− y ∂

∂x=

∂φ.

(7.48)

The 3 rotational Killing vectors are correspondingly:

rotation about x-axis: − sinφgθ − cot θ cosφgφ ,

rotation about y-axis: cosφgθ − cot θ sinφgφ ,

rotation about z-axis: gφ .

(7.49)

You can check that the action of the x and y rotational Killing vectors on the metric does not kill the

metric. For example, iLxgφφ = 2r2 cosφ sin θ cos θ does not vanish. This example shows that a more powerful

and general condition, described in the next section 7.22, is needed to establish whether a quantity is or is

not a Killing vector.

Because spherical symmetry does not define a unique azimuthal axis gφ, its scalar product with itself

gφ · gφ = gφφ = −r2 sin2θ is not a coordinate-invariant scalar. However, the sum of the scalar products of

the 3 rotational Killing vectors is rotationally invariant, and is therefore a coordinate-invariant scalar

(− sinφgθ − cot θ cosφgφ)2 + (cosφgθ − cot θ sinφgφ)2 + g2φ = gθθ + (cot2θ + 1)gφφ = −2r2 . (7.50)

This shows that the circumferential radius r is a scalar, as you would expect.

7.22 Killing equation

As seen in the previous section, a Killing vector does not always kill the metric in a given coordinate system.

This is not really surprising given the arbitrariness of coordinates in GR. What is true is that a quantity is

a Killing vector if and only if there exists a coordinate system such that the Killing vector kills the metric

in that system.

Suppose that in some coordinate system the metric is independent of the coordinate φ. In problem set 2

you showed that in such a case the covariant component uφ of the 4-velocity along a geodesic is constant

uφ = constant . (7.51)

Equivalently

ξνuν = constant (7.52)

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94 Schwarzschild Black Hole

where ξν is the associated Killing vector, whose only non-zero component is ξφ = 1 in this particular

coordinate system. The converse is also true: if ξνuν = constant along all geodesics, then ξν is a Killing

vector. The constancy of ξνuν along all geodesics is equivalent to the condition that its proper time derivative

vanish along all geodesics

dξνuν

dτ= 0 . (7.53)

But this is equivalent to

0 = uµDµ(ξνuν) = uµuνDµξν =1

2uµuν(Dµξν +Dνξµ) (7.54)

where the second equality follows from the geodesic equation, uµDµuν = 0, and the last equality is true

because of the symmetry of uµuν in µ ↔ ν. A necessary and sufficient condition for equation (7.54) to be

true for all geodesics is that

Dµξν +Dνξµ = 0 (7.55)

which is Killing’s equation. This equation is the desired necessary and sufficient condition for ξν to be a

Killing vector. It is a generally covariant equation, valid in any coordinate system.

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8

Reissner-Nordstrom Black Hole

The Reissner-Nordstrom geometry, discovered independently by Hans Reissner in 1916, Hermann Weyl in

1917, and Gunnar Nordstrom in 1918, describes the unique spherically symmetric static solution for a black

hole with mass and electric charge in asymptotically flat spacetime.

8.1 Reissner-Nordstrom metric

The Reissner-Nordstrom metric for a black hole of mass M and electric charge Q is, in geometric units

c = G = 1,

ds2 = −(

1− 2M

r+Q2

r2

)

dt2 +

(

1− 2M

r+Q2

r2

)−1

dr2 + r2do2 (8.1)

which looks like the Schwarzschild metric with the replacement

M →M(r) = M − Q2

2r. (8.2)

In fact equation (8.2) has a coordinate independent interpretation as the mass M(r) interior to radius r,

which here is the mass M at infinity, minus the mass in the electric field E = Q/r2 outside r∫ ∞

r

E2

8π4πr2dr =

∫ ∞

r

Q2

8πr44πr2dr =

Q2

2r. (8.3)

This seems like a Newtonian calculation of the energy in the electric field, but it turns out to be valid also

in general relativity.

Real astronomical black holes probably have very little electric charge, because the Universe as a whole

appears almost electrically neutral (and Maxwell’s equations in fact require that the Universe in its entirety

should be exactly electrically neutral), and a charged black hole would quickly neutralize itself. It would

probably not neutralize itself completely, but have some small residual positive charge, because protons

(positive charge) are more massive than electrons (negative charge), so it is slightly easier for a black hole

to accrete protons than electrons.

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96 Reissner-Nordstrom Black Hole

Nevertheless, the Reissner-Nordstrom solution is of more than passing interest because its internal geom-

etry resembles that of the Kerr solution for a rotating black hole.

Concept question 8.1 What is the charge Q in standard (gaussian) units?

8.2 Energy-momentum tensor

The Einstein tensor of the Reissner-Nordstrom metric (8.1) is diagonal, with elements given by

Gνµ =

Gtt 0 0 0

0 Grr 0 0

0 0 Gθθ 0

0 0 0 Gφφ

= 8π

−ρ 0 0 0

0 pr 0 0

0 0 p⊥ 0

0 0 0 p⊥

=Q2

r4

−1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

. (8.4)

The trick of writing one index up and the other down on the Einstein tensor Gνµ partially cancels the

distorting effect of the metric, yielding the proper energy density ρ, the proper radial pressure pr, and

transverse pressure p⊥, up to factors of ±1. A more systematic way to extract proper quantities is to work

in the tetrad formalism, but this will do for now.

The energy-momentum tensor is that of a radial electric field

E =Q

r2. (8.5)

Notice that the radial pressure pr is negative, while the transverse pressure p⊥ is positive. It is no coincidence

that the sum of the energy density and pressures is twice the energy density, ρ+ pr + 2p⊥ = 2ρ.

The negative pressure, or tension, of the radial electric field produces a gravitational repulsion that domi-

nates at small radii, and that is responsible for much of the strange phenomenology of the Reissner-Nordstrom

geometry. The gravitational repulsion mimics the centrifugal repulsion inside a rotating black hole, for which

reason the Reissner-Nordstrom geometry is often used a surrogate for the rotating Kerr-Newman geometry.

At this point, the statements that the energy-momentum tensor is that of a radial electric field, and that

the radial tension produces a gravitational repulsion that dominates at small radii, are true but unjustified

assertions.

8.3 Weyl tensor

As with the Schwarzschild geometry (indeed, any spherically symmetric geometry), only 1 of the 10 inde-

pendent spin components of the Weyl tensor is non-vanishing, the real spin-0 component, the Weyl scalar

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8.4 Horizons 97

C. The Weyl scalar for the Reissner-Nordstrom geometry is

C = − M

r3+Q2

r4. (8.6)

The Weyl scalar goes to infinity at zero radius

C →∞ as r → 0 (8.7)

signalling the presence of a real singularity at zero radius.

8.4 Horizons

The Reissner-Nordstrom geometry has not one but two horizons. The horizons occur where an object at

rest in the geometry, dr = dθ = dφ = 0, follows a null geodesic, ds2 = 0, which occurs where

1− 2M

r+Q2

r2= 0 . (8.8)

This is a quadratic equation in r, and it has two solutions, an outer horizon r+ and an inner horizon r−

r± = M ±√

M2 −Q2 . (8.9)

It is straightforward to check that the Reissner-Nordstrom time coordinate t is timelike outside the outer

horizon, r > r+, spacelike between the horizons r− < r < r+, and again timelike inside the inner horizon

r < r−. Conversely, the radial coordinate r is spacelike outside the outer horizon, r > r+, timelike between

the horizons r− < r < r+, and spacelike inside the inner horizon r < r−.

The physical meaning of this strange behaviour is akin to that of the Schwarzschild geometry. As in the

Schwarzschild geometry, outside the outer horizon space is falling at less than the speed of light; at the outer

horizon space hits the speed of light; and inside the outer horizon space is falling faster than light. But

a new ingredient appears. The gravitational repulsion caused by the negative pressure of the electric field

slows down the flow of space, so that it slows back down to the speed of light at the inner horizon. Inside

the inner horizon space is falling at less than the speed of light.

8.5 Gullstrand-Painleve metric

Deeper insight into the Reissner-Nordstrom geometry comes from examining its Gullstrand-Painleve metric.

The Gullstrand-Painleve metric for the Reissner-Nordstrom geometry is the same as that for the Schwarz-

schild geometry

ds2 = − dt2ff + (dr − β dtff)2 + r2do2 . (8.10)

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98 Reissner-Nordstrom Black Hole

The velocity β is again the escape velocity, but this is now

β = ∓√

2M(r)

r(8.11)

where M(r) = M − Q2/2r is the interior mass already given as equation (8.2). Horizons occur where the

magnitude of the velocity β equals the speed of light

|β| = 1 (8.12)

which happens at the outer and inner horizons r = r+ and r = r−, equation (8.9).

The Gullstrand-Painleve metric once again paints the picture of space falling into the black hole. Outside

the outer horizon r+ space falls at less than the speed of light, at the horizon space falls at the speed of

light, and inside the horizon space falls faster than light. But the gravitational repulsion produced by the

tension of the radial electric field starts to slow down the inflow of space, so that the infall velocity reaches

a maximum at r = Q2/M . The infall slows back down to the speed of light at the inner horizon r−. Inside

the inner horizon, the flow of space slows all the way to zero velocity, β = 0, at the turnaround radius

r0 =Q2

2M. (8.13)

Space then turns around, the velocity β becoming positive, and accelerates back up to the speed of light.

Space is now accelerating outward, to larger radii r. The outfall velocity reaches the speed of light at the

inner horizon r−, but now the motion is outward, not inward. Passing back out through the inner horizon,

space is falling outward faster than light. This is not the black hole, but an altogether new piece of spacetime,

a white hole. The white hole looks like a time-reversed black hole. As space falls outward, the gravitational

repulsion produced by the tension of the radial electric field declines, and the outflow slows. The outflow

slows back to the speed of light at the outer horizon r+ of the white hole. Outside the outer horizon of the

white hole is a new universe, where once again space is flowing at less than the speed of light.

What happens inward of the turnaround radius r0, equation (8.13)? Inside this radius the interior mass

M(r), equation (8.2), is negative, and the velocity β is imaginary. The interior massM(r) diverges to negative

infinity towards the central singularity at r → 0. The singularity is timelike, and infinitely gravitationally

repulsive, unlike the central singularity of the Schwarzschild geometry. Is it physically realistic to have a

singularity that has infinite negative mass and is infinitely gravitationally repulsive? Undoubtedly not.

8.6 Complete Reissner-Nordstrom geometry

As with the Schwarzschild geometry, it is possible to go through the steps: Reissner-Nordstrom coordinates→Eddington-Finkelstein coordinates → Kruskal-Szekeres coordinates → Penrose coordinates. The conclusion

of these constructions is that the Reissner-Nordstrom geometry can be analytically continued, and the

complete analytic continuation consists of an infinite ladder of universes and parallel universes connected to

each other by black hole → wormhole → white hole tunnels. I like to call the various pieces of spacetime

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8.6 Complete Reissner-Nordstrom geometry 99

r = ∞

r =∞

r =∞

r = ∞

r =−∞

r = −∞

r = −∞

r =−∞

Para

llel Ant

ihor

izon Antihorizon

Parallel Universe Universe

Parallel HorizonHor

izon

Black Hole

Inne

r Horiz

on

Parallel Inner Horizon

Wor

mho

le ParallelW

ormhole

AntiverseParallel

Antiverse

White HoleInner Antihorizon

Para

llel In

ner Ant

ihor

izon

New Parallel Universe New Universe

Figure 8.1 Penrose diagram of the complete Reissner-Nordstrom geometry.

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100 Reissner-Nordstrom Black Hole

“Universe”, “Parallel Universe”, “Black Hole”, “Wormhole”, “Parallel Wormhole”, and “White Hole”. These

pieces repeat in an infinite ladder.

The Wormhole and Parallel Wormhole contain separate central singularities, the “Singularity” and the

“Parallel Singularity”, which are oppositely charged. If the black hole is positively charged as measured by

observers in the Universe, then it is negatively charged as measured by observers in the Parallel Universe,

and the Wormhole contains a positive charge singularity while the Parallel Wormhole contains a negative

charge singularity.

Where does the electric charge of the Reissner-Nordstrom geometry “actually” reside? This comes down

to the question of how observers detect the presence of charge. Observers detect charge by the electric field

that it produces. Equip all (radially moving) observers with a gyroscope that they orient consistently in

the same radial direction, which can be taken to be towards the black hole as measured by observers in

the Universe. Observers in the Parallel Universe find that their gyroscope is pointed away from the black

hole. Inside the black hole, observers from either Universe agree that the gyroscope is pointed towards the

Wormhole, and away from the Parallel Wormhole. All observes agree that the electric field is pointed in the

same radial direction. Observers who end up inside the Wormhole measure an electric field that appears to

emanate from the Singularity, and which they therefore attribute to charge in the Singularity. Observers

who end up inside the Parallel Wormhole measure an electric field that appears to emanate in the opposite

direction from the Parallel Singularity, and which they therefore attribute to charge of opposite sign in the

Parallel Singularity. Strange, but all consistent.

8.7 Antiverse: Reissner-Nordstrom geometry with negative mass

It is also possible to consider the Reissner-Nordstrom geometry for negative values of the radius r. I call the

extension to negative r the “Antiverse”. There is also a “Parallel Antiverse”.

Changing the sign of r in the Reissner-Nordstrom metric (8.1) is equivalent to changing the sign of the

mass M . Thus the Reissner-Nordstrom metric with negative r describes a charged black hole of negative

mass

M < 0 . (8.14)

The negative mass black hole is gravitationally repulsive at all radii, and it has no horizons.

8.8 Ingoing, outgoing

The black hole in the Reissner-Nordstrom geometry is bounded at its inner edge by not one but two inner

horizons. The two distinct horizons play a crucial role in the mass inflation instability described in §8.9

below.

The inner horizons can be called ingoing and outgoing. Persons freely falling in the Black Hole region are

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8.9 Mass inflation instability 101

all moving inward in coordinate radius r, but they may be moving either forward or backward in Reissner-

Nordstrom coordinate time t. In the Black Hole region, the conserved energy along a geodesic is positive if

the time coordinate t is decreasing, negative if the time coordinate t is increasing1. Persons with positive

energy are ingoing, while persons with negative energy are outgoing. Both ingoing and outgoing persons

fall inward, to smaller radii, but ingoing persons think that the inward direction is towards the Wormhole,

while outgoing persons think that the inward direction is in the opposite direction, towards the Parallel

Wormhole. Ingoing persons fall through the ingoing inner horizon, while outgoing persons fall through the

outgoing inner horizon.

Coordinate time t moves forwards in the Universe and Wormhole regions, and geodesics have positive

energy in these regions. Conversely, coordinate time t moves backwards in the Parallel Universe and Parallel

Wormhole regions, and geodesics have negative energy in these regions. Of course, all observers, whereever

they may be, always perceive their own proper time to be moving forward in the usual fashion, at the rate

of one second per second.

8.9 Mass inflation instability

Roger Penrose (1968) first pointed out that a person passing through the outgoing inner horizon (also

called the Cauchy horizon) of the Reissner-Nordstrom geometry would see the outside Universe infinitely

blueshifted, and he suggested that this would destabilize the geometry. Perturbation theory calculations,

starting with Simpson & Penrose (1973) and culminating with Chandrasekhar and Hartle (1982), confirmed

that waves become infinitely blueshifted as they approach the outgoing inner horizon, and that their energy

density diverges. The perturbation theory calculations were widely construed as indicating that the Reissner-

Nordstrom geometry was “unstable”, although the precise nature of this instability remained obscure.

It was not until a seminal paper by Poisson & Israel (1990) that the true nonlinear nature of the instability

at the inner horizon was clarified. Poisson & Israel showed that the Reissner-Nordstrom geometry is subject

to an exponentially growing instability which they dubbed mass inflation. The term refers to the fact that

the interior mass M(r) grows exponentially during mass inflation. The interior mass M(r) has the property

of being a gauge-invariant, scalar quantity, so it has a physical meaning independent of the coordinate system.

What causes mass inflation? Actually it has nothing to do with mass: the inflating mass is just a symptom

of the underlying cause. What causes mass inflation is relativistic counter-streaming between ingoing and

outgoing streams. As the Penrose diagram of the Reissner-Nordstrom geometry shows, ingoing and outgoing

streams must drop through separate ingoing and outgoing inner horizons into separate pieces of spacetime,

the Wormhole and the Parallel Wormhole. The regions of spacetime must be separate because coordinate

time t is timelike in both regions, but going in opposite directions in the two regions, forward in the Wormhole,

backward in the Parallel Wormhole. In other words, ingoing and outgoing streams cannot co-exist in the

1 The apparently wrong direction of time comes from requiring that t → ∞ for ingoing trajectories near the horizon, whetherjust above or just below the horizon, while t → −∞ for outgoing trajectories near the horizon, whether just above or justbelow the horizon. It would be unconventional, but consistent, to flip the sign of time t between the horizons so thatpositive energy increased with time, and negative energy decreased with time.

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102 Reissner-Nordstrom Black Hole

Universe

Horiz

on

Black HoleIn

goin

gin

ner ho

rizon

Outgoinginner horizon

IngoingOut

goin

g

t

t

t t

Figure 8.2 Penrose diagram illustrating why the Reissner-Nordstrom geometry is subject to the mass infla-tion instability. Ingoing and outgoing streams just outside the inner horizon must pass through separateingoing and outgoing inner horizons into causally separated pieces of spacetime where the timelike timecoordinate t goes in opposite directions. To accomplish this, the ingoing and outgoing streams must exceedthe speed of light through each other, which physically they cannot do. The mass inflation instability isdriven by the pressure of the relativistic counter-streaming between ingoing and outgoing streams. Theinset shows the direction of coordinate time t in the various regions. Proper time of course always increasesupward in a Penrose diagram.

same subluminal region of spacetime because they would have to be moving in opposite directions in time,

which cannot be.

In the Reissner-Nordstrom geometry, ingoing and outgoing streams resolve their differences by exceeding

the speed of light relative to each other, and passing into causally separated regions. As the ingoing and

outgoing streams drop through their respective inner horizons, they each see the other stream infinitely

blueshifted.

In reality however, this cannot occur: ingoing and outgoing streams cannot exceed the speed of light

relative to each other. Instead, as the ingoing and outgoing streams move ever faster through each other

in their effort to drop through the inner horizon, their counter-streaming generates a radial pressure. The

pressure, which is positive, exerts an inward gravitational force. As the counter-streaming approaches the

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8.10 Inevitability of mass inflation 103

speed of light, the gravitational force produced by the counter-streaming pressure eventually exceeds the

gravitational force produced by the background Reissner-Nordstrom geometry. At this point, mass inflation

begins.

The gravitational force produced by the counter-streaming is inwards, but, in the strange way that general

relativity operates, the inward direction is in opposite directions for the ingoing streams, towards the black

hole for the ingoing stream, and away from the black hole for the outgoing stream. Consequently the counter-

streaming pressure simply accelerates the ingoing and outgoing streams ever faster through each other. The

result is an exponential feedback instability. The increasing pressure accelerates the streams faster through

each other, which increases the pressure, which increases the acceleration.

The interior mass is not the only thing that increases exponentially during mass inflation. The proper

density and pressure, and the Weyl scalar (all gauge-invariant scalars) exponentiate together.

Exercise 8.2 Show that, in the Reissner-Nordstrom geometry, the blueshift of a photon with energy

vt = ±1 and angular momentum per unit energy v⊥ = J observed by observer on a geodesic with energy

per unit mass ut = E and angular momentum per unit mass u⊥ = L is

uµvµ =

something

B. (8.15)

Argue that the blueshift diverges at the horizon for ingoing observers observing outgoing photons, and for

outgoing observers observing ingoing photons.

8.10 Inevitability of mass inflation

Mass inflation requires the simultaneous presence of both ingoing and outgoing streams near the inner

horizon. Will that happen in real black holes? Any real black hole will of course accrete matter from its

surroundings, so certainly there will be a stream of one kind or another (ingoing or outgoing) inside the

black hole. But is it guaranteed that there will also be a stream of the other kind? The answer is probably.

One of the remarkable features of the mass inflation instability is that, as long as ingoing and outgoing

streams are both present, the smaller the perturbation the more violent the instability. That is, if say the

outgoing stream is reduced to a tiny trickle compared to the ingoing stream (or vice versa), then the length

scale (and time scale) over which mass inflation occurs gets shorter. During mass inflation, as the counter-

streaming streams drop through an interval ∆r of circumferential radius, the interior mass M(r) increases

exponentially with length scale l

M(r) ∝ e∆r/l . (8.16)

It turns out that the inflationary length scale l is proportional to the accretion rate

l ∝ M , (8.17)

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104 Reissner-Nordstrom Black Hole

so that smaller accretion rates produce more violent inflation. Physically, the smaller accretion rate, the

closer the streams must approach the inner horizon before the pressure of their counter-streaming begins to

dominate the gravitational force. The distance between the inner horizon and where mass inflation begins

effectively sets the length scale l of inflation.

Given this feature of mass inflation, that the tinier the perturbation the more rapid the growth, it seems

almost inevitable that mass inflation must occur inside real black holes. Even the tiniest piece of stuff going

the wrong way is apparently enough to trigger the mass inflation instability.

One way to avoid mass inflation inside a real black hole is to have a large level of dissipation inside the

black hole, sufficient to reduce the charge (or spin) to zero near the singularity. In that case the central

singularity reverts to being spacelike, like the Schwarzschild singularity. While the electrical conductivity of

a realistic plasma is more than adequate to neutralize a charged black hole, angular momentum transport

is intrinsically a much weaker process, and it is not clear whether the dissipation of angular momentum

might be large enough to eliminate the spin near the singularity of a rotating black hole. There has been no

research on the latter subject.

8.11 The black hole particle accelerator

A good way to think conceptually about mass inflation is that it acts like a particle accelerator. The counter-

streaming pressure accelerates ingoing and outgoing streams through each other at an exponential rate, so

that a Lagrangian gas element spends equal amounts of proper time accelerating through equal decades of

counter-streaming velocity. The center of mass energy easily exceeds the Planck energy, where quantum

mechanics presumably comes into play.

Mass inflation is expected to occur just above the inner horizon of a black hole. In a realistic rotating

astronomical black hole, the inner horizon is likely to be at a considerable fraction of the radius of the

outer horizon. Thus the black hole accelerator operates not near a central singularity, but rather at a

macroscopically huge scale. This machine is truly monstrous.

Undoubtedly much fascinating physics occurs in the black hole particle accelerator. The situation is far

more extreme than anywhere else in our Universe today. Who knows what Nature does there? To my

knowledge, there has been no research on the subject.

8.12 The X point

The point in the Reissner-Nordstrom geometry where the ingoing and outgoing inner horizons intersect, the

X point, is a special one. This is the point through which geodesics of zero energy must pass. Persons with

zero energy who reach the X point see both ingoing and outgoing streams, coming from opposite directions,

infinitely blueshifted.

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8.13 Extremal Reissner-Nordstrom geometry 105

r =∞

r =∞

r =∞

r = ∞

r = ∞

r = −∞

r = −∞

r =−∞

r =−∞

r =−∞

Antihorizon

Universe

Horiz

onWor

mho

leAntiverse

New Universe

Figure 8.3 Penrose diagram of the extremal Reissner-Nordstrom geometry.

8.13 Extremal Reissner-Nordstrom geometry

So far the discussion of the Reissner-Nordstrom geometry has centered on the case Q < M (or more generally,

|Q| < |M |) where there are separate outer and inner horizons. In the special case that the charge and mass

are equal,

Q = M , (8.18)

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106 Reissner-Nordstrom Black Hole

the inner and outer horizons merge into one, r+ = r−, equation (8.9). This special case describes the

extremal Reissner-Nordstrom geometry.

The extremal Reissner-Nordstrom geometry proves to be of particular interest in quantum gravity because

its Hawking temperature is zero, and in string theory because extremal black holes arise as solutions under

certain duality transformations.

The Penrose diagram of the extremal Reissner-Nordstrom geometry is different from that of the standard

Reissner-Nordstrom geometry.

8.14 Reissner-Nordstrom geometry with charge exceeding mass

The Reissner-Nordstrom geometry with charge greater than mass,

Q > M , (8.19)

has no horizons. The change in geometry from an extremal black hole, with horizon at finite radius r+ =

r− = M , to one without horizons is discontinuous. This suggests that there is no way to pack a black hole

with more charge than its mass. Indeed, if you try to force additional charge into an extremal black hole,

then the work needed to do so increases its mass so that the charge Q does not exceed its mass M .

Real fundamental particles nevertheless have charge far exceeding their mass. For example, the charge-

to-mass ratio of a proton ise

mp≈ 1018 (8.20)

where e is the square root of the fine-structure constant α ≡ e2/~c ≈ 1/137, and mp ≈ 10−19 is the mass of

the proton in Planck units. However, the Schwarzschild radius of such a fundamental particle is far tinier

than its Compton wavelength ∼ ~/m (or its classical radius e2/m = α~/m), so quantum mechanics, not

general relativity, governs the structure of these fundamental particles.

8.15 Reissner-Nordstrom geometry with imaginary charge

It is possible formally to consider the Reissner-Nordstrom geometry with imaginary charge Q

Q2 < 0 . (8.21)

This is completely unphysical. If charge were imaginary, then electromagnetic energy would be negative.

However, the Reissner-Nordstrom metric with Q2 < 0 is well-defined, and it is possible to calculate

geodesics in that geometry. What makes the geometry interesting is that the singularity, instead of being

gravitationally repulsive, becomes gravitationally attractive. Thus particles, instead of bouncing off the

singularity, are attracted to it, and it turns out to be possible to continue geodesics through the singularity.

Mathematically, the geometry can be considered as the Kerr-Newman geometry in the limit of zero spin. In

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8.15 Reissner-Nordstrom geometry with imaginary charge 107

r = ∞

r =∞

r =∞

r = ∞

r =−∞

r = −∞

r = −∞

r =−∞

SingularityPara

llel Ant

ihor

izon Antihorizon

Parallel Universe Universe

Parallel HorizonHor

izon

Black Hole

Singularity (r = 0)Black Hole

Inne

r Horiz

on

Parallel Inner Horizon

AntiverseParallel

Antiverse

White Hole

SingularityWhite Hole

Inner Antihorizon

Para

llel In

ner Ant

ihor

izon

New Parallel Universe New Universe

i

i

i

Figure 8.4 Penrose diagram of the Reissner-Nordstrom geometry with imaginary charge Q. If charge wereimaginary, then electromagnetic energy would be negative, which is completely unphysical. But the metricis well-defined, and the spacetime is fun.

the Kerr-Newman geometry, geodesics can pass from positive to negative radius r, and the passage through

the singularity of the Reissner-Nordstrom geometry can be regarded as this process in the limit of zero spin.

Suffice to say that it is intriguing to see what it looks like to pass through the singularity of a charged

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108 Reissner-Nordstrom Black Hole

black hole of imaginary charge, however unrealistic. The Penrose diagram is even more eventful than that

for the usual Reissner-Nordstrom geometry.

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9

Kerr-Newman Black Hole

The geometry of a stationary, rotating, uncharged black hole in asymptotically flat empty space was dis-

covered unexpectedly by Roy Kerr in 1963. Kerr’s (2007) own account of the history of the discovery is at

http://arxiv.org/abs/0706.1109. You can read in that paper that the discovery was not mere chance:

Kerr used sophisticated mathematical methods to make it. The extension to a rotating electrically charged

black hole was made shortly thereafter by Ted Newman (Newman et al. 1965). Newman told me (private

communication 2009) that, after seeing Kerr’s work, he quickly realized that the extension to a charged black

hole was straightforward. He set the problem to the graduate students in his relativity class, who became

coauthors of Newman et al. (1965).

The importance of the Kerr-Newman geometry stems in part from the no-hair theorem, which states

that this geometry is the unique end state of spacetime outside the horizon of an undisturbed black hole in

asymptotically flat space.

9.1 Boyer-Lindquist metric

The Boyer-Linquist metric of the Kerr-Newman geometry is

ds2 = − ∆

ρ2

(

dt− a sin2θ dφ)2

+ρ2

∆dr2 + ρ2dθ2 +

R4 sin2θ

ρ2

(

dφ− a

R2dt)2

(9.1)

where R and ρ are defined by

R ≡√

r2 + a2 , ρ ≡√

r2 + a2 cos2θ , (9.2)

and ∆ is the horizon function defined by

∆ ≡ R2 − 2Mr +Q2 . (9.3)

At large radius r, the Boyer-Linquist metric is

ds2 → −(

1− 2M

r

)

dt2 +

(

1 +2M

r

)

dr2 + r2(

dθ2 + sin2θ dφ2)

− 4aM sin2θ

rdtdφ . (9.4)

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110 Kerr-Newman Black Hole

Comparison of this metric to the metric of a weak field establishes that M is the mass of the black hole and

a is its angular momentum per unit mass. For positive a, the black hole rotates right-handedly about its

polar axis θ = 0.

The Boyer-Linquist line-element (9.1) defines not only a metric but also a tetrad. The Boyer-Linquist

coordinates and tetrad are carefully chosen to exhibit the symmetries of the geometry. In the locally inertial

frame defined by the Boyer-Linquist tetrad, the energy-momentum tensor (which is non-vanishing for charged

Kerr-Newman) and the Weyl tensor are both diagonal. These assertions becomes apparent only in the tetrad

frame, and are obscure in the coordinate frame.

9.2 Oblate spheroidal coordinates

Boyer-Linquist coordinates r, θ, φ are oblate spheroidal coordinates (not polar coordinates). Corresponding

Cartesian coordinates are

x = R sin θ cosφ ,

y = R sin θ sinφ ,

z = r cos θ .

(9.5)

Surfaces of constant r are confocal oblate spheroids, satisfying

x2 + y2

r2 + a2+z2

r2= 1 . (9.6)

Equation (9.6) implies that the spheroidal coordinate r is given in terms of x, y, z by the quadratic equation

r4 − r2(x2 + y2 + z2 − a2)− a2z2 = 0 . (9.7)

9.3 Time and rotation symmetries

The Boyer-Linquist metric coefficients are independent of the time coordinate t and of the azimuthal angle

φ. This shows that the Kerr-Newman geometry has time translation symmetry, and rotational symmetry

about its azimuthal axis. The time and rotation symmetries means that the tangent vectors gt and gφ in

Boyer-Linquist coordinates are Killing vectors. It follows that their scalar products

gt · gt = gtt = − 1

ρ2

(

∆− a2 sin2θ)

,

gt · gφ = gtφ = − a sin2θ

ρ2

(

R2 −∆)

,

gφ · gφ = gφφ =sin2θ

ρ2

(

R4 − a2 sin2θ∆)

, (9.8)

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9.4 Ring singularity 111

are all gauge-invariant scalar quantities. As will be seen below, gtt = 0 defines the boundary of ergospheres,

gtφ = 0 defines the turnaround radius, and gφφ = 0 defines the boundary of the toroidal region containing

closed timelike curves.

The Boyer-Linquist time t and azimuthal angle φ are arranged further to satisfy the condition that gt and

gφ are each orthogonal to both gr and gθ.

9.4 Ring singularity

The Kerr-Newman geometry contains a ring singularity where the Weyl tensor (9.21) diverges, ρ = 0, or

equivalently at

r = 0 and θ = π/2 . (9.9)

The ring singularity is at the focus of the confocal ellipsoids of the Boyer-Linquist metric. Physically, the

singularity is kept open by the centrifugal force.

9.5 Horizons

The horizon of a Kerr-Newmman black hole rotates, as observed by a distant observer, so it is incorrect to

try to solve for the location of the horizon by assuming that the horizon is at rest. The worldline of a photon

that sits on the horizon, battling against the inflow of space, remains at fixed radius r and polar angle θ, but

it moves in time t and azimuthal angle φ. The photon’s 4-velocity is vµ = vt, 0, 0, vφ, and the condition

that it is on a null geodesic is

0 = vµvµ = gµνv

µvν = gtt(vt)2 + 2 gtφ v

tvφ + gφφ(vφ)2 . (9.10)

This equation has solutions provided that the determinant of the 2× 2 matrix of metric coefficients in t and

φ is less than or equal to zero (why?). The determinant is

gttgφφ − g2tφ = − sin2θ∆ (9.11)

where ∆ is the horizon function defined above, equation (9.3). Thus if ∆ ≥ 0, then there exist null geodesics

such that a photon can be instantaneously at rest in r and θ, whereas if ∆ < 0, then no such geodesics exist.

The boundary

∆ = 0 (9.12)

defines the location of horizons. With ∆ given by equation (9.3), equation (9.12) gives outer and inner

horizons at

r± = M ±√

M2 −Q2 − a2 . (9.13)

Between the horizons ∆ is negative, and photons cannot be at rest. This is consistent with the picture that

space is falling faster than light between the horizons.

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112 Kerr-Newman Black Hole

Ergosphere

CTCs

Rot

atio

nax

is

Inner horizon

Outer horizon

r = 0Ringsingularity

Ergosphere

CTCs

Rot

atio

nax

is

Inner horizon

Outer horizon

Turnaround

r = 0Ringsingularity

Figure 9.1 Geometry of (upper) a Kerr black hole with spin parameter a = 0.96M , and (lower) a Kerr-Newman black hole with charge Q = 0.8M and spin parameter a = 0.56M . The upper half of each diagramshows r ≥ 0, while the lower half shows r ≤ 0, the Antiverse. The outer and inner horizons are confocaloblate spheroids whose focus is the ring singularity. For the Kerr geometry, the turnaround radius is atr = 0. CTCs are closed timelike curves.

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9.6 Angular velocity of the horizon 113

Figure 9.2 Not a mouse’s eye view of a snake coming down its mousehole, uhoh. Contours of constant ρ,and their normals, in Boyer-Linquist coordinates, in a Kerr black hole of spin parameter a = 0.96M . Thethicker contours are the outer and inner horizons, which are confocal spheroids with the ring singularity attheir focus.

9.6 Angular velocity of the horizon

The Boyer-Linquist metric (9.1) has been cunningly written so that you can read off the angular velocity of

the horizon as observed by observers at rest at infinity. The horizon is at dr = dθ = 0 and ∆ = 0, and then

the null condition ds2 = 0 implies that the angular velocity is

dt=

a

R2. (9.14)

The derivative is with respect to the proper time t of observers at rest at infinity, so this is the angular

velocity observed by such observers.

9.7 Ergospheres

There are finite regions, just outside the outer horizon and just inside the inner horizon, within which the

worldline of an object at rest, dr = dθ = dφ = 0, is spacelike. These regions, called ergospheres, are places

where nothing can remain at rest (the place where little children come from). Objects can escape from within

the outer ergosphere (whereas they cannot escape from within the outer horizon), but they cannot remain

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114 Kerr-Newman Black Hole

at rest there. A distant observer will see any object within the outer ergosphere being dragged around by

the rotation of the black hole. The direction of dragging is the same as the rotation direction of the black

hole in both outer and inner ergospheres.

The boundary of the ergosphere is at

gtt = 0 (9.15)

which occurs where

∆ = a2 sin2θ . (9.16)

Equation (9.16) has two solutions, the outer and inner ergospheres. The outer and inner ergospheres touch

respectively the outer and inner horizons at the poles, θ = 0 and π.

9.8 Antiverse

The surface at zero radius, r = 0, forms a disk bounded by the ring singularity. Objects can pass through

this disk into the region at negative radius, r < 0, the Antiverse.

The Boyer-Lindquist metric (9.1) is unchanged by a symmetry transformation that simultaneously flips

the sign both of the radius and mass, r → −r and M → −M . Thus the Boyer-Linquist geometry at

negative r with positive mass is equivalent to the geometry at positive r with negative mass. In effect, the

Boyer-Linquist metric with negative r describes a rotating black hole of negative mass

M < 0 . (9.17)

9.9 Closed timelike curves

Inside the inner horizon there is a toroidal region around the ring singularity within which the light cone in

t-φ coordinates opens up to the point that φ as well as t are timelike coordinates. The direction of increasing

proper time along t is t increasing, and along φ is φ decreasing, which is retrograde. Within the toroidal

region, there exist timelike trajectories that go either forwards or backwards in coordinate time t as they wind

retrograde around the toroidal tunnel. Because the φ coordinate is periodic, these timelike curves connect

not only the past to the future (the usual case), but also the future to the past, which violates causality. In

particular, as first pointed out by Carter (1968), there exist closed timelike curves (CTCs), trajectories

that connect to themselves, connecting their own future to their own past, and repeating interminably, like

Sisyphus pushing his rock up the mountain.

The boundary of this toroidal region is at

gφφ = 0 (9.18)

which occurs whereR4

∆= a2 sin2θ . (9.19)

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9.9 Closed timelike curves 115

r = ∞

r =∞

r =∞

r = ∞

r =−∞

r = −∞

r = −∞

r =−∞

White HolePa

ralle

l Antih

oriz

on Antihorizon

Parallel Universe Universe

Parallel HorizonHor

izon

Black Hole

Inne

r Horiz

on

Parallel Inner Horizon

Wor

mho

le ParallelW

ormhole

AntiverseParallel

Antiverse

White HoleInner Antihorizon

Para

llel In

ner Ant

ihor

izon

New Parallel Universe New Universe

Figure 9.3 Penrose diagram of the Kerr-Newman geometry. The diagram is similar to that of the Reissner-Nordstrom geometry, except that it is possible to pass through the disk at r = 0 from the Wormhole regioninto the Antiverse region. This Penrose diagram, which represents a slice at fixed θ and φ, does not capturethe full richness of the geometry, which contains closed timelike curves in a torus around the ring singularity.

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116 Kerr-Newman Black Hole

In the uncharged Kerr geometry the CTC torus is entirely at negative radius, r < 0, but in the Kerr-Newman

geometry the CTC torus extends to positive radius.

9.10 Energy-momentum tensor

The Einstein tensor of the Kerr-Newman geometry in Boyer-Linquist coordinates is a bit of a mess, so I won’t

write it down. The trick of raising one index, which for the Reissner-Nordstrom metric brought the Einstein

tensor to diagonal form, equation (8.4), fails for Boyer-Linquist (because the Boyer-Linquist metric is not

diagonal). The problem is endemic to the coordinate approach to general relativity. When we have done

tetrads we will find that, in the Boyler-Linquist tetrad, the Einstein tensor is diagonal, and that the proper

density ρ, the proper radial pressure pr, and the proper transverse pressure p⊥ in that frame are (do not

confuse the notation ρ for proper density with the radial parameter ρ, equation (9.2), of the Boyer-Linquist

metric)

ρ = −pr = p⊥ =Q2

8πρ4. (9.20)

This looks like the energy-momentum tensor (8.4) of the Reissner-Nordstrom geometry with the replacement

r → ρ. The energy-momentum is that of an electric field produced by a charge Q seemingly located at the

singularity.

9.11 Weyl tensor

The Weyl tensor of the Kerr-Newman geometry in Boyer-Linquist coordinates is likewise a mess. After

tetrads, we will find that the 10 components of the Weyl tensor can be decomposed into 5 complex components

of spin 0, ±1, and ±2. In the Boyer-Linquist tetrad, the only non-vanishing component is the spin-0

component, the Weyl scalar C, but in contrast to the Schwarzschild and Reissner-Nordstrom geometries the

spin-0 component is complex, not real:

C = − 1

(r − ia cos θ)3

(

M − Q2

r + ia cos θ

)

. (9.21)

9.12 Electromagnetic field

The expression for the electromagnetic field in Boyer-Linquist coordinates is again a mess. After tetrads, we

will discover that, in the Boyer-Linquist tetrad, the electromagnetic field is purely radial, and the electro-

magnetic potential has only a time component. For subsequent reference, the electromagnetic potential Aµ

in the Boyer-Linquist coordinate (not tetrad) frame is

Aµ =Qr

ρ2

1, 0, 0, −a sin2θ

. (9.22)

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9.13 Doran coordinates 117

9.13 Doran coordinates

For the Kerr-Newman geometry, the analog of the Gullstrand-Painleve metric is the Doran (2000) metric

ds2 = − dt2ff +

[

ρ

Rdr − βR

ρ

(

dtff − a sin2θdφff

)

]2

+ ρ2dθ2 +R2 sin2θ dφ2ff (9.23)

where the free-fall time tff and azimuthal angle φff are related to the Boyer-Linquist time t and azimuthal

angle φ by

dtff = dt− β

1− β2dr , dφff = φ− aβ

R2(1− β2)dr . (9.24)

The free-fall time tff is the proper time experienced by persons who free-fall from rest at infinity, with zero

angular momentum. They follow trajectories of fixed θ and φff , with radial velocity dr/dtff = β. In other

words, the 4-velocity uν ≡ dxν/dτ of such free-falling observers is

utff = 1 , ur = β , uθ = 0 , uφff = 0 . (9.25)

For the Kerr-Newman geometry, the velocity β is

β = ∓√

2Mr −Q2

R(9.26)

where the ∓ sign is − (infalling) for black hole solutions, and + (outfalling) for white hole solutions.

Horizons occur where the magnitude of the velocity β equals the speed of light

β = ∓1 . (9.27)

The boundaries of ergospheres occur where the velocity is

β = ∓ ρR. (9.28)

The turnaround radius is where the velocity is zero

β = 0 . (9.29)

The region containing closed timelike curves is bounded by the imaginary velocity

β = iρ

a sin θ. (9.30)

9.14 Extremal Kerr-Newman geometry

The Kerr-Newman geometry is called extremal when the outer and inner horizons coincide, r+ = r−, which

occurs where

M2 = Q2 + a2 . (9.31)

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118 Kerr-Newman Black Hole

Rot

atio

nax

is

Inner horizon

Outer horizon

Figure 9.4 Geometry of a Kerr black hole with spin parameter a = 0.96M . The arrows show the velocityβ in the Doran metric. The flow follows lines of constant θ, which form nested hyperboloids orthogonal toand confocal with the nested spheroids of constant r.

Figure 9.5 illustrates the structure of an extremal Kerr (uncharged) black hole, and an extremal Kerr-Newman

(charged) black hole.

9.15 Trajectories of test particles in the Kerr-Newman geometry

Geodesics of test particles in the Kerr-Newman geometry have the expected three constants of motion

associated with conservation of energy, conservation of azimuthal angular momentum, and conservation of

rest mass. Remarkably, Carter (1968) was able to show by separation of variables in the Hamilton-Jacobi

equation that a fourth integral of motion exists, the Carter integral, so that there is a complete set of

four integrals of motion. Moreover, the complete set of integrals exists not only for uncharged particles

(geodesics), but also for charged particles.

The Hamilton-Jacobi method is the most powerful known method for solving equations of motion. For a

test particle it starts with conservation of rest mass m,

gµνpµpν = −m2 , (9.32)

where pµ ≡ mdxµ/dτ is the usual coordinate 4-momentum of the particle. Hamilton-Jacobi replaces the

usual momentum pµ in favour of the generalized momentum πµ = pµ + qAµ, so that the equation (9.32) of

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9.15 Trajectories of test particles in the Kerr-Newman geometry 119

Ergosphere

CTCs

Rot

atio

nax

is

Horizon

r = 0Ringsingularity

Ergosphere

CTCs

Rot

atio

nax

is

Horizon

Turnaround

r = 0Ringsingularity

Figure 9.5 Geometry of (upper) an extremal (a = M) Kerr black hole, and (lower) an extremal Kerr-Newmanblack hole with charge Q = 0.8M and spin parameter a = 0.6M .

conservation of rest mass becomes

gµν (πµ − qAµ) (πν − qAν) = −m2 . (9.33)

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120 Kerr-Newman Black Hole

Finally, Hamilton-Jacobi replaces the generalized momentum with its derivative with respect to the action,

πµ = ∂S/∂xµ,

gµν

(

∂S

∂xµ− qAµ

)(

∂S

∂xν− qAν

)

= −m2 . (9.34)

Equation (9.34) is the Hamilton-Jacobi equation for a test particle of rest mass m and charge q moving in a

prescribed background with metric gµν and electromagnetic potential Aµ.

For the Boyer-Linqust geometry, two integrals of motion follow immediately from the fact that the metric

is independent of time t and azimuthal angle φ. These imply conservation of energy E and azimuthal angular

momentum Lz,

πt =∂S

∂t= −E , πφ =

∂S

∂φ= Lz . (9.35)

Thus for Boyer-Linquist the Hamiton-Jacobi equation (9.34) becomes

gtt(−E− qAt)2 + 2gtφ(−E− qAt)(Lz − qAφ)+ gφφ(Lz− qAφ)2 + grr

(

∂S

∂r

)2

+ gθθ

(

∂S

∂θ

)2

= −m2 . (9.36)

Substituting in the Boyer-Linquist metric and the electromagnetic potential (9.22) brings equation (9.36) to

− P2

∆+

(

aE sin θ − Lz

sin θ

)2

+ ∆

(

∂S

∂r

)2

+

(

∂S

∂θ

)2

= −m2ρ2 , (9.37)

where P , a function of radius r, is

P ≡ ER2 − aLz − qQr . (9.38)

Separating variables in equation (9.37) gives

−∆

(

∂S

∂r

)2

+P 2

∆−m2r2 =

(

∂S

∂θ

)2

+

(

aE sin θ − Lz

sin θ

)2

+m2a2 cos2θ = K , (9.39)

with K a separation constant. The separation constant K provides the desired fourth integral of motion, and

the solution is now essentially complete. The separated equation (9.39) implies

πr =∂S

∂r=

√R

∆, (9.40a)

πθ =∂S

∂θ=√

Θ , (9.40b)

where R and Θ are radial and angular potentials

R ≡ P 2 −(

K +m2r2)

∆ , (9.41a)

Θ ≡ K −(

aE sin θ − Lz

sin θ

)2

−m2a2 cos2θ . (9.41b)

The middle expression of equation (9.39) shows that the constant K is necessarily positive, hitting zero for

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9.15 Trajectories of test particles in the Kerr-Newman geometry 121

a massless particle, m = 0, moving along the polar axis, Lz = 0 and θ = 0 or π. However, it is common to

replace the constant K by the Carter integral Q

K = Q+ (aE − Lz)2 , (9.42)

which has the property that Q = 0 for orbits in the equatorial plane, θ = π/2. In terms of the Carter integral

Q, the radial and angular potentials R and Θ are

R = P 2 −[

Q+ (aE − Lz)2 +m2r2

]

∆ , (9.43a)

Θ = Q− cos2θ

[

a2(m2 − E2) +L2

z

sin2θ

]

. (9.43b)

Converting the generalized covariant momenta, equations (9.35) and (9.40), to ordinary contravariant mo-

menta, pµ = gµν(πν − qAµ), yields

pt =1

ρ2

[

PR2

∆− a

(

aE sin2θ − Lz

)

]

, (9.44a)

pr =1

ρ2

√R , (9.44b)

pθ =1

ρ2

√Θ , (9.44c)

pφ =1

ρ2

[

aP

∆− aE +

Lz

sin2θ

]

. (9.44d)

Equations (9.44) give the general solution for the 4-momentum pµ of a test particle of mass m and charge q

moving in the Kerr-Newman geometry, in terms of its integrals of motion E, Lz, and Q.

The following exercise shows that among ideal black holes, the Schwarzschild geometry is exceptional, not

typical, in having a gravitationally attractive singularity.

Exercise 9.1 Near the Kerr-Newman ring singularity. Explore the behaviour of trajectories of test

particles in the vicinity of the Kerr-Newman singularity, where ρ → 0. Under what conditions does a test

particle reach the singularity?

1. Argue that for a particle to reach the singularity at θ = π/2, positivity of the angular potential Θ requires

that

Q ≥ 0 . (9.45)

2. Argue that for a particle to reach the singularity at r = 0, positivity of the radial potential R requires

that

Q2(aE − Lz)2 + (Q2 + a2)Q ≤ 0 . (9.46)

3. Schwarzschild case: show that if Q = 0 and a = 0, then a particle reaches the singularity provided that

the mass of the black hole is positive, M > 0.

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122 Kerr-Newman Black Hole

4. Reissner-Nordstrom case: show that if Q2 > 0 and a = 0, then a particle can reach the singularity only if

it has zero angular momentum, Q = Lz = 0, and if the particle’s charge-to-mass exceeds unity,

q2

m2≥ 1 . (9.47)

5. Kerr case: show that if Q = 0 but a2 > 0, then a particle can reach the singularity only if Q = 0, and

provided that the mass of the black hole is positive, M > 0.

6. Kerr-Newman case: show that if Q2 > 0 and a2 > 0, then a particle can reach the singularity only if

Lz = aE and Q = 0, and if the particle’s charge-to-mass is large enough,

q2

m2≥ Q2 + a2

Q2, (9.48)

which generalizes the Reissner-Nordstrom condition (9.47).

9.16 Penrose process

Trajectories in the Kerr-Newman geometry can have negative energy E outside the horizon. It is possible to

reduce the mass M of the black hole by dropping negative energy particles into the black hole. This process

of extracting mass-energy from the black hole is called the Penrose process.

Exercise 9.2 Negative energy trajectories outside the horizon. Under what conditions can test

particles have negative energy trajectories, E < 0, outside the horizon?

1. Argue that outside the horizon, the positivity of the horizon function ∆, and of the radial and angular

potentials R and Θ, equations (9.41) implies that P , equation (9.38), satisfies

P 2 ≥(

K +m2r2)

∆ ≥[

(

aE sin θ − Lz

sin θ

)2

+m2ρ2

]

∆ . (9.49)

2. Argue that the condition (9.49) implies by continuity that for a massive particle P must be strictly positive

outside the horizon. Extend your argument to a massless particle by taking a massless particle as a massive

particle in the limit of large energy.

3. Argue that the positivity of P implies that aLz + qQr must be negative for the energy E to be negative.

Show that, more stringently, negative E requires that

aLz + qQr ≤ −√

(

L2z

sin2θ+m2ρ2

)

∆ . (9.50)

4. Argue that for an uncharged particle, q = 0, negative energy trajectories exist only inside the ergosphere.

5. Do negative energy trajectories exist outside the ergosphere for a charged particle?

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9.17 Constant latitude trajectories in the Kerr-Newman geometry 123

6. For the Penrose process to work, the negative energy particle must fall through the horizon, where ∆ = 0.

Does this happen?

Exercise 9.3 When can objects go forwards or backwards in time t?

9.17 Constant latitude trajectories in the Kerr-Newman geometry

A trajectory is at constant latitude if it is at constant polar angle θ,

θ = constant . (9.51)

Constant latitude orbits occur where the angular potential Θ, equation (9.43b), not only vanishes, but is an

extremum,

Θ =dΘ

dθ= 0 , (9.52)

the derivative being taken with the constants of motion E, Lz, and Q of the orbit being held fixed. The

condition Θ = 0 simply sets the value of the Carter integral Q. Solving dΘ/dθ = 0 yields the condition

between energy E and angular momentum Lz

E = ±√

1 +L2

z

a2 sin4θ. (9.53)

Solutions at any polar angle θ and any angular momentum Lz exist, ranging from E = ±1 at Lz = 0, to

E = ±Lz/(a sin2θ) at Lz → ±∞. The solutions with Lz = 0 are those of the freely-falling observers that

define the Doran coordinate system, §9.13. The solutions with Lz →∞ define the principal null congruences

discussed in §9.18.

9.18 Principal null congruence

A congruence is a space-filling, non-overlapping set of geodesics. In the Kerr-Newman geometry there is

a special set of null geodesics, the ingoing and outgoing principal null congruences, with respect to

which the symmetries of the geometry are especially apparent. Photons that hold steady on the horizon are

members of the outgoing principal null congruence. The energy-momentum tensor is diagonal in a locally

inertial frame aligned with the ingoing or outgoing principal null congruence. The Weyl tensor, decomposed

into spin components in the locally inertial frame of the principal null congruences, contains only spin-0

components.

The Boyer-Linquist metric (9.1) is specifically constructed so that the Boyer Linquist tetrad is aligned with

the principal null tetrad. Along the principal null congruences, the final two terms of the Boyer-Linquist

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124 Kerr-Newman Black Hole

line element (9.1) vanish

dθ = dφ− a

R2dt = 0 . (9.54)

Solving the null condition ds2 = 0 on the rest of the metric yields the photon 4-velocity vµ ≡ dxµ/dλ on the

principal null congruences

vt =R2

∆, vr = ±1 , vθ = 0 , vφ =

a

∆. (9.55)

In the regions outside the outer horizon or inside the inner horizon, the ± sign in front of vr is + for outgoing,

− for ingoing geodesics. Between the outer and inner horizons, vr is negative in the Black Hole region, and

positive in the White Hole region, while vt and vφ are negative for ingoing, positive for outgoing geodesics.

The angular momentum per unit energy Jz ≡ Lz/|E| of photons along the principal null congruences is not

zero, but is

Jz = a sin2θ (9.56)

with the same sign for both ingoing and outgoing geodesics.

9.19 Circular orbits in the Kerr-Newman geometry

An orbit can be termed circular if it is at constant radius r,

r = constant . (9.57)

It is convenient to call such an orbit circular even if the orbit is at finite inclination (not confined to the

equatorial plane) about a rotating black hole, and therefore follows the surface of a spheroid (in Boyer-

Lindquist coordinates).

Orbits turn around in r, reaching periapsis or apoapsis, where the radial potential R, equation (9.43a),

vanishes. Circular orbits occur where the radial potential R not only vanishes, but is an extremum,

R =dRdr

= 0 , (9.58)

the derivative being taken with the constants of motion E, Lz, and Q of the orbit being held fixed. Circular

orbits may be either stable or unstable. The stability of a circular orbit is determined by the sign of the

second derivative of the potential

d2Rdr2

, (9.59)

with + for stable, − for unstable circular orbits. Marginally stable orbits occur where d2R/dr2 = 0.

Circular orbits occur not only in the equatorial plane, but at general inclinations. The inclination of an

orbit can be characterized by the minimum polar angle θmin to which it extends. An astronomer would call

π/2− θmin the inclination angle of the orbit. It is convenient to define an inclination parameter α by

α ≡ cos2θmin , (9.60)

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9.19 Circular orbits in the Kerr-Newman geometry 125

which lies in the interval [0, 1]. Equatorial orbits, at θ = π/2, correspond to α = 0, while polar orbits, those

that go over the poles at θ = 0 and π, correspond to α = 1.

9.19.1 General solution for circular orbits

The general solution for circular orbits of a test particle of arbitrary electric charge q in the Kerr-Newman

geometry is as follows.

The rest mass m of the test particle can be set equal to unity, m = 1, without loss of generality. Circular

orbits of particles with zero rest mass, m = 0, discussed later in this section, occur in cases where the circular

orbits for massive particles attain infinite energy and angular momentum.

In the radial potential R, equation (9.43a), eliminate the Carter integral Q in favour of the inclination

parameter α, equation (9.60), using equation (9.43b)

Q = α

[

a2(1− E2) +L2

z

1− α

]

. (9.61)

Furthermore, eliminate the energy E in favour of P , equation (9.38). The radial derivatives dnR/drn must

be taken before E is replaced by P , since E is a constant of motion, whereas P varies with r. The physical

motivation for replacing E with P lies in the sign of P . Solutions with positive P correspond to orbits in the

Universe, Wormhole, or Antiverse parts of the Kerr-Newman geometry in the Penrose diagram of Figure 9.3,

while solutions with negative P correspond to orbits in their Parallel counterparts. If only the Universe

region is considered, then P is necessarily positive. By contrast, the energy E can be either positive or

negative in the same region of the Kerr-Newman geometry (the energy E is negative for orbits of sufficiently

large negative angular momentum Lz inside the ergosphere of the Universe).

The condition R = 0 is a quadratic equation in Lz, whose solutions are

Lz =1

r2 + a2α

[

a(1− α)(P + qQr)±R2√

(1− α) [P 2/∆− (r2 + a2α)]]

. (9.62)

Substituting the two (±) expressions (9.62) for Lz into dR/dr, and setting the product of the resulting two

expressions for dR/dr equal to zero, yields a quartic equation for P/∆:

p0 + p1(P/∆) + p2(P/∆)2 + p3(P/∆)3 + p4(P/∆)4 = 0 , (9.63)

with coefficients

p0 ≡ r2(r2 + a2α)2 , (9.64a)

p1 ≡ − 2qQr(r2 − a2α)(r2 + a2α) , (9.64b)

p2 ≡ − 2r2(r2 + a2α)(r2 − 3Mr + 2Q2 + a2α+ a2αM/r) + q2Q2(r2 − a2α)2 , (9.64c)

p3 ≡ 2qQr(r2 − a2α)(r2 − 3Mr + 2Q2 + 2a2 − a2α+ a2αM/r) , (9.64d)

p4 ≡[

r6 − 6Mr5 + (9M2+4Q2+2a2α)r4 − 4M(3Q2+a2)r3

+ (4Q4−6a2αM+4a2Q2+a4α2)r2 + 2a2α(2Q2+2a2−a2α)Mr + a4α2M2]

. (9.64e)

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126 Kerr-Newman Black Hole

The quartic (9.63) is the condition for an orbit at radius r to be circular. Physical solutions must be real.

The quartic (9.63) has either zero, two, or four real solutions at any one radius r. Numerically, it is better

to solve the quartic (9.63) for the reciprocal ∆/P rather than P/∆, since the vanishing of 1/P defines the

location of circular orbits of massless particles.

−3 −2 −1 0 1 2 3 4 5 6 7 8 9−3

−2

−1

0

1

2

3

Radius r/M

∆/P

Outer

horizon

Innerhorizon

Figure 9.6 Values of ∆/P for circular orbits at radius r of a charged particle about a Kerr-Newman blackhole. The values ∆/P are real roots of the quartic (9.63); there are either zero, two, or four real roots atany one radius. The parameters are representative: a particle of charge-to-mass q/m = 2.4 on an orbit ofinclination parameter α = 0.5 about a black hole of charge Q = 0.5M and spin parameter a = 0.5M . Solid(green) lines indicate stable orbits; dashed (brown) lines indicate unstable orbits. Positive ∆/P orbits occurin Universe, Wormhole, and Antiverse regions; negative ∆/P orbits occur in their Parallel counterparts;zero ∆/P orbits are null. The fact that the particle is charged breaks the symmetry between positive andnegative ∆/P . If the charge of the particle were flipped, q/m = −2.4, then the diagram would be reflectedabout the horizontal axis (the sign of ∆/P would flip).

The angular momentum Lz, energy E, and stability d2R/dr2 of a circular orbit are, in terms of a solution

P/∆ of the quartic (9.63),

Lz = ± 1

r2 + a2α

(1− α) [l−1(∆/P ) + l0 + l1(P/∆) + l2(P/∆)2] , (9.65a)

E =1

2[(∆/P ) + qQ/r + (1−M/r)(P/∆)] , (9.65b)

d2Rdr2

=2

(r2 + a2α)2[

q−1(∆/P ) + q0 + q1(P/∆) + q2(P/∆)2]

, (9.65c)

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9.19 Circular orbits in the Kerr-Newman geometry 127

where the coefficients li and qi are

l−1 = qQrR2(r2 + a2α) , (9.66a)

l0 = −R2(r2 + a2α)(2Mr −Q2)− q2Q2(r4 − a4α) , (9.66b)

l1 = − qQr[

2r4 − 5Mr3 + 3(Q2+a2)r2 − a2(1+α)Mr + a2(Q2+αQ2+a2−a2α)

+ 3a4αM/r − a4α(Q2+a2)/r2]

, (9.66c)

l2 =[

3Mr3 − 2Q2r2 + a2(1+α)Mr − a2(1+α)Q2 − a4αM/r]

∆ , (9.66d)

and

q−1 = 2qQr(r2 − a2α)(r2 + a2α) , (9.67a)

q0 = − 4(r2 + a2α)(Mr3 −Q2r2 − a2αMr)− q2Q2(r2 − a2α)2 , (9.67b)

q1 = − qQr[

r4 − 4Mr3 + 3(Q2+a2−2a2α)r2 + 12a2αMr − a2α(6Q2+6a2−a2α)

− a4α2(Q2+a2)/r2]

, (9.67c)

q2 =(

3Mr3 − 4Q2r2 − 6a2αMr − a4α2M/r)

∆ . (9.67d)

The sign of the angular momentum Lz in equation (9.65a) should be chosen such that the relations (9.38)

for P and (9.65b) for E hold. This choice of sign becomes ambiguous for a = 0; but this is as it should be,

since either sign of Lz is valid for a = 0, where the black hole is spherically symmetric, and therefore defines

no preferred direction.

The expressions (9.62) and (9.65a) for Lz are equal on a circular orbit. The advantage of the latter

expression (9.65a) will become apparent below, where it is found that for particles of zero electric charge,

q = 0, one circular orbit is always prograde, aLz > 0, while the other is always retrograde, aLz < 0.

For non-zero a, the reality of a solution P/∆ of the quartic (9.63) is a necessary and sufficient condition for

a corresponding circular orbit to exist. In particular, the argument of the square root in the expression (9.65a)

for Lz is guaranteed to be positive. For zero a, however, the quartic (9.65a), which reduces in this case to

the square of a quadratic, admits real solutions that do not correspond to a circular orbit. For these invalid

solutions, the argument of the square root in the expression (9.65a) for Lz is negative. Thus for zero a, a

necessary and sufficient condition for a circular orbit to exist is that the solutions for both P/∆ and Lz be

real.

9.19.2 Circular orbits for massless particles

Circular orbits for massless particles, m = 0, or null circular orbits, follow from the solutions for massive

particles in the case where the energy and angular momentum on the circular orbit become infinite, which

occurs when P → ±∞. Except at horizons, where ∆ = 0, the solution for P from the quartic (9.63) diverges

when the ratio p4/p0 of the highest to lowest order coefficients vanishes. The ratio p4/p0, equations (9.64),

factors as

p4/p0 =F+F−

(r2 + a2α)2, (9.68)

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128 Kerr-Newman Black Hole

where

F± ≡ r2 − 3Mr + 2Q2 + a2α(1 +M/r) ± 2a√

(1− α)(Mr −Q2 − a2αM/r) . (9.69)

A null circular orbit thus occurs at a radius r such that

F+ = 0 or F− = 0 , (9.70)

with + for prograde (aLz > 0) orbits, − for retrograde (aLz < 0) orbits. The location of null circular

orbits are independent of the charge q of the particle, since F± are independent of charge q. The angular

momentum Jz per unit energy on the null circular orbit is, from equations (9.65a) and (9.65b) in the limit

P → ±∞,

Jz ≡ Lz/|E| = ±2√

(1− α)l2(r2 + a2α)2(1−M/r)

. (9.71)

The case where F+ or F− vanishes at a horizon is special. This occurs when the black hole is extremal,

M2 = Q2 + a2. A circular orbit exists at the horizon of an extremal black hole provided that the charge

squared Q2 and inclination parameter α are not too large, the precise condition being

a4α2 + 6(Q2+a2)α− (Q2+a2)(Q2−3a2) ≤ 0 . (9.72)

The circular orbit is non-null, since the vanishing of ∆/P no longer implies that P diverges if ∆ = 0, as is

true on the horizon. A careful analysis shows that the limiting value of P/√

∆ is finite for a circular orbit

at the horizon of an extremal black hole, so in fact P = 0 for such an orbit.

Since there are null geodesics, the ingoing or outgoing principal null geodesics, that hold steady on the

horizon, one might have expected that there would always be solutions for null circular orbits on the horizon,

but this is false. The resolution of the paradox is that massless particles experience no proper time along

their geodesics. If a massive particle is put on the horizon on a relativistic geodesic, then the massive particle

necessarily falls off the horizon in a finite proper time: it is impossible for the geodesic to hold steady on

the horizon. The only exception is that, as discussed in the previous paragraph, an extremal black hole may

have circular orbits at its horizon; but these orbits have P = 0, and are not null.

9.19.3 Circular orbits for particles with zero electric charge

For a particle with zero electric charge, q = 0, the quartic condition (9.63) for a circular orbit reduces to a

quadratic in (P/∆)2. Solving the quadratic for the reciprocal (∆/P )2 yields two possible solutions

(∆/P )2 =F±

r2 + a2α, (9.73)

where F± are defined by equation (9.69), with + for prograde (aLz > 0) orbits, − for retrograde (aLz < 0)

orbits. The sign of P is positive in the Universe, Wormhole, and Antiverse of Figure 9.3, negative in their

Parallel counterparts. For zero electric charge, the expressions (9.65) for the angular momentum Lz, energy

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9.19 Circular orbits in the Kerr-Newman geometry 129

E, and stability d2R/dr2 of a circular orbit simplify to

Lz = ± 1

r2 + a2α

(1− α) [l0 + l2(P/∆)2] , (9.74a)

E =1

2[(∆/P ) + (1 −M/r)(P/∆)] , (9.74b)

d2Rdr2

=2

(r2 + a2α)2[

q0 + q2(P/∆)2]

. (9.74c)

The coefficients li and qi in equations (9.74) reduce from the expressions (9.66) and (9.67) to

l0 = −R2(r2 + a2α)(2Mr −Q2) , (9.75a)

l2 =[

3Mr3 − 2Q2r2 + a2(1+α)Mr − a2(1+α)Q2 − a4αM/r]

∆ , (9.75b)

and

q0 = − 4(r2 + a2α)(Mr3 −Q2r2 − a2αMr) , (9.76a)

q2 =(

3Mr3 − 4Q2r2 − 6a2αMr − a4α2M/r)

∆ . (9.76b)

.0 .5 1.0 1.5 2.0−2.0

−1.5

−1.0

−.5

.0

.5

1.0

1.5

2.0

Radius r/M

P

Outer

horizon

Innerhorizon

Figure 9.7 Values of P for circular orbits at radius r in the equatorial plane of a near-extremal Kerr blackhole, with black hole spin parameter a = 0.999M . The diagram illustrates that as the orbital radius rapproaches the horizon, P first approaches zero, but then increases sharply to infinity, corresponding to nullcircular orbits. In the case of an exactly extremal black hole, P goes as to zero at the horizon, there is noincrease of P to infinity, and no null circular orbit. Solid (green) lines indicate stable orbits; dashed (brown)lines indicate unstable orbits.

9.19.4 Equatorial circular orbits in the Kerr geometry

The case of greatest practical interest to astrophysicists is that of circular orbits in the equatorial plane of

an uncharged black hole, the Kerr geometry.

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130 Kerr-Newman Black Hole

For circular orbits in the equatorial plane, α = 0, of an uncharged black hole, Q = 0, the solution (9.73)

simplifies to

(∆/P )2 =F±r2

(9.77)

where F±, equation (9.69), reduce to

F± ≡ r2 − 3Mr ± 2a√Mr , (9.78)

with + for prograde (aLz > 0) orbits, − for retrograde (aLz < 0) orbits.

As discussed above, null circular orbits occur where F± = 0, except in the special case that the circular

orbit is at the horizon, which occurs when the black hole is extremal. In the limit where the Kerr black hole

is near but not exactly extremal, a → |M |, null circular orbits occur at r → M (prograde) and r → 4M

(retrograde). For an exactly extremal Kerr black hole, a = |M |, the (prograde) circular orbit at the horizon

is no longer null. The situation of a near extremal Kerr black hole is illustrated by Figure 9.7.

It is generally argued that the inner edge of an accretion disk is likely to occur at the innermost stable

equatorial circular orbit. An orbit at this point has marginal stability, d2R/dr2 = 0. Simplifying the stability

d2R/dr2 from equation (9.74c) to the case of equatorial orbits, α = 0, and zero black hole charge, Q = 0,

yields the condition of marginal stability

r2 − 6Mr − 3a2 ± 8a√Mr = 0 . (9.79)

The + (prograde) orbit has the smaller radius, and so defines the innermost stable circular orbit. For an

extremal Kerr black hole, a = |M |, marginally stable circular equatorial orbits are at r = M (prograde) and

r = 9M (retrograde).

9.19.5 Circular orbits in the Reissner-Nordstrom geometry

Circular orbits of particles in the Reissner-Nordstrom geometry follow from those in the Kerr-Newman

geometry in the limit of a non-rotating black hole, a = 0. For a non-rotating black hole, an orbit can be

taken without loss of generality to circulate right-handedly in the equatorial plane, θ = π/2, so that α = 0

and the azimuthal angular momentum Lz equals the positive total angular momentum L. For non-equatorial

orbits, the relation between azimuthal and total angular momentum is Lz = ±√

1− αL.

For a non-rotating black hole, a = 0, the quartic condition (9.63) for a circular orbit of a particle of rest

mass m = 1 and electric charge q reduces to the square of a quadratic,

r2 − qQr(P/∆)−(

r2 − 3Mr + 2Q2)

(P/∆)2 = 0 . (9.80)

Solving the quadratic (9.80) for the reciprocal ∆/P yields two solutions

∆/P =qQ

2r±√

1− 3M

r+

2Q2

r2+q2Q2

4r2. (9.81)

The sign of P is positive in the Universe, Wormhole, and Antiverse parts of the Reissner-Nordstrom geometry

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9.19 Circular orbits in the Kerr-Newman geometry 131

in the Penrose diagram of Figure 8.1, negative in their Parallel counterparts. The angular momentum L,

energy E, and stability d2R/dr2 of a circular orbit are, in terms of a solution ∆/P of the quadratic (9.80),

L =√

P 2/∆− r2 , (9.82)

E =P

r2+qQ

r, (9.83)

d2Rdr2

= 2(

r2 − 6Mr + 5Q2 + q2Q2)

− 2

(

1− 6M

r+

6Q2

r2

)

P 2

∆. (9.84)

For massless particles, circular orbits occur where the solution (9.81) for ∆/P vanishes, which occurs when

r2 − 3Mr + 2Q2 = 0 , (9.85)

independent of the charge q of the particle. The condition (9.85) is consistent with the Kerr-Newman

condition for a null circular orbit, the vanishing of F± given by equation (9.69). However, for Kerr-Newman,

the argument of the square root on the right hand side of equation (9.69) for F± must be positive, even in

the limit of infinitesimal a. In the limit of small a, this requires that Mr −Q2 ≥ 0. If the charge Q of the

Reissner-Nordstrom black hole lies in the standard range 0 ≤ Q2 ≤ M2, then one of the solutions of the

quadratic (9.85) lies outside the outer horizon, while the other lies between the outer and inner horizons.

As one might hope, the additional condition Mr −Q2 ≥ 0 eliminates the undesirable solution between the

horizons, leaving only the solution outside the horizon, which is

r =3M

2

(

1 +

1− 8Q2

9

)

for 0 ≤ Q2 ≤M2 . (9.86)

In (unphysical) cases Q2 < 0 or M2 < Q2 ≤ (9/8)M2, both solutions of equation (9.85) are valid.

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PART FOUR

HOMOGENEOUS, ISOTROPIC COSMOLOGY

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Concept Questions

1. What does it mean that the Universe is expanding?

2. Does the expansion affect the solar system or the Milky Way?

3. How far out do you have to go before the expansion is evident?

4. What is the Universe expanding into?

5. In what sense is the Hubble constant constant?

6. Does our Universe have a center, and if so where is it?

7. What evidence suggests that the Universe at large is homogeneous and isotropic?

8. How can the CMB be construed as evidence for homogeneity and isotropy given that it provides information

only about a 2D surface on the sky?

9. What is thermodynamic equilibrium? What evidence suggests that the early Universe was in thermody-

namic equilibrium?

10. What are cosmological parameters?

11. What cosmological parameters can or cannot be measured from the power spectrum of fluctuations of the

CMB?

12. FRW Universes are characterized as closed, flat, or open. Does flat here mean the same as flat Minkowski

space?

13. What is it that astronomers call dark matter?

14. What is the primary evidence for the existence of non-baryonic cold dark matter?

15. How can astronomers detect dark matter in galaxies or clusters of galaxies?

16. How can cosmologists claim that the Universe is dominated by not one but two distinct kinds of mysterious

mass-energy, dark matter and dark energy, neither of which has been observed in the laboratory?

17. What key property or properties distinguish dark energy from dark matter?

18. Does the Universe conserve entropy?

19. Does the annihilation of electron-positron pairs into photons generate entropy in the early Universe, as its

temperature cools through 1 MeV?

20. How does the wavelength of light change with the expansion of the Universe?

21. How does the temperature of the CMB change with the expansion of the Universe?

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136 Concept Questions

22. How does a blackbody (Planck) distribution change with the expansion of the Universe? What about a

non-relativistic distribution? What about a semi-relativistic distribution?

23. What is the horizon of our Universe?

24. What happens beyond the horizon of our Universe?

25. What caused the Big Bang?

26. What happened before the Big Bang?

27. What will be the fate of the Universe?

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What’s important?

1. The CMB indicates that the early (≈ 400,000 year old) Universe was (a) uniform to a few ×10−5, and (b)

in thermodynamic equilibrium. This indicates that

the Universe was once very simple .

It is this simplicity that makes it possible to model the early Universe with some degree of confidence.

2. The power spectrum of fluctuations of the CMB has enabled precise measurements of cosmological pa-

rameters, excepting the Hubble constant.

3. There is a remarkable concordance of evidence from a broad range of astronomical observations — su-

pernovae, big bang nucleosynthesis, the clustering of galaxies, the abundances of clusters of galaxies,

measurements of the Hubble constant from Cepheid variables, the ages of the oldest stars.

4. Observational evidence is consistent with the predictions of the theory of inflation in its simplest form —

the expansion of the Universe, the spatial flatness of the Universe, the near uniformity of temperature

fluctuations of the CMB (the horizon problem), the presence of acoustic peaks and troughs in the power

spectrum of fluctuations of the CMB, the near power law shape of the power spectrum at large scales, its

spectral index (tilt), the gaussian distribution of fluctuations at large scales.

5. What is non-baryonic dark matter?

6. What is dark energy? What is its equation of state w ≡ p/ρ, and how does w evolve with time?

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10

Homogeneous, Isotropic Cosmology

10.1 Observational basis

Since 1998, observations have converged on a Standard Model of Cosmology, a spatially flat universe domi-

nated by dark energy and by non-baryonic dark matter.

1. The Hubble diagram (distance versus redshift) of galaxies indicates that the Universe is expanding (Hubble

1929).

2. The Cosmic Microwave Background (CMB).

• Near black body spectrum, with T0 = 2.725± 0.001 K (Fixsen & Mather 2002).

• Dipole ⇒ the solar system is moving at 365 kms−1 through the CMB.

• After dipole subtraction, the temperature of the CMB over the sky is uniform to a few parts in 105.

• The power spectrum of temperature T fluctuations shows a scale-invariant spectrum at large scales,

and prominent acoustic peaks at smaller scales. Allows measurement of the amplitude As and tilt ns of

primordial fluctuations, the curvature density Ωk, and the proper densities Ωch2 of non-baryonic cold

dark matter and Ωbh2 of baryons. Does not measure Hubble constant h ≡ H0/(100 kms−1 Mpc−1).

• The power spectra of E and B polarization fluctuations, and the various cross power spectra (only T -E

should be non-vanishing).

3. The Hubble diagram of Type Ia (thermonuclear) supernovae indicates that the Universe is accelerating.

This points to the dominance of gravitationally repulsive dark energy, with ΩΛ ≈ 0.75. The amount of

dark energy is consistent with observations from the CMB indicating that the Universe is spatially flat,

Ω ≈ 1, and observations from CMB, galaxy clustering, and clusters of galaxies indicating that the density

in gravitationally attractive matter is only Ωm ≈ 0.25.

4. Observed abundances of light elements H, D, 3He, He, and Li are consistent with the predictions of big

bang nucleosynthesis (BBN) provideed that Ωb ≈ 0.04, in good agreement with measurements from the

CMB.

5. The clustering of matter (dark and bright) shows a power spectrum in good agreement with the Standard

Model:

• galaxies;

• the Lyman alpha forest;

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10.2 Cosmological Principle 139

• gravitational lensing.

Historically, the principle evidence for non-baryonic cold dark matter is comparison between the power

spectra of galaxies versus CMB. How can tiny fluctuations in the CMB grow into the observed fluctuations

in matter today in only the age of the Universe? Answer: non-baryonic dark matter that begins to cluster

before Recombination, when the CMB was released.

6. The abundance of galaxy clusters as a function of redshift.

7. The ages of the oldest stars, in globular clusters. The Hubble constant yields an estimate of the age of

the Universe that is older with dark energy than without. The ages of the oldest stars agree with the age

of the Universe with dark enery, but are older than the Universe without dark energy.

8. Ubiquitous evidence for dark matter, deduced from sizes and velocities (or in the case of gravitational

lensing, the gravitational potential) of various objects.

• The Local Group of galaxies.

• Rotation curves of spiral galaxies.

• The temperature and distribution of x-ray gas in elliptical galaxies.

• The temperature and distribution of x-ray gas in clusters of galaxies.

• Gravitational lensing by clusters of galaxies.

9. The Bullet cluster is a rare example that supports the notion that the dark matter is non-baryonic. In the

Bullet cluster, two clusters recently passed through each other. The baryonic matter, as measured from

x-ray emission of hot gas, appears displaced from the dark matter, as measured from weak gravitational

lensing.

10.2 Cosmological Principle

The cosmological principle states that the Universe at large is

• homogeneous (has spatial translation symmetry),

• isotropic (has spatial rotation symmetry).

The primary evidence for this is the uniformity of the temperature of the CMB, which, after subtraction

of the dipole produced by the motion of the solar system through the CMB, is constant over the sky to a

few parts in 105. Confirming evidence is the statistical uniformity of the distribution of galaxies over large

scales.

The cosmological principle allows that the Universe evolves in time, as observations surely indicate — the

Universe is expanding, galaxies, quasars, and galaxy clusters evolve with redshift, and the temperature of

the CMB is undoubtedly decreasing as the Universe expands.

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140 Homogeneous, Isotropic Cosmology

10.3 Friedmann-Robertson-Walker metric

Universes satisfying the cosmological principle are described by the Friedmann-Robertson-Walker (FRW)

metric, equation (10.25) below. The metric, and the associated Einstein equations, which are known as the

Friedmann equations, are set forward in the next several sections, §§10.4–10.9.

10.4 Spatial part of the FRW metric: informal approach

The cosmological principle implies that

the spatial part of the FRW metric is a 3D hypersphere (10.1)

where in this context the term hypersphere is to be construed as including not only cases of positive curvature,

which have finite positive radius of curvature, but also cases of zero and negative curvature, which have

infinite and imaginary radius of curvature.

w

r = Rsinχ

R

xy

r// =

RχRdχ

Rsinχdφ

χ

φ

Figure 10.1 Embedding diagram of the FRW geometry.

Figure 10.1 shows an embedding diagram of a 3D hypersphere in 4D Euclidean space. The horizontal

directions in the diagram represent the normal 3 spatial x, y, z dimensions, with one dimension z suppressed,

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10.4 Spatial part of the FRW metric: informal approach 141

while the vertical dimension represents the 4th spatial dimension w. The 3D hypersphere is a set of points

x, y, z, w satisfying(

x2 + y2 + z2 + w2)1/2

= R = constant . (10.2)

An observer is sitting at the north pole of the diagram, at 0, 0, 0, 1. A 2D sphere (which forms a 1D circle

in the embedding diagram of Figure 10.1) at fixed distance surrounding the observer has geodesic distance

r‖ defined by

r‖ ≡ proper distance to sphere measured along a radial geodesic , (10.3)

and circumferential radius r defined by

r ≡(

x2 + y2 + z2)1/2

, (10.4)

which has the property that the proper circumference of the sphere is 2πr. In terms of r‖ and r, the spatial

metric is

dl2 = dr2‖ + r2do2 (10.5)

where do2 ≡ dθ2 + sin2θ dφ2 is the metric of a unit 2-sphere.

Introduce the angle χ illustrated in the diagram. Evidently

r‖ = Rχ ,

r = R sinχ . (10.6)

In terms of the angle χ, the spatial metric is

dl2 = R2(

dχ2 + sin2χdo2)

(10.7)

which is one version of the spatial FRW metric. The metric resembles the metric of a 2-sphere of radius R,

which is not surprising since the same construction, with Figure 10.1 interpreted as the embedding diagram

of a 2D sphere in 3D, yields the metric of a 2-sphere. Indeed, the construction iterates to give the metric of

an n-dimensional sphere of arbitrarily many dimensions n.

Instead of the angle χ, the metric can be expressed in terms of the circumferential radius r. It follows

from equations (10.6) that

r‖ = R sin−1(r/R) (10.8)

whence

dr‖ =dr

1− r2/R2

=dr√

1−Kr2(10.9)

where K is the curvature

K ≡ 1

R2. (10.10)

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142 Homogeneous, Isotropic Cosmology

In terms of r, the spatial FRW metric is then

dl2 =dr2

1−Kr2 + r2do2 . (10.11)

The embedding diagram Figure 10.1 is a nice prop for the imagination, but it is not the whole story. The

curvature K in the metric (10.11) may be not only positive, corresponding to real finite radius R, but also

zero or negative, corresponding to infinite or imaginary radius R. The possibilities are called closed, flat,

and open:

K

> 0 closed R real ,

= 0 flat R→∞ ,

< 0 open R imaginary .

(10.12)

10.5 Comoving coordinates

The metric (10.11) is valid at any single instant of cosmic time t. As the Universe expands, the 3D spatial

hypersphere (whether closed, flat, or open) expands. In cosmology it is highly advantageous to work in

comoving coordinates that expand with the Universe. Why? First, it is helpful conceptually and math-

ematically to think of the Universe as at rest in comoving coordinates. Second, linear perturbations, such

as those in the CMB, have wavelengths that expand with the Universe, and are therefore fixed in comoving

coordinates.

In practice, cosmologists introduce the cosmic scale factor a(t)

a(t) ≡ measure of the size of the Universe, expanding with the Universe (10.13)

which is proportional to but not necessarily equal to the radius R of the Universe. The cosmic scale factor

a can be normalized in any arbitrary way. The most common convention adopted by cosmologists is to

normalize it to unity at the present time,

a0 = 1 , (10.14)

where the 0 subscript conventionally signifies the present time.

Comoving geodesic and circumferential radial distances x‖ and x are defined in terms of the proper geodesic

and circumferential radial distances r‖ and r by

ax‖ ≡ r‖ , ax ≡ r . (10.15)

Objects expanding with the Universe remain at fixed comoving positions x‖ and x. In terms of the comoving

circumferential radius x, the spatial FRW metric is

dl2 = a2

(

dx2

1− κx2+ x2do2

)

, (10.16)

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10.6 Spatial part of the FRW metric: more formal approach 143

where the curvature constant κ, a constant in time and space, is related to the curvature K, equation (10.10),

by

κ ≡ a2K . (10.17)

Alternatively, in terms of the geodesic comoving radius x‖, the spatial FRW metric is

dl2 = a2(

dx2‖ + x2do2

)

, (10.18)

where

x =

sin(κ1/2x‖)

κ1/2κ > 0 closed ,

x‖ κ = 0 flat ,

sinh(|κ|1/2x‖)

|κ|1/2κ < 0 open .

(10.19)

For some purposes it is convenient to normalize the cosmic scale factor a so that κ = 1, 0, or −1. In this

case the spatial FRW metric may be written

dl2 = a2(

dχ2 + x2do2)

, (10.20)

where

x =

sin(χ) κ = 1 closed ,

χ κ = 0 flat ,

sinh(χ) κ = −1 open .

(10.21)

Exercise 10.1 By a suitable transformation of the comoving radial coordinate x, bring the spatial FRW

metric (10.16) to the “isotropic” form

dl2 =a2

(

1 + 14κX

2)2

(

dX2 +X2do2)

. (10.22)

What is the relation between X and x?

10.6 Spatial part of the FRW metric: more formal approach

A more formal approach to the derivation of the spatial FRW metric from the cosmological principle starts

with the proposition that the spatial components Gij of the Einstein tensor at fixed scale factor a (all time

derivatives of a set to zero) should be proportional to the metric tensor

Gij = K gij (i, j = 1, 2, 3) . (10.23)

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144 Homogeneous, Isotropic Cosmology

Without loss of generality, the spatial metric can be taken to be of the form

dl2 = f(r) dr2 + r2do2 . (10.24)

Imposing the condition (10.23) on the metric (10.24) recovers the spatial FRW metric (10.11).

10.7 FRW metric

The full Friedmann-Robertson-Walker spacetime metric is

ds2 = − dt2 + a(t)2(

dx2

1− κx2+ x2do2

)

(10.25)

where t is cosmic time, which is the proper time experienced by comoving observers, who remain at rest

in comoving coordinates dx = dθ = dφ = 0. Any of the alternative versions of the comoving spatial FRW

metric, equations (10.16), (10.18), (10.20), or (10.22). may be used as the spatial part of the FRW spacetime

metric (10.25).

10.8 Einstein equations for FRW metric

The Einstein equations for the FRW metric (10.25) are

−Gtt = 3

(

κ

a2+a2

a2

)

= 8πGρ ,

Gxx = Gθ

θ = Gφφ = − κ

a2− a2

a2− 2 a

a= 8πGp , (10.26)

where overdots represent differentiation with respect to cosmic time t, so that for example a ≡ da/dt. Note

the trick of one index up, one down, to remove, modulo signs, the distorting effect of the metric on the

Einstein tensor. The Einstein equations (10.26) rearrange to give Friedmann’s equations

a2

a2=

8πGρ

3− κ

a2,

a

a= −4πG

3(ρ+ 3p) .

(10.27)

Friedmann’s two equations (10.27) are fundamental to cosmology. The first one relates the curvature κ of

the Universe to the expansion rate a/a and the density ρ. The second one relates the acceleration a/a to

the density ρ plus 3 times the pressure p.

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10.9 Newtonian “derivation” of Friedmann equations 145

10.9 Newtonian “derivation” of Friedmann equations

10.9.1 Energy equation

Model a piece of the Universe as a ball of radius a and mass M = 43πρa

3. Consider a small mass m attracted

by this ball. Conservation of the kinetic plus potential energy of the small mass m implies

1

2ma2 − GMm

a= −κmc

2

2, (10.28)

where the quantity on the right is some constant whose value is not determined by this Newtonian treatment,

but which GR implies is as given. The energy equation (10.28) rearranges to

a2

a2=

8πGρ

3− κc2

a2, (10.29)

which reproduces the first Friedmann equation.

10.9.2 First law of thermodynamics

For adiabatic expansion, the first law of thermodynamics is

dE + p dV = 0 . (10.30)

With E = ρV and V = 43πa

3, the first law (10.30) becomes

d(ρa3) + p da3 = 0 , (10.31)

or, with the derivative taken with respect to cosmic time t,

ρ+ 3(ρ+ p)a

a= 0 . (10.32)

Differentiating the first Friedmann equation in the form

a2 =8πGρa2

3− κc2 (10.33)

gives

2aa =8πG

3

(

ρa2 + 2ρaa)

, (10.34)

and substituting ρ from the first law (10.32) reduces this to

2aa =8πG

3aa (− ρ− 3p) . (10.35)

Hence

a

a= −4πG

3(ρ+ 3p) , (10.36)

which reproduces the second Friedmann equation.

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146 Homogeneous, Isotropic Cosmology

10.9.3 Comment on the Newtonian derivation

The above Newtonian derivation of Friedmann’s equations is only heuristic. A different result could have

been obtained if different assumptions had been made. If for example the Newtonian gravitational force law

ma = −GMm/a2 were taken as correct, then it would follow that a/a = − 43πGρ, which is missing the all

important 3p contribution (without which there would be no inflation or dark energy) to Friedmann’s second

equation.

It is notable that the first law of thermodynamics is built in to the Friedmann equations. This implies that

entropy is conserved in FRW Universes. This remains true even when the mix of particles changes, as happens

for example during the epoch of electron-positron annihilation, or during big bang nucleosynthesis. How

then does entropy increase in the real Universe? Through fluctuations away from the perfect homogeneity

and isotropy assumed by the FRW metric.

10.10 Hubble parameter

The Hubble parameter H(t) is defined by

H ≡ a

a. (10.37)

The Hubble parameter H varies in cosmic time t, but is constant in space at fixed cosmic time t.

The value of the Hubble parameter today is called the Hubble constant H0 (the subscript 0 signifies

the present time). The Hubble constant is measured from Cepheids and Type Ia supernova to be (Riess et

al. 2005, astro-ph/0503159)

H0 = 73± 4(stat)± 5(sys) kms−1 Mpc−1 . (10.38)

The distance d to an object that is receding with the expansion of the universe is proportional to the

cosmic scale factor, d ∝ a, and its recession velocity v is consequently proportional to a. The result is

Hubble’s law relating the recession velocity v and distance d of distant objects

v = H0d . (10.39)

Since it takes light time to travel from a distant object, and the Hubble parameter varies in time, the linear

relation (10.39) breaks down at cosmological distances.

We, in the Milky Way, reside in an overdense region of the Universe that has collapsed out of the general

Hubble expansion of the Universe. The local overdense region of the Universe that has just turned around

from the general expansion and is beginning to collapse for the first time is called the Local Group of

galaxies. The Local Group consists of about 40 or so galaxies, mostly dwarf and irregular galaxies. It

contains two major spiral galaxies, Andromeda (M31) and the Milky Way, and one mid-sized spiral galaxy

Triangulum (M33). The Local Group is about 1 Mpc in radius.

Page 159: General Relativity, Black Holes, And Cosmology

10.11 Critical density 147

Because of the ubiquity of the Hubble constant in cosmological studies, cosmologists often parameterize

it by the quantity h defined by

h ≡ H0

100 km s−1 Mpc−1 . (10.40)

10.11 Critical density

The critical density ρcrit is defined to be the density required for the Universe to be flat, κ = 0. According

to the first of Friedmann equations (10.27), this sets

ρcrit ≡3H2

8πG. (10.41)

The critical density ρcrit, like the Hubble parameter H , evolves with time.

10.12 Omega

Cosmologists designate the ratio of the actual density ρ of the Universe to the critical density ρcrit by the

fateful letter Ω, the final letter of the Greek alphabet,

Ω ≡ ρ

ρcrit. (10.42)

With no subscript, Ω denotes the total mass-energy density in all forms. A subscript x on Ωx denotes

mass-energy density of type x.

The curvature density ρk, which is not really a form of mass-energy but it is sometimes convenient to treat

Table 10.1 Cosmic inventory

Species (2008)

Dark energy (Λ) ΩΛ 0.72 ± 0.02Non-baryonic cold dark matter (CDM) Ωc 0.234 ± 0.02Baryonic matter Ωb 0.046 ± 0.002Neutrinos Ων < 0.014Photons (CMB) Ωγ 5 × 10−5

Total Ω 1.005 ± 0.006

Curvature Ωk −0.005 ± 0.006

Page 160: General Relativity, Black Holes, And Cosmology

148 Homogeneous, Isotropic Cosmology

it as though it were, is defined by

ρk ≡ −3κc2

8πGa2(10.43)

and correspondingly Ωk ≡ ρk/ρcrit. According to the first of Friedmann’s equations (10.27), the curvature

density Ωk satisfies

Ωk = 1− Ω . (10.44)

Table 10.1 gives 2008 measurements of Ω in various species, obtained by combining 5-year WMAP CMB

measurements with a variety of other astronomical evidence, including supernovae, big bang nucleosynthesis,

galaxy clustering, weak lensing, and local measurements of the Hubble constant H0.

10.13 Redshifting

The spatial translation symmetry of the FRW metric implies conservation of generalized momentum. As you

will show in a problem set, a particle that moves along a geodesic in the radial direction, so that dθ = dφ = 0,

has 4-velocity uν satisfying

ux‖= constant . (10.45)

This conservation law implies that the proper momentum p‖ of a radially moving particle decays as

p‖ ≡ madx‖dτ∝ 1

a, (10.46)

which is true for both massive and massless particles.

It follows from equation (10.46) that light observed on Earth from a distant object will be redshifted by

a factor

1 + z =a0

a, (10.47)

where a0 is the present day cosmic scale factor. Cosmologists often refer to the redshift of an epoch, since

the cosmological redshift is an observationally accessible quantity that uniquely determines the cosmic time

of emission.

10.14 Types of mass-energy

The energy-momentum tensor Tµν of an FRW Universe is necessarily homogeneous and isotropic, by as-

sumption of the cosmological principle, taking the form (note yet again the trick of one index up and one

Page 161: General Relativity, Black Holes, And Cosmology

10.15 Evolution of the cosmic scale factor 149

down to remove the distorting effect of the metric)

T µν =

T tt 0 0 0

0 T rr 0 0

0 0 T θθ 0

0 0 0 T φφ

=

−ρ 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

. (10.48)

Table 10.2 gives equations of state p/ρ for generic species of mass-energy, along with (ρ + 3p)/ρ, which

determines the gravitational attraction per unit energy, and how the mass-energy varies with cosmic scale

factor, ρ ∝ an.

Table 10.2 Properties of universes dominated by various species

Species p/ρ (ρ+ 3p)/ρ ρ ∝

Radiation 1/3 2 a−4

Matter 0 1 a−3

Curvature “−1/3” “0” a−2

Vaccum −1 −2 a0

As commented in §23.16 above, the first law of thermodynamics for adiabatic expansion is built into Fried-

mann’s equations. In fact the law represents covariant conservation of energy-momentum for the system as

a whole

DµTµν = 0 . (10.49)

As long as species do not convert into each other (for example, no annihilation), covariant energy-momentum

conservation holds individually for each species, so the first law applies to each species individually, deter-

mining how its energy density ρ varies with cosmic scale factor a. Figure 10.2 illustrates how the energy

densities ρ of various species evolve as a function of scale factor a.

10.15 Evolution of the cosmic scale factor

Given how the energy density ρ of each species evolves with cosmic scale factor a, the first Friedmann

equation then determines how the cosmic scale factor a(t) itself evolves with cosmic time t. The evolution

equation for a(t) can be cast as an equation for the Hubble parameter H ≡ a/a, which in view of the

definition (10.41) of the critical density can be written

H(t)

H0=

[

ρcrit(t)

ρcrit(t0)

]1/2

. (10.50)

Page 162: General Relativity, Black Holes, And Cosmology

150 Homogeneous, Isotropic Cosmology

Cosmic scale factor a

Mas

s-en

ergy

dens

ity

ρ

ργ ∝ a−4

ρm ∝ a−3

ρk ∝ a−2

ρΛ = constant

Figure 10.2 Behavior of the mass-energy density ρ of various species as a function of cosmic time t.

Given the definition (10.43) of the curvature density as the critical density minus the total density, the

critical density ρcrit is itself the sum of the densities ρ of all species including the curvature density

ρcrit = ρk +∑

species x

ρx . (10.51)

Integrating equation (10.50) gives cosmic time t as a function of cosmic scale factor a

t =

da

aH. (10.52)

For example, in the case that the density is comprised of radiation, matter, and vacuum, the critical density

is

ρcrit = ργ + ρm + ρk + ρΛ , (10.53)

and equation (10.50) is

H(t)

H0=(

Ωγa−4 + Ωma

−3 + Ωka−2 + ΩΛ

)1/2, (10.54)

where Ωx represents its value at the present time. The time t, equation (10.52), is then

t =1

H0

da

a (Ωγa−4 + Ωma−3 + Ωka−2 + ΩΛ)1/2

, (10.55)

Page 163: General Relativity, Black Holes, And Cosmology

10.16 Conformal time 151

which is an elliptical integral of the 3rd kind.

If one single species in particular dominates the mass-energy density, then equation (10.55) integrates

easily to give the results in the following table.

Table 10.3 Evolution of cosmic scale factor in universes dominated by various species

Dominant Species a ∝

Radiation t1/2

Matter t2/3

Curvature tVaccum eHt

10.16 Conformal time

Especially when doing cosmological perturbation theory, it is convenient to use conformal time η defined

by (with units c temporarily restored)

a dη ≡ c dt (10.56)

with respect to which the FRW metric is

ds2 = a(η)2(

− dη2 +dx2

1− κx2+ x2do2

)

. (10.57)

The term conformal refers to a metric that is multiplied by an overall factor, the conformal factor. In the

FRW metric (10.57), the cosmic scale factor a is the conformal factor.

Conformal time η has the property that the speed of light is one in conformal coordinates: light moves

unit comoving distance per unit conformal time. In particular, light moving radially towards an observer at

x‖ = 0, with dθ = dφ = 0, satisfies

dx‖dη

= −1 . (10.58)

10.17 Looking back along the lightcone

Since light moves at unit velocity in conformal coordinates, an object at geodesic distance x‖ that emits

light at conformal time ηemit is observed at conformal time ηobs given by

x‖ = ηobs − ηemit . (10.59)

Page 164: General Relativity, Black Holes, And Cosmology

152 Homogeneous, Isotropic Cosmology

The comoving geodesic distance x‖ to an object is

x‖ =

∫ ηobs

ηemit

dη =

∫ tobs

temit

c dt

a=

∫ aobs

aemit

c da

a2H=

∫ z

0

c dz

H, (10.60)

where the last equation assumes the relation 1 + z = 1/a, valid as long as a is normalized to unity at the

observer (us) at the present time aobs = a0 = 1. In the case that the density is comprised of (curvature and)

radiation, matter, and vacuum, equation (10.60) gives

x‖ =c

H0

∫ 1

1/(1+z)

da

a2 (Ωγa−4 + Ωma−3 + Ωka−2 + ΩΛ)1/2

, (10.61)

which is an elliptical integral of the 1st kind. Given the geodesic comoving distance x‖, the circumferential

comoving distance x then follows as

x =sinh(Ω

1/2k H0x‖/c)

Ω1/2k H0/c

. (10.62)

To second order in redshift z,

x ≈ x‖ ≈c

H0

[

z − z2(

Ωγ + 34Ωm + 1

2Ωk

)

+ ...]

. (10.63)

The geodesic and circumferential distances x‖ and x differ at order z3.

10.18 Horizon

Light can come from no more distant point than the Big Bang. This distant point defines the horizon of

our Universe, which is located at infinite redshift, z =∞. Equation (10.60) gives the geodesic distance from

us at redshift zero to the horizon as

x‖(horizon) =

∫ ∞

0

c dz

H(10.64)

where again the cosmic scale factor has been normalized to unity at the present time, a0 = 1.

Equation (10.64) formally defines the event horizon of the Universe, but the cosmological scale over which

objects can continue to affect each other causally is typically smaller than this (much smaller, post-inflation).

It is thus common to define the cosmological horizon distance at any time as

cosmological horizon distance ≡ c

H(10.65)

which is roughly the scale over which objects can remain in causal contact.

Exercise 10.2 Then versus now.

Page 165: General Relativity, Black Holes, And Cosmology

10.18 Horizon 153

10−40 10−30 10−20 10−10 100 1010 102010−40

10−30

10−20

10−10

100

1010

1020

103010−50 10−40 10−30 10−20 10−10 100 1010

Age of the Universe (seconds)

Age of the Universe (years)

Size

ofth

eU

nive

rse

(met

ers)

Figure 10.3 Cosmic scale factor a and cosmological horizon distance c/H as a function of cosmic time t.

1. Prove that∫ ∞

0

xn−1 dx

ex + 1=(

1− 21−n)

∫ ∞

0

xn−1 dx

ex − 1. (10.66)

[Hint: Use the fact that (ex + 1)(ex − 1) = (e2x − 1).] Hence argue that the ratios of energy, entropy,

and number densities of relativistic fermionic (f) to relativistic bosonic (b) species in thermodynamic

equilibrium at the same temperature are

ρf

ρb=sf

sb=

7

8,

nf

nb=

3

4. (10.67)

[Hint: The proper entropy density of each relativistic species is s = (ρ+ p)/T = (4/3)ρ/T .]

2. Weak interactions were fast enough to keep neutrinos in thermodynamic equilibrium with photons, elec-

trons, and positrons up to just before ee annihilation, but then neutrinos decoupled. Argue that conser-

vation of comoving entropy implies

a3T 3(

gγ +7

8ge

)

= T 3γ gγ , (10.68a)

a3T 3 gν = T 3ν gν , (10.68b)

Page 166: General Relativity, Black Holes, And Cosmology

154 Homogeneous, Isotropic Cosmology

10−40 10−30 10−20 10−10 100 1010 1020

10−5

100

105

1010

1015

1020

1025

1030

1035

10−50 10−40 10−30 10−20 10−10 100 1010

Age of the Universe (years)

Age of the Universe (seconds)

Rad

iati

onT

empe

ratu

reof

the

Uni

vers

e(K

elvi

n)

10−40 10−30 10−20 10−10 100 1010 1020

10−30

10−20

10−10

100

1010

1020

1030

1040

1050

1060

1070

1080

1090

10100

10−50 10−40 10−30 10−20 10−10 100 1010

Age of the Universe (years)

Age of the Universe (seconds)

Mas

s-E

nerg

yD

ensi

tyof

the

Uni

vers

e(k

g/m

3 )

Figure 10.4 (Top) Temperature T , and (bottom) mass-energy density ρ, of the Universe as a function ofcosmic time t.

Page 167: General Relativity, Black Holes, And Cosmology

10.18 Horizon 155

where the left hand sides refer to quantities before ee annihilation, which happened at T ∼ 1 MeV ≈ 1010 K,

and the right hand sides to quantities after ee annihilation (including today). Deduce the ratio of neutrino

to photon temperatures today,

Tγ. (10.69)

Does the temperature ratio (10.69) depend on the number of neutrino types? What is the neutrino

temperature today in K, if the photon temperature today is 2.725 K?

3. The energy, entropy, and number densities of relativistic particles today are, with units restored (energy

density ρ in units energy/volume; entropy and number density s and n in units 1/volume),

ρ = gρ,0

π2(kT0)4

30c3~3, s = gs,0

2π2(kT0)3

45c3~3, n = gn,0

ζ(3)(kT0)3

π2c3~3, (10.70)

where T0 = 2.725 K is the CMB temperature today, ζ(3) = 1.2020569 is a Riemann zeta function, and gρ,0 ,

gs,0 , and gn,0 denote the energy-, entropy-, and number-weighted effective number of relativistic species

today, normalized to 1 per bosonic degree of freedom. What are the arithmetic values of gρ,0 , gs,0 , and

gn,0 if the relativistic species consist of photons and three species of neutrino? What is the energy density

Ωr of relativistic particles today relative to the critical density? [Hint: Don’t forget to take into account

the fact that the neutrino temperature today differs from the photon temperature.]

4. Evidence for neutrino oscillations from the MINOS experiment (2008, http://www-numi.fnal.gov/

PublicInfo/forscientists.html) indicates that at least one neutrino type has mass mν>∼ 0.05 eV. At

what redshift zν would such a neutrino become non-relativistic? If neutrinos are non-relativistic, what is

the neutrino density Ων relative to the critical density, in terms of the sum of the neutrino masses∑

mν?

Which of the effective number of relativistic species today gρ,0 , gs,0 , and gn,0 is changed if some neutrinos

are non-relativistic today?

5. Use entropy conservation to argue that the ratio of the photon temperature T at redshift z in the early

Universe to the photon temperature T0 today is

T

T0= (1 + z)

(

gs,0

gs

)1/3

. (10.71)

What is gs in terms of the numbers gb and gf of relativistic boson and fermion types, if all species were

at the same temperature T ?

Solution. The ratio of neutrino to photon temperatures post ee annihilation is

Tγ=

(

gγ + 78 ge

)1/3

=

(

4

11

)1/3

. (10.72)

The Cosmic Neutrino Background temperature is

Tν =

(

4

11

)1/3

2.725 K = 1.945 K . (10.73)

With 2 bosonic degrees of freedom from photons, and 6 fermionic degrees of freedom from 3 relativistic

Page 168: General Relativity, Black Holes, And Cosmology

156 Homogeneous, Isotropic Cosmology

neutrino types, the effective energy-, entropy-, and number-weighted number of relativistic degrees of freedom

is

gρ,0 = gγ +

(

)47

8gν = 2 +

(

4

11

)4/37

86 = 3.36 , (10.74a)

gs,0 = gγ +

(

)37

8gν = 2 +

4

11

7

86 =

43

11= 3.91 , (10.74b)

gn,0 = gγ +

(

)33

4gν = 2 +

4

11

3

46 =

40

11= 3.64 . (10.74c)

The redshift at which a neutrino of mass mν becomes non-relativistic is

1 + zν =mν

Tν= 300

( mν

0.05 eV

)

. (10.75)

If some neutrinos are non-relativistic, then the neutrino density Ων today is related to the sum∑

mν of

neutrino masses by

Ων =8πG

mνnν

3H20

= 5.3× 10−4

( ∑

0.05 eV

)(

h

0.71

)−2

. (10.76)

gρ,0 is changed if some neutrinos are non-relativistic today, but gs,0 and gn,0 remain unchanged. If just one

of the neutrino types is massive, and the other two are relativistic, then

gρ,0 = 2 +

(

4

11

)4/37

84 = 2.91 . (10.77)

The radiation density Ωr today, including photons and neutrinos, is

Ωr =8πGρr

3c2H20

= 1.236× 10−5gρ,0h−2 = 8.2× 10−5

( gρ,0

3.36

)

(

h

0.71

)−2

. (10.78)

If the temperatures of all species are equal, then the entropy-weighted effective number of relativistic species

is

gs = gb +7

8gf . (10.79)

Page 169: General Relativity, Black Holes, And Cosmology

PART FIVE

TETRAD APPROACH TO GENERAL RELATIVITY

Page 170: General Relativity, Black Holes, And Cosmology
Page 171: General Relativity, Black Holes, And Cosmology

Concept Questions

1. The vierbein has 16 degrees of freedom instead of the 10 degrees of freedom of the metric. What do the

extra 6 degrees of freedom correspond to?

2. Tetrad transformations are defined to be Lorentz transformations. Don’t general coordinate transfor-

mations already include Lorentz transformations as a particular case, so aren’t tetrad transformations

redundant?

3. What does coordinate gauge-invariant mean? What does tetrad gauge-invariant mean?

4. Is the coordinate metric gµν tetrad gauge-invariant?

5. What does a directed derivative ∂m mean physically?

6. Is the directed derivative ∂m coordinate gauge-invariant?

7. Is the tetrad metric γmn coordinate gauge-invariant? Is it tetrad gauge-invariant?

8. What is the tetrad-frame 4-velocity um of a person at rest in an orthonormal tetrad frame?

9. If the tetrad frame is accelerating (not in free-fall), which of the following is true/false?

a. Does the tetrad-frame 4-velocity um of a person continuously at rest in the tetrad frame change with

time? ∂0um = 0? D0u

m = 0?

b. Do the tetrad axes γγm change with time? ∂0γγm = 0? D0γγm = 0?

c. Does the tetrad metric γmn change with time? ∂0γmn = 0? D0γmn = 0?

d. Do the covariant components um of the 4-velocity of a person continuously at rest in the tetrad frame

change with time? ∂0um = 0? D0um = 0?

10. Suppose that p = γγmpm is a 4-vector. Is the proper rate of change of the proper components pm measured

by an observer equal to the directed time derivative ∂0pm or to the covariant time derivative D0p

m? What

about the covariant components pm of the 4-vector? [Hint: The proper contravariant components of the

4-vector measured by an observer are pm ≡ γγm · p where γγm are the contravariant locally inertial rest

axes of the observer. Similarly the proper covariant components are pm ≡ γγm · p.]

11. A person with two eyes separated by proper distance δξn observes an object. The observer observes the

photon 4-vector from the object to be pm. The observer uses the difference δpm in the two 4-vectors

detected by the two eyes to infer the binocular distance to the object. Is the difference δpm in photon

Page 172: General Relativity, Black Holes, And Cosmology

160 Concept Questions

4-vectors detected by the two eyes equal to the directed derivative δξn∂npm or to the covariant derivative

δξnDnpm?

12. Suppose that pm is a tetrad 4-vector. Parallel-transport the 4-vector by an infinitesimal proper distance

δξn. Is the change in pm measured by an ensemble of observers at rest in the tetrad frame equal to the

directed derivative δξn∂npm or to the covariant derivative δξnDnp

m? [Hint: What if “rest” means that the

observer at each point is separately at rest in the tetrad frame at that point? What if “rest” means that

the observers are mutually at rest relative to each other in the rest frame of the tetrad at one particular

point?]

13. What is the physical significance of the fact that directed derivatives fail to commute?

14. Physically, what do the tetrad connection coefficients Γkmn mean?

15. What is the physical significance of the fact that Γkmn is antisymmetric in its first two indices (if the

tetrad metric γmn is constant)?

16. Are the tetrad connections Γkmn coordinate gauge-invariant?

Page 173: General Relativity, Black Holes, And Cosmology

What’s important?

This part of the notes describes the tetrad formalism of GR.

1. Why tetrads? Because physics is clearer in a locally inertial frame than in a coordinate frame.

2. The primitive object in the tetrad formalism is the vierbein emµ, in place of the metric in the coordinate

formalism.

3. Written suitably, for example as equation (11.9), a metric ds2 encodes not only the metric coefficients gµν ,

but a full (inverse) vierbein emµ, through ds2 = γmn e

mµdx

µ enνdx

ν .

4. The tetrad road from vierbein to energy-momentum is similar to the coordinate road from metric to

energy-momentum, albeit a little more complicated.

5. In the tetrad formalism, the directed derivative ∂m is the analog of the coordinate partial derivative ∂/∂xµ

of the coordinate formalism. Directed derivatives ∂m do not commute, whereas coordinate derivatives

∂/∂xµ do commute.

Page 174: General Relativity, Black Holes, And Cosmology

11

The tetrad formalism

11.1 Tetrad

A tetrad (greek foursome) γγm(x) is a set of axes

γγm ≡ γγ0,γγ1,γγ2,γγ3 (11.1)

attached to each point xµ of spacetime. The common case is that of an orthonormal tetrad, where the axes

form a locally inertial frame at each point, so that the scalar products of the axes constitute the Minkowski

metric ηmn

γγm · γγn = ηmn . (11.2)

However, other tetrads prove useful in appropriate circumstances. There are spinor tetrads, null tetrads

(notably the Newman-Penrose double null tetrad), and others (indeed, the basis of coordinate tangent

vectors gµ is itself a tetrad). In general, the tetrad metric is some symmetric matrix γmn

γγm · γγn ≡ γmn . (11.3)

Andrew’s convention:

latin (black) dummy indices label tetrad frames.

greek (brown) dummy indices label coordinate frames.

Why introduce tetrads?

1. The physics is more transparent when expressed in a locally inertial frame (or some other frame adapted

to the physics), as opposed to the coordinate frame, where Salvador Dali rules.

2. If you want to consider spin- 12 particles and quantum physics, you better work with tetrads.

3. For good reason, much of the GR literature works with tetrads, so it’s useful to understand them.

11.2 Vierbein

The vierbein (German four-legs, or colloquially, critter) emµ is defined to be the matrix that transforms

between the tetrad frame and the coordinate frame (note the placement of indices: the tetrad index m comes

Page 175: General Relativity, Black Holes, And Cosmology

11.3 The metric encodes the vierbein 163

first, then the coordinate index µ)

γγm = emµ gµ . (11.4)

The vierbein is a 4× 4 matrix, with 16 independent components. The inverse vierbein emµ is defined to be

the matrix inverse of the vierbein emµ, so that

emµ em

ν = δνµ , em

µ enµ = δn

m . (11.5)

Thus equation (11.4) inverts to

gµ = emµ γγm . (11.6)

11.3 The metric encodes the vierbein

The scalar spacetime distance is

ds2 = gµν dxµ dxν = gµ · gν dx

µ dxν = γmn em

µ en

ν dxµ dxν (11.7)

from which it follows that the coordinate metric gµν is

gµν = γmn em

µ en

ν . (11.8)

The shorthand way in which metric’s are commonly written encodes not only a metric but also an inverse

vierbein, hence a tetrad. For example, the Schwarzschild metric

ds2 = −(

1− 2M

r

)

dt2 +

(

1− 2M

r

)−1

dr2 + r2dθ2 + r2 sin2θ dφ2 (11.9)

takes the form (11.7) with an orthonormal (Minkowski) tetrad metric γmn = ηmn, and an inverse vierbein

encoded in the differentials

e0µdxµ =

(

1− 2M

r

)1/2

dt , (11.10a)

e1µdxµ =

(

1− 2M

r

)−1/2

dr , (11.10b)

e2µdxµ = r dθ , (11.10c)

e3µdxµ = r sin θ dφ , (11.10d)

Explicitly, the inverse vierbein of the Schwarzschild metric is the diagonal matrix

emµ =

(1− 2M/r)1/2 0 0 0

0 (1− 2M/r)−1/2 0 0

0 0 r 0

0 0 0 r sin θ

, (11.11)

Page 176: General Relativity, Black Holes, And Cosmology

164 The tetrad formalism

and the corresponding vierbein is (note that, because the tetrad index is always in the first place and the

coordinate index is always in the second place, the matrices as written are actually inverse transposes of

each other, not just inverses)

emµ =

(1− 2M/r)−1/2 0 0 0

0 (1− 2M/r)1/2 0 0

0 0 1/r 0

0 0 0 1/(r sin θ)

. (11.12)

Concept question 11.1 Schwarzschild vierbein. The components e0t and e1

r of the Schwarzschild

vierbein (11.12) are imaginary inside the horizon. What does this mean? Is the vierbein still valid inside

the horizon? ⋄

11.4 Tetrad transformations

Tetrad transformations are transformations that preserve the fundamental property of interest, for example

the orthonormality, of the tetrad. For all tetrads of interest in these notes, which includes not only orthonor-

mal tetrads, but also spinor tetrads and null tetrads (but not coordinate-based tetrads), tetrad transfor-

mations are Lorentz transformations. Hereafter these notes will presume that a tetrad transformation is a

Lorentz transformation. The Lorentz transformation may be, and usually is, a different transformation at

each point. Tetrad transformations rotate the tetrad axes γγk at each point by a Lorentz transformation

Lkm, while keeping the background coordinates xµ unchanged:

γγk → γγ′k = Lk

m γγm . (11.13)

In the case that the tetrad axes γγk are orthonormal, with a Minkowski metric, the Lorentz transformation

matrices Lkm in equation (11.13) take the familiar special relativistic form, but the linear matrices Lk

m in

equation (11.13) signify a Lorentz transformation in any case.

In all the cases of interest, including orthonormal, spinor, and null tetrads, the tetrad metric γmn is

constant. Lorentz transformations are precisely those transformations that leave the tetrad metric unchanged

γ′kl = γγ′k · γγ′

l = LkmLl

n γγm · γγn = LkmLl

n γmn = γkl . (11.14)

Exercise 11.2 Generators of Lorentz transformations are antisymmetric. From the condition that

the tetrad metric γmn is unchanged by a Lorentz transformation, show that the generator of an infinitesimal

Lorentz transformation is an antisymmetric matrix. Is this true only for an orthonormal tetrad, or is it true

more generally?

Solution. An infinitesimal Lorentz transformation is the sum of the unit matrix and an infinitesimal piece

∆Lkm, the generator of the infinitesimal Lorentz transformation,

Lkm = δm

k + ∆Lkm . (11.15)

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11.5 Tetrad Tensor 165

Under such an infinitesimal Lorentz transformation, the tetrad metric transforms to

γ′kl = (δmk + ∆Lk

m)(δnl + ∆Ll

n)γmn ≈ γkl + ∆Lkl + ∆Llk , (11.16)

which by proposition equals the original tetrad metric γkl, equation (11.14). It follows that

∆Lkl + ∆Llk = 0 , (11.17)

that is, the generator ∆Lkl is antisymmetric, as claimed. ⋄

11.5 Tetrad Tensor

In general, a tetrad-frame tensor Akl...mn... is an object that transforms under tetrad (Lorentz) transforma-

tions (11.13) as

A′kl...mn... = Lk

aLlb ... Lm

cLnd ... Aab...

cd... . (11.18)

11.6 Raising and lowering indices

In the coordinate approach to GR, coordinate indices were lowered and raised with the coordinate metric

gµν and its inverse gµν . In the tetrad formalism there are two kinds of indices, tetrad indices and coordinate

indices, and they flip around as follows:

1. Lower and raise coordinate indices with the coordinate metric gµν and its inverse gµν ;

2. Lower and raise tetrad indices with the tetrad metric γmn and its inverse γmn;

3. Switch between coordinate and tetrad frames with the vierbein emµ and its inverse em

µ.

The kinds of objects for which this flippery is valid are called tensors. Tensors with only tetrad indices,

such as the tetrad axes γγm or the tetrad metric γmn, are called tetrad tensors, and they remain unchanged

under coordinate transformations. Tensors with only coordinate indices, such as the coordinate tangent

axes gµ or the coordinate metric gµν , are called coordinate tensors, and they remain unchanged under tetrad

transformations. Tensors may also be mixed, such as the vierbein emµ. And of course just because something

has an index, greek or latin, does not make it a tensor: a tensor is a tensor if any only if it transforms like

a tensor.

11.7 Gauge transformations

Gauge transformations are transformations of the coordinates or tetrad. Such transformations do not

change the underlying spacetime.

Quantities that are unchanged by a coordinate transformation are coordinate gauge-invariant. Quan-

tities that are unchanged under a tetrad transformation are tetrad gauge-invariant. For example, tetrad

tensors are coordinate gauge-invariant, while coordinate tensors are tetrad gauge-invariant.

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166 The tetrad formalism

Tetrad transformations have the 6 degrees of freedom of Lorentz transformations, with 3 degrees of freedom

in spatial rotations, and 3 more in Lorentz boosts. General coordinate transformations have 4 degrees of

freedom. Thus there are 10 degrees of freedom in the choice of tetrad and coordinate system. The 16 degrees

of freedom of the vierbein, minus the 10 degrees of freedom from the transformations of the tetrad and

coordinates, leave 6 physical degrees of freedom in spacetime, the same as in the coordinate approach to GR,

which is as it should be.

11.8 Directed derivatives

Directed derivatives ∂m are defined to be the directional derivatives along the axes γγm

∂m ≡ γγm · ∂ = γγm · gµ ∂

∂xµ= em

µ ∂

∂xµis a tetrad 4-vector . (11.19)

The directed derivative ∂m is independent of the choice of coordinates, as signalled by the fact that it has

only a tetrad index, no coordinate index.

Unlike coordinate derivatives ∂/∂xµ, directed derivatives ∂m do not commute. Their commutator is

[∂m, ∂n] =

[

emµ ∂

∂xµ, en

ν ∂

∂xν

]

= emµ ∂en

ν

∂xµ

∂xν− en

ν ∂emµ

∂xν

∂xµ

= (dknm − dk

mn) ∂k is not a tetrad tensor (11.20)

where dlmn ≡ γlk dkmn is the vierbein derivative

dlmn ≡ γlk ek

κ enν ∂em

κ

∂xνis not a tetrad tensor . (11.21)

Since the vierbein and inverse vierbein are inverse to each other, an equivalent definition of dlmn in terms of

the inverse vierbein is

dlmn ≡ − γlk emµ en

ν ∂ek

µ

∂xνis not a tetrad tensor . (11.22)

11.9 Tetrad covariant derivative

The derivation of tetrad covariant derivatives Dm follows precisely the analogous derivation of coordinate

covariant derivatives Dµ. The tetrad-frame formulae look entirely similar to the coordinate-frame formu-

lae, with the replacement of coordinate partial derivatives by directed derivatives, ∂/∂xµ → ∂m, and the

replacement of coordinate-frame connections by tetrad-frame connections Γκµν → Γk

mn. There are two things

to be careful about: first, unlike coordinate partial derivatives, directed derivatives ∂m do not commute;

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11.9 Tetrad covariant derivative 167

and second, neither tetrad-frame nor coordinate-frame connections are tensors, and therefore it should be

no surprise that the tetrad-frame connections Γlmn are not related to the coordinate-frame connections

Γλµν by the ‘usual’ vierbein transformations. Rather, the tetrad and coordinate connections are related by

equation (11.32).

If Φ is a scalar, then ∂mΦ is a tetrad 4-vector. The tetrad covariant derivative of a scalar is just the

directed derivative

DmΦ = ∂mΦ is a tetrad 4-vector . (11.23)

If Am is a tetrad 4-vector, then ∂nAm is not a tetrad tensor, and ∂nAm is not a tetrad tensor. But the

4-vector A = γγmAm, being by construction invariant under both tetrad and coordinate transformations, is

a scalar, and its directed derivative is therefore a 4-vector

∂nA = ∂n(γγmAm) is a tetrad 4-vector

= γγm∂nAm + (∂nγγm)Am

= γγm∂nAm + Γk

mnγγk Am (11.24)

where the tetrad-frame connection coefficients, Γkmn, also known as Ricci rotation coefficients (or, in

the context of Newman-Penrose tetrads, spin coefficients) are defined by

∂nγγm ≡ Γkmn γγk is not a tetrad tensor . (11.25)

Equation (11.24) shows that

∂nA = γγk(DnAk) is a tetrad tensor (11.26)

where DnAk is the covariant derivative of the contravariant 4-vector Ak

DnAk ≡ ∂nA

k + ΓkmnA

m is a tetrad tensor . (11.27)

Similarly,

∂nA = γγk(DnAk) (11.28)

where DnAk is the covariant derivative of the covariant 4-vector Ak

DnAk ≡ ∂nAk − ΓmknAm is a tetrad tensor . (11.29)

In general, the covariant derivative of a tensor is

DaAkl...mn... = ∂aA

kl...mn... + Γk

baAbl...mn... + Γl

baAkb...mn... + ...− Γb

maAkl...bn... − Γb

naAkl...mb... − ... (11.30)

with a positive Γ term for each contravariant index, and a negative Γ term for each covariant index.

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168 The tetrad formalism

11.10 Relation between tetrad and coordinate connections

The relation between the tetrad connections Γkmn and their coordinate counterparts Γκ

µν follows from

Γkmnγγk = ∂nγγm = en

ν ∂emκgκ

∂xνis not a tetrad tensor

= enν ∂em

κ

∂xνgκ + en

ν emκ ∂gκ

∂xν

= dkmn ekκ gκ + en

ν emκ Γλ

κν gλ . (11.31)

Thus the relation is

Γlmn − dlmn = elλ em

µ enν Γλµν is not a tetrad tensor (11.32)

where

Γlmn ≡ γlk Γkmn . (11.33)

11.11 Torsion tensor

The torsion tensor Smkl , which GR assumes to vanish, is defined in the usual way by the commutator of

the covariant derivative acting on a scalar Φ

[Dk, Dl] Φ = Smkl ∂mΦ is a tetrad tensor . (11.34)

The expression (11.29) for the covariant derivatives coupled with the commutator (11.20) of directed deriva-

tives shows that the torsion tensor is

Smkl = Γm

kl − Γmlk − dm

kl + dmlk is a tetrad tensor (11.35)

where dmkl are the vierbein derivatives defined by equation (11.21). The torsion tensor Sm

kl is antisymmetric

in k ↔ l, as is evident from its definition (11.34).

11.12 No-torsion condition

GR assumes vanishing torsion. Then equation (11.35) implies the no-torsion condition

Γmkl − dmkl = Γmlk − dmlk is not a tetrad tensor . (11.36)

In view of the relation (11.32) between tetrad and coordinate connections, the no-torsion condition (11.36) is

equivalent to the usual symmetry condition Γµκλ = Γµλκ on the coordinate frame connections, as it should

be.

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11.13 Antisymmetry of the connection coefficients 169

11.13 Antisymmetry of the connection coefficients

The directed derivative of the tetrad metric is

∂nγlm = ∂n(γγl · γγm)

= γγl · ∂nγγm + γγm · ∂nγγl

= Γlmn + Γmln . (11.37)

In most cases of interest, including orthonormal, spinor, and null tetrads, the tetrad metric is chosen to be a

constant. For example, if the tetrad is orthonormal, then the tetrad metric is the Minkowski metric, which

is constant, the same everywhere. If the tetrad metric is constant, then all derivatives of the tetrad metric

vanish, and then equation (11.37) shows that the tetrad connections are antisymmetric in their first two

indices

Γlmn = −Γmln . (11.38)

This antisymmetry reflects the fact that Γlmn is the generator of a Lorentz transformation for each n.

11.14 Connection coefficients in terms of the vierbein

In the general case of non-constant tetrad metric, and non-vanishing torsion, the following manipulation

∂nγlm + ∂mγln − ∂lγmn = Γlmn + Γmln + Γlnm + Γnlm − Γmnl − Γnml (11.39)

= 2 Γlmn − Slmn − Smnl − Snml − dlmn + dlnm − dmnl + dmln − dnml + dnlm

implies that the tetrad connections Γlmn are given in terms of the derivatives ∂nγlm of the tetrad metric,

the torsion Slmn, and the vierbein derivatives dlmn by

Γlmn = 12 (∂nγlm + ∂mγln − ∂lγmn + Slmn + Smnl + Snml

+ dlmn − dlnm + dmnl − dmln + dnml − dnlm) is not a tetrad tensor . (11.40)

If torsion vanishes, as GR assumes, and if furthermore the tetrad metric is constant, then equation (11.40)

simplifies to the following expression for the tetrad connections in terms of the vierbein derivatives dlmn

defined by (11.21)

Γlmn = 12 (dlmn − dlnm + dmnl − dmln + dnml − dnlm) is not a tetrad tensor . (11.41)

This is the formula that allows connection coefficients to be calculated from the vierbein.

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170 The tetrad formalism

11.15 Riemann curvature tensor

The Riemann curvature tensor Rklmn is defined in the usual way by the commutator of the covariant

derivative acting on a contravariant 4-vector. In the presence of torsion,

[Dk, Dl]Am ≡ SnklDnAm +RklmnA

n is a tetrad tensor . (11.42)

If torsion vanishes, as GR assumes, then the definition (11.42) reduces to

[Dk, Dl]Am ≡ RklmnAn is a tetrad tensor . (11.43)

The expression (11.29) for the covariant derivative coupled with the torsion equation (11.34) yields the

following formula for the Riemann tensor in terms of connection coefficients, for the general case of non-

vanishing torsion:

Rklmn = ∂kΓmnl − ∂lΓmnk + ΓamlΓank − Γa

mkΓanl + (Γakl − Γa

lk − Sakl)Γmna is a tetrad tensor . (11.44)

The formula has extra terms (Γakl − Γa

lk − Sakl)Γmna compared to the usual formula for the coordinate-frame

Riemann tensor Rκλµν . If torsion vanishes, as GR assumes, then

Rklmn = ∂kΓmnl − ∂lΓmnk + ΓamlΓank − Γa

mkΓanl + (Γakl − Γa

lk)Γmna is a tetrad tensor . (11.45)

The symmetries of the tetrad-frame Riemann tensor are the same as those of the coordinate-frame Riemann

tensor. For vanishing torsion, these are

R([kl][mn]) , (11.46)

Rklmn +Rknlm +Rkmnl = 0 . (11.47)

Exercise 11.3 Riemann tensor. From the definition (11.42), derive the expression (11.44) for the

Riemann tensor. Show that, in addition to the antisymmetry in kl which follows immediately from the

definition (11.42), the Riemann tensor Rklmn is antisymmetric in the indices mn. [Hint: Start by expanding

out the definition (11.42) using the definition (11.30) of the covariant derivative. You will find it easier to

derive an expression for the Riemann tensor with one index raised, such as Rklmn, but you should resist the

temptation to leave it there, because the symmetries of the Riemann tensor are obscured when one index is

raised. To switch to all lowered indices, you will need to convert terms such as ∂kΓnml by

∂kΓnml = ∂k(γnpΓpml) = γnp ∂kΓpml + Γpml ∂kγ

np . (11.48)

You should show that the directed derivative ∂kγnp in this expression is related to tetrad connections through

a formula similar to equation (11.37)

∂kγnp = −Γnp

k − Γpnk , (11.49)

which you should recognize as equivalent to Dkγnp = 0. To complete the derivation, show that

∂k(Γmnl + Γnml)− ∂l(Γmnk + Γnmk) = [∂k, ∂l]γmn = (Γalk − Γa

kl + Sakl)(Γmna + Γnma) . (11.50)

The antisymmetry of Rklmn in mn follows from equation (11.50).] ⋄

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11.16 Ricci, Einstein, Bianchi 171

11.16 Ricci, Einstein, Bianchi

The usual suite of formulae leading to Einstein’s equations apply. Since all the quantities are tensors, and

all the equations are tensor equations, their form follows immediately from their coordinate counterparts.

Ricci tensor:

Rkm ≡ γlnRklmn . (11.51)

Ricci scalar:

R ≡ γkmRkm . (11.52)

Einstein tensor:

Gkm ≡ Rkm − 12 γkm R . (11.53)

Einstein’s equations:

Gkm = 8πGTkm . (11.54)

Bianchi identities:

DkRlmnp +DlRmknp +DmRklnp = 0 , (11.55)

which most importantly imply covariant conservation of the Einstein tensor, hence conservation of energy-

momentum

DkTkm = 0 . (11.56)

11.17 Electromagnetism

11.17.1 Electromagnetic potential and field

The electromagnetic field is derivable from an electromagnetic 4-potential Am,

Am = (φ,A) . (11.57)

The electromagnetic field is a bivector field (an antisymmetric tensor) Fmn,

Fmn ≡ DnAm −DmAn . (11.58)

If torsion vanishes, as general relativity assumes, then the covariant derivatives in the definition (11.58) can

be replaced by directed derivatives,

Fmn = ∂nAm − ∂mAn . (11.59)

If the tetrad is orthonormal, then the traditional description of the electromagnetic field in terms of electric

and magnetic fields E and B is valid:

E = − ∂tA−∇φ , B = ∇×A , (11.60)

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172 The tetrad formalism

where ∇ ≡ ∂x, ∂y, ∂z denotes the spatial tetrad directed derivative 3-vector. In an orthonormal tetrad

γγt,γγx,γγy,γγz, the 6 components of the electromagnetic field Fmn are related to the electric and magnetic

fields E = Ex, Ey, Ez and B = Bx, By, Bz by

Fmn =

0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

, Fmn =

0 −Ex −Ey −Ez

Ex 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

. (11.61)

11.17.2 Lorentz force law

In the presence of an electromagnetic field Fmn, the general relativistic equation of motion for the 4-velocity

um ≡ dxm/dτ of a particle of mass m and charge q is modified by the addition of a Lorentz force qFnmun

mDum

Dτ= qFn

mun . (11.62)

In the absence of gravitational fields, so D/Dτ = d/dτ , and with um = ut1,v where v is the 3-velocity,

the spatial components of equation (11.62) reduce to [note that d/dt = (1/ut) d/dτ ]

mdutv

dt= q (E + v ×B) (11.63)

which is the classical special relativistic Lorentz force law. The signs in the expression (11.61) for Fmn in

terms of E = Ex, Ey, Ez and B = Bx, By, Bz are arranged to agree with the classical law (11.63).

11.17.3 Maxwell’s equations

The source-free Maxwell’s equations are

DlFmn +DmFnl +DnFlm = 0 , (11.64)

while the sourced Maxwell’s equations are

DmFmn = 4πjn , (11.65)

where jn is the electric 4-current. The sourced Maxwell’s equations (11.65) coupled with the antisymmetry

of the electromagnetic field tensor Fmn ensure conservation of electric charge

Dnjn = 0 . (11.66)

In flat space with a Minkowski metric, the covariant derivatives simplify to ordinary derivatives, Dm →∂m → ∂/∂xm, and the source-free Maxwell’s equations (11.64) reduce to the traditional form

∇ ·B = 0 , ∇×E +∂B

∂t= 0 , (11.67)

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11.17 Electromagnetism 173

while the sourced Maxwell’s equations reduce to

∇ ·E = 4πq , ∇×B − ∂E

∂t= 4πj , (11.68)

where the electric charge density q and the electric current density j are the time and space components of

the electric 4-current

jn = q, j . (11.69)

11.17.4 Electromagnetic energy-momentum tensor

The energy-momentum tensor Tmne of an electromagnetic field Fmn is

Tmne =

1

(

FmkF

nk − 1

4γmnFklF

kl

)

. (11.70)

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12

∗More on the tetrad formalism

This chapter presents some more advanced aspects of the tetrad formalism. It discusses spinor tetrads

(§12.1) and Newman-Penrose tetrads (§12.2). The chapter goes on to show how the fields that describe

electromagnetic (§12.3) and gravitational (§12.4) waves have a natural and insightful complex structure that

is brought out in a Newman-Penrose tetrad. The Newman-Penrose formalism provides a natural context for

the Petrov classification of the Weyl tensor (§12.5), and for the derivation of the Raychaudhuri equations

(§12.6) which imply the focussing theorem (§12.7) that is a key ingredient of the Penrose-Hawking singularity

theorems.

12.1 Spinor tetrad formalism

In quantum mechanics, fundamental particles have spin. The 3 generations of leptons (electrons, muons,

tauons, and their respective neutrino partners) and quarks (up, strange, top, and their down, charm, and

bottom partners) have spin 12 (in units ~ = 1). The carrier particles of the electromagnetic force (photons),

the weak force (the W± and Z bosons), and the colour force (the 8 gluons), have spin 1. The carrier of the

gravitational force, the graviton, is expected to have spin 2, though as of 2010 no gravitational wave, let

alone its quantum, the graviton, has been detected.

General relativity is a classical, not quantum, theory. Nevertheless the spin properties of classical waves,

such as electromagnetic or gravitational waves, are already apparent classically.

12.1.1 Spinor tetrad

A systematic way to project objects into spin components is to work in a spinor tetrad. As will become

apparent below, equation (12.5), spin describes how an object transforms under rotation about some preferred

axis. In the case of an electromagnetic or gravitational wave, the natural preferred axis is the direction of

propagation of the wave. With respect to the direction of propagation, electromagnetic waves prove to have

two possible spins, or helicities, ±1, while gravitational waves have two possible spins, or helicities, ±2. A

preferred axis might also be set by an experimenter who chooses to measure spin along some particular

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12.1 Spinor tetrad formalism 175

direction. The following treatment takes the preferred direction to lie along the z-axis γγz, but there is no

loss of generality in making this choice.

Start with an orthonormal tetrad γγt,γγx,γγy,γγz. If the preferred tetrad axis is the z-axis γγz , then the

spinor tetrad axes γγ+,γγ− are defined to be complex combinations of the transverse axes γγx,γγy,

γγ+ ≡ 1√2(γγx + iγγy) , (12.1a)

γγ− ≡ 1√2(γγx − iγγy) . (12.1b)

The tetrad metric of the spinor tetrad γγt,γγz,γγ+,γγ− is

γmn =

−1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (12.2)

Notice that the spinor axes γγ+,γγ− are themselves null, γγ+ ·γγ+ = γγ− ·γγ− = 0, whereas their scalar product

with each other is non-zero γγ+ · γγ− = 1. The null character of the spinor axes is what makes spin especially

well-suited to describing fields, such as electromagnetism and gravity, that propagate at the speed of light.

An even better trick in dealing with fields that propagate at the speed of light is to work in a Newman-Penrose

tetrad, §12.2, in which all 4 tetrad axes are taken to be null.

12.1.2 Transformation of spin under rotation about the preferred axis

Under a right-handed rotation by angle χ about the preferred axis γγz, the transverse axes γγx and γγy

transform as SIGN?!

γγx → cosχγγx − sinχγγy ,

γγy → sinχγγx + cosχγγy . (12.3)

It follows that the spinor axes γγ+ and γγ− transform under a right-handed rotation by angle χ about γγz as

γγ± → e±iχ γγ± . (12.4)

The transformation (12.4) identifies the spinor axes γγ+ and γγ− as having spin +1 and −1 respectively.

12.1.3 Spin

More generally, an object can be defined as having spin s if it varies by

esiχ (12.5)

under a right-handed rotation by angle χ about the preferred axis γγz . Thus an object of spin s is unchanged

by a rotation of 2π/s about the preferred axis. A spin-0 object is symmetric about the γγz axis, unchanged

by a rotation of any angle about the axis. The γγz axis itself is spin-0, as is the time axis γγt.

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176 ∗More on the tetrad formalism

The components of a tensor in a spinor tetrad inherit spin properties from that of the spinor basis. The

general rule is that the spin s of any tensor component is equal to the number of + covariant indices minus

the number of − covariant indices:

spin s = number of + minus − covariant indices . (12.6)

12.1.4 Spin flip

Under a reflection through the y-axis, the spinor axes swap:

γγ+ ↔ γγ− , (12.7)

which may also be accomplished by complex conjugation. Reflection through the y-axis, or equivalently

complex conjugation, changes the sign of all spinor indices of a tensor component

+↔ − . (12.8)

In short, complex conjugation flips spin, a pretty feature of the spinor formalism.

12.1.5 Spin versus spherical harmonics

In physical problems, such as in cosmological perturbations, or in perturbations of spherical black holes, or

in the hydrogen atom, spin often appears in conjunction with an expansion in spherical harmonics. Spin

should not be confused with spherical harmonics.

Spin and spherical harmonics appear together whenever the problem at hand has a symmetry under the 3D

special orthogonal group SO(3) of spatial rotations (special means of unit determinant; the full orthogonal

group O(3) contains in addition the discrete transformation corresponding to reflection of one of the axes,

which flips the sign of the determinant). Rotations in SO(3) are described by 3 Euler angles θ, φ, χ. Spin

is associated with the Euler angle χ. The usual spherical harmonics Yℓm(θ, φ) are the spin-0 eigenfunctions

of SO(3). The eigenfunctions of the full SO(3) group are the spin harmonics SIGN?

sYℓm(θ, φ, χ) = Θℓms(θ, φ, χ)eimφeisχ . (12.9)

12.1.6 Spinor components of the Einstein tensor

With respect to a spinor tetrad, the components of the Einstein tensor Gmn are

Gmn =

Gtt Gtz Gt+ Gt−

Gtz Gzz Gz+ Gz−

Gt+ Gz+ G++ G+−

Gt− Gz− G+− G−−

. (12.10)

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12.2 Newman-Penrose tetrad formalism 177

From this it is apparent that the 10 components of the Einstein tensor decompose into 4 spin-0 components,

4 spin-±1 components, and 2 spin-±2 components:

−2 : G−− ,

−1 : Gt− , Gz− ,

0 : Gtt , Gtz , Gzz , G+− ,

+1 : Gt+ , Gz+ ,

+2 : G++ .

(12.11)

The 4 spin-0 components are all real; in particular G+− is real since G∗+− = G−+ = G+−. The 4 spin-±1 and

2 spin-±2 components comprise 3 complex components

G∗++

= G−− , G∗t+ = Gt− , G∗

z+= Gz− . (12.12)

In some contexts, for example in cosmological perturbation theory, REALLY? the various spin components

are commonly referred to as scalar (spin-0), vector (spin-±1), and tensor (spin-±2).

12.2 Newman-Penrose tetrad formalism

The Newman-Penrose formalism (E. T. Newman & R. Penrose, 1962, “An Approach to Gravitational Ra-

diation by a Method of Spin Coefficients,” J. Math. Phys. 3, 566–579; E. (Ted) Newman & R. Penrose,

2009, “Spin-coefficient formalism,” Scholarpedia, 4(6), 7445, http://www.scholarpedia.org/article/

Newman-Penrose_formalism) provides a particularly powerful way to deal with fields that propagate at the

speed of light. The Newman-Penrose formalism adopts a tetrad in which the two axes γγv (outgoing) and

γγu (ingoing) along the direction of propagation are chosen to be lightlike, while the two axes γγ+ and γγ−

transverse to the direction of propagation are chosen to be spinor axes.

Sadly, the literature on the Newman-Penrose formalism is characterized by an arcane and random notation

whose principal purpose seems to be to perpetuate exclusivity for an old-boys club of people who understand

it. This is unfortunate given the intrinsic power of the formalism. A. Held (1974, “A formalism for the

investigation of algebraically special metrics. I,” Commun. Math. Phys. 37, 311–326) comments that the

Newman-Penrose formalism presents “a formidable notational barrier to the uninitiate.” For example, the

tetrad connections Γkmn are called “spin coefficients,” and assigned individual greek letters that obscure

their transformation properties. Do not be fooled: all the standard tetrad formalism presented in Chapter 11

carries through unaltered. One ill-born child of the notation that persists in widespread use is ψ2−s for the

spin s component of the Weyl tensor, equations (12.49).

Gravitational waves are commonly characterized by the Newman-Penrose (NP) components of the Weyl

tensor. The NP components of the Weyl tensor are sometimes referred to as the NP scalars. The designation

as NP scalars is potentially misleading, because the NP components of the Weyl tensor form a tetrad-frame

tensor, not a set of scalars (though of course the tetrad-frame Weyl tensor is, like any tetrad-frame quantity, a

coordinate scalar). The NP components do become proper quantities, and in that sense scalars, when referred

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178 ∗More on the tetrad formalism

to the frame of a particular observer, such as a gravitational wave telescope, observing along a particular

direction. However, the use of the word scalar to describe the components of a tensor is unfortunate.

12.2.1 Newman-Penrose tetrad

A Newman-Penrose tetrad γγv,γγu,γγ+,γγ− is defined in terms of an orthonormal tetrad γγt,γγx,γγy,γγz by

γγv ≡ 1√2(γγt + γγz) , (12.13a)

γγu ≡ 1√2(γγt − γγz) , (12.13b)

γγ+ ≡ 1√2(γγx + iγγy) , (12.13c)

γγ− ≡ 1√2(γγx − iγγy) , (12.13d)

or in matrix form

γγv

γγu

γγ+

γγ−

=1√2

1 0 0 1

1 0 0 −1

0 1 i 0

0 1 −i 0

=

γγt

γγx

γγy

γγz

. (12.14)

Just as each of the spinor axes γγ+ and γγ− individually specifies not one but two distinct directions γγx and

γγy, so also each of the null axes γγv and γγv individually specifies not one but two distinct directions γγt and

γγz. The ingoing null axis γγu may be obtained from the outgoing null axis γγv, and vice versa, by the parity

operation of inverting all the spatial axes.

All four tetrad axes are null

γγv · γγv = γγu · γγu = γγ+ · γγ+ = γγ− · γγ− = 0 . (12.15)

In a profound sense, the null, or lightlike, character of each the four NP axes explains why the NP formalism is

well adapted to treating fields that propagate at the speed of light. The tetrad metric of the Newman-Penrose

tetrad γγv,γγu,γγ+,γγ− is

γmn =

0 −1 0 0

−1 0 0 0

0 0 0 1

0 0 1 0

. (12.16)

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12.3 Electromagnetic field tensor 179

12.2.2 Boost weight

A boost along the γγz axis multiplies the outgoing and ingoing axes γγv and γγu by a blueshift factor ǫ and its

reciprocal

γγv → ǫγγv ,

γγu → (1/ǫ)γγu . (12.17)

If the observer boosts by velocity v in the γγz direction away from the source, then the blueshift factor is the

special relativistic Doppler shift factor

ǫ =

(

1− v1 + v

)1/2

. (12.18)

The exponent n of the power ǫn by which an object changes under a boost along the γγz axis is called its

boost weight. Thus γγv has boost weight +1, and γγu has boost weight −1. The spinor axes γγ± both have

boost weight 0. The NP components of a tensor inherit their boost weight properties from those of the NP

basis. The general rule is that the boost weight n of any tensor component is equal to the number of v

covariant indices minus the number of u covariant indices:

boost weight n = number of v minus u covariant indices . (12.19)

12.3 Electromagnetic field tensor

12.3.1 Complexified electromagnetic field tensor

The electromagnetic field Fmn is a bivector, and as such has a natural complex structure. The real part

of the electromagnetic bivector field is the electric field E, which changes sign under spatial inversion,

while the imaginary part is the magnetic field B, which remains unchanged under spatial inversion. In an

orthonormal tetrad γγt,γγx,γγy,γγz, the electromagnetic bivector can be written as though it were a vector

with 6 components:

F =(

E B)

=(

Ex Ey Ez Bx By Bz

)

=(

Ftx Fty Ftz Fzy Fxz Fyx

)

. (12.20)

The bivector has 3 electric and 3 magnetic components:

electric bivector indices: tx, ty, tz ,

magnetic bivector indices: zy, xz, yx .(12.21)

The natural complex structure motivates defining the complexified electromagnetic field tensor Fkl to be

the complex combination

Fkl ≡1

2(Fkl + ∗Fkl) =

1

2

(

δmk δ

nl +

i

2εkl

mn

)

Fmn is a tetrad tensor , (12.22)

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180 ∗More on the tetrad formalism

where ∗Fkl denotes the Hodge dual of Fkl

∗Fkl ≡i

2εkl

mn Fmn . (12.23)

Here εklmn is the totally antisymmetric tensor (see Exercise 12.1). The overall factor of 12 on the right hand

sides of equations (12.22) is introduced so that the complexification operator Pmnkl ≡ 1

2 (δmk δ

nl + i

2 εklmn)

is a projection operator, satisfying P 2 = P . The definitions (12.22) and (12.23) of the complexified and

dual electromagnetic field tensors Fkl and ∗Fkl are valid in any frame, not just an orthonormal frame or

a Newman-Penrose frame. In an orthonormal frame, the dual ∗F has the structure, in the same notation

as (12.20),

∗F = i(

B −E)

. (12.24)

In an orthonormal frame the complexified electromagnetic field F , equation (12.22), then has the structure

F = 12

(

1 −i)

(E + iB) . (12.25)

Thus the complexified electromagnetic field effectively embodies the electric and magnetic fields in the

complex 3-vector combination E + iB. The complexified electromagnetic field is self-dual

∗F = F . (12.26)

One advantage of this approach is that Maxwell’s equations (11.64) and (11.65) combine into a single

complex equation

DmFmn = 4πjn (12.27)

whose real and imaginary parts represent respectively the source and source-free Maxwell’s equations.

In some cases, such as the Kerr-Newman geometry expressed in the Boyer-Lindquist orthonormal tetrad,

equation (15.33), the complexified electromagnetic field makes manifest the inner elegance of the geometry.

Exercise 12.1 Totally antisymmetric tensor. In an orthonormal tetrad γγm where γγ0 points to the

future and γγ1, γγ2, γγ3 are right-handed, the contravariant totally antisymmetric tensor εklmn is defined by

(this is the opposite sign from MTW’s notation)

εklmn ≡ [klmn] (12.28)

and hence

εklmn = −[klmn] (12.29)

where [klmn] is the totally antisymmetric symbol

[klmn] ≡

+1 if klmn is an even permutation of 0123 ,

−1 if klmn is an odd permutation of 0123 ,

0 if klmn are not all different .

(12.30)

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12.3 Electromagnetic field tensor 181

Argue that in a general basis gµ the contravariant totally antisymmetric tensor εκλµν is

εκλµν = ekκel

λemµen

ν εklmn = e [κλµν] (12.31)

while its covariant counterpart is

εκλµν = −(1/e) [κλµν] (12.32)

where e ≡ |emµ| is the determinant of the vierbein. ⋄

12.3.2 Newman-Penrose components of the electromagnetic field

With respect to a NP null tetrad γγv,γγu,γγ+,γγ−, equation (12.13), the electromagnetic field Fmn has 3

distinct complex components, here denoted φs, of spins respectively s = −1, 0, and +1 in accordance with

the rule (12.6):

−1 : φ−1 ≡ Fu− ,

0 : φ0 ≡ 12 (Fuv + F+−) ,

+1 : φ1 ≡ Fv+ .

(12.33)

The complex conjugates φ∗s of the 3 NP components of the electromagnetic field are

φ∗−1 = Fu+ ,

φ∗0 = 12 (Fuv − F+−) ,

φ∗1 = Fv− ,

(12.34)

whose spins have the opposite sign, in accordance with the rule (12.8) that complex conjugation flips spin.

The above convention that the index s on the NP component φs labels its spin differs from the standard

convention, where the spin s component is capriciously denoted φ1−s (e.g. S. Chandrasekhar, 1983, The

Mathematical Theory of Black Holes, Clarendon Press, Oxford, 1983):

−1 : φ2 ,

0 : φ1 ,

+1 : φ0 .

(standard convention, not followed here) (12.35)

In terms of the electric and magnetic fields E and B in the parent orthonormal tetrad of the NP tetrad,

the 3 complex NP components φs of the electromagnetic field are

φ−1 = 12 [Ex + iBx − i(Ey + iBy)] ,

φ0 = 12 (Ez + iBz) ,

φ1 = 12 [Ex + iBx + i(Ey + iBy)] .

(12.36)

Equations (12.36) show that the NP components of the electromagnetic field contain the electric and mag-

netic fields in the complex combination E + iB, just like the complexified electromagnetic field Fkl, equa-

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182 ∗More on the tetrad formalism

tion (12.25). Explicitly, the NP components are related to the components Fkl of the complexified electro-

magnetic field by

φ−1 = Ftx − iFty ,

φ0 = Ftz ,

φ1 = Ftx + iFty .

(12.37)

Part of the power of the NP formalism arises from the fact that it exploits the natural complex structure of

the electromagnetic bivector field.

12.3.3 Newman-Penrose components of the complexified electromagnetic field

The non-vanishing NP components of the complexified electromagnetic field Fkl defined by equation (12.22)

are

Fu− = φ−1 ,

Fuv = F+− = φ0 ,

Fv+ = φ1 ,

(12.38)

whereas components with bivector indices v− or u+ vanish,

Fv− = Fu+ = 0 . (12.39)

The rule that complex conjugation flips spin fails here because the complexification operator in equa-

tion (12.22) breaks the rule. Equations (12.38) and (12.39) show that the complexified electromagnetic

field in an NP tetrad contains just 3 distinct non-vanishing complex components, and those components are

precisely equal to the complex spin components φs.

12.3.4 Propagating components of electromagnetic waves

An oscillating electric charge emits electromagnetic waves. Similarly, an electromagnetic wave incident on an

electric charge causes it to oscillate. An electromagnetic wave moving away from a source is called outgoing,

while a wave moving towards a source is called ingoing.

It can be shown that only the spin −1 NP component φ−1 of an outgoing electromagnetic wave propagates,

carrying electromagnetic energy to infinity:

φ−1 : propagating, outgoing . (12.40)

This propagating, outgoing −1 component has spin −1, but its complex conjugate has spin +1, so effec-

tively both spin components, or helicities, or circular polarizations, of an outgoing electromagnetic wave are

embodied in the single complex component φ−1. The remaining 2 complex NP components φ0 and φ1 of an

outgoing wave are short range, describing the electromagnetic field near the source.

Similarly, only the spin +1 component φ1 of an ingoing electromagnetic wave propagates, carrying energy

from infinity:

φ1 : propagating, ingoing . (12.41)

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12.4 Weyl tensor 183

The isolation of each propagating mode into a single complex NP mode, incorporating both helicities, is

simpler than the standard picture of oscillating orthogonal electric and magnetic fields.

12.4 Weyl tensor

The Weyl tensor is the trace-free part of the Riemann tensor,

Cklmn ≡ Rklmn − 12 (γkmRln − γknRlm + γlnRkm − γlmRkn) + 1

6 (γkmγln − γknγlm)R . (12.42)

By construction, the Weyl tensor vanishes when contracted on any pair of indices. Whereas the Ricci and

Einstein tensors vanish identically in any region of spacetime containing no energy-momentum, Tmn = 0,

the Weyl tensor can be non-vanishing. Physically, the Weyl tensor describes tidal forces and gravitational

waves.

12.4.1 Complexified Weyl tensor

The Weyl tensor is is, like the Riemann tensor, a symmetric matrix of bivectors. Just as the electromagnetic

bivector Fkl has a natural complex structure, so also the Weyl tensor Cklmn has a natural complex structure.

The properties of the Weyl tensor emerge most plainly when that complex structure is made manifest.

In an orthonormal tetrad γγt,γγx,γγy,γγz, the Weyl tensor Cklmn can be written as a 6 × 6 symmetric

bivector matrix, organized as a 2× 2 matrix of 3× 3 blocks, with the structure

C =

(

CEE CEB

CBE CBB

)

=

Ctxtx Ctxty Ctxtz Ctxzy Ctxxz Ctxyx

Ctytx ... ... ... ... ...

Ctztx ... ... ... ... ...

Czytx ... ... ... ... ...

Cxztx ... ... ... ... ...

Cyxtx ... ... ... ... ...

, (12.43)

where E denotes electric indices, B magnetic indices, per the designation (12.21). The condition of being

symmetric implies that the 3 × 3 blocks CEE and CBB are symmetric, while CBE = C⊤EB. The cyclic

symmetry (11.47) of the Riemann, hence Weyl, tensor implies that the off-diagonal 3 × 3 block CEB (and

likewise CBE) is traceless.

The natural complex structure motivates defining a complexified Weyl tensor Cklmn by

Cklmn ≡1

4

(

δpkδ

ql +

i

2εkl

pq

)(

δrmδ

sn +

i

2εmn

rs

)

Cpqrs is a tetrad tensor (12.44)

analogously to the definition (12.22) of the complexified electromagnetic field. The definition (12.44) of the

complexified Weyl tensor Cklmn is valid in any frame, not just an orthonormal frame. In an orthonormal

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184 ∗More on the tetrad formalism

frame, if the Weyl tensor Cklmn is organized according to the structure (12.43), then the complexified Weyl

tensor Cklmn defined by equation (12.44) has the structure

C =1

4

(

1 −i−i −1

)

(CEE − CBB + i CEB + i CBE) . (12.45)

Thus the independent components of the complexified Weyl tensor Cklmn constitute a 3 × 3 complex sym-

metric traceless matrix CEE − CBB + i(CEB + CBE), with 5 complex degrees of freedom. Although the

complexified Weyl tensor Cklmn is defined, equation (12.44), as a projection of the Weyl tensor, it nevertheless

retains all the 10 degrees of freedom of the original Weyl tensor Cklmn.

The same complexification projection operator applied to the trace (Ricci) parts of the Riemann tensor

yields only the Ricci scalar multiplied by that unique combination of the tetrad metric that has the symme-

tries of the Riemann tensor. Thus complexifying the trace parts of the Riemann tensor produces nothing

useful.

12.4.2 Newman-Penrose components of the Weyl tensor

With respect to a NP null tetrad γγv,γγu,γγ+,γγ−, equation (12.13), the Weyl tensor Cklmn has 5 distinct

complex components, here denoted ψs, of spins respectively s = −2, −1, 0, +1, and +2:

−2 : ψ−2 ≡ Cu−u− ,

−1 : ψ−1 ≡ Cuvu− = C+−u− ,

0 : ψ0 ≡ 12 (Cuvuv + Cuv+−) = 1

2 (C+−+− + Cuv+−) = Cv+−u ,

+1 : ψ1 ≡ Cvuv+ = C−+v+ ,

+2 : ψ2 ≡ Cv+v+ .

(12.46)

The complex conjugates ψ∗s of the 5 NP components of the Weyl tensor are:

ψ∗−2 = Cu+u+ ,

ψ∗−1 = Cuvu+ = C−+u+ ,

ψ∗0 = 1

2 (Cuvuv + Cuv−+) = 12 (C−+−+ + Cuv−+) = Cv−+u ,

ψ∗1 = Cvuv− = C+−v− ,

ψ∗2 = Cv−v− .

(12.47)

whose spins have the opposite sign, in accordance with the rule (12.8) that complex conjugation flips spin.

The above expressions (12.46) and (12.47) account for all the NP components Cklmn of the Weyl tensor but

four, which vanish identically:

Cv+v− = Cu+u− = Cv+u+ = Cv−u− = 0 . (12.48)

The above convention that the index s on the NP component ψs labels its spin differs from the standard con-

vention, where the spin s component of the Weyl tensor is impenetrably denoted ψ2−s (e.g. S. Chandrasekhar

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12.4 Weyl tensor 185

1983):

−2 : ψ4 ,

−1 : ψ3 ,

0 : ψ2 ,

+1 : ψ1 ,

+2 : ψ0 .

(standard convention, not followed here) (12.49)

With respect to a triple of bivector indices ordered as u−, uv,+v, the NP components of the Weyl tensor

constitute the 3× 3 complex symmetric matrix IS THIS NP OR COMPLEX NP?

Cklmn =

ψ−2 ψ−1 ψ0

ψ−1 ψ0 ψ1

ψ0 ψ1 ψ2

. (12.50)

12.4.3 Newman-Penrose components of the complexified Weyl tensor

The non-vanishing NP components of the complexified Weyl tensor Cklmn defined by equation (12.44) are

Cu−u− = ψ−2 ,

Cuvu− = C+−u− = ψ−1 ,

Cuvuv = C+−+− = Cuv+− = Cv+−u = ψ0 ,

Cvuv+ = C−+v+ = ψ1 ,

Cv+v+ = ψ2 .

(12.51)

whereas any component with either of its two bivector indices equal to v− or u+ vanishes. As with the

complexified electromagnetic field, the rule that complex conjugation flips spin fails here because the com-

plexification operator breaks the rule. Equations (12.51) show that the complexified Weyl tensor in an NP

tetrad contains just 5 distinct non-vanishing complex components, and those components are precisely equal

to the complex spin components ψs.

12.4.4 Components of the complexified Weyl tensor in an orthonormal tetrad

The complexified Weyl tensor forms a 3 × 3 complex symmetric traceless matrix in any frame, not just an

NP frame. In an orthonormal frame, with respect to a triple of bivector indices tx, ty, tz, the complexified

Weyl tensor Cklmn can be expressed in terms of the NP spin components ψs as

Cklmn =

ψ012 (ψ1 − ψ−1) − i

2 (ψ1 + ψ−1)12 (ψ1 − ψ−1) − 1

2ψ0 + 14 (ψ2 + ψ−2) − i

4 (ψ2 − ψ−2)

− i2 (ψ1 + ψ−1) − i

4 (ψ2 − ψ−2) − 12ψ0 − 1

4 (ψ2 + ψ−2)

. (12.52)

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186 ∗More on the tetrad formalism

12.4.5 Propagating components of gravitational waves

For outgoing gravitational waves, only the spin −2 component ψ−2 (the one conventionally called ψ4) prop-

agates, carrying gravitational waves from a source to infinity:

ψ−2 : propagating, outgoing . (12.53)

This propagating, outgoing −2 component has spin −2, but its complex conjugate has spin +2, so effectively

both spin components, or helicities, or circular polarizations, of an outgoing gravitational wave are embodied

in the single complex component. The remaining 4 complex NP components (spins −1 to 2) of an outgoing

gravitational wave are short range, describing the gravitational field near the source.

Similarly, only the spin +2 component ψ2 of an ingoing gravitational wave propagates, carrying energy

from infinity:

ψ2 : propagating, ingoing . (12.54)

12.5 Petrov classification of the Weyl tensor

As seen above, the complexified Weyl tensor is a complex symmetric traceless 3 × 3 matrix. If the matrix

were real symmetric (or complex Hermitian), then standard mathematical theorems would guarantee that

it would be diagonalizable, with a complete set of eigenvalues and eigenvectors. But the Weyl matrix is

complex symmetric, and there is no such theorem.

The mathematical theorems state that a matrix is diagonalizable if and only if it has a complete set of

linearly independent eigenvectors. Since there is always at least one distinct linearly independent eigenvector

associated with each distinct eigenvalue, if all eigenvalues are distinct, then necessarily there is a complete

set of eigenvectors, and the Weyl tensor is diagonalizable. However, if some of the eigenvalues coincide, then

there may not be a complete set of linearly independent eigenvectors, in which case the Weyl tensor is not

diagonalizable.

The Petrov classification, tabulated in Table 12.1, classifies the Weyl tensor in accordance with the number

of distinct eigenvalues and eigenvectors. The normal form is with respect to an orthonormal frame aligned

with the eigenvectors to the extent possible. The tetrad with respect to which the complexified Weyl tensor

takes its normal form is called the Weyl principal tetrad. The Weyl principal tetrad is unique except in

cases D, O, and N. For Types D and N, the Weyl principal tetrad is unique up to Lorentz transformations

that leave the eigen-bivector γγtz unchanged, which is to say, transformations generated by the Lorentz rotor

exp(ζγγtz) where ζ is complex.

The Kerr-Newman geometry is Type D. General spherically symmetric geometries are Type D. The

Friedmann-Robertson-Walker geometry is Type O. Plane gravitational waves are Type N.

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12.6 Raychaudhuri equations and the Sachs optical scalars 187

expansion θ rotation ω shear σ

Figure 12.1 Illustrating how the Sachs optical scalars, the expansion θ, the rotation ω, and the shear σ,defined by equations (12.60), characterize the rate at which a bundle of light rays changes shape as itpropagates. The bundle of light is coming vertically upward out of the paper.

12.6 Raychaudhuri equations and the Sachs optical scalars

Consider a light ray. Let the γγv null axis lie along the worldline of the light ray. Choose the Newman-

Penrose tetrad so that the tetrad axes γγm are parallel-transported along the path of the light ray. You can

think of a bunch of observers arrayed along the ray each observing the same image, unprecessed, unboosted,

Table 12.1 Petrov classification of the Weyl tensor

Petrov Distinct Distinct Normal formtype eigenvalues eigenvectors of the complexified Weyl tensor

I 3 3

0

@

ψ0 0 00 − 1

2ψ0 + 1

2ψ2 0

0 0 − 12ψ0 − 1

2ψ2

1

A

D 2 3

0

@

ψ0 0 00 − 1

2ψ0 0

0 0 − 12ψ0

1

A

II 2 2

0

@

ψ0 0 00 − 1

2ψ0 + 1

4ψ2 − i

4ψ2

0 − i4ψ2 − 1

2ψ0 − 1

4ψ2

1

A

O 1 3

0

@

0 0 00 0 00 0 0

1

A

N 1 2

0

@

0 0 00 1

4ψ2 − i

4ψ2

0 − i4ψ2 − 1

4ψ2

1

A

III 1 1

0

@

0 12ψ1 − i

2ψ1

12ψ1 0 0

− i2ψ1 0 0

1

A

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188 ∗More on the tetrad formalism

unredshifted. Mathematically, this means that the tetrad axes along the worldline of the light ray satisfy

∂vγγm = 0 . (12.55)

By definition of the tetrad connections, this is equivalent to the conditions that the Newman-Penrose tetrad

connections with final index v all vanish

Γkmv = 0 . (12.56)

The conditions (12.56) constitute a set of 6 conditions which define the Lorentz transformation of the tetrad

axes along the worldline of the light ray. Given the conditions (12.56), the usual expression (11.45) for the

Riemann tensor implies that the rate of change ∂vΓmnl of each of the 18 remaining tetrad connections Γmnl,

those with final index l 6= v, along the worldline of the light ray satisfies

∂vΓmnl + ΓkvlΓmnk = Rvlmn . (12.57)

As commented after the definition (12.13) of the NP tetrad, the null axis γγv ≡ (γγt + γγz)/√

2 defines not a

single direction, but rather a 2D surface spanned by the two directions γγt and γγz. Orthogonal to this surface

is the 2D surface spanned by the transverse axes γγx and γγy, or equivalently by the spinor axes γγ+ and γγ−.

Of the 18 tetrad connections Γmnl with l 6= v, four are embodied in the extrinsic curvature Kab defined by

Kab ≡ γγa · ∂bγγv = Γavb for a, b = +,− . (12.58)

The extrinsic curvature (12.58) describes how the null axis γγv varies over the 2D surface spanned by the

transverse axes γγ+ and γγ−. For the extrinsic curvature Kab, the evolution equations (12.57) become

∂vKab +K+

bKa+ +K−bKa− = Rvbav . (12.59)

The Sachs optical scalars constitute the components of the extrinsic curvatureKab defined by equation (12.58).

Conventionally, the Sachs optical scalars consist of the expansion θ, the rotation ω, and the complex shear

σ, defined in terms of the extrinsic curvature (12.58) by

θ + iω ≡ K+− , (12.60a)

σ ≡ K++ , (12.60b)

whose complex conjugates are θ − iω ≡ K−+ and σ∗ = K−−. Resolved into real and imaginary parts, the

definitions (12.60) of the Sachs optical scalars are

θ ≡ 12 (K+− +K−+) , (12.61a)

ω ≡ 12i (K+− −K−+) , (12.61b)

Reσ ≡ 12 (K++ +K−−) , (12.61c)

Imσ ≡ 12i (K++ −K−−) . (12.61d)

Physically, the Sachs scalars characterize how the shape of a bundle of light rays evolves as it propagates,

as illustrated in Figure 12.1. The expansion represents how fast the bundle expands, the rotation how fast

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12.7 Focussing theorem 189

it rotates, and the shear how fast its ellipticity is changing. The amplitude and phase of the complex shear

represent the amplitude and phase of the major axis of the shear ellipse.

The derivation from equations (12.59) of the evolutionary equations governing the Sachs scalars θ, ω, and

σ is left as Exercise 12.2. The equation (12.62b) for the shear σ shows that the shear changes only in the

presence of a non-vanishing Weyl tensor, that is, in the presence of tidal forces. The equation (12.63b) for

the rotation ω shows that if the rotation is initially zero, then it will remain zero along the path of the light

bundle. Thus geodesic motion cannot by itself generate rotation from nothing; but non-geodesic processes,

such the electromagnetic scattering of light, can generate rotation. The equation (12.63a) for the expansion

θ is commonly called the Raychaudhuri equation (A. K. Raychaudhuri, 1955, “Relativistic cosmology,

I,” Phys. Rev. 98, 1123). The Raychaudhuri equation is the basis of the focussing theorem, §12.7, which is

a central ingredient of the Penrose-Hawking singularity theorems.

Exercise 12.2 Raychaudhuri equations

Show that equations (12.59) imply the following evolutionary equations for the Sachs scalars defined by (12.60):

(∂v + θ + iω)(θ + iω) + σσ∗ + 12Gvv = 0 , (12.62a)

(∂v + 2θ)σ + Cv+v+ = 0 , (12.62b)

where Gmn and Cklmn denote the Einstein and Weyl tensors as usual. Show that the first (12.62a) of these

equations is equivalent to the two equations

∂vθ + (θ2 − ω2) + σσ∗ + 12Gvv = 0 , (12.63a)

(∂v + 2θ)ω = 0 . (12.63b)

12.7 Focussing theorem

The Raychaudhuri equation (12.63a) provides the basis for the focussing theorem, which is a key ingredient

of the singularity theorems introduced by Penrose (R. Penrose, 1965, “Gravitational collapse and spacetime

singularities,” Phys. Rev. Lett. 14, 57–59) and elaborated extensively by Hawking (S. W. Hawking and G.

F. R. Ellis, 1975, The large scale structure of space-time, Cambridge University Press).

If the rotation ω vanishes, then equation (12.63a) for the expansion θ simplifies to

∂vθ + θ2 + σσ∗ + 12Gvv = 0 . (12.64)

The terms θ2 and σσ∗ are necessarily positive. The NP component Gvv of the Einstein tensor is related to

the components in the parent orthonormal tetrad by

Gvv = 12Gtt +Gtz + 1

2Gzz . (12.65)

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190 ∗More on the tetrad formalism

Boosted along the z-direction into the center-of-mass frame, where Gtz = 0, equation (12.65) reduces to

Gvv = 4π(ρ+ pz) (12.66)

where ρ is the energy density and pz the pressure along the z-direction. If it is true that

ρ+ pz ≥ 0 , (12.67)

then the rotation-free Raychaudhuri equation (12.64) shows that the expansion θ must always decrease.

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13

∗The 3+1 (ADM) formalism

Einstein’s equations constitute a set of 10 coupled second-order partial differential equations. Solving these

equations in a general fashion presents a formidable challenge. The 3+1, or ADM, formalism devised by

R. Arnowitt, S. Deser, & C. W. Misner (1959, Phys. Rev. 116, 1322–1330; 1962, “The dynamics of general

relativity,” in Gravitation: an introduction to current research, ed. L. Witten, 227–265) offers an insightful

and systematic way to proceed. The formalism is widely used in numerical general relativity. For reviews,

see L. Lehner (2001, “Numerical relativity: a review,” CQG 18, R25–86, gr-qc/0106072), and H. Shinkai

(2008, “Formulations of the Einstein equations for numerical simulations,” APCTP winter school on black

hole astrophysics, arXiv:0805.0068).

The central idea of the ADM formalism is to recast the Einstein equations into Hamiltonian form. The

Hamiltonian approach identifies “canonical momenta” conjugate to the “coordinates,” and converts the

equations of motion from second order partial differential equations in the coordinates into coupled first

order partial differential equations in the coordinates and momenta. The Hamiltonian H of a system is

its “energy” expressed in terms of the coordinates and momenta. In quantum mechanics, equating the

time translation operator to the Hamiltonian operator, i~ ∂/∂t = H , determines the evolution in time t of

the system. To implement the Hamiltonian approach in general relativity, it is necessary to identify one

coordinate, the time coordinate t, as having a special status. The system of Einstein (and other) equations

is evolved by integrating from one spacelike hypersurface of constant time, t = constant, to the next.

The 3+1 formalism provides answers to several basic questions about the dynamical structure of Einstein’s

equations:

1. Are there natural coordinates for the gravitational field, and what are they? Answer: Yes. The natural

coordinates are the 6 components of the spatial metric gαβ on the hypersurfaces of constant time. The

3+1 formalism shows that only these 6 of the 10 components of the 4D metric gµν are governed by time

evolution equations. The remaining 4 degrees of freedom in the metric represent gauge freedoms associated

with general coordinate transformations.

2. What happens to the 4 degrees of freedom associated with general coordinate transformations? Answer:

They are accomodated into the lapse α and shift βα, which express the rate at which the unit timelike

normal γγ0 to the spatial hypersurfaces of constant time marches through the coordinates, equations (13.9).

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192 ∗The 3+1 (ADM) formalism

In the 3+1 formalism, the lapse and shift are specifiable arbitrarily, and are not governed by time evolution

equations.

3. What are the 6 momenta conjugate to the 6 coordinates gαβ? Answer: The momenta are, up to a factor,

the components of the trace-modified extrinsic curvature Kαβ − gαβK, equation (13.36). The extrinsic

curvature Kαβ , whose tetrad-frame expression is defined by equation (13.12b), is a symmetric 3×3 matrix

that describes how the unit timelike normal γγ0 varies over the spatial hypersurfaces of constant time.

4. What is the structure of the 10 Einstein equations in the 3+1 formalism? Answer: The 6 spatial com-

ponents of the Einstein equations provide dynamical time evolution equations for the 6 momenta. The

4 remaining components of the Einstein equations, the time-time and time-space components, prove to

be constraint equations, called the Hamiltonian (or scalar) and momentum (or vector) constraints. The

constraint equations specify conditions that must be arranged to be satisfied on the initial hypersurface

of constant time, but thereafter the constraints are automatically satisfied (modulo numerical error and

instabilities).

5. How is covariant conservation of energy-momentum expressed? Answer: The fact that the Hamiltonian

and momentum constraints continue to be satisfied as time advances expresses covariant conservation of

energy-momentum, as guaranteed by the contracted Bianchi identities.

In this chapter, the coordinate time index is t, while the tetrad time index is 0. Early-alphabet greek

(brown) letters α, β, ..., denote 3D spatial coordinate indices, while mid-alphabet greek letters κ, λ, ...,

denote 4D spacetime coordinate indices. Early-alphabet latin (black) letters a, b, ..., denote 3D spatial

tetrad indices, while mid-alphabet latin (black) letters k, l, ..., denote 4D spacetime tetrad indices.

13.1 ADM tetrad

The ADM formalism splits the spacetime coordinates xµ into a time coordinate t and spatial coordinates

xα, α = 1, 2, 3,

xµ ≡ t, xα . (13.1)

At each point of spacetime, the spacelike hypersurface of constant time t has a unique future-pointing unit

normal γγ0, defined to have unit length and to be orthogonal to the spatial tangent axes gα,

γγ0 · γγ0 = −1 , γγ0 · gα = 0 α = 1, 2, 3 . (13.2)

The central element of the ADM approach is to work in a tetrad frame γγm consisting of this time axis γγ0,

together with three spatial tetrad axes γγa that are orthogonal to the tetrad time axis γγ0, and therefore lie

in the 3D spatial hypersurface of constant time,

γγ0 · γγa = 0 a = 1, 2, 3 . (13.3)

The tetrad metric γmn in the ADM formalism is thus

γmn =

( −1 0

0 γab

)

, (13.4)

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13.2 Traditional ADM approach 193

and the inverse tetrad metric γmn is correspondingly

γmn =

( −1 0

0 γab

)

, (13.5)

whose spatial part γab is the inverse of γab. Given the conditions (13.2) and (13.3), the vierbein emµ and

inverse vierbein emµ take the form

emµ =

(

1/α βα/α

0 eaα

)

, emµ =

(

α 0

−eaαβ

α eaα

)

, (13.6)

where α and βα are the lapse and shift (see next paragraph), and eaα and ea

α represent the spatial vierbein

and inverse vierbein, which are inverse to each other, eaαeb

α = δca. The ADM metric is

ds2 = −α2dt2 + gαβ (dxα − βαdt)(

dxβ − ββdt)

, (13.7)

where gαβ is the spatial coordinate metric

gαβ = γabea

αebβ . (13.8)

Essentially all the tetrad formalism developed in Chapter 11 carries through, subject only to the condi-

tions (13.2) and (13.3).

The vierbein coefficient α is called the lapse, while βα is called the shift. Physically, the lapse α is the

rate at which the proper time τ of the tetrad rest frame elapses per unit coordinate time t, while the shift βα

is the velocity at which the tetrad rest frame moves through the spatial coordinates xα per unit coordinate

time t,

α =dτ

dt, βα =

dxα

dt. (13.9)

These relations (13.9) follow from the fact that the 4-velocity in the tetrad rest frame is by definition

um ≡ 1, 0, 0, 0, so the coordinate 4-velocity uµ ≡ emµum of the tetrad rest frame is

dxµ

dτ≡ uµ = et

µ =1

α1, βα . (13.10)

13.2 Traditional ADM approach

The traditional ADM approach sets the spatial tetrad axes γγa equal to the spatial coordinate tangent axes

gα,

γγa = gα (traditional ADM) , (13.11)

equivalent to choosing the spatial vierbein to be the unit matrix, eaα = δα

a . The traditional ADM approach

may be termed semi-tetrad, since it works with a tetrad time axis γγ0 together with coordinate spatial axes

gα. It is natural however to extend the ADM approach into a full tetrad approach, allowing the spatial

tetrad axes γγa to be chosen more generally, subject only to the condition (13.3) that they be orthogonal to

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194 ∗The 3+1 (ADM) formalism

the tetrad time axis, and therefore lie in the hypersurface of constant time t. For example, the spatial tetrad

γγa can be chosen to form 3D orthonormal axes, γab ≡ γγa · γγb = δab, so that the full 4D tetrad metric γmn is

Minkowski.

This chapter follows the full tetrad approach to the ADM formalism, but all the results hold for traditional

case where the spatial tetrad axes are set equal to the coordinate spatial axes, equation (13.11).

13.3 Spatial tetrad vectors and tensors

Since the tetrad time axis γγ0 in the ADM formalism is defined uniquely by the choice of hypersurfaces

of constant time t, there is no freedom of tetrad transformations of the time axis distinct from temporal

coordinate transformations (no distinct freedom of Lorentz boosts). However, there is still freedom of tetrad

transformations of the spatial tetrad axes (spatial rotations).

A covariant spatial tetrad vector Aa is defined in the usual way as a vector that transforms like the

spatial tetrad axes γγa. Similarly a covariant spatial tetrad tensor is a tensor that transforms like products

of the spatial tetrad axes γγa. Indices on spatial tetrad vectors and tensors are raised with the inverse spatial

tetrad metric γγab, and lowered with the spatial tetrad metric γγab.

A temporal coordinate transformation changes the hypersurface of constant time t, and therefore changes

its unit normal, the tetrad time axis γγ0, and correspondingly all the spatial tetrad axes γγa.

13.4 ADM connections, gravity, and extrinsic curvature

Since the tetrad time axis γγ0 is a spatial tetrad scalar, its directed time derivative ∂0γγ0 is a spatial tetrad

scalar, while its directed spatial derivatives ∂bγγ0 form a spatial tetrad vector. It follows that the connections

Γm0n defined by the directed derivatives of γγ0 are spatial tetrad tensors. These connections play an important

role in the ADM formalism, and they are given special names and symbols, the gravity κa, and the extrinsic

curvature Kab (the remaining components of the directed derivatives of γγ0 vanish, Γ000 = Γ00a = 0):

κa ≡ γγa · ∂0γγ0 = Γa00 is a spatial tetrad vector , (13.12a)

Kab ≡ γγa · ∂bγγ0 = Γa0b is a spatial tetrad tensor . (13.12b)

The gravity κa is justly named because the geodesic equation shows that it is minus the acceleration expe-

rienced in the tetrad rest frame, where um = 1, 0, 0, 0,dua

dτ= −κa . (13.13)

The extrinsic curvature Kab describes how the unit normal γγ0 to the 3-dimensional spatial hypersurface

changes over the hypersurface, and can therefore be regarded as embodying the curvature of the 3-dimensional

spatial hypersurface embedded in the 4-dimensional spacetime. From equation (11.40) with vanishing torsion,

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13.5 ADM Riemann, Ricci, and Einstein tensors 195

it follows that the gravity and the extrinsic curvature are

κa = d00a , (13.14a)

Kab = 12 (∂0γab − dab0 − dba0 + da0b + db0a) , (13.14b)

where the relevant vierbein derivatives dlmn are

d00a =1

α∂aα , da0b =

1

αeaα ∂bβ

α , dab0 = − γac ebβ ∂0e

cβ . (13.15)

Equation (13.14b) shows that the extrinsic curvature is symmetric,

Kab = Kba . (13.16)

The non-vanishing tetrad connections are, from the general formula (11.40) with vanishing torsion,

Γa00 = −Γ0a0 = κa , (13.17a)

Γa0b = −Γ0ab = Kab , (13.17b)

Γab0 = Kab + dab0 − da0b , (13.17c)

Γabc = same as eq. (11.40) . (13.17d)

The connections (13.17a) and (13.17b) form, as commented above, a spatial tetrad vector κa and tensor

Kab, but the remaining connections (13.17c) and (13.17d) are not spatial tetrad tensors. Note that the

purely spatial tetrad connections Γabc, like the spatial tetrad axes γγa, transform under temporal coordinate

transformations despite the absence of temporal indices.

13.5 ADM Riemann, Ricci, and Einstein tensors

The ADM Riemann tensor Rklmn inherits from the standard tetrad formalism the property of being a full

tetrad tensor, its components transforming like products of the tetrad axes γγm. Of course, since the ADM

tetrad time axis γγ0 is tied to the coordinates, a tetrad transformation of the time axis requires a simultaneous

coordinate transformation consistent with it. The usual expression (11.45) for the tetrad-frame Riemann

tensor with vanishing torsion yields the ADM Riemann tensor

R0a0b = −D0Kab −KacKcb +

1

αDaDbα , (13.18a)

R0abc = DcKab −DbKac , (13.18b)

Rabcd = KacKbd −KadKbc +R(3)abcd . (13.18c)

Here R(3)abcd is the spatial tetrad Riemann tensor considered confined to the 3D spatial hypersurface (given by

equation (11.45) with all time components discarded), and Dm denotes the usual 4D tetrad-frame covariant

derivative, with the understanding that when acting on a spatial tensor such as Kab, all time components

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196 ∗The 3+1 (ADM) formalism

of the spatial tensor are to be considered equal to zero. Thus the 4D tetrad covariant derivative of a spatial

tetrad vector Aa is

DmAa = ∂mAa − ΓbamAb , (13.19)

in which the possible Γ0amA0 term is considered to vanish. When acting on a spatial tetrad tensor, the spatial

part Da of the 4D tetrad covariant derivative Dm involves only spatial connections, and is identical to the

3D spatial tetrad covariant derivative D(3) considered confined to the 3D spatial hypersurface,

Da ≡ D(3)a when acting on a spatial tetrad tensor . (13.20)

The covariant tetrad time derivative D0 acting on a spatial tetrad tensor yields a 3D spatial tetrad tensor,

but the full 4D tetrad covariant derivative Dm acting on a spatial tetrad tensor does not yield a 4D tetrad

tensor. It is important to bear these fine distinctions in mind when integrating by parts, as done in §13.6.

The Ricci tensor Rkm ≡ γlnRklmn is, like the Riemann tensor, a tetrad tensor. Its components are

R00 = − ∂0K −KabKab +

1

αDaD

aα , (13.21a)

R0a = DbKab − ∂aK , (13.21b)

Rab = D0Kab +KKab −1

αDaDbα+R

(3)ab , (13.21c)

where K ≡ γabKab is the trace of the extrinsic curvature, and R(3)ac ≡ γbdR

(3)abcd is the Ricci tensor confined

to the 3D spatial hypersurface. The Ricci scalar R ≡ γkmRkm is

R = 2 ∂0K +K2 +KabKab − 2

αDaD

aα+R(3) , (13.22)

where R(3) ≡ γabR(3)ab is the Ricci scalar confined to the 3D spatial hypersurface.

The Einstein tensor Gkm ≡ Rkm − 12γkmR is

G00 = 12

(

K2 −KabKab +R(3)

)

, (13.23a)

G0a = DbKab −DaK , (13.23b)

Gab = D0Kab +KKab −1

αDaDbα− γab

(

∂0K + 12K

2 + 12KcdK

cd − 1

αDcD

)

+G(3)ab , (13.23c)

where G(3)ab ≡ R

(3)ab − 1

2γabR(3) is the Einstein tensor confined to the 3D spatial hypersurface.

13.6 ADM action

In this chapter up to this point, the Einstein tensor and other quantities have been expressed in terms of

other things, but no equation of motion has been invoked. The ADM philosophy is to derive the gravitational

equations of motion — Einstein’s equations — in Hamiltonian form. The starting point is to write down the

gravitational action, extremization of which will yield equations of motion.

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13.6 ADM action 197

The Hilbert gravitational action Sg is

Sg =

∫ tf

ti

Lg dx4 =1

16π

∫ tf

ti

Rdx4 , (13.24)

with scalar Lagrangian Lg equal to a normalization factor times the Ricci scalar R,

Lg =1

16πR . (13.25)

The integration in the action (13.24) is over a 4-volume from an initial hypersurface of constant time ti to

a final hypersurface of constant time tf . The integration measure dx4 in the integral (13.24) denotes the

scalar 4-volume element1. With respect to coordinates t, xα, the scalar 4-volume element is

dx4 = αdtdx30 = (α/e) dt d3x , (13.26)

where dx30 is the tetrad time component (the component along the timelike normal γγ0 to the spatial hy-

persurface) of the vector 3-volume element dx3k, and e = |ea

β | is the determinant of the spatial vierbein, so

that 1/e = |eaβ | is the determinant of the inverse spatial vierbein. Instead of the scalar Lagrangian Lg, it is

equally possible to work with the Lagrangian density Lg,

Sg =

∫ tf

ti

Lg dt d3x with Lg ≡ Lg α/e . (13.27)

Either way, the important thing is to be careful to get factors in the integration measure right when inte-

grating by parts. The following development works with the scalar Lagrangian Lg, in which case a term

integrates over a scalar 4-volume V to a 3D surface integral over the 3-boundary ∂V of the volume provided

that the term is a covariant 4-divergence,∫

V

D ·A dx4 =

∂V

A · dx3 . (13.28)

The least action principle demands that the conditions on the initial and final hypersurfaces ti and tf be

considered fixed, and asserts that the path followed by the system between the fixed initial and final conditions

is such that the action is minimized. The equations of motion that result from extremizing the action are

unaffected by terms in the scalar Lagrangian that are covariant 4-divergences, since these integrate to surface

terms that are asserted to be fixed, and therefore unchanged by variation. The ADM expression (13.22) for

the Ricci scalar involves two terms, 2∂0K and (2/α)DaDaα, that contain second derivatives of the vierbein

coefficients, and therefore demand to be integrated by parts to bring the Lagrangian into Hamiltonian form,

depending only on first derivatives. To integrate the ∂0K term by parts, it is necessary to express it in terms

of a covariant 4-divergence, which is accomplished by

∂0K = DmKm −K2 , (13.29)

1 Technically, the scalar volume element dx4 is the quadvector, or pseudoscalar, 4-volume element, the differential 4-form14!

εklmn dxk∧ dxl

∧ dxm∧ dxn. Likewise the vector 3-volume element dx3

k is the trivector, or pseudovector, 3-volume

element, the differential 3-form 13!

εklmn dxl∧ dxm

∧ dxn.

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198 ∗The 3+1 (ADM) formalism

where Km ≡ K, 0, 0, 0. Similarly, the (1/α)DaDaα term is converted to a 4-divergence by

1

αDaD

aα = Dmκm , (13.30)

where κm ≡ 0, κa with κa defined by equation (13.12a).

Inserting the ADM expression (13.22) for the Ricci scalar into the Hilbert action (13.24), and integrating

the ∂0K and (1/α)DaDaα terms by parts, yields the ADM gravitational action

Sg =1

[∫

K dx30

]tf

ti

+1

∂V

κa dx3a +

1

16π

∫ tf

ti

(

KabKab −K2 +R(3)

)

dx4 . (13.31)

For the purpose of extremizing the action, the surface terms can be discarded. Thus the action to be

extremized is the one with the ADM Lagrangian

LADM =1

16π

(

KabKab −K2 +R(3)

)

. (13.32)

According to the usual procedure, conjugate momenta are obtained as partial derivatives of the Lagrangian

with respect to velocities. In the present instance, the velocities are time derivatives of the coordinates. Now

the only things containing time derivatives in the ADM Lagrangian (13.32) are those contained in the terms

involving the extrinsic curvature Kab (the spatial Ricci scalar R(3) contains no time derivatives). From the

expression (13.14b) for the extrinsic curvature, together with equations (13.15), the time derivatives in the

extrinsic curvature are the directed time derivatives ∂0γab of the spatial tetrad metric, and the directed time

derivatives ∂0ecβ of the spatial inverse vierbein. However, these time derivatives appear only the combination

∂0γab − dab0 − dba0 = eaαeb

β∂0(γcd ecα e

dβ) = ea

αebβ ∂0gαβ . (13.33)

Thus the ADM Lagrangian picks out the natural coordinates as being the spatial components gαβ of the

coordinate metric, since only time derivatives of these appear in the ADM Lagrangian. An expression for

the extrinsic curvature that demonstrates explicitly its dependence on the time derivatives of the spatial

coordinate metric gαβ is, from manipulating equation (13.14b),

Kab =1

(

eaαeb

β ∂gαβ

∂t−Daebt −Dbeat

)

. (13.34)

Conjugate momenta are obtained by differentiating the Lagrangian with respect to the velocities. The

derivatives of the extrinsic curvature with respect to the velocities gαβ ≡ ∂gαβ/∂t are

∂Kab

∂gαβ=

1

2αea

αebβ . (13.35)

The conjugate momenta παβ are therefore

παβ ≡ α ∂LADM

∂gαβ=

1

16π

(

Kαβ − gαβK)

, (13.36)

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13.7 ADM equations of motion 199

in which the factor of α is introduced to convert the conjugate momenta into a tensor, as opposed to a tensor

density. Projected into the tetrad frame, the conjugate momenta are

πab = eaαe

bβ π

αβ =1

16π

(

Kab − γabK)

. (13.37)

The ADM Lagrangian (13.32) can be rewritten in terms of the conjugate momenta (13.37) as

LADM = 2Kabπab − G0

0

8π, (13.38)

in which the 3D Ricci scalar R(3) has been eliminated in favour of the time-time component G00 of the tetrad-

frame Einstein tensor by equation (13.23a). Substituting the expression (13.34) for the extrinsic curvature

Kab brings the ADM Lagrangian to

LADM =1

α

(

eaαeb

β ∂gαβ

∂t−Daebt −Dbeat

)

πab − G00

8π. (13.39)

The two terms Daebt and Dbeat can be combined into one because of the symmetry of πab, and integrated

by parts to give the ADM action

SADM ≡∫ tf

ti

LADM dx4 = −2

∂V

ebtπab dx3

a +

∫ tf

ti

(

eaαeb

β ∂gαβ

∂tπab + 2 eatDbπ

ab − αG00

)

dtdx30 .

(13.40)

Once again, for the purposes of extremizing the action, the surface term can be discarded. Eliminating the

Dbπab term in favour of the time-space part G0

a of the tetrad-frame Einstein tensor by equation (13.23b)

produces

SADM =

∫ tf

ti

(

eaαeb

β ∂gαβ

∂tπab − ea

tG0

a

8π− e0t

G00

)

dtdx30 . (13.41)

This is the ADM gravitational action in desired Hamiltonian form. Compactly,

SADM =

∫ tf

ti

(

∂gαβ

∂tπαβ −H

)

dtdx30 (13.42)

where H is the Hamiltonian

H ≡ emtG

0m

8π=G0

t

8π. (13.43)

13.7 ADM equations of motion

Equations of motion follow from extremizing the ADM action (13.42) in Hamiltonian form. To accomplish

this, the Hamiltonian H must be expressed in terms of the coordinates and momenta. The expression (13.42)

for the ADM action shows that the natural dynamical coordinates and momenta of the system are the

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200 ∗The 3+1 (ADM) formalism

components of the spatial coordinate-frame metric gαβ and their conjugate momenta παβ . In terms of these,

the Hamiltonian is

H =1

(

αG00 + βαG0

α

)

= α

16π[

παβπαβ − (πγ

γ )2]

− R(3)

16π

− βα

2Dβπαβ

, (13.44)

where the expressions for G00 and G0

α are from equations (13.23a) and (13.23b) recast into coordinate-frame

quantities, the extrinsic curvature Kαβ being eliminated in favour of the momenta παβ by equation (13.36).

The expression (13.44) for the Hamiltonian depends not only on the coordinates gαβ and momenta παβ , but

also on the lapse α and shift βα, which are independent of gαβ and παβ . Consequently the lapse and shift must

also be treated as additional coordinates. The Hamiltonian (13.44) depends on the lapse and shift linearly,

the quantities in braces in equation (13.44) being independent of the lapse and shift. The Hamiltonian (13.44)

also depends on spatial derivatives ∂gαβ/∂xγ of the coordinates through its dependence on the Riemann

3-scalar R(3) and on the spatial connections Γαβγ associated with the covariant spatial coordinate derivative

Dβ in the term Dβπαβ . However, the spatial derivatives ∂gαβ/∂xγ are to be considered as determined by

the coordinates gαβ as a function of the spatial coordinates xγ on a spatial hypersurface of constant time t,

not as independent coordinates to be varied separately.

Variation of the ADM action (13.42) with respect to the lapse α, the shift βα, the coordinates gαβ, and

the momenta παβ , gives

δSADM =

[∫

παβδgαβ dx30

]tf

ti

(13.45)

+

∫ tf

ti

∂H

∂αδα+

∂H

∂βαδβα −

(

∂παβ

∂t+

∂H

∂gαβ

)

δgαβ +

(

∂gαβ

∂t− ∂H

∂παβ

)

δπαβ

dtdx30 .

The least action principle asserts that the action is minimized along the actual path taken by the system

between fixed initial and final conditions at ti and tf . Setting the variation (13.45) of the action equal to

zero with respect to arbitrary variations δα and δβα of the lapse and shift yields the constraint equations

∂H

∂α= 0 ,

∂H

∂βα= 0 . (13.46)

Setting the variation (13.45) of the action equal to zero with respect to arbitrary variations δgαβ and δπαβ

of the coordinates and momenta yields Hamilton’s equations

∂παβ

∂t= − ∂H

∂gαβ,

∂gαβ

∂t=

∂H

∂παβ. (13.47)

13.8 Constraints and energy-momentum conservation

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14

∗The geometric algebra

The geometric algebra is an intuitively appealing formalism that draws together several mathematical threads

relevant to special and general relativity:

1. What is the best way to conceptualize Lorentz transformations (§14.6), and to implement them on a

computer (§14.20)?

2. How is it that bivectors in 4D spacetime have a natural complex structure, which has been seen to be the

heart of the Newman-Penrose formalism? See §14.17.

3. How can spin- 12 objects be incorporated into general relativity? What is a Dirac spinor? See §14.23.

4. How and why do differential forms work?

This chapter starts by setting up the geometric algebra in n-dimensional Euclidean space Rn, then gener-

alizes to Minkowski space, where the geometric algebra is called the spacetime algebra. The 4D spacetime

algebra proves to be identical to the Clifford algebra of the Dirac γ-matrices (which explains the adoption

of the symbol γγm to denote the basis vectors of a tetrad). Although the formalism is presented initially

in Euclidean or Minkowski space, everything generalizes immediately to general relativity, where the basis

vectors γγm form the basis of an orthonormal tetrad at each point of spacetime.

One convention adopted here, which agrees with the convention adopted by OpenGL and the computer

graphics industry, but is opposite to the standard physics convention, is that a rotor R rotates a multivector

a as a→ RaR, equation (14.35). This, along with the standard definition (14.16) for the pseudoscalar, has

the consequence that a right-handed rotation corresponds to R = eiθ/2 with θ increasing, and that rotations

accumulate to the right, that is, a rotation R followed by a rotation S is the product RS. By contrast, in the

standard physics convention a → RaR, a right-handed rotation corresponds to R = e−iθ/2, and rotations

accumulate to the left, that is, R followed by S is SR. The convention adopted here also means that a Weyl

or Dirac spinor ϕ is isomorphic to a scaled reverse rotor R, not to R as in the standard physics convention.

In this chapter, boldface denotes a multivector. A rotor is written in normal (not bold) face as a reminder

that, even though a rotor is an even member of the geometric algebra, it can also be regarded as a spin- 12

object with a transformation law (14.37) different from that (14.35) of multivectors. Later Latin indices

m,n, ... run over both time and space indices 0, 1, 2, 3, while earlier Latin indices i, j, k run over spatial

indices 1, 2, 3 only.

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202 ∗The geometric algebra

a

θa

b

a

b

c

Figure 14.1 Multivectors of grade 1, 2, and 3: a vector a (left), a bivector a∧ b (middle), and a trivectora∧ b∧ c (right).

14.1 Products of vectors

In 3-dimensional Euclidean space R3, there are two familiar ways of taking the product of two vectors, the

scalar product and the vector product.

1. The scalar product a · b, also known as the dot product or inner product, of two vectors a and b is

a scalar of magnitude |a| |b| cos θ, where |a| and |b| are the lengths of the two vectors, and θ the angle

between them. The scalar product is commutative, a · b = b · a.

2. The vector product, a×b, also known as the cross product, is a vector of magnitude |a| |b| sin θ, directed

perpendicular to both a and b, such that a, b, and a× b form a right-handed set. The vector product is

anticommutative, a× b = −b× a.

The definition of the scalar product continues to work fine in a Euclidean space of any dimension, but the

definition of the vector product works only in three dimensions, because in two dimensions there is no vector

perpendicular to two vectors, and in four or more dimensions there are many vectors perpendicular to two

vectors. It is therefore useful to define a more general version, the outer product (H. Grassmann, 1862 Die

Ausdehnungslehre, Berlin) that works in Euclidean space Rn of any dimension.

3. The outer product a∧ b, also known as the wedge product, of two vectors a and b is a bivector, a

multivector of dimension 2, or grade 2. The bivector a∧b is the directed 2-dimensional area, of magnitude

|a| |b| sin θ, of the parallelogram formed by the vectors a and b, as illustrated in Figure 14.1. The bivector

has an orientation, or handedness, defined by circulating the parallelogram first along a, then along b.

The outer product is anticommutative, a∧ b = −b∧a, like its forebear the vector product.

The outer product can be repeated, so that (a∧ b)∧ c is a trivector, a directed volume, a multivector

of grade 3. The magnitude of the trivector is the volume of the parallelepiped defined by the vectors a, b,

and c, illustrated in Figure 14.1. The outer product is by construction associative, (a∧ b)∧ c = a∧ (b∧ c).

Associativity, together with anticommutativity of bivectors, implies that the trivector a∧b∧ c is totally

antisymmetric under permutations of the three vectors, that is, it is unchanged under even permutations, and

changes sign under odd permutations. The ordering of an outer product thus defines one of two handednesses.

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14.2 Geometric product 203

It is a familiar concept that a vector a can be regarded as a geometric object, a directed length, independent

of the coordinates used to describe it. The components of a vector change when the reference frame changes,

but the vector itself remains the same physical thing. In the same way, a bivector a∧ b is a directed area,

and a trivector a∧b∧ c is a directed volume, both geometric objects with a physical meaning independent

of the coordinate system.

In two dimensions the triple outer product of any three vectors is zero, a∧ b∧ c = 0, because the volume

of a parallelepiped confined to a plane is zero. More generally, in n-dimensional space Rn, the outer product

of n+ 1 vectors is zero

a1 ∧a2 ∧ · · · ∧an+1 = 0 (n dimensions) . (14.1)

14.2 Geometric product

The inner and outer products offer two different ways of multiplying vectors. However, by itself neither

product conforms to the usual desideratum of multiplication, that the product of two elements of a set be

an element of the set. Taking the inner product of a vector with another vector lowers the dimension by

one, while taking the outer product raises the dimension by one.

H. Grassmann H. (1877, “Der ort der Hamilton’schen quaternionen in der audehnungslehre,” Math. Ann.

12, 375) and W. K. Clifford (1878, “Applications of Grassmann’s extensive algebra,” Am. J. Math. 1, 350)

resolved the problem by defining a multivector as any linear combination of scalars, vectors, bivectors, and

objects of higher grade. Let γγ1,γγ2, ...,γγn form an orthonormal basis for n-dimensional Euclidean space Rn.

A multivector in n = 2 dimensions is then a linear combination of

1 ,

1 scalar

γγ1 , γγ2 ,

2 vectors

γγ1 ∧γγ2 ,

1 bivector(14.2)

forming a linear space of dimension 1 + 2 + 1 = 4 = 22. Similarly, a multivector in n = 3 dimensions is a

linear combination of

1 ,

1 scalar

γγ1 , γγ2 , γγ3 ,

3 vectors

γγ1 ∧γγ2 , γγ2 ∧γγ3 , γγ3 ∧γγ1 ,

3 bivectors

γγ1 ∧γγ2 ∧γγ3 ,

1 trivector(14.3)

forming a linear space of dimension 1 + 3 + 3 + 1 = 8 = 23. In general, multivectors in n dimensions form a

linear space of dimension 2n, with n!/[m!(n−m)!] distinct basis elements of grade m.

A multivector a in n-dimensional Euclidean space Rn can thus be written as a linear combination of basis

elements

a =∑

distinct i,j,...,m⊆1,2,...,naij...m γγi ∧γγj ∧ ...∧γγm (14.4)

the sum being over all 2n distinct subsets of 1, 2, ..., n. The index on each component aij...m is a totally

antisymmetric quantity, reflecting the total antisymmetry of γγi ∧γγj ∧ ...∧γγm.

The point of introducing multivectors is to allow multiplication to be defined so that the product of two

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204 ∗The geometric algebra

multivectors is a multivector. The key trick is to define the geometric product ab of two vectors a and b

to be the sum of their inner and outer products:

ab = a · b + a∧ b . (14.5)

That is a seriously big trick, and if you buy a ticket to it, you are in for a seriously big ride. As a particular

example of (14.5), the geometric product of any element γγi of the orthonormal basis with itself is a scalar,

and with any other element of the basis is a bivector:

γγiγγj =

1 (i = j)

γγi ∧γγj (i 6= j) .(14.6)

Conversely, the rules (14.6), plus distributivity, imply the multiplication rule (14.5). A generalization of the

rule (14.6) completes the definition of the geometric product:

γγiγγj ...γγm = γγi ∧γγj ∧ ...∧γγm (i, j, ...,m all distinct) . (14.7)

The rules (14.6) and (14.7), along with the usual requirements of associativity and distributivity, combined

with commutativity of scalars and anticommutativity of pairs of γγi, uniquely define multiplication over the

space of multivectors. For example, the product of the bivector γγ1 ∧γγ2 with the vector γγ1 is

(γγ1 ∧γγ2)γγ1 = γγ1γγ2γγ1 = −γγ2γγ1γγ1 = −γγ2 . (14.8)

Sometimes it is convenient to denote the wedge product (14.7) of distinct basis elements by the abbreviated

symbol γγij...m

γγij...m ≡ γγi ∧γγj ∧ ...∧γγm (i, j, ...,m all distinct) . (14.9)

By construction, γγij...m is antisymmetric in its indices. The product of two general multivectors a = aαγγα

and b = bαγγα, with paired indices implicitly summed over distinct subsets of 1, ..., n, is

ab = aαbβγγαγγβ . (14.10)

Does the geometric algebra form a group under multiplication? No. One of the defining properties of a

group is that every element should have an inverse. But, for example,

(1 + γγ1)(1− γγ1) = 0 (14.11)

shows that neither 1 + γγ1 nor 1− γγ1 has an inverse.

14.3 Reverse

The reverse of any basis element is defined to be the reversed product

γγi ∧γγj ∧ ...∧γγm ≡ γγm ∧ ...∧γγj ∧γγi . (14.12)

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14.4 The pseudoscalar and the Hodge dual 205

The reverse a of any multivector a is the multivector obtained by reversing each of its components. Reversion

leaves unchanged all multivectors whose grade is 0 or 1, modulo 4, and changes the sign of all multivectors

whose grade is 2 or 3, modulo 4. For example, scalars and vectors are unchanged by reversion, but bivectors

and trivectors change sign. Reversion satisfies

a + b = a + b , (14.13)

ab = ba . (14.14)

Among other things, it follows that the reverse of any product of multivectors is the reversed product, as

you would hope:

ab ... c = c ... ba . (14.15)

14.4 The pseudoscalar and the Hodge dual

Orthogonal to any m-dimensional subspace of n-dimensional space is an (n−m)-dimensional space, called

the Hodge dual space. For example, the Hodge dual of a bivector in 2 dimensions is a 0-dimensional

object, a pseudoscalar. Similarly, the Hodge dual of a bivector in 3 dimensions is a 1-dimensional object, a

pseudovector.

Define the pseudoscalar in in n dimensions to be

in ≡ γγ1 ∧γγ2 ∧ ...∧γγn (14.16)

with reverse

in = (−)[n/2]γγ1 ∧γγ2 ∧ ...∧γγn . (14.17)

The quantity [n/2] in equation (14.17) signifies the largest integer less than or equal to n/2. The square of

the pseudoscalar is

i2n = (−)[n/2] =

1 if n = (0 or 1) modulo 4

−1 if n = (2 or 3) modulo 4 .(14.18)

The pseudoscalar anticommutes (commutes) with vectors a, that is, with multivectors of grade 1, if n is

even (odd):

ina = −ain if n is even

ina = ain if n is odd .(14.19)

This implies that the pseudoscalar in commutes with all even grade elements of the geometric algebra, and

that it anticommutes (commutes) with all odd elements of the algebra if n is even (odd).

Exercise 14.1 Prove that the only multivectors that commute with all elements of the algebra are linear

combinations of the scalar 1 and, if n is odd, the pseudoscalar in. ⋄

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206 ∗The geometric algebra

The Hodge dual ∗a of a multivector a in n dimensions is defined by premultiplication by the pseudoscalar

in,

∗a ≡ ina . (14.20)

In 3 dimensions, the Hodge duals of the basis vectors γγi are the bivectors

i3γγ1 = γγ2 ∧γγ3 , i3γγ2 = γγ3 ∧γγ1 , i3γγ3 = γγ1 ∧γγ2 . (14.21)

Thus in 3 dimensions the bivector a∧ b is seen to be the pseudovector Hodge dual to the familiar vector

product a× b:

a∧ b = i3 a× b . (14.22)

14.5 Reflection

Multiplying a vector (a multivector of grade 1) by a vector shifts the grade (dimension) of the vector by

±1. Thus, if one wants to transform a vector into another vector (with the same grade, one), at least two

multiplications by a vector are required.

The simplest non-trivial transformation of a vector a is

n : a→ nan (14.23)

in which the vector a is multiplied on both left and right with a unit vector n. If a is resolved into components

a‖ and a⊥ respectively parallel and perpendicular to n, then the transformation (14.23) is

n : a‖ + a⊥ → a‖ − a⊥ (14.24)

which represents a reflection of the vector a through the axis n, a reversal of all components of the vector

n

a

−nan

nan

Figure 14.2 Reflection of a vector a through axis n.

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14.6 Rotation 207

perpendicular to n, as illustrated by Figure 14.2. Note that −nan is the reflection of a through the

hypersurface normal to n, a reversal of the component of the vector parallel to n.

The operation of left- and right-multiplying by a unit vector n reflects not only vectors, but multivectors

a in general:

n : a→ nan . (14.25)

For example, the product ab of two vectors transforms as

n : ab→ n(ab)n = (nan)(nbn) (14.26)

which works because n2 = 1.

A reflection leaves any scalar λ unchanged, n : λ→ nλn = λn2 = λ. Geometrically, a reflection preserves

the lengths of, and angles between, all vectors.

14.6 Rotation

Two successive reflections yield a rotation. Consider reflecting a vector a (a multivector of grade 1) first

through the unit vector m, then through the unit vector n:

mn : a→ nmamn . (14.27)

Any component a⊥ of a simultaneously orthogonal to both m and n (i.e. m ·a⊥ = n ·a⊥ = 0) is unchanged

by the transformation (14.27), since each reflection flips the sign of a⊥:

mn : a⊥ → nma⊥mn = −na⊥n = a⊥ . (14.28)

Rotations inherit from reflections the property of preserving the lengths of, and angles between, all vectors.

Thus the transformation (14.27) must represent a rotation of those components a‖ of a lying in the 2-dim-

ensional plane spanned by m and n, as illustrated by Figure 14.3. To determine the angle by which the

plane is rotated, it suffices to consider the case where the vector a‖ is equal to m (or n, as a check). It is

not too hard to figure out that, if the angle from m to n is θ/2, then the rotation angle is θ in the same

sense, from m to n.

mamam

n

nmamnθ

Figure 14.3 Rotation of a vector a by the bivector mn. Baffled? Hey, draw your own picture.

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208 ∗The geometric algebra

For example, if m and n are parallel, so that m = ±n, then the angle between m and n is θ/2 = 0 or π,

and the transformation (14.27) rotates the vector a‖ by θ = 0 or 2π, that is, it leaves a‖ unchanged. This

makes sense: two reflections through the same plane leave everything unchanged. If on the other hand m

and n are orthogonal, then the angle between them is θ/2 = ±π/2, and the transformation (14.27) rotates

a‖ by θ = ±π, that is, it maps a‖ to −a‖.

The rotation (14.27) can be abbreviated

R : a→ RaR (14.29)

where R = mn is called a rotor, and R = nm is its reverse. Rotors are unimodular, satisfying RR =

RR = 1. According to the discussion above, the transformation (14.29) corresponds to a rotation by angle

θ in the m–n plane if the angle from m to n is θ/2. Then m · n = cos θ/2 and m∧n = (γγ1 ∧γγ2) sin θ/2,

where γγ1 and γγ2 are two orthonormal vectors spanning the m–n plane, oriented so that the angle from γγ1

to γγ2 is positive π/2 (i.e. γγ1 is the x-axis and γγ2 the y-axis). Note that the outer product γγ1 ∧γγ2 is invariant

under rotations in the m–n plane, hence independent of the choice of orthonormal basis vectors γγ1 and γγ2.

It follows that the rotor R corresponding to a right-handed rotation by θ in the γγ1–γγ2 plane is given by

R = cosθ

2+ (γγ1 ∧γγ2) sin

θ

2. (14.30)

It is straightforward to check that the rotor (14.30) rotates the basis vectors γγi as

R : γγ1 → Rγγ1R = γγ1 cos θ + γγ2 sin θ , (14.31a)

R : γγ2 → Rγγ2R = γγ2 cos θ − γγ1 sin θ , (14.31b)

R : γγi → RγγiR = γγi (i 6= 1, 2) , (14.31c)

which indeed corresponds to a right-handed rotation by angle θ in the γγ1–γγ2 plane. The inverse rotation is

R : a→ RaR (14.32)

with

R = cosθ

2− (γγ1 ∧γγ2) sin

θ

2. (14.33)

A rotation of the form (14.30), a rotation in a single plane, is called a simple rotation.

A rotation first by R and then by S transforms a vector a as

RS : a→ SRaRS = RS aRS . (14.34)

Thus the composition of two rotations, first R and then S, is given by their geometric product RS. In three

dimensions or less, all rotations are simple, but in four dimensions or higher, compositions of simple rotations

can yield rotations that are not simple. For example, a rotation in the γγ1–γγ2 plane followed by a rotation in

the γγ3–γγ4 plane is not equivalent to any simple rotation. However, it will be seen in §14.17 that bivectors

in the 4D spacetime algebra have a natural complex structure, which allows 4D spacetime rotations to take

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14.7 A rotor is a spin- 12 object 209

a simple form similar to (14.30), but with complex angle θ and two orthogonal planes of rotation combined

into a complex pair of planes.

Simple rotors are both even and unimodular, and composition preserves those properties. A rotor R is

defined in general to be any even, unimodular (RR = 1) element of the geometric algebra. The set of rotors

defines a group the rotor group, also referred to here as the rotation group. A rotor R rotates not only

vectors, but multivectors a in general:

R : a→ RaR . (14.35)

For example, the product ab of two vectors transforms as

R : ab→ R(ab)R = (RaR)(RbR) (14.36)

which works because RR = 1.

Concept question 14.2 If vectors rotate twice as fast as rotors, do bivectors rotate twice as fast as

vectors? What happens to a bivector when you rotate it by π radians? Construct a mental picture of a

rotating bivector. ⋄

To summarize, the characterization of rotations by rotors has considerable advantages. Firstly, the trans-

formation (14.35) applies to multivectors a of arbitrary grade in arbitrarily many dimensions. Secondly,

the composition law is particularly simple, the composition of two rotations being given by their geometric

product. A third advantage is that rotors rotate not only vectors and multivectors, but also spin- 12 objects

— indeed rotors are themselves spin- 12 objects — as might be suspected from the intriguing factor of 1

2 in

front of the angle θ in equation (14.30).

14.7 A rotor is a spin-12

object

A rotor was defined in the previous section, §14.6, as an even, unimodular element of the geometric algebra.

As a multivector, a rotor R would transform under a rotation by the rotor S as R → SRS. As a rotor,

however, the rotor R transforms under a rotation by the rotor S as

S : R→ RS , (14.37)

according to the transformation law (14.34). That is, composition in the rotor group is defined by the

transformation (14.37): R rotated by S is RS.

The expression (14.30) for a simple rotation in the γγ1–γγ2 plane shows that the rotor corresponding to a

rotation by 2π is −1. Thus under a rotation (14.37) by 2π, a rotor R changes sign:

2π : R→ −R . (14.38)

A rotation by 4π is necessary to bring the rotor R back to its original value:

4π : R→ R . (14.39)

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210 ∗The geometric algebra

Thus a rotor R behaves like a spin- 12 object, requiring 2 full rotations to restore it to its original state.

The two different transformation laws for a rotor — as a multivector, and as a rotor — describe two

different physical situations. The transformation of a rotor as a multivector answers the question, what is

the form of a rotor R rotated into another, primed, frame? In the unprimed frame, the rotor R transforms

a multivector a to RaR. In the primed frame rotated by rotor S from the unprimed frame, a′ = SaS, the

transformed rotor is SRS, since

a′ = SaS → SRaRS = SRSa′SRS . (14.40)

By contrast, the transformation (14.37) of a rotor as a rotor answers the question, what is the rotor corre-

sponding to a rotation R followed by a rotation S?

14.8 A multivector rotation is an active rotation

In most of the rest of this book, indices indicate how an object transforms, so that the notation

amγγm (14.41)

indicates a scalar, an object that is unchanged by a transformation, because the transformation of the con-

travariant vector am cancels against the corresponding transformation of the covariant vector γγm. However,

the transformation (14.35) of a multivector is to be understood as an active transformation that rotates the

basis vectors γγα while keeping the coefficients aα fixed, as opposed to a passive transformation that rotates

the tetrad while keeping the thing itself unchanged. Thus a multivector a ≡ aαγγα (implicit summation over

α ⊆ 1, ..., n) is not a scalar under the transformation (14.35), but rather transforms to the multivector

a′ ≡ aαγγ′α given by

R : aαγγα → aαRγγαR = aαγγ′α . (14.42)

An explicit example is the transformation (14.31) of the tetrad axes γγi under a right-handed rotation by

angle θ.

14.9 2D rotations and complex numbers

Section 14.6 identified the rotation group in n dimensions with the geometric subalgebra of even, unimodular

multivectors. In two dimensions, the even grade multivectors are linear combinations of the basis set

1 ,

1 scalar

i2 ,

1 bivector (pseudoscalar)(14.43)

forming a linear space of dimension 2. The sole bivector is the pseudoscalar i2 ≡ γγ1 ∧γγ2, equation (14.16),

the highest grade element in 2 dimensions. The rotor R that generates a right-handed rotation by angle θ

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14.9 2D rotations and complex numbers 211

is, according to equation (14.30),

R = eθ/2 = ei2 θ/2 = cosθ

2+ i2 sin

θ

2, (14.44)

where θ = i2 θ is the bivector of magnitude θ.

Since the square of the pseudoscalar i2 is minus one, the pseudoscalar resembles the pure imaginary i, the

square root of −1. Sure enough, the mapping

i2 ↔ i (14.45)

defines an isomorphism between the algebra of even grade multivectors in 2 dimensions and the field of

complex numbers

a+ i2b↔ a+ i b . (14.46)

With the isomorphism (14.46), the rotor R that generates a right-handed rotation by angle θ is equivalent

to the complex number

R = eiθ/2 . (14.47)

The associated reverse rotor R is

R = e−iθ/2 , (14.48)

the complex conjugate of R. The group of 2D rotors is isomorphic to the group of complex numbers of unit

magnitude, the unitary group U(1),

2D rotors↔ U(1) . (14.49)

Let z denote an even multivector, equivalent to some complex number by the isomorphism (14.46). Ac-

cording to the transformation formula (14.35), under the rotationR = eiθ/2, the even multivector, or complex

number, z transforms as

R : z → e−iθ/2z eiθ/2 = e−iθ/2eiθ/2z = z (14.50)

which is true because even multivectors in 2 dimensions commute, as complex numbers should. Equa-

tion (14.50) shows that the even multivector, or complex number, z is unchanged by a rotation. This might

seem strange: shouldn’t the rotation rotate the complex number z by θ in the Argand plane? The an-

swer is that the rotation R : a → RaR rotates vectors γγ1 and γγ2 (Exercise 14.3), as already seen in the

transformation (14.31). The same rotation leaves the scalar 1 and the bivector i2 ≡ γγ1 ∧γγ2 unchanged. If

temporarily you permit yourself to think in 3 dimensions, you see that the bivector γγ1 ∧γγ2 is Hodge dual to

the pseudovector γγ1 × γγ2, which is the axis of rotation and is itself unchanged by the rotation, even though

the individual vectors γγ1 and γγ2 are rotated.

Exercise 14.3 Confirm that a right-handed rotation by angle θ rotates the axes γγi by

R : γγ1 → e−iθ/2γγ1 eiθ/2 = γγ1 cos θ + γγ2 sin θ , (14.51a)

R : γγ2 → e−iθ/2γγ2 eiθ/2 = γγ2 cos θ − γγ1 sin θ , (14.51b)

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212 ∗The geometric algebra

in agreement with (14.31). The important thing to notice is that the pseudoscalar i2, hence i, anticommutes

with the vectors γγi. ⋄

14.10 Quaternions

A quaternion can be regarded as a kind of souped-up complex number

q = a+ ıb1 + b2 + kb3 , (14.52)

where a and bi (i = 1, 2, 3) are real numbers, and the three imaginary numbers ı, , k, also denoted ı1, ı2, ı3here for convenience and brevity, are defined to satisfy1

ı2 = 2 = k2 = −ık = −1 . (14.53)

Remark the dotless ı, to distinguish these quaternionic imaginaries from other possible imaginaries. A

consequence of equations (14.53) is that each pair of imaginary numbers anticommutes:

ı = −ı = −k , k = −k = −ı , kı = −ık = − . (14.54)

A quaternion (14.52) can be expressed compactly as a sum of its scalar, a, and vector (actually pseudovector,

as will become apparent below from the isomorphism (14.68)), ı · b, parts

q = a+ ı · b , (14.55)

where ı is shorthand for the triple of quaternionic imaginaries,

ı ≡ ı, , k ≡ ı1, ı2, ı3 , (14.56)

and where b ≡ b1, b2, b3, and ı · b ≡ ıibi (implicit summation over i = 1, 2, 3) is the usual Euclidean dot

product. A fundamentally useful formula, which follows from the defining equations (14.53), is

(ı · a)(ı · b) = −a · b− ı · (a× b) (14.57)

where a × b is the usual 3D vector product. The product of two quaternions p ≡ a+ ı · b and q ≡ c+ ı · dcan thus be written

pq = (a+ ı · b)(c + ı · d) = ac− b · d + ı · (ad + cb− b× d) . (14.58)

The quaternionic conjugate q of a quaternion q ≡ a + ı · b is (the overbar symbol ¯ for quaternionic

conjugation distinguishes it from the asterisk symbol ∗ for complex conjugation)

q = a− ı · b . (14.59)

1 The choice ık = 1 in the definition (14.53) is the opposite of the conventional definition ijk = −1 famously carved by W.R. Hamilton in the stone of Brougham Bridge while walking with his wife along the Royal Canal to Dublin on 16 October1843 (S. O’Donnell, 1983, William Rowan Hamilton: Portrait of a Prodigy, Boole Press, Dublin). To map to Hamilton’sdefinition, you can take ı = −i, = −j, k = −k, or alternatively ı = i, = −j, k = k, or ı = k, = j, k = i. The adoptedchoice ık = 1 has the merit that it avoids a treacherous minus sign in the isomorphism (14.68) between 3-dimensionalpseudovectors and quaternions. The present choice also conforms to the convention used by OpenGL and other computergraphics programs.

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14.11 3D rotations and quaternions 213

The quaternionic conjugate of a product is the reversed product of quaternionic conjugates

pq = qp (14.60)

just like reversion in the geometric algebra, equation (14.14) (the choice of the same symbol, an overbar,

to represent both reversion and quaternionic conjugation is not coincidental). The magnitude |q| of the

quaternion q ≡ a+ ı · b is

|q| = (qq)1/2 = (qq)1/2 = (a2 + b · b)1/2 = (a2 + b21 + b22 + b23)1/2 . (14.61)

The inverse q−1 of the quaternion, satisfying qq−1 = q−1q = 1, is

q−1 = q/(qq) = (a− ı · b)/(a2 + b · b) = (a− ı1b1 − ı2b2 − ı3b3)/(a2 + b21 + b22 + b23) . (14.62)

14.11 3D rotations and quaternions

As before, the rotation group is the group of even, unimodular multivectors of the geometric algebra. In

three dimensions, the even grade multivectors are linear combinations of the basis set

1 ,

1 scalar

i3γγ1 , i3γγ2 , i3γγ3 ,

3 bivectors (pseudovectors)(14.63)

forming a linear space of dimension 4. The three bivectors are pseudovectors, equation (14.21). The squares

of the pseudovector basis elements are all minus one,

(i3γγ1)2 = (i3γγ2)

2 = (i3γγ3)2 = −1 , (14.64)

and they anticommute with each other,

(i3γγ1)(i3γγ2) = −(i3γγ2)(i3γγ1) = −i3γγ3 ,

(i3γγ2)(i3γγ3) = −(i3γγ3)(i3γγ2) = −i3γγ1 , (14.65)

(i3γγ3)(i3γγ1) = −(i3γγ1)(i3γγ3) = −i3γγ2 .

The rotor R that generates a rotation by angle θ right-handedly about unit vector n in 3 dimensions is,

according to equation (14.30),

R = eθ/2 = ei3 n θ/2 = cosθ

2+ i3 n sin

θ

2. (14.66)

where θ is the bivector

θ ≡ i3 n θ (14.67)

of magnitude θ and unit vector direction n ≡ γγini.

Comparison of equations (14.64) and (14.65) to equations (14.53) and (14.54), shows that the mapping

i3γγi ↔ ıi (i = 1, 2, 3) (14.68)

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214 ∗The geometric algebra

defines an isomorphism between the space of even multivectors in 3 dimensions and the non-commutative

division algebra of quaternions

a+ i3γγibi ↔ a+ ıibi . (14.69)

With the equivalence (14.69), the rotor R that generates a rotation by angle θ right-handedly about unit

vector n in 3 dimensions is equivalent to the quaternion

R = eθ/2 = eı·n θ/2 = cosθ

2+ ı · n sin

θ

2, (14.70)

where θ is the pseudovector quaternion

θ ≡ ı · n θ ≡ (ı1n1 + ı2n2 + ı3n3) θ (14.71)

whose magnitude is |θ| = θ and whose unit direction is θ ≡ θ/θ = ı · n. The associated reverse rotor R is

R = e−θ/2 = e−ı·n θ/2 = cosθ

2− ı · n sin

θ

2, (14.72)

the quaternionic conjugate of R.

The group of rotors is isomorphic to the group of unit quaternions, quaternions q = a+ ı1b1 + ı2b2 + ı3b3satisfying qq = a2 + b21 + b22 + b23 = 1. Unit quaternions evidently define a unit 3-sphere in the 4-dimensional

space of coordinates a, b1, b2, b3. From this it is apparent that the rotor group in 3 dimensions has the

geometry of a 3-sphere S3.

Exercise 14.4 This exercise is a precursor to Exercise 14.15. Let b ≡ γγibi be a 3D vector, a multivector of

grade 1 in the 3D geometric algebra. Use the quaternionic composition rule (14.57) to show that the vector

b transforms under a right-handed rotation by angle θ about unit direction n = γγini as

R : b→ R bR = b + 2 sinθ

2n×

(

cosθ

2b + sin

θ

2n× b

)

. (14.73)

Here the cross-product n × b denotes the usual vector product, which is dual to the bivector product

n∧ b, equation (14.22). Suppose that the quaternionic components of the rotor R are w, x, y, z, that is,

R = eı·n θ/2 = w+ı1x+ı2y+ı3z. Show that the transformation (14.73) is (note that the 3×3 rotation matrix

is written to the right of the vector, in accordance with the computer graphics convention that rotations

accumulate to the right — opposite to the physics convention; to recover the physics convention, take the

transpose):

R :(

b1 b2 b2)

→(

b1 b2 b3)

w2+x2−y2−z2 2(xy+wz) 2(zx−wy)2(xy−wz) w2−x2+y2−z2 2(yz+wx)

2(zx+wy) 2(yz−wx) w2−x2−y2+z2

. (14.74)

Confirm that the 3× 3 rotation matrix on the right hand side of the transformation (14.74) is an orthogonal

matrix (its inverse is its transverse) provided that the rotor is unimodular, RR = 1, so that w2+x2+y2+z2 =

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14.12 Pauli matrices 215

1. As a simple example, show that the transformation (14.74) in the case of a right-handed rotation by angle

θ about the 3-axis (the 1–2 plane), where w = cos θ2 and z = sin θ

2 , is

R :(

b1 b2 b2)

→(

b1 b2 b3)

cos θ sin θ 0

− sin θ cos θ 0

0 0 1

. (14.75)

14.12 Pauli matrices

The Pauli matrices σ ≡ σi ≡ σ1, σ2, σ3 form a vector of 2 × 2 complex (with respect to a quantum-

mechanical imaginary i) matrices whose three components are each traceless (Tr σi = 0), Hermitian (σ†i = σi),

and unitary (σ†i σi = 1, no implicit summation):

σ1 ≡(

0 1

1 0

)

, σ2 ≡(

0 −ii 0

)

, σ3 ≡(

1 0

0 −1

)

. (14.76)

The Pauli matrices anticommute with each other

σ1σ2 = −σ1σ2 = iσ3 , σ2σ3 = −σ3σ2 = iσ1 , σ3σ1 = −σ1σ3 = iσ2 . (14.77)

The particular choice (14.76) of Pauli matrices is conventional but not unique: any three traceless, Hermitian,

unitary, anticommuting 2× 2 complex matrices will do. The product of the 3 Pauli matrices is i times the

unit matrix,

σ1σ2σ3 = i

(

1 0

0 1

)

. (14.78)

The multiplication rules of the Pauli matrices σi are identical to those of the basis vectors γγi of the 3D

geometric algebra. If the scalar 1 in the geometric algebra is identified with the unit 2× 2 matrix, and the

pseudoscalar i3 is identified with the imaginary i times the unit matrix, then the 3D geometric algebra is

isomorphic to the algebra generated by the Pauli matrices, the Pauli algebra, through the mapping

1↔(

1 0

0 1

)

, γγi ↔ σi , i3 ↔ i

(

1 0

0 1

)

. (14.79)

The 3D pseudoscalar i3 commutes with all elements of the 3D geometric algebra.

Concept question 14.5 The Pauli matrices are traceless, Hermitian, unitary, and anticommuting. What

do these properties correspond to in the geometric algebra? Are all these properties necessary for the Pauli

algebra to be isomorphic to the 3D geometric algebra? Are the properties sufficient?

The rotation group is the group of even, unimodular multivectors of the geometric algebra. The isomor-

phism (14.79) establishes that the rotation group is isomorphic to the group of complex 2 × 2 matrices of

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216 ∗The geometric algebra

the form

a+ iσ · b , (14.80)

with a, bi (i = 1, 3) real, and with the unimodular condition requiring that a2+b ·b = 1. It is straightforward

to check (Exercise 14.6) that the group of such matrices constitutes the group of unitary complex 2 × 2

matrices of unit determinant, the special unitary group SU(2). The isomorphisms

a+ i3γγibi ↔ a+ ıibi ↔ a+ iσibi (14.81)

have thus established isomorphisms between the group of 3D rotors, the group of unit quaternions, and the

special unitary group of complex 2× 2 matrices

3D rotors↔ unit quaternions↔ SU(2) . (14.82)

An isomorphism that maps a group into a set of matrices, such that group multiplication corresponds to

ordinary matrix multiplication, is called a representation of the group. The representation of the rotation

group as 2×2 complex matrices may be termed the Pauli representation. The Pauli representation is the

lowest dimensional representation of the 3D rotation group.

Exercise 14.6 Translate a rotor into an element of SU(2). Show that the rotor R = ei3 n θ corre-

sponding to a right-handed rotation by angle θ about unit axis n ≡ n1, n2, n3 is equivalent to the special

unitary 2× 2 matrix

R↔(

cos θ2 + in3 sin θ

2 (n2 + in1) sin θ2

(−n2 + in1) sin θ2 cos θ

2 − in3 sin θ2

)

. (14.83)

Show that the reverse rotor R is equivalent to the Hermitian conjugate R† of the corresponding 2×2 matrix.

Show that the determinant of the matrix equals RR, which is 1. ⋄

14.13 Pauli spinors

In the Pauli representation, spin- 12 objects ϕ are Pauli spinors, 2-dimensional complex (with respect to i)

vectors

ϕ =

(

ϕ↑

ϕ↓

)

(14.84)

that are rotated by pre-multiplying by elements of the special unitary group SU(2). According to the

equivalence (14.83), a rotation by 2π is represented by minus the unit matrix,( −1 0

0 −1

)

. (14.85)

Consequently a rotation by 2π changes the sign of a Pauli spinor ϕ. A rotation by 4π is required to rotate

a Pauli spinor back to its original value. Thus a Pauli spinor indeed behaves like a spin- 12 object.

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14.13 Pauli spinors 217

In quantum mechanics, the Pauli matrices σi, equations (14.76), provide a representation of the spin

operator s ≡ si ≡ s1, s2, s3 (equation (14.86) is in units ~ = 1; in standard units, s = ~

2σ)

s = 12 σ . (14.86)

The eigenvectors of the spin operator s · ζ projected along any axis ζ define objects of definite spin ± 12

measured along that axis. Each Pauli matrix σi has two eigenvalues ±1, and thus each spin operator

component si has two eigenvalues ± 12 . The eigenvectors of si are spin-up (eigenvalue + 1

2 ) and spin-down

(eigenvalue − 12 ), as measured along the i-axis. In particular, the normalized eigenvectors of σ3 are

↑ ≡(

1

0

)

, ↓ ≡(

0

1

)

, (14.87)

satisfying

s3 ↑ = 12 ↑ , s3 ↓ = − 1

2 ↓ . (14.88)

The Pauli spinor ϕ, equation (14.84), can thus be expressed

ϕ = ϕ↑ ↑ + ϕ↓ ↓ (14.89)

where ϕ↑ and ϕ↓ are the complex amplitudes along the up and down directions of the 3-axis (the z-axis).

Essential to quantum mechanics is the existence of an inner product. The inner product of two Pauli

spinors ϕ and ψ is the product ϕ†ψ of the Hermitian conjugate of ϕ with ψ. The Hermitian conjugate ϕ† of

the Pauli spinor ϕ (14.84) is

ϕ† =(

ϕ∗↑ ϕ∗

)

. (14.90)

The magnitude squared of the spinor ϕ is the real number

|ϕ|2 = ϕ†ϕ = |ϕ↑|2 + |ϕ↓|2 . (14.91)

In quantum mechanics, the magnitude squared of the spinor |ϕ|2 is interpreted as the total probability (or

probability density) of the particle. The two parts |ϕ↑|2 and |ϕ↓|2 are the probabilities of the particle being

in the up and down states. The probabilities in the up and down states depend on the direction along which

the spin is measured, but the total probability |ϕ|2 is independent on the choice of direction.

Exercise 14.7 Orthonormal eigenvectors of the spin operator. Show that the orthonormal eigen-

vectors ↑ζ and ↓ζ of the spin operator s · ζ projected along the unit direction ζ ≡ ζ1, ζ2, ζ3 are

↑ζ =1

2(1 + ζ3)

(

1 + ζ3ζ1 + iζ2

)

, ↓ζ =1

2(1− ζ3)

( −1 + ζ3ζ1 + iζ2

)

. (14.92)

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218 ∗The geometric algebra

14.14 Pauli spinors as scaled 3D rotors, or quaternions

Rotors and Pauli spinors both behave like spin- 12 objects, requiring a rotation of 4π to bring them full circle.

In fact a Pauli spinor is equivalent (14.96) to a (reverse) 3D rotor R scaled by a positive real scalar λ.

Consequently a Pauli spinor is equivalent (14.97) to a quaternion. The 2 complex degrees of freedom of the

Pauli spinor are equivalent to the 4 real degrees of freedom of the quaternion.

Start with the eigenequation (14.88) for the unit spin-up eigenvector ↑ in the 3-direction (z-direction),

s3 ↑ = 12 ↑ . (14.93)

If the spin operator s3 is rotated by rotor R, and the spin-up eigenvector ↑ is pre-multiplied by R, then the

eigenequation (14.93) transforms into an eigenequation for the unit Pauli spinor R ↑:

(Rs3R) (R ↑) = 12 (R ↑) (14.94)

(notice that the isomorphism (14.79) between the geometric algebra and the Pauli algebra guarantees that

the rule s3 → Rs3R for rotating the spin operator s3 is valid regardless of whether s3 and R are considered

as elements of the geometric algebra or as 2 × 2 complex matrices in the Pauli algebra). The Pauli spinor

↑, equation (14.87), is normalized to unit magnitude, and the rotated Pauli spinor R ↑ in equation (14.94)

is likewise of unit magnitude. A general Pauli spinor ϕ is the product of a real scalar λ and a rotated unit

spinor R ↑,ϕ = λR ↑ . (14.95)

The real scalar λ can be taken without loss of generality to be positive, since any minus sign can be absorbed

into a rotation by 2π of the rotor R. It is straightforward to check (Exercise 14.8) that any Pauli spinor ϕ

can be expressed in the form (14.95). Equation (14.95) establishes an equivalence between Pauli spinors and

reversed 3D rotors R scaled by a positive real scalar λ

ϕ↔ λR . (14.96)

Given the equivalence (14.82) between 3D rotors and unit quaternions, it follows that Pauli spinors are

equivalent to reverse quaternions (see Exercises 14.8 and 14.9 for the precise translation)

Pauli spinors↔ reverse quaternions . (14.97)

The equivalence means that there is a one-to-one correspondence between Pauli spinors and reverse quater-

nions, and that they transform in the same way under 3D rotations.

The Hermitian conjugate ϕ† of the Pauli spinor is

ϕ† = ↑†λR , (14.98)

where ↑†=(

1 0)

is the Hermitian conjugate of the spin-up eigenvector ↑. The squared amplitude of the

Pauli spinor

ϕ†ϕ = λ2 (14.99)

is the probability (or probability density) of the particle, which is unchanged by a rotation.

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14.15 Spacetime algebra 219

One is used to thinking of a Pauli spinor as an instrinsically quantum mechanical object. The equiva-

lence (14.96) between Pauli spinors and scaled reverse rotors shows that Pauli spinors also have a classical

interpretation: they encode a real amplitude λ, and a rotation R. This provides a mathematical basis for

the idea that, through their spin, fundamental particles “know” about the rotational structure of space.

The spin axis ζ of a Pauli spinor χ is the direction along which the Pauli spinor is pure up. For example,

the spin axis of the of the spin-up eigenvector ↑ is the positive 3-axis (the z-axis), while the spin axis of the

of the spin-down eigenvector ↓ is the negative 3-axis. In general, the spin axis of a Pauli spinor (14.95) is

the unit direction ζ of the rotated 3-axis,

Rs3R = s · ζ . (14.100)

The rotor S corresponding to a right-handed rotation by angle ζ about the spin axis ζ is S = ei3ζ/2 where

ζ = ζζ. Such a rotation transforms R → SR, hence transforms the Pauli spinor (14.95) as ϕ → Sϕ.

Rotating the Pauli spinor ϕ right-handedly by angle ζ about its spin axis leaves the spin axis unchanged,

but multiplies the spinor by a phase e−iζ/2,

e−i3ζ/2 ϕ = e−iζ/2 ϕ . (14.101)

Exercise 14.8 Translate a Pauli spinor into a quaternion. Given any Pauli spinor ϕ ≡(

ϕ↑

ϕ↓

)

,

show that the corresponding scaled reverse rotor λR in the Pauli representation (14.76) is the unitary 2× 2

matrix

λR =

(

ϕ↑ −ϕ∗↓

ϕ↓ ϕ∗↑

)

. (14.102)

Show that the corresponding real quaternion is

q = λR = Reϕ↑, Imϕ↓,−Reϕ↓, Imϕ↑ . (14.103)

Exercise 14.9 Translate a quaternion into a Pauli spinor. Show that if the rotor R corresponds

to a right-handed rotation by angle θ about unit axis n ≡ n1, n2, n3, then the corresponding scaled Pauli

spinor ϕ ≡ λR ↑ is, from (14.83),

ϕ ≡ λR ↑ = λ

(

cos θ2 − in3 sin θ

2

(n2 − in1) sin θ2

)

. (14.104)

14.15 Spacetime algebra

So far this chapter has concerned itself with ordinary n-dimensional Euclidean space, in which the length

squared of a vector is the sum of the squares of its components. In special relativity, however, the scalar

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220 ∗The geometric algebra

length s of a spacetime interval t, x, y, x is given by s2 = −t2 + x2 + y2 + z2. Happily, all the results of

previous sections hold with scarcely a change of stride.

Let γγm (m = 0, 1, 2, 3) denote an orthonormal basis of spacetime, with γγ0 representing the time axis, and

γγi (i = 1, 2, 3) the spatial axes. Geometric multiplication in the spacetime algebra is defined by

γγmγγn = γγm · γγn + γγm ∧γγn (14.105)

in the usual way. The key difference between the spacetime basis γγm and Euclidean bases is that scalar

products of the basis vectors γγm form the Minkowski metric ηmn,

γγm · γγn = ηmn (14.106)

whereas scalar products of Euclidean basis elements γγi formed the unit matrix, γγi ·γγj = δij , equation (14.6).

In less abbreviated form, equations (14.105) state that the geometric product of each basis element with

itself is

−γγ20 = γγ2

1 = γγ22 = γγ2

3 = 1 , (14.107)

while geometric products of different basis elements γγm anticommute

γγmγγn = −γγnγγm = γγm ∧γγn (m 6= n) . (14.108)

In the Dirac theory of relativistic spin- 12 particles, the Dirac γ-matrices are required to satisfy

γγm,γγn = 2 ηmn (14.109)

where denotes the anticommutator, γγm,γγn ≡ γγmγγn + γγnγγm. The multiplication rules (14.109) for

the Dirac γ-matrices are the same as those for geometric multiplication in the spacetime algebra, equa-

tions (14.107) and (14.108).

A 4-vector a, a multivector of grade 1 in the geometric algebra of spacetime, is

a = γγmam = γγ0a

0 + γγ1a1 + γγ2a

2 + γγ3a3 . (14.110)

Such a 4-vector a would be denoted 6a in the Dirac slash notation. The product of two 4-vectors a and b is

ab = a · b + a∧ b = ambnγγm · γγn + ambnγγm ∧γγn = ambnηmn + 12a

mbn[γγm,γγn] . (14.111)

It is convenient to denote three of the six bivectors of the spacetime algebra by σi,

σi ≡ γγ0γγi (i = 1, 2, 3) . (14.112)

The symbol σi is used because the algebra of bivectors σi is isomorphic to the algebra of Pauli matrices σi.

The triple of bivectors σi will often be denoted shorthandedly by the symbol σ

σ ≡ σ1,σ2,σ3 . (14.113)

The pseudoscalar, the highest grade basis element of the spacetime algebra, is denoted I

γγ0γγ1γγ2γγ3 = σ1σ2σ3 = I . (14.114)

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14.16 Complex quaternions 221

The pseudoscalar I satisfies

I2 = −1 , Iγγm = −γγmI , Iσi = σiI . (14.115)

The basis elements of the 4-dimensional spacetime algebra are then

1 ,

1 scalar

γγm ,

4 vectors

σi , Iσi ,

6 bivectors

Iγγm ,

4 pseudovectors

I ,

1 pseudoscalar(14.116)

forming a linear space of dimension 1 + 4 + 6 + 4 + 1 = 16 = 24. The reverse is defined in the usual

way, equation (14.12), leaving unchanged multivectors of grade 0 or 1, modulo 4, and changing the sign of

multivectors of grade 2 or 3, modulo 4:

1 = 1 , γγm = γγm , σi = −σi , Iσi = −Iσi , Iγγm = −Iγγm , I = I . (14.117)

The mapping

γγ(3)i ↔ σi (i = 1, 2, 3) (14.118)

(the superscript (3) distinguishes the 3D basis vectors from the 4D spacetime basis vectors) defines an iso-

morphism between the 8-dimensional geometric algebra (14.3) of 3 spatial dimensions and the 8-dimensional

even spacetime subalgebra. Among other things, the isomorphism (14.118) implies the equivalence of the

3D spatial pseudoscalar i3 and the 4D spacetime pseudoscalar I

i3 ↔ I (14.119)

since i3 = γγ1γγ2γγ3 and I = σ1σ2σ3.

14.16 Complex quaternions

A complex quaternion (also called a biquaternion by W. R. Hamilton) is a quaternion

q = a+ ıibi = a+ ı · b (14.120)

in which the four coefficients a, bi (i = 1, 2, 3) are each complex numbers

a = aR + IaI , bi = bi,R + Ibi,I . (14.121)

The imaginary I is taken to commute with each of the quaternionic imaginaries ıi. The choice of symbol

I is deliberate: in the isomorphism (14.133) between the even spacetime algebra and complex quaternions,

the commuting imaginary I is isomorphic to the spacetime pseudoscalar I.

All of the equations in §14.10 on real quaternions remain valid without change, including the multiplication,

conjugation, and inversion formulae (14.57)–(14.62). In the quaternionic conjugate q of a complex quaternion

q ≡ a+ ı · b,

q = a− ı · b , (14.122)

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222 ∗The geometric algebra

the complex coefficients a and b are not conjugated with respect to the complex imaginary I. The magnitude

|q| of a complex quaternion q ≡ a+ ı · b,

|q| = (qq)1/2 = (qq)1/2 = (a2 + b · b)1/2 = (a2 + b21 + b22 + b23)1/2 , (14.123)

is a complex number, not a real number. The complex conjugate q∗ of the complex quaternion is

q∗ = a∗ + ı · b∗ , (14.124)

in which the complex coefficients a and b are conjugated with respect to the imaginary I, but the quaternionic

imaginaries ı are not conjugated.

A non-zero complex quaternion can have zero magnitude (unlike a real quaternion), in which case it is

null. The null condition qq = a2 + b21 + b22 + b23 = 0 is a complex condition. The product of two null complex

quaternions is a null quaternion. Under multiplication, null quaternions form a 6-dimensional subsemigroup

(not a subgroup, because null quaternions do not have inverses) of the 8-dimensional semigroup of complex

quaternions.

Exercise 14.10 Show that any non-trivial null complex quaternion q can be written uniquely in the form

q = (1 + Iı · n)p , (14.125)

where p is a real quaternion, and n is a unit real 3-vector. Equivalently,

q = p(1 + Iı · n′) , (14.126)

where n′ is the unit real 3-vector

n′ =pnp

|p|2 . (14.127)

Solution. Write the null quaternion q as

q = p+ Ir (14.128)

where p and r are real quaternions, both of which must be non-zero if q is non-trivial. Then equation (14.125)

is true with

ı · n =rp

|p|2. (14.129)

The null condition is qq = 0. The vanishing of the real part, Re (qq) = pp− rr = 0, shows that |p|2 = |r|2.The vanishing of the imaginary (I) part, Im (qq) = rp+ pr = rp+ rp = 0 shows that the rp must be a pure

quaternionic imaginary, since the quaternionic conjugate of rp is minus itself, so rp/ |p|2 must be of the form

ı ·n. Its squared magnitude ı · n ı · n = rp pr/ |p|4 = |r|2/ |p|2 = 1 is unity, so n must be a unit 3-vector. It

follows immediately from the manner of construction that the expression (14.125) is unique, as long as q is

non-trivial. ⋄

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14.17 Lorentz transformations and complex quaternions 223

14.17 Lorentz transformations and complex quaternions

Lorentz transformations are rotations of spacetime. Such rotations correspond, in the usual way, to even,

unimodular elements of the geometric algebra of spacetime. The basis elements of the even spacetime algebra

are

1 ,

1 scalar

σi , Iσi ,

6 bivectors

I ,

1 pseudoscalar(14.130)

forming a linear space of dimension 1 + 6 + 1 = 8 over the real numbers. However, it is more elegant to

treat the even spacetime algebra as a linear space of dimension 8 ÷ 2 = 4 over complex scalars of the form

λ = λR + IλI . The pseudoscalar I qualifies as a scalar because it commutes with all elements of the even

spacetime algebra, and it qualifies as an imaginary because I2 = −1. It is convenient to take the basis

elements of the even spacetime algebra over the complex numbers to be

1 ,

1 scalar

Iσi ,

3 bivectors(14.131)

forming a linear space of dimension 1+3 = 4. The reason for choosing Iσi rather than σi as the elements of

the basis (14.131) is that the basis is 1, Iσi is equivalent to the basis (14.63) of the even algebra of 3-dim-

ensional Euclidean space through the isomorphism (14.118) and (14.119). This basis in turn is equivalent to

the quaternionic basis 1, ıi through the isomorphism (14.68):

Iσi ↔ i3γγ(3)i ↔ ıi (i = 1, 2, 3) . (14.132)

In other words, the even spacetime algebra is isomorphic to the algebra of quaternions with complex coeffi-

cients:

a+ Iσ · b↔ a+ ı · b (14.133)

where a = aR + IaI is a complex number, b = bR + IbI , is a triple of complex numbers, σ is the triple of

bivectors σi, and ı is the triple of quaternionic imaginaries.

The isomorphism (14.133) between even elements of the spacetime algebra and complex quaternions implies

that the group of Lorentz rotors, which are unimodular elements of the even spacetime algebra, is isomorphic

to the group of unimodular complex quaternions

spacetime rotors↔ unit complex quaternions . (14.134)

In §14.11 it was found that the group of 3D spatial rotors is isomorphic to the group of unimodular real

quaternions. Thus Lorentz transformations are mathematically equivalent to complexified spatial rotations.

A Lorentz rotor can be written as a complex quaternion in what looks like the same form as the expres-

sion (14.70) for a 3D spatial rotor, with the difference that the rotation angle θ is complex, and the axis n

of rotation is likewise complex. Thus

R = eθ/2 = eı·n θ/2 = cosθ

2+ ı · n sin

θ

2(14.135)

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224 ∗The geometric algebra

where θ is the bivector complex quaternion

θ ≡ ı · n θ ≡ (ı1n1 + ı2n2 + ı3n3) θ (14.136)

whose complex magnitude is |θ| ≡ (θθ)1/2 = θ and whose complex unit direction is θ ≡ θ/θ ≡ ı·n. The angle

θ = θR + IθI is a complex angle, and n = nR + InI is a complex-valued unit 3-vector, satisfying n · n = 1.

The condition n ·n = 1 of unit normalization is equivalent to the two conditions nR ·nR −nI ·nI = 1 and

2nR ·nI = 0 on the real and imaginary parts of n ·n. The complex angle θ has 2 degrees of freedom, while

the complex unit vector has 4 degrees of freedom, so the Lorentz rotor R has 6 degrees of freedom, which

is the correct number of degrees of freedom of the group of Lorentz transformations. The associated reverse

rotor R is

R = e−θ/2 = e−ı·n θ/2 = cosθ

2− ı · n sin

θ

2(14.137)

the quaternionic conjugate of R. Note that θ and n in equation (14.137) are not conjugated with respect to

the imaginary I. The sine and cosine of the complex angle θ appearing in equations (14.135) and (14.137)

are related to its real and imaginary parts in the usual way,

cosθ

2= cos

θR

2cosh

θI

2− I sin

θR

2sinh

θI

2, sin

θ

2= sin

θR

2cosh

θI

2+ I cos

θR

2sinh

θI

2. (14.138)

In the case of a pure spatial rotation, the angle θ = θR and axis n = nR in the rotor (14.135) are both

real. The rotor corresponding to a pure spatial rotation by angle θR right-handedly about unit real axis nR

is

R = eı·nR θR/2 = cosθR

2+ ı · nR sin

θR

2. (14.139)

A Lorentz boost is a change of velocity in some direction, without any spatial rotation, and represents

a rotation of spacetime about some time-space plane. For example, a Lorentz boost along the 1-axis (the

x-axis) is a rotation of spacetime in the 0–1 plane (the t–x plane). In the case of a pure Lorentz boost, the

angle θ = IθI is pure imaginary, but the axis n = nR remains pure real. The rotor corresponding to a boost

by velocity v = tanh θI in unit real direction nR is

R = eı·nR IθI/2 = coshθI

2+ ı · nR I sinh

θI

2. (14.140)

Exercise 14.11 Factor a general Lorentz rotor R = eı·n θ/2 into the product UL of a pure spatial rotation

U followed by a pure Lorentz boost L. Do the two factors commute? ⋄Exercise 14.12 Show that the geometry of the group of Lorentz rotors is the product of the geometries

of the spatial rotation group and the boost group, which is a 3-sphere times Euclidean 3-space, S3 × R3. ⋄

14.18 Spatial Inversion (P ) and Time Inversion (T )

Spatial inversion, or P for parity, is the operation of reflecting all spatial coordinates while keeping the time

coordinate unchanged. Spatial inversion may be accomplished by reflecting the spatial vector basis elements

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14.19 Electromagnetic field bivector 225

γγi → −γγi, while keeping the time vector basis element γγ0 unchanged. This results in σ → −σ and I → −I.The equivalence Iσ ↔ ı means that the quaternionic imaginary ı is unchanged. Thus, if multivectors in

the geometric spacetime algebra are written as linear combinations of products of γγ0, ı, and I, then spatial

inversion P corresponds to the transformation

P : γγ0 → γγ0 , ı→ ı , I → −I . (14.141)

In other words spatial inversion may be accomplished by the rule, take the complex conjugate (with respect

to I) of a multivector.

Time inversion, or T , is the operation of reversing time while keeping all spatial coordinates unchanged.

Time inversion may be accomplished by reflecting the time vector basis element γγ0 → −γγ0, while keeping the

spatial vector basis elements γγi unchanged. As with spatial inversion, this results in σ → −σ and I → −I,which keeps Iσ hence ı unchanged. If multivectors in the geometric spacetime algebra are written as linear

combinations of products of γγ0, ı, and I, then time inversion T corresponds to the transformation

T : γγ0 → −γγ0 , ı→ ı , I → −I . (14.142)

For any multivector, time inversion corresponds to the instruction to flip γγ0 and take the complex conjugate

(with respect to I).

The combined operation PT of inverting both space and time corresponds to

PT : γγ0 → −γγ0 , ı→ ı , I → I . (14.143)

For any multivector, spacetime inversion corresponds to the instruction to flip γγ0, while keeping ı and I

unchanged.

14.19 Electromagnetic field bivector

The electromagnetic field tensor Fmn can be expressed as the bivector

F = 12F

mnγγm ∧γγn , (14.144)

the factor of 12 compensating for the double-counting over indices m and n (the 1

2 could be omitted if the

counting were over distinct bivector indices only). In terms of the electric and magnetic fields E and B, and

the bivector basis elements σ ≡ σ1,σ2,σ3 defined by equation (14.112), the electromagnetic field bivector

F is

F = −σ · (E + IB) . (14.145)

14.20 How to implement Lorentz transformations on a computer

The advantages of quaternions for implementing spatial rotations are well-known to 3D game programmers.

Compared to standard rotation matrices, quaternions offer increased speed and require less storage, and

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226 ∗The geometric algebra

their algebraic properties simplify interpolation and splining. Complex quaternions retain similar advantages

for implementing Lorentz transformations. They are fast, compact, and straightforward to interpolate or

spline (Exercises 14.13 and 14.14). Moreover, since complex quaternions contain real quaternions, Lorentz

transformations can be implemented simply as an extension of spatial rotations in 3D programs that use

quaternions to implement spatial rotations.

Lorentz rotors, 4-vectors, spacetime bivectors, and spinors (spin- 12 objects) can all be implemented as

complex quaternions. A complex quaternion

q = w + ı1x+ ı2y + ı3z (14.146)

with complex coefficients w, x, y, z can be stored as the 8-component object

q =

wR xR yR zR

wI xI yI zI

. (14.147)

Actually, OpenGL and other computer software store the scalar (w) component of a quaternion in the last

(fourth) place, but here the scalar components are put in the zeroth position to conform to standard physics

convention. The quaternion conjugate q of the quaternion (14.147) is

q =

wR −xR −yR −zR

wI −xI −yI −zI

, (14.148)

while its complex conjugate q∗ is

q∗ =

wR xR yR zR

−wI −xI −yI −zI

. (14.149)

A Lorentz rotor R corresponds to a complex quaternion of unit modulus. The unimodular condition RR =

1, a complex condition, removes 2 degrees of freedom from the 8 degrees of freedom of complex quaternions,

leaving the Lorentz group with 6 degrees of freedom, which is as it should be. Spatial rotations correspond

to real unimodular quaternions, and account for 3 of the 6 degrees of freedom of Lorentz transformations.

A spatial rotation by angle θ right-handedly about the 1-axis (the x-axis) is the real Lorentz rotor

R = cos(θ/2) + ı1 sin(θ/2) , (14.150)

or, stored as a complex quaternion,

R =

cos(θ/2) sin(θ/2) 0 0

0 0 0 0

. (14.151)

Lorentz boosts account for the remaining 3 of the 6 degrees of freedom of Lorentz transformations. A Lorentz

boost by velocity v, or equivalently by boost angle θ = atanh(v), along the 1-axis (the x-axis) is the complex

Lorentz rotor

R = cosh(θ/2) + Iı1 sinh(θ/2) , (14.152)

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14.20 How to implement Lorentz transformations on a computer 227

or, stored as a complex quaternion,

R =

cosh(θ/2) 0 0 0

0 sinh(θ/2) 0 0

. (14.153)

The rule for composing Lorentz transformations is simple: a Lorentz transformation R followed by a Lorentz

transformation S is just the product RS of the corresponding complex quaternions.

The inverse of a Lorentz rotor R is its quaternionic conjugate R.

Any even multivector q is equivalent to a complex quaternion by the isomorphism (14.133). According to

the usual transformation law (14.35) for multivectors, the rule for Lorentz transforming an even multivector

q is

R : q → RqR (even multivector) . (14.154)

The transformation (14.154) instructs to multiply three complex quaternions R, q, and R, a one-line expres-

sion in a c++ program.

As an example of an even multivector, the electromagnetic field F , equation (14.145), is a bivector in the

spacetime algebra. In view of the isomorphism (14.133), the electromagnetic field bivector F can be written

as the complex quaternion

F =

0 −B1 −B2 −B3

0 E1 E2 E3

. (14.155)

Under the parity transformation P (14.141), the electric field E changes sign, whereas the magnetic field B

does not, which is as it should be:

P : E → −E , B → B . (14.156)

According to the rule (14.154), the electromagnetic field bivector F Lorentz transforms as F → RFR, which

is a powerful and elegant way to Lorentz transform the electromagnetic field.

A 4-vector a ≡ γγmam is a multivector of grade 1 in the spacetime algebra. A general odd multivector in

the spacetime algebra is the sum of a vector (grade 1) part a and a pseudovector (grade 3) part Ib = Iγγmbm.

The odd multivector can be written as the product of the time basis vector γγ0 and an even multivector q

a + Ib = γγ0q = γγ0

(

a0 + Iıiai − Ib0 + ıib

i)

. (14.157)

By the isomorphism (14.133), the even multivector q is equivalent to the complex quaternion

q =

a0 b1 b2 b3

−b0 a1 a2 a3

. (14.158)

According to the usual transformation law (14.35) for multivectors, the rule for Lorentz transforming the

odd multivector γγ0q is

R : γγ0q → Rγγ0qR = γγ0R∗qR . (14.159)

In the last expression of (14.159), the factor γγ0 has been brought to the left, to be consistent with the

convention (14.157) that an odd multivector is γγ0 on the left times an even multivector on the right. Notice

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228 ∗The geometric algebra

that commuting γγ0 through R converts the latter to its complex conjugate (with respect to I) R∗, which

is true because γγ0 commutes with the quaternionic imaginary ı, but anticommutes with the pseudoscalar

I. Thus if the components of an odd multivector are stored as a complex quaternion (14.158), then that

complex quaternion q Lorentz transforms as

R : q → R∗qR (odd multivector) . (14.160)

The rule (14.160) again instructs to multiply three complex quaternions R∗, q, and R, a one-line expression in

a c++ program. The transformation rule (14.160) for an odd multivector encoded as a complex quaternion

differs from that (14.154) for an even multivector in that the first factor R is complex conjugated (with

respect to I).

A vector a differs from a pseudovector Ib in that the vector a changes sign under a parity transformation

P whereas the pseudovector Ib does not. However, the behaviour of a pseudovector under a normal Lorentz

transformation (which preserves parity) is identical to that of a vector. Thus in practical situations two

4-vectors a and b can be encoded into a single complex quaternion (14.158), and Lorentz transformed

simultaneously, enabling two transformations to be done for the price of one.

Finally, a Dirac spinor is equivalent to a complex quaternion q (§14.23). It Lorentz transforms as

R : q → Rq (spinor) . (14.161)

Exercise 14.13 Interpolate a Lorentz transformation. Argue that the interpolating Lorentz rotor

R(x) that corresponds to uniform rotation and acceleration between initial and final Lorentz rotors R0 and

R1 as the parameter x varies uniformly from 0 to 1 is

R(x) = R0 exp [x ln(R1/R0)] . (14.162)

What are the exponential and logarithm of a complex quaternion in terms of its components? Address the

issue of the multi-valued character of the logarithm. ⋄

Exercise 14.14 Spline a Lorentz transformation. A spline is a polynomial that interpolates between

two points with given values and derivatives at the two points. Confirm that the cubic spline of a real

function f(x) with given initial and final values f0 and f1 and given initial and final derivatives f ′0 and f ′

1

at x = 0 and x = 1 is

f(x) = f0 + f ′0x+ [3(f1 − f0)− 2f ′

0 − f ′1]x

2 + [2(f0 − f1) + f ′0 + f ′

1]x3 . (14.163)

The case in which the derivatives at the endpoints are set to zero, f ′0 = f ′

1 = 0, is called the “natural” spline.

Argue that a Lorentz rotor can be splined by splining the quaternionic components of the logarithm of the

Lorentz rotor. ⋄

Exercise 14.15 The wrong way to implement a Lorentz transformation. The purpose of this

exercise is to persuade you that Lorentz transforming a 4-vector by the rule (14.160) is a much better idea

than Lorentz transforming by multiplying by an explicit 4× 4 matrix. Suppose that the Lorentz rotor R is

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14.21 Dirac matrices 229

the complex quaternion

R =

wR xR yR zR

wI xI yI zI

. (14.164)

Show that the Lorentz transformation (14.160) transforms the 4-vector components am = a0, a1, a2, a3 as

(note that the 4× 4 rotation matrix is written to the right of the 4-vector in accordance with the computer

graphics convention that rotations accumulate to the right — opposite to the physics convention; to recover

the physics convention, take the transpose):

R :(

a0 a1 a2 a3)

→(

a0 a1 a2 a3)

|w|2 + |x|2 + |y|2 + |z|2 2 (wRxI − wIxR + yRzI − yIzR)

2 (wRxI − wIxR − yRzI + yIzR) |w|2 + |x|2 − |y|2 − |z|22 (wRyI − wIyR − zRxI + zIxR) 2 (xRyR + xIyI − wRzR − wIzI)

2 (wRzI − wIzR − xRyI + xIyR) 2 (zRxR + zIxI + wRyR + wIyI)

2 (wRyI − wIyR + zRxI − zIxR) 2 (wRzI − wIzR + xRyI − xIyR)

2 (xRyR + xIyI + wRzR + wIzI) 2 (zRxR + zIxI − wRyR − wIyI)

|w|2 − |x|2 + |y|2 − |z|2 2 (yRzR + yIzI + wRxR + wIxI)

2 (yRzR + yIzI − wRxR − wIxI) |w|2 − |x|2 − |y|2 + |z|2

, (14.165)

where | | signifies the absolute value of a complex number, as in |w|2 = w2R +w2

I . As a simple example, show

that the transformation (14.165) in the case of a Lorentz boost by velocity v along the 1-axis, where the

rotor R takes the form (14.153), is

R :(

a0 a1 a2 a3)

→(

a0 a1 a2 a3)

γ γv 0 0

γv γ 0 0

0 0 1 0

0 0 0 1

, (14.166)

with γ the familiar Lorentz gamma factor

γ = cosh θ =1

(1 − v2)1/2, γv = sinh θ =

v

(1− v2)1/2. (14.167)

14.21 Dirac matrices

The multiplication rules (14.105) for the basis vectors γγm of the spacetime algebra are identical to the

rules (14.109) governing the Clifford algebra of the Dirac γ-matrices used in the Dirac theory of relativistic

spin- 12 particles.

The Dirac γ-matrices are conventionally represented by 4 × 4 complex matrices. To ensure consistency

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230 ∗The geometric algebra

between the relativistic and quantum mechanical ways of taking the scalar (inner) product, it is desirable to

require that taking the Hermitian conjugate of any of the basis vectors γγm be equivalent to raising its index,

γγ†m = γγm . (14.168)

Given the requirement (14.168), the matrices representing the basis vectors γγm must be traceless (because

a trace is a scalar, and the basis vectors cannot contain any scalar part), Hermitian or anti-Hermitian as

the self-product of the matrix is ±1 (so that γγ†m = γγm = ηmnγγn), and unitary and anticommuting (so that

γγ†m · γγn = γγm · γγn = δm

n ). The precise choice of matrices is not fundamental: any set of 4 matrices satisfying

these conditions will do.

The high-energy physics community conventionally adopts the +−−− metric signature, which is opposite

to the convention adopted here. With the high-energy +−−− signature, the standard convention for the

Dirac γ-matrices is

γγ0 =

(

1 0

0 −1

)

, γγi =

(

0 σi

−σi 0

)

, (14.169)

where 1 denotes the unit 2 × 2 matrix, and σi denote the three 2× 2 Pauli matrices (14.76). The choice of

γγ0 as a diagonal matrix is motivated by Dirac’s discovery that eigenvectors of the time basis vector γγ0 with

eigenvalues of opposite sign define particles and antiparticles in their rest frames (see §14.22). To convert to

the −+++ metric signature adopted here while retaining the conventional set of eigenvectors, an additional

factor of i must be inserted into the γ-matrices:

γγ0 = i

(

1 0

0 −1

)

, γγi = i

(

0 σi

−σi 0

)

, (14.170)

In the representation of equations (14.169) or (14.170), the bivectors σi and Iσi and the pseudoscalar I of

the spacetime algebra are

σi =

(

0 −σi

−σi 0

)

, Iσi = i

(

σi 0

0 σi

)

, I = −i(

0 1

1 0

)

, (14.171)

whose representation as matrices is the same for either signature−+++ or +−−−. The Hermitian conjugates

of the bivector and pseudoscalar basis elements are

σ†i = σi , (Iσi)

† = −Iσi , I† = −I . (14.172)

The conventional chiral matrix γ5 of Dirac theory is defined by

γ5 ≡ iγγ0γγ1γγ2γγ3 = iI =

(

0 1

1 0

)

, (14.173)

whose representation is again the same for either signature −+++ or +−−−. The chiral matrix is Hermitian

γ†5 = γ5 . (14.174)

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14.22 Dirac spinors 231

14.22 Dirac spinors

In the Dirac theory of relativistic spin- 12 particles, a Dirac spinor ϕ is represented as a 2-component column

vector of Pauli spinors ϕ⇑ and ϕ⇓, comprising 4 complex (with respect to i) components and hence 8 degrees

of freedom,

ϕ =

(

ϕ⇑

ϕ⇓

)

=

ϕ⇑↑

ϕ⇑↓

ϕ⇓↑

ϕ⇓↓

. (14.175)

The Dirac γ-matrices operate by pre-multiplication on Dirac spinors ϕ, yielding other Dirac spinors.

In the Dirac representation (14.170), the four unit Dirac spinors

⇑↑ =

1

0

0

0

, ⇑↓ =

0

1

0

0

, ⇓↑ =

0

0

1

0

, ⇓↓ =

0

0

0

1

, (14.176)

are eigenvectors of the time basis vector γγ0 and of the bivector Iσ3, with ⇑ and ⇓ denoting eigenvectors of

γγ0, and ↑ and ↓ eigenvectors of Iσ3,

γγ0 ⇑ = i⇑ , γγ0 ⇓ = −i⇓ , Iσ3 ↑ = i ↑ , Iσ3 ↓ = −i ↓ . (14.177)

The bivector Iσ3 is the generator of a spatial rotation about the 3-axis (z-axis), equation (14.132). The four

eigenvectors (14.177) form an orthonormal basis

A pure spin-up state ↑ can be rotated into a pure spin-down state ↓, or vice versa, by a spatial rotation

about the 1-axis or 2-axis. By contrast, a pure time-up state ⇑ cannot be rotated into a pure time-down

state ⇓, or vice versa, by any Lorentz transformation. Consider for example trying to rotate the pure time-up

spin-up ⇑↑ state into any combination of pure time-down ⇓ states. According to the expression (14.192), the

Dirac spinor ϕ obtained by Lorentz transforming the ⇑↑ state is pure ⇓ only if the corresponding complex

quaternion q is pure imaginary. But a pure imaginary quaternion has negative squared magnitude qq, so

cannot be equivalent to any rotor of unit magnitude.

Thus the pure time-up and pure time-down states ⇑ and ⇓ are distinct states that cannot be transformed

into each other by any Lorentz transformation. The two states represent distinct species, particles and

antiparticles.

Although a pure time-up state cannot be transformed into a pure time-down state or vice versa by any

Lorentz transformation, the time-up and time-down eigenstates ⇑ and ⇓ do mix under Lorentz transforma-

tions. The manner in which Dirac spinors transform is described in §14.23.

The choice of time-axis γγ0 and spin-axis γγ3 with respect to which the eigenvectors are defined can of course

be adjusted arbitrarily by a Lorentz boost and a spatial rotation. The eigenvectors of a particular time-axis

γγ0 correspond to particles and antiparticles that are at rest in that frame. The eigenvectors associated with

a particular spin-axis γγ3 correspond to particles or antiparticles that are pure spin-up or pure spin-down in

that frame.

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232 ∗The geometric algebra

14.23 Dirac spinors as complex quaternions

In §14.14 it was found that a spin- 12 object in 3D space, a Pauli spinor, is isomorphic to a scaled 3D reverse

rotor, or real quaternion. In the relativistic theory, the corresponding spin- 12 object, a Dirac spinor ϕ, is

isomorphic (14.181) to a complex quaternion. The 4 complex degrees of freedom of the Dirac spinor ϕ are

equivalent to the 8 degrees of freedom of a complex quaternion. A physically interesting complication arises

in the relativistic case because a non-trivial Dirac spinor can be null, with zero magnitude, whereas any non-

trivial Pauli spinor is necessarily non-null. The case of non-null (massive) and null (massless) Dirac spinors

are considered respectively in §14.24 and §14.25. The present section establishes an isomorphism (14.181)

between Dirac spinors and complex quaternions that is valid in general, regardless of whether the Dirac

spinor is null or not.

If a is a spacetime multivector, equivalent to an element of the Clifford algebra of Dirac γ-matrices, then

under rotation by Lorentz rotor R, the multivector a operating on the Dirac spinor ϕ transforms as

R : aϕ→ (RaR)(Rϕ) = Raϕ . (14.178)

This shows that a Dirac spinor ϕ Lorentz transforms, by construction, as

R : ϕ→ Rϕ . (14.179)

The rule (14.179) is precisely the transformation rule for reverse spacetime rotors under Lorentz trans-

formations: under a rotation by rotor R, a reverse rotor S transforms as S → RS. More generally, the

transformation law (14.179) holds for any linear combination of Dirac spinors ϕ. The isomorphism (14.134)

between spacetime rotors and unit quaternions shows that unit Dirac spinors are isomorphic to unit (reverse)

complex quaternions. The algebra of linear combinations of unit complex quaternions is just the algebra of

complex quaternions. Thus the algebra of Dirac spinors is isomorphic to the algebra of (reverse) complex

quaternions. Specifically, any Dirac spinor ϕ can be expressed uniquely in the form of a 4× 4 matrix q, the

Dirac representation of a reverse complex quaternion q, acting on the time-up spin-up eigenvector ⇑↑ (the

precise translation between Dirac spinors and complex quaternions is left as Exercises 14.16 and 14.17):

ϕ = q ⇑↑ . (14.180)

In this section (including the Exercises) the 4× 4 matrix q is written in boldface to distinguish it from the

quaternion q that it represents; but the distinction is not fundamental, so the temporary boldface notation

is dropped in subsequent sections. The equivalence (14.180) establishes that Dirac spinors are isomorphic to

reverse complex quaternions

ϕ↔ q . (14.181)

The isomorphism means that there is a one-to-one correspondence between Dirac spinors ϕ and reverse

complex quaternions q, and that they transform in the same way under Lorentz transformations.

Notwithstanding the isomorphism (14.181), Dirac spinors differ from complex quaternions in that they

have an additional structure that is essential to quantum mechanics, an inner product ϕ†1ϕ2 of two Dirac

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14.23 Dirac spinors as complex quaternions 233

spinors ϕ1 and ϕ2. The inner product ϕ†1ϕ2 is a complex (with respect to i) number. The Hermitian

conjugate ϕ† of a Dirac spinor ϕ (14.180) is defined to be

ϕ† = (⇑↑)†q† , (14.182)

where (⇑↑)† =(

1 0 0 0)

is the Hermitian conjugate of the time-up spin-up eigenvector ⇑↑, and q† is

the Hermitian conjugate of the matrix q. A related spinor is the the reverse, or adjoint, spinor ϕ, defined to

be

ϕ = (⇑↑)†q , (14.183)

where q is the reverse of the matrix q. The Hermitian conjugate spinor ϕ† is related to the adjoint spinor

ϕ by (Exercise 14.18)

ϕ† = iϕγγ0 . (14.184)

The product ϕϕ of the adjoint spinor ϕ with ϕ is a Lorentz-invariant scalar, as follows from the Lorentz

invariance of qq. On the other hand, the product ϕ†ϕ of the Hermitian conjugate spinor ϕ† with ϕ is the

time component of a 4-vector

iϕγγmϕ . (14.185)

Exercise 14.16 Translate a Dirac spinor into a complex quaternion. Given any Dirac spinor

ϕ =

ϕ⇑↑

ϕ⇑↓

ϕ⇓↑

ϕ⇓↓

, (14.186)

show that the corresponding reverse complex quaternion q, and the equivalent 4× 4 matrix q in the Dirac

representation (14.170), such that ϕ = q⇑↑, are (the complex conjugates ϕ∗a of the components ϕa of the

spinor are with respect to the quantum mechanical imaginary i)

q =

Reϕ⇑↑ Imϕ⇑↓ −Reϕ⇑↓ Imϕ⇑↑

−Imϕ⇓↑ Reϕ⇓↓ Imϕ⇓↓ Reϕ⇓↑

↔ q =

ϕ⇑↑ −ϕ∗⇑↓ ϕ⇓↑ ϕ∗

⇓↓

ϕ⇑↓ ϕ∗⇑↑ ϕ⇓↓ −ϕ∗

⇓↑

ϕ⇓↑ ϕ∗⇓↓ ϕ⇑↑ −ϕ∗

⇑↓

ϕ⇓↓ −ϕ∗⇓↑ ϕ⇑↓ ϕ∗

⇑↑

. (14.187)

Show that the complex quaternion q (the reverse of q), and the equivalent 4× 4 matrix q (the reverse of q)

in the Dirac representation (14.170), are

q =

Reϕ⇑↑ −Imϕ⇑↓ Reϕ⇑↓ −Imϕ⇑↑

−Imϕ⇓↑ −Reϕ⇓↓ −Imϕ⇓↓ −Reϕ⇓↑

↔ q =

ϕ∗⇑↑ ϕ∗

⇑↓ −ϕ∗⇓↑ −ϕ∗

⇓↓

−ϕ⇑↓ ϕ⇑↑ −ϕ⇓↓ ϕ⇓↑

−ϕ∗⇓↑ −ϕ∗

⇓↓ ϕ∗⇑↑ ϕ∗

⇑↓

−ϕ⇓↓ ϕ⇓↑ −ϕ⇑↓ ϕ⇑↑

. (14.188)

Conclude that the reverse spinor ϕ ≡ (⇑↑)†q is

ϕ ≡ (⇑↑)†q =(

ϕ∗⇑↑ ϕ∗

⇑↓ −ϕ∗⇓↑ −ϕ∗

⇓↓

)

. (14.189)

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234 ∗The geometric algebra

Exercise 14.17 Translate a complex quaternion into a Dirac spinor. Show that the complex

quaternion q ≡ w + ıx+ y + kz is equivalent in the Dirac representation (14.170) to the 4× 4 matrix q

q =

wR xR yR zR

wI xI yI zI

↔ q =

wR + izR ixR + yR −iwI + zI xI − iyI

ixR − yR wR − izR xI + iyI −iwI − zI

−iwI + zI xI − iyI wR + izR ixR + yR

xI + iyI −iwI − zI ixR − yR wR − izR

. (14.190)

Show that the reverse quaternion q, the complex conjugate (with respect to I) quaternion q∗, and the reverse

complex conjugate (with respect to I) quaternion q∗ are respectively equivalent to the 4× 4 matrices

q ↔ q ≡ −γγ0q†γγ0 , (14.191a)

q∗ ↔ q† = −γγ0qγγ0 , (14.191b)

q∗ ↔ q† = −γγ0qγγ0 . (14.191c)

Conclude that the Dirac spinor ϕ ≡ q ⇑↑ corresponding to the reverse complex quaternion q is

ϕ ≡ q ⇑↑ =

wR − izR

−ixR + yR

−iwI − zI

−xI − iyI

, (14.192)

that the reverse spinor ϕ ≡ (⇑↑)†q is

ϕ ≡ (⇑↑)†q =(

wR + izR ixR + yR −iwI + zI xI − iyI

)

, (14.193)

and that the Hermitian conjugate spinor ϕ† ≡ (⇑↑)†q† is

ϕ† ≡ (⇑↑)†q† =(

wR + izR ixR + yR iwI − zI −xI + iyI

)

. (14.194)

Hence conclude that ϕϕ and ϕ†ϕ are respectively the real part of, and the absolute value of, the complex

magnitude squared qq ≡ λ2 of the complex quaternion q,

ϕϕ = λ2R − λ2

I , (14.195a)

ϕ†ϕ = λ2R + λ2

I , (14.195b)

with

λ2R = w2

R + x2R + y2

R + z2R , (14.196a)

λ2I = w2

I + x2I + y2

I + z2I , (14.196b)

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14.24 Non-null Dirac spinor — particle and antiparticle 235

Exercise 14.18 Relation between ϕ† and ϕ. Confirm equation (14.184) by showing from equa-

tion (14.191b) that

ϕ† = i(⇑↑)†qγγ0 . (14.197)

14.24 Non-null Dirac spinor — particle and antiparticle

A non-null, or massive, Dirac spinor ϕ, one for which ϕϕ 6= 0, is isomorphic (14.181) to a non-null reverse

complex quaternion q, which can be factored as a non-zero complex scalar times a unit quaternion, a rotor.

Thus a non-null Dirac spinor can be expressed as the product of a complex scalar λ = λR +IλI and a reverse

Lorentz rotor R, acting on the time-up spin-up eigenvector ⇑↑,

ϕ = λR⇑↑ . (14.198)

The complex scalar λ can be taken without loss of generality to lie in the right hemisphere of the complex

plane (positive real part), since a minus sign can be absorbed into a spatial rotation by 2π of the rotor

R. There is no further ambiguity in the decomposition (14.198) into scalar and rotor, because the squared

magnitude λRλR = λ2 of the scaled rotor λR is the same for any decomposition.

The fact that a non-null Dirac spinor ϕ encodes a Lorentz rotor shows that a non-null Dirac spinor in

some sense “knows” about the Lorentz structure of spacetime. It is intriguing that the Lorentz structure of

spacetime is built in to a non-null Dirac particle.

As discussed in §14.22, a pure time-up eigenvector ⇑ represents a particle in its own rest frame, while a pure

time-down eigenvector ⇓ represents an antiparticle in its own rest frame. The time-up spin-up eigenvector ⇑↑is by definition (14.198) equivalent to the unit scaled rotor, λR = 1, so in this case the scalar λ is pure real.

Lorentz transforming the eigenvector multiplies it by a rotor, but leaves the scalar λ unchanged, therefore

pure real. Conversely, if the time-up spin-up eigenvector ⇑↑ is multiplied by the imaginary I, then according

to the expression (14.192) the resulting spinor can be Lorentz transformed into a pure ⇓ state, corresponding

to a pure antiparticle. Thus one may conclude that the real and imaginary parts (with respect to I)

of the complex scalar λ = λR + IλI correspond respectively to particles and antiparticles. The

magnitude squared ϕ†ϕ of the Dirac spinor, equation (14.195b),

ϕ†ϕ = |λ|2 = λ2R + λ2

I , (14.199)

is the sum of the probabilities λ2R of particles and λ2

I of antiparticles.

Among other things, the decomposition of a Dirac spinor into its particle and antiparticle parts shows

that multiplying a non-null Dirac spinor by the pseudoscalar I converts a particle to an antiparticle, and

vice versa.

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236 ∗The geometric algebra

14.25 Null Dirac Spinor

A null spinor is a spinor ϕ whose magnitude is zero,

ϕϕ = 0 . (14.200)

Such a spinor is equal to a null complex quaternion q acting on the time-up spin-up eigenvector ⇑↑,

ϕ = q ⇑↑ . (14.201)

Physically, a null spinor represents a spin- 12 particle moving at the speed of light. A non-trivial null spinor

must be moving at the speed of light because if it were not, then there would be a rest frame where the rotor

part of the spinor ϕ = λR⇑↑ would be unity, R = 1, and the spinor, being non-trivial, λ 6= 0, would not be

null. The null condition (14.200) is a complex constraint, which eliminates 2 of the 8 degrees of freedom of

a complex quaternion, so that a null spinor has 6 degrees of freedom.

Any non-trivial null complex quaternion q can be written uniquely as the product of a null factor

(1 + Iı ·n)/√

2 and a real quaternion λU (Exercise 14.10):

q =(1 + Iı · n)√

2λU . (14.202)

Here n is a unit real 3-vector, λ is a positive real scalar, and U is a purely spatial (i.e. real, with no I part)

rotor. The factor of 1/√

2 is inserted for normalization purposes. Physically, equation (14.202) contains the

instruction to boost to light speed in the direction n, then scale by the real scalar λ and rotate spatially by

U . The 2 + 1 + 3 = 6 degrees of freedom from the real unit vector n, the real scalar λ, and the spatial rotor

U in the expression (14.202) are precisely the number needed to specify a null quaternion. One might have

thought that the boost factor 1 + Iı · n in equation (14.202) would change under a Lorentz transformation,

but in fact it is Lorentz-invariant. For if the boost factor 1+Iı ·n is transformed (multiplied) by any complex

quaternion p+ Ir, then the result

(1 + Iı · n)(p+ Ir) = (1 + Iı · n)(p− ı · n r) (14.203)

is the same unchanged boost factor 1+Iı·n multiplied by a purely spatial transformation, the real quaternion

p−ı·n r. Equation (14.203) is true because (ı·n)2 = −1. Since the boost factor 1+Iı·n is Lorentz-invariant,

Lorentz transforming the null quaternion q (14.202) probes only 4 of the 6 degrees of freedom of the group

of null quaternions.

The null Dirac spinor ϕ corresponding to the reverse q of the null complex quaternion q, equation (14.202),

is

ϕ ≡ q ⇑↑ = λU(1 − Iı · n)√

2⇑↑ . (14.204)

It is natural to choose basis vectors of the representation to be eigenvectors of the Lorentz-invariant boost

factor 1− Iı ·n. In the Dirac representation (14.170), the basis spinors (14.176) are eigenvectors of Iσ3, and

it natural to choose the 3-direction to be in either the positive or negative n direction, in which case

(1− Iı · n)⇑↑ = (1∓ Iı3)⇑↑ = (1± σ3)⇑↑ = (⇑ ∓ ⇓) ↑ . (14.205)

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14.26 Chiral decomposition of a Dirac spinor 237

The null basis vectors are left- and right-handed chiral eigenvectors, eigenvectors of the chiral operator γ5

with eigenvalues ∓ respectively,

γ5(⇑∓⇓) ↑√

2= ∓ (⇑∓⇓) ↑√

2. (14.206)

The 1/√

2 factor ensures unit normalization

[

(⇑∓⇓) ↑√2

]†(⇑∓⇓) ↑√

2= 1 . (14.207)

A general null Dirac spinor ϕ, equation (14.204), is the real scaled spatial reverse rotor λU acting on one of

the two null chiral basis spinors, either left-handed (⇑−⇓) ↑/√

2, or right-handed (⇑+⇓) ↑/√

2,

ϕL = λU(⇑−⇓) ↑√

2, ϕR = λU

(⇑+⇓) ↑√2

. (14.208)

A left- or right-handed null Dirac spinor is called a Weyl spinor.

Concept question 14.19 The null boost factor (1 + Iı · n) in a null quaternion, equation (14.202), is

Lorentz-invariant, as shown by equation (14.203) (which you should confirm). Consequently a null Dirac

spinor has a Lorentz-invariant boost axis n. Does a null 4-vector have a Lorentz-invariant axis? What does

it mean physically that a null Dirac spinor has a Lorentz-invariant boost axis n?

14.26 Chiral decomposition of a Dirac spinor

A general (non-null or null) Dirac spinor ϕ can be decomposed into a sum of left- and right-handed chiral

components

ϕ = ϕL + ϕR , (14.209)

that are eigenvectors of the chiral operator γ5,

γ5ϕLR

= ∓ϕLR. (14.210)

The left- and right-handed chiral components can be projected out by applying the chiral projection operators12 (1∓ γ5) (which are projection operators because their squares are themselves):

1

2(1∓ γ5)ϕ = ϕL

R. (14.211)

The decomposition into chiral components is Lorentz-invariant because the pseudoscalar I, hence the chiral

operator γ5 ≡ iI, is Lorentz-invariant, which is true because the pseudoscalar I commutes with any Lorentz

rotor. The chiral projection operators are null, because the reverse of the chiral operator is γ5 = iI = −iI =

−γ5, and its square is one, γ25 = 1, so

(

1 + γ5

)

(1 + γ5) = (1− γ5)(

1− γ5

)

= (1− γ5) (1 + γ5) = 0 . (14.212)

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238 ∗The geometric algebra

Consequently each of the chiral components is null, hence massless,

ϕLϕL = ϕRϕR = 0 . (14.213)

Since a pure left- or right-handed spinor must be null, a non-null Dirac particle cannot be purely left- or

right-handed.

14.27 Dirac equation

The Dirac equation is the relativistic quantum mechanical wave equation for spin- 12 particles. By itself, the

Dirac equation does not provide a consistent theory of relativistic quantum mechanics, because in relativistic

quantum mechanics there is no such thing as a single particle that evolves in isolation. Rather, a “funda-

mental” particle such as an electron is dressed in a sea of particle-antiparticle pairs polarized out of the

vacuum by the presence of the electron. Nevertheless, the Dirac equation is a fundamental building block

for the quantum field theory of spin- 12 particles.

The Dirac theory starts with the momentum 4-vector in the form p = γγmpm, where γγm are not only the

basis vectors of an orthonormal tetrad, but also the basis vectors of the spacetime (Clifford) algebra. In the

Dirac slash notation p =6p, but the notation is superfluous here. For a particle of rest mass m, the geometric

square of the momentum is

pp = γγmγγnpmpn = pnpn = −m2 . (14.214)

The vanishing sum pp +m2 factors as

pp +m2 = (p + im)(p− im) = 0 . (14.215)

The factorization provides the motivation for the Dirac wave equation for a free relativistic spin- 12 particle

or antiparticle,

(p− im)ϕ = 0 (particle) , (14.216a)

(p + im)ϕ = 0 (antiparticle) , (14.216b)

in which ϕ is a Dirac spinor, and the momentum operator p ≡ γγmpm should be replaced, according to the

usual rules of quantum mechanics, by minus i times the gradient operator ∂ = γγm∂m,

p = −i∂ . (14.217)

With respect to locally inertial coordinates xm ≡ t, xi,

p0 = −p0 = i∂

∂t, pi = pi = −i ∂

∂xi. (14.218)

With the replacement (14.217), the Dirac equations (14.216) are

(∂ +m)ϕ = 0 (particle) , (14.219a)

(∂ −m)ϕ = 0 (antiparticle) . (14.219b)

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14.28 Antiparticles are negative mass particles moving backwards in time 239

The reason for choosing opposite signs for the mass m in the particle and antiparticle equations is discussed

further in §14.28. For now, two comments can be made about the different choice of sign. Firstly, as seen in

§14.24, particles and antiparticles belong to distinct representations that do not mix under Lorentz transfor-

mations, so it is consistent to allow them to satisfy different equations. Secondly, the factorization (14.215)

involves factors with both signs of m, so it is reasonable — one could say demanded — that both equations

would occur.

In flat (Minkowski) space, the Dirac wave equations (14.219) for a free particle or antiparticle are most

easily solved by Fourier transforming with respect to space and time. The differential wave equations (14.219)

then revert to being algebraic equations (14.216), with p being the momentum of the corresponding Fourier

mode. If the Dirac spinor ϕ is a particle as opposed to an antiparticle, so that ϕ is (up to an irrelevant real

scale factor) R⇑ where ⇑ is any rest-frame particle eigenvector, then the following calculation

pϕ = (Rmγγ0R)(R⇑) = mRγγ0⇑ = imR⇑ = imϕ (particle) , (14.220)

confirms that the particle Dirac equation (14.216a) recovers the expected momentum p = Rmγγ0R, which

is the rest frame momentum p ≡ pmγγm = mγγ0 Lorentz transformed into the tetrad frame. Likewise, if the

Dirac spinor ϕ is an antiparticle as opposed to a particle, so that ϕ is (up to an irrelevant real scale factor)

R⇓ where ⇓ is any rest-frame antiparticle eigenvector, then

pϕ = (Rmγγ0R)(R⇓) = mRγγ0⇓ = −imR⇓ = −imϕ (antiparticle) (14.221)

confirms that the antiparticle Dirac equation (14.216b) again recovers the expected momentum p = Rmγγ0R.

The sign flip between the particle and antiparticle equations (14.220) and (14.221) occurs because the

time basis vector γγ0 yields opposite signs when acting on a rest-frame particle eigenvector ⇑ versus rest-

frame antiparticle eigenvector ⇓, equation (14.177). The same sign flip occurs if the rest-frame antiparticle

eigenvector is taken to be I⇑ (per §14.24) instead of ⇓, since γγ0 anticommutes with I.

Fourier-transformed back into real space, the free Dirac spinor wavefunctions ϕ in flat space are, for either

particles or antiparticles, FREQUENCY HAS WRONG SIGN

ϕ = ϕ0 eipmxm

, (14.222)

where pm is the momentum, satisfying p = γγmpm, and ϕ0 is the value of the Dirac spinor at the origin

xm = 0. Regarded as an element of the spacetime algebra, the exponential factor eipmxm

is a scalar, so it

commutes with ϕ0, so it does not matter on which side of ϕ0 the exponential is placed.

14.28 Antiparticles are negative mass particles moving backwards in time

The original Dirac treatment took the particle Dirac equation (14.219a) as describing both particles and

antiparticles. This led to solutions in which the free-wave factor contained not only a positive frequency

component, as in equation (14.222), but also a negative frequency component e−ipmxm

. These negative

frequency components were interpreted as indicating an antiparticle, with negative mass m.

In the original Dirac theory, the prediction of particles with negative rest mass was problematic: where are

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240 ∗The geometric algebra

such particles? And if they existed, why wouldn’t pairs of positive and negative mass particles spontaneously

pop out of the vacuum, causing a catastrophic breakdown of the vacuum? To solve the problem, Dirac

proposed that all negative energy states of the vacuum are already occupied, and that antiparticles correspond

to holes in the negative energy sea.

Dirac’s conundrum was eventually solved by Feynman, who realised that anti-particles are equivalent

to negative mass particles moving backwards in time. A negative mass particle moving backwards in

time looks like a positive mass particle moving forwards in time. Feynman’s solution obviates the need for

any negative energy sea of antiparticles. Feynman’s dictum corresponds mathematically to choosing particle

and antiparticle spinors not only to yield opposite signs when acted on by the time basis vector γγ0 in their

rest frames, as in Dirac theory, but also to have the opposite sign of mass m in the Dirac equations (14.219a).

This is the approach adopted in the previous section, §14.27.

14.29 Dirac equation with electromagnetism

The Dirac equation for a spin- 12 particle of charge e moving in an external electromagnetic field with potential

A ≡ Amγγm is given by the same Dirac equations (14.216), but now the momentum p is rewritten in terms

of the canonical momentum π,

p = π + eA , (14.223)

and it is the canonical momentum π that is replaced by minus i times the gradient operator ∂ ≡ γγm∂m,

π = −i∂ . (14.224)

Consequently the Dirac equation for charged particle or antiparticle is

(∂ + ieA +m)ϕ = 0 (particle) , (14.225a)

(∂ + ieA−m)ϕ = 0 (antiparticle) . (14.225b)

Equation (14.225b) appears to describe an antiparticle as having mass −m opposite to that of a particle,

and charge e the same as that of a particle. If the antiparticle is interpreted as having negative mass

moving backwards in time, then the antiparticle has positive mass m moving forwards in time, and charge

−e opposite to that of a particle.

Equations (14.225) are not easy to solve in general. Analytic solutions exist in some cases, such as when

the electromagnetic field consists of a uniform magnetic field B.

14.30 CPT

It was seen in §14.24 that multiplying a non-null Dirac spinor ϕ by the pseudoscalar I converts a particle

spinor into an antiparticle spinor, and vice versa. This operation is conventionally called CPT ,

CPT : ϕ→ Iϕ . (14.226)

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14.31 Charge conjugation C 241

The operation is called CPT because it is conventionally parsed into 3 distinct discrete transformations C,

P , and T , discussed in turn below. In the spacetime algebra, the CPT operation flips the sign of all the

spacetime axes γγm,

CPT : γγm → IγγmI−1 = −γγm , (14.227)

which is true because the pseudoscalar I anti-commutes with each of the basis vectors γγm. The CPT

operation leaves all even multivectors unchanged since I commutes with all even multivectors. In particular,

CPT is Lorentz invariant, since I commutes with Lorentz rotors.

The CPT operation converts the particle Dirac equation (14.225a) into the antiparticle Dirac equa-

tion (14.225b):

CPT : −I(∂ + ieA +m)ϕ = (∂ + ieA−m)Iϕ . (14.228)

Equation (14.228) shows that the spinor Iϕ satisfies the Dirac equation (14.225b) for an antiparticle, con-

sistent with conclusion of §14.24 that multiplying a non-null Dirac spinor by I converts a particle into an

antiparticle.

14.31 Charge conjugation C

A Dirac spinor, non-null or null, contains two distinct components that remain separate under Lorentz

transformations. For a non-null spinor the two components are particles and antiparticles. For a null spinor

the two components are the left- and right-handed chiralities. For a non-null spinor, the CPT operation

of multiplying the spinor by I converts a particle into an antiparticle and versa. But for a null spinor,

multiplying by I = −iγ5 leaves left- and right-handed particles as they are: it does not transform opposite

chiralities into each other.

A charge conjugation operation C can be defined with the property that it converts particles of one type

into the opposite type for both non-null and null spinors: particles into antiparticles and vice versa, and

left-handed into right-handed chiralities and vice versa.

In quantum mechanics, it is natural to regard particles and antiparticles as belonging to complex (with

respect to i) conjugate representations. The charge conjugation operator C is defined by the requirement

that it transforms the spacetime basis vectors γγm to their complex conjugates (with respect to i)

C : γγm → CγγmC−1 = γγ∗

m . (14.229)

In the Dirac representation (14.170), the condition (14.229) requires that C commute with γγ2, but anticom-

mute with γγ0, γγ1, and γγ3. A suitable matrix is γγ2 itself,

C = γγ2 . (14.230)

Charge conjugation of a Dirac spinor ϕ is accomplished by taking the complex conjugate (with respect to i)

of the spinor, and multiplying by the charge conjugation operator C:

C : ϕ→ Cϕ∗ . (14.231)

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242 ∗The geometric algebra

The charge-conjugates Cϕ∗ of the non-null basis eigenvectors (14.176) in the Dirac representation are

C(⇑↑)∗ = ⇓↓ , C(⇑↓)∗ = −⇓↑ , C(⇓↑)∗ = −⇑↓ , C(⇓↓)∗ = ⇑↑ , (14.232)

while the charge-conjugates of the null chiral basis eigenvectors (14.206) are

C

[

(⇑ ∓ ⇓) ↑√2

]∗= ± (⇑ ± ⇓) ↓√

2. (14.233)

Equations (14.232) and (14.233) show that charge conjugation not only converts rest-frame particle eigen-

vectors ⇑ into rest-frame antiparticle eigenvectors ⇓, and vice versa, but also converts a left-handed null

eigenvector into a right-handed null eigenvector and vice versa. Since the operation of complex conjuga-

tion commutes with Lorentz transformation (because Lorentz rotors R are independent of the quantum

mechanical imaginary i), it is true in general that charge conjugation switches particles into antiparticles,

and left-handed into right-handed chiralities.

The complex conjugate (with respect to i) of the charged particle Dirac equation (14.225a) is

(∂∗ − ieA∗ +m)ϕ∗ = 0 . (14.234)

Complex conjugation leaves the components ∂m and Am of the gradient operator and electromagnetic po-

tential unchanged, but conjugates the basis vectors γγm. Left-multiplying equation (14.234) by C, and

commuting C through the wave operator, yields

(∂ − ieA +m)Cϕ∗ = 0 . (14.235)

Thus the charge-conjugated spinor Cϕ∗ satisfies the Dirac equation (14.225a) for a particle with the same

mass m but opposite charge −e.

14.32 Parity reversal P

The parity operation P is the operation of reversing all the spatial axes, while keeping the time axis un-

changed,

P : γγm → PγγmP−1 =

γγm m = 0 ,

−γγm m = 1, 2, 3 .(14.236)

A suitable matrix is the time axis γγ0

P = γγ0 . (14.237)

Parity reversal transforms a Dirac spinor ϕ as

P : ϕ→ Pϕ . (14.238)

Parity reversal commutes with spatial rotations, but not with Lorentz boosts. Parity reversal transforms the

left- and right-handed chiral components of a Dirac spinor into each other:

P : ϕL ↔ ϕR , (14.239)

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14.33 Time reversal T 243

which is true because parity reversal flips the sign of the pseudoscalar, PIP−1 = −I, hence also the sign of

the chiral operator γ5 ≡ iI.

14.33 Time reversal T

The Wigner time reversal operation T is conventionally defined so that CPT is the product of the three

operations C, P , and T . This requires that time-reversal transforms a Dirac spinor ϕ as

T : ϕ→ Tϕ∗ (14.240)

with

T = γγ1γγ3 , (14.241)

so that

CPT = γγ2γγ0γγ1γγ3 = I . (14.242)

14.34 Majorana spinor

The charge conjugation operation considered in §14.31 switched left- and right-handed null spinors into each

other. However, it is possible for a null spinor to be its own antiparticle. In this case the complex (with

respect to i) conjugate representation is itself, rather than being a distinct representation. A null spinor

which is its own antiparticle is a Majorana spinor.

For a Majorana spinor, complex conjugation should leave the representation unchanged; that is, the γ-

matrices should be real, in contrast to the Dirac representation (14.170), where the γ-matrices are complex.

A suitable real representation of the γ-matrices is

γγ0 =

(

0 iσ2

iσ2 0

)

, γγ1 =

(

σ1 0

0 σ1

)

, γγ2 =

(

0 −iσ2

iσ2 0

)

, γγ3 =

(

σ3 0

0 σ3

)

. (14.243)

The chiral matrix γ5 ≡ iI is

γ5 =

(

σ2 0

0 σ2

)

. (14.244)

14.35 Covariant derivatives revisited

Under a Lorentz transformation by rotor R, any multivector a transforms as a → RaR. The covariant

derivative D ≡ γγmDm must transform likewise as

R : D → RDR . (14.245)

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244 ∗The geometric algebra

14.36 General relativistic Dirac equation

To convert the Dirac equations (14.219) or (14.225) into general relativistic equations, the derivative ∂ ≡γγm∂m must be converted to a covariant derivative.

14.37 3D Vectors as rank-2 spinors

THIS NEEDS TO BE CONVERTED FROM 3D TO SPACETIME.

Concept question 14.20 One is used to thinking of a spin- 12 particle as in some sense the square root of

a spin-1 particle, a vector. How is this concept compatible with the idea that a spin- 12 object is a (scaled)

rotor?

A Pauli spinor ϕ contains two complex components ϕa, where the index a runs over the two indices ↑ and

↓. The Pauli spinor is a spinor of rank 1, having one spinor index. Under a rotation by rotor R, the spinor

ϕ transforms as ϕ→ Rϕ. Under a rotation, the spinor components ϕa of the Pauli spinor transform as

ϕa → Rbaϕb (14.246)

where Rba is the special unitary 2 × 2 matrix representing the reverse rotor R. The rotor R itself has

components Rab. Note the placement of indices: for the rotor R, the first index is down and the second up,

while for the reverse rotor R, the first index is up and the second down. The rotor and its reverse are inverse

to each other, satisfying RR = RcaRc

b = δba.

The Hermitian conjugate spinor ϕ† transforms in the opposite way ϕ† → ϕ†R, and can therefore be written

as a spinor with contravariant (raised) index, ϕ† = ϕa. The contravariant spinor components ϕa transform

as

ϕa → ϕbRba . (14.247)

The product of a Hermitial conjugate spinor χ† with another spinor ϕ defines their scalar product, which is

unchanged by a rotation,

χ†ϕ→ χ†RRϕ = χ†ϕ = χaϕa . (14.248)

Explicitly, the scalar product χaϕa is the complex number

χaϕa = χ∗↑ϕ↑ + χ∗

↓ϕ↓ . (14.249)

An element a of the 3D geometric algebra transforms under rotation as a→ RaR. The behaviour under

rotation shows that a 3D multivector is a rank-2 spinor a = aab, a complex 2×2 matrix, with the first index

being covariant (lowered), and the second being contravariant (raised). The spinor components aab of the

multivector transform as

aab → Rc

a acdRd

b . (14.250)

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14.37 3D Vectors as rank-2 spinors 245

A 3D multivector a contains 8 real components, comprising a scalar, a vector, a pseudovector, and a pseu-

doscalar. All 8 complex components are represented by the complex 2 × 2 spinor aab. The scalar and

pseudoscalar components are contained in the complex trace aaa of the spinor. The rotational transforma-

tion a → RaR preserves the grade of the multivector, so scalars transform to scalars, vectors to vectors,

pseudovectors to pseudovectors, and pseudoscalars to pseudoscalars.

A reversed multivector a transforms in the same way a→ RaR as the parent multivector a. Consquently,

like the multivector a, the reversed multivector a transforms like a rank-2 spinor with one covariant (lowered)

and one contravariant (raised) index. It is consistent to write the reversed spinor as the original spinor with

the first index raised and the second lowered, a = aab.

The above has shown that a 3D multivector a is represented naturally as a rank-2 spinor aab with one

covariant and one contravariant index. This might seem strange: one might have expected that a vector — a

spin-1 particle — might be represented as a rank-2 spinor aab with two covariant indices — a sum of tensor

products ϕ⊗ χ = ϕaχb of two spin- 12 particles. Under a rotation, the components aab of a covariant rank-2

spinor transform as

aab → RcaR

db acd . (14.251)

One aspect of this transformation is straightforward: the following combination of tensor products of spinors

is invariant under rotation, and is therefore a scalar:

↑ ⊗ ↓ − ↓ ⊗ ↑ . (14.252)

The remaining tensor combinations ↑ ⊗ ↑, ↓ ⊗ ↓, and ↑ ⊗ ↓ + ↓ ⊗ ↑ provide a basis for vectors, but under

rotation they transform into complex, not real, linear combinations of each other. This contrasts with the

representation of multivectors a by spinors aab with one covariant and one contravariant index, where the

grade-preserving property of the rotational transformation a → RaR ensures that vectors rotate into real,

not complex, linear combinations of each other.

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PART SIX

BLACK HOLE INTERIORS

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Concept Questions

1. Explain how the equation for the Gullstrand-Painleve metric (15.16) encodes not merely a metric but a

full vierbein.

2. In what sense does the Gullstrand-Painleve metric (15.16) depict a flow of space? [Are the coordinates

moving? If not, then what is moving?]

3. If space has no substance, what does it mean that space falls into a black hole?

4. Would there be any gravitational field in a spacetime where space fell at constant velocity instead of

accelerating?

5. In spherically symmetric spacetimes, what is the most important Einstein equation, the one that causes

Reissner-Nordstrom black holes to be repulsive in their interiors, and causes mass inflation in non-empty

(non Reissner-Nordstrom) charged black holes?

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What’s important?

1. The tetrad formalism provides a firm mathematical foundation for the concept that space falls faster than

light inside a black hole.

2. Whereas the Kerr-Newman geometry of an ideal rotating black hole contains inside its horizon wormhole

and white hole connections to other universes, real black holes are subject to the mass inflation stability

discovered by Eric Poisson & Werner Israel (1990, “Internal structure of black holes,” Phys. Rev. D 41,

1796-1809).

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15

Black hole waterfalls

15.1 Tetrads move through coordinates

As already discussed in §11.3, the way in which metrics are commonly written, as a (weighted) sum of squares

of differentials,

ds2 = γmn em

µ en

ν dxµdxν , (15.1)

encodes not only a metric gµν = γmn em

µ en

ν , but also an inverse vierbein emµ, and consequently a vierbein

emµ, and associated tetrad γγm. Most commonly the tetrad metric is orthonormal (Minkowski), γmn = ηmn,

but other tetrad metrics, such as Newman-Penrose, occur. Usually it is self-evident from the form of the

line-element what the tetrad metric γmn is in any particular case.

If the tetrad is orthonormal, γmn = ηmn, then the 4-velocity um of an object at rest in the tetrad, or

equivalently the 4-velocity of the tetrad rest frame itself, is

um = 1, 0, 0, 0 . (15.2)

The tetrad-frame 4-velocity (15.2) of the tetrad rest frame is transformed to a coordinate-frame 4-velocity

uµ in the usual way, by applying the vierbein,

dxµ

dτ≡ uµ = em

µum = e0µ . (15.3)

Equation (15.3) says that the tetrad rest frame moves through the coordinates at coordinate 4-velocity given

by the zero’th row of the vierbein, dxµ/dτ = e0µ. The coordinate 4-velocity uµ is related to the lapse α and

shift βα in the ADM formalism by uµ = 1, βα/α, equation (13.10).

The idea that locally inertial frames move through the coordinates provides the simplest way to conceptu-

alize black holes. The motion of locally inertial frames through coordinates is what is meant by the “dragging

of inertial frames” around rotating masses.

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252 Black hole waterfalls

Figure 15.1 The fish upstream can make way against the current, but the fish downstream is swept to thebottom of the waterfall. Art by Wildrose Hamilton.

15.2 Gullstrand-Painleve waterfall

The Gullstrand-Painleve metric is a version of the metric for a spherical (Schwarzschild or Reissner-Nordstrom)

black hole discovered in 1921 independently by Allvar Gullstrand (1922, “Allgemeine Losung des statischen

Einkorperproblems in der Einsteinschen Gravitationstheorie,” translated by http://babelfish.altavista.

com/tr as “General solution of the static body problem in Einstein’s Gravitation Theory,” Arkiv. Mat. As-

tron. Fys. 16(8), 1–15) and Paul Painleve (1921, “La mecanique classique et la theorie de la relativite”, C.

R. Acad. Sci. (Paris) 173, 677–680). Although Gullstrand’s paper was published in 1922, after Painleve’s, it

appears that Gullstrand’s work has priority. Gullstrand’s paper was dated 25 May 1921, whereas Painleve’s

is a write up of a presentation to the Academie des Sciences in Paris on 24 October 1921. Moreover, Gull-

strand seems to have had a better grasp of what he had discovered than Painleve, for Gullstrand recognized

that observables such as the redshift of light from the Sun are unaffected by the choice of coordinates in

the Schwarzschild geometry, whereas Painleve, noting that the spatial metric was flat at constant free-fall

time, dtff = 0, concluded in his final sentence that, as regards the redshift of light and such, “c’est pure

imagination de pretendre tirer du ds2 des consequences de cette nature”.

Although neither Gullstrand nor Painleve understood it, their metric paints a picture of space falling like

a river, or waterfall, into a spherical black hole, Figure 15.1. The river has two key features: first, the river

flows in Galilean fashion through a flat Galilean background, equation (15.19); and second, as a freely-falling

fishy swims through the river, its 4-velocity, or more generally any 4-vector attached to it, evolves by a

series of infinitesimal Lorentz boosts induced by the change in the velocity of the river from place to place,

equation (15.24). Because the river moves in Galilean fashion, it can, and inside the horizon does, move

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15.2 Gullstrand-Painleve waterfall 253

faster than light through the background coordinates. However, objects moving in the river move according

to the rules of special relativity, and so cannot move faster than light through the river.

15.2.1 Gullstrand-Painleve tetrad

The Gullstrand-Painleve metric (7.26) is

ds2 = − dt2ff + (dr − β dtff)2 + r2(dθ2 + sin2θ dφ2) , (15.4)

where β is defined to be the radial velocity of a person who free-falls radially from rest at infinity,

β =dr

dτ=

dr

dtff, (15.5)

and tff is the free-fall time, the proper time experienced by a person who free-falls from rest at infinity. The

radial velocity β is the (apparently) Newtonian escape velocity

β = ∓√

2M(r)

r, (15.6)

where M(r) is the interior mass within radius r, and the sign is − (infalling) for a black hole, + (outfalling)

for a white hole. For the Schwarzschild or Reissner-Nordstrom geometry the interior mass M(r) is the mass

M at infinity minus the mass Q2/2r in the electric field outside r,

M(r) = M − Q2

2r. (15.7)

Figure 15.2 illustrates the velocity fields in Schwarzschild and Reissner-Nordstrom black holes. Horizons

occur where the radial velocity β equals the speed of light

β = ∓1 , (15.8)

with − for black hole solutions, + for white hole solutions. The phenomenology of Schwarzschild and

Reissner-Nordstrom black holes has already been explored in Chapters 7 and 8.

Exercise 15.1 Schwarzschild to Gullstrand-Painleve. Show that the Schwarzschild metric transforms

into the Gullstrand-Painleve metric under the coordinate transformation of the time coordinate

dtff = dt− β

1− β2dr . (15.9)

Exercise 15.2 Radial free-fall from rest. Confirm that β given by equation (15.6) is indeed the

velocity (15.5) of a person who free-falls radially from rest at infinity in the Reissner-Nordstrom geometry.

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254 Black hole waterfalls

Horizon

Inner horizon

Outer horizon

Turnaround

Figure 15.2 Radial velocity β in (upper panel) a Schwarzschild black hole, and (lower panel) a Reissner-Nordstrom black hole with electric charge Q = 0.96.

The Gullstrand-Painleve line-element (15.4) encodes an inverse vierbein with an orthonormal tetrad metric

γmn = ηmn through

e0µ dxµ = dtff , (15.10a)

e1µ dxµ = dr − β dtff , (15.10b)

e2µ dxµ = r dθ , (15.10c)

e3µ dxµ = r sin θ dφ . (15.10d)

Explicitly, the inverse vierbein emµ of the Gullstrand-Painleve line-element (15.4), and the corresponding

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15.2 Gullstrand-Painleve waterfall 255

vierbein emµ, are the matrices

emµ =

1 0 0 0

−β 1 0 0

0 0 r 0

0 0 0 r sin θ

, emµ =

1 β 0 0

0 1 0 0

0 0 1/r 0

0 0 0 1/(r sin θ)

. (15.11)

According to equation (15.3), the coordinate 4-velocity of the tetrad frame through the coordinates is

dtffdτ

,dr

dτ,dθ

dτ,dφ

= uµ = e0µ = 1, β, 0, 0 , (15.12)

consistent with the claim (15.5) that β represents a radial velocity, while tff coincides with the proper time

in the tetrad frame.

The tetrad and coordinate axes γγm and gµ are related to each other by the vierbein and inverse vierbein

in the usual way, γγm = emµ gµ and gµ = em

µ γγm. The Gullstrand-Painleve orthonormal tetrad axes γγm are

thus related to the coordinate axes gµ by

γγ0 = gtff + βgr , γγ1 = gr , γγ2 = gθ/r , γγ3 = gφ/(r sin θ) . (15.13)

Physically, the Gullstrand-Painleve-Cartesian tetrad (15.13) are the axes of locally inertial orthonormal

frames (with spatial axes γγi oriented in the polar directions r, θ, φ) attached to observers who free-fall

radially, without rotating, starting from zero velocity and zero angular momentum at infinity. The fact

that the tetrad axes γγm are parallel-transported, without precessing, along the worldlines of the radially

free-falling observers can be confirmed by checking that the tetrad connections Γnm0 with final index 0 all

vanish, which implies that

dγγm

dτ= ∂0γγm ≡ Γn

m0γγn = 0 . (15.14)

That the proper time derivative d/dτ in equation (15.14) of a person at rest in the tetrad frame, with

4-velocity (15.2), is equal to the directed time derivative ∂0 follows from

d

dτ= uµ ∂

∂xµ= um∂m = ∂0 . (15.15)

15.2.2 Gullstrand-Painleve-Cartesian tetrad

The manner in which the Gullstrand-Painleve line-element depicts a flow of space into a black hole is eluci-

dated further if the line-element is written in Cartesian rather than spherical polar coordinates. Introduce a

Cartesian coordinate system xµ ≡ tff , xi ≡ tff , x, y, z. The Gullstrand-Painleve metric in these Cartesian

coordinates is

ds2 = − dt2ff + δij(dxi − βidtff)(dxj − βjdtff) , (15.16)

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256 Black hole waterfalls

with implicit summation over spatial indices i, j = 1, 2, 3. The βi in the metric (15.16) are the components

of the radial velocity expressed in Cartesian coordinates

βi = βx

r,y

r,z

r

. (15.17)

The inverse vierbein emµ and vierbein em

µ encoded in the Gullstrand-Painleve-Cartesian line-element (15.16)

are

emµ =

1 0 0 0

−β1 1 0 0

−β2 0 1 0

−β3 0 0 1

, emµ =

1 β1 β2 β3

0 1 0 0

0 0 1 0

0 0 0 1

. (15.18)

The tetrad axes γγm of the Gullstrand-Painleve-Cartesian line-element (15.16) are related to the coordinate

tangent axes gµ by

γγ0 = gtff + βigi , γγi = gi , (15.19)

and conversely the coordinate tangent axes gµ are related to the tetrad axes γγm by

gtff = γγ0 − βiγγi , gi = γγi . (15.20)

Note that the tetrad-frame contravariant components βi of the radial velocity coincide with the coordinate-

frame contravariant components βi; for clarification of this point see the more general equation (15.48)

for a rotating black hole. The Gullstrand-Painleve-Cartesian tetrad axes (15.19) are the same as the tetrad

axes (15.13), but rotated to point in Cartesian directions x, y, z rather than in polar directions r, θ, φ. Like the

polar tetrad, the Cartesian tetrad axes γγm are parallel-transported, without precessing, along the worldlines

of radially free-falling observers, as can be confirmed by checking once again that the tetrad connections

Γnm0 with final index 0 all vanish.

Remarkably, the transformation (15.19) from coordinate to tetrad axes is just a Galilean transformation

of space and time, which shifts the time axis by velocity β along the direction of motion, but which leaves

unchanged both the time component of the time axis and all the spatial axes. In other words, the black

hole behaves as if it were a river of space that flows radially inward through Galilean space and time at the

Newtonian escape velocity.

15.2.3 Gullstrand-Painleve fishies

The Gullstrand-Painleve line element paints a picture of locally inertial frames falling like a river of space into

a spherical black hole. What happens to fishies swimming in that river? Of course general relativity supplies

a mathematical answer in the form of the geodesic equation of motion (15.21). Does that mathematical

answer lead to further conceptual insight?

Consider a fishy swimming in the Gullstrand-Painleve river, with some arbitrary tetrad-frame 4-velocity

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15.2 Gullstrand-Painleve waterfall 257

um, and consider a tetrad-frame 4-vector pk attached to the fishy. If the fishy is in free-fall, then the geodesic

equation of motion for pk is as usual

dpk

dτ+ Γk

mnunpm = 0 . (15.21)

As remarked in §11.13, for a constant (for example Minkowski) tetrad metric, as here, the tetrad connections

Γkmn constitute a set of four generators of Lorentz transformations, one in each of the directions n. In

particular Γkmnu

n is the generator of a Lorentz transformation along the path of a fishy moving with 4-

velocity un. In a small (infinitesimal) time δτ , the fishy moves a proper distance δξn ≡ unδτ relative to the

infalling river. This proper distance δξn = enνδx

ν = δnν (δxν − βνδtff) = δxn − βnδτ equals the distance

δxn moved relative to the background Gullstrand-Painleve-Cartesian coordinates, minus the distance βnδτ

moved by the river. The geodesic equation (15.21) says that the change δpk in the tetrad 4-vector pk in the

time δτ is

δpk = −Γkmnδξ

npm . (15.22)

Equation (15.22) describes an infinitesimal Lorentz transformation −Γkmnδξ

n of the 4-vector pk.

Equation (15.22) is quite general in general relativity: it says that as a 4-vector pk free-falls through a

system of locally inertial tetrads, it finds itself Lorentz-transformed relative those tetrads. What is special

about the Gullstrand-Painleve-Cartesian tetrad is that the tetrad-frame connections, computed by the usual

formula (11.41), are given by the coordinate gradient of the radial velocity (the following equation is valid

component-by-component despite the non-matching up-down placement of indices)

Γ0ij = Γi

0j =∂βi

∂xj(i, j = 1, 2, 3) . (15.23)

The same property, that the tetrad connections are a pure coordinate gradient, holds also for the Doran-

Cartesian tetrad for a rotating black hole, equation (15.51). With the connections (15.23), the change

δpk (15.22) in the tetrad 4-vector is

δp0 = − δβi pi , δpi = − δβi p0 , (15.24)

where δβi is the change in the velocity of the river as seen in the tetrad frame,

δβi = δξj ∂βi

∂xj. (15.25)

But equation (15.24) is nothing more than an infinitesimal Lorentz boost by a velocity change δβi. This

shows that a fishy swimming in the river follows the rules of special relativity, being Lorentz boosted by tidal

changes δβi in the river velocity from place to place.

Is it correct to interpret equation (15.25) as giving the change δβi in the river velocity seen by a fishy? Of

course general relativity demands that equation (15.25) be mathematically correct; the issue is merely one

of interpretation. Shouldn’t the change in the river velocity really be

δβi ?= δxν ∂β

i

∂xν, (15.26)

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258 Black hole waterfalls

where δxν is the full change in the coordinate position of the fishy? No. Part of the change (15.26) in the

river velocity can be attributed to the change in the velocity of the river itself over the time δτ , which is

δxνriver∂β

i/∂xν with δxνriver = βνδτ = βνδtff . The change in the velocity relative to the flowing river is

δβi = (δxν − δxνriver)

∂βi

∂xν= (δxν − βνδtff)

∂βi

∂xν(15.27)

which reproduces the earlier expression (15.25). Indeed, in the picture of fishies being carried by the river,

it is essential to subtract the change in velocity of the river itself, as in equation (15.27), because otherwise

fishies at rest in the river (going with the flow) would not continue to remain at rest in the river.

15.3 Boyer-Lindquist tetrad

The Boyer-Lindquist metric for an ideal rotating black hole was explored already in Chapter 9. With the

tetrad formalism in hand, the advantages of the Boyer-Lindquist tetrad for portraying the Kerr-Newman

geometry become manifest. With respect to the orthonormal Boyer-Lindquist tetrad, the electromagnetic

field is purely radial, and the energy-momentum and Weyl tensors are diagonal. The Boyer-Lindquist tetrad

is aligned with the principal (ingoing or outgoing) null congruences.

The Boyer-Lindquist orthonormal tetrad is encoded in the Boyer-Lindquist metric

ds2 = − ∆

ρ2

(

dt− a sin2θ dφ)2

+ρ2

∆dr2 + ρ2dθ2 +

R4 sin2θ

ρ2

(

dφ− a

R2dt)2

(15.28)

where

R ≡√

r2 + a2 , ρ ≡√

r2 + a2 cos2θ , ∆ ≡ R2 − 2Mr +Q2 = R2(1− β2) . (15.29)

Explicitly, the vierbein emµ of the Boyer-Lindquist orthonormal tetrad is

emµ =

1

ρ

R2/√

∆ 0 0 a/√

0√

∆ 0 0

0 0 1 0

a sin θ 0 0 1/ sin θ

, (15.30)

with inverse vierbein emµ

emµ =

√∆/ρ 0 0 − a sin2θ

√∆/ρ

0 ρ/√

∆ 0 0

0 0 ρ 0

− a sin θ/ρ 0 0 R2 sin θ/ρ

. (15.31)

With respect to the Boyer-Lindquist tetrad, only the time component At of the electromagnetic potential

Am is non-vanishing,

Am =

Qr

ρ√

∆, 0, 0, 0

. (15.32)

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15.4 Doran waterfall 259

Only the radial components Er and Br of the electric and magnetic fields are non-vanishing, and they are

given by the complex combination

Er + i Br =Q

(r − ia cos θ)2, (15.33)

or explicitly

Er =Q(

r2−a2 cos2θ)

ρ4, Br =

2Qar cos θ

ρ4. (15.34)

The electrogmagnetic field (15.33) satisfies Maxwell’s equations (11.64) and (11.65) with zero electric charge

and current, jn = 0, except at the singularity ρ = 0.

The non-vanishing components of the tetrad-frame Einstein tensor Gmn are

Gmn =Q2

ρ4

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

, (15.35)

which is the energy-momentum tensor of the electromagnetic field. The non-vanishing components of the

tetrad-frame Weyl tensor Cklmn are

− 12 Ctrtr = 1

2 Cθφθφ = Ctθtθ = Ctφtφ = −Crθrθ = −Crφrφ = ReC , (15.36a)

12 Ctrθφ = Ctθrφ = −Ctφrθ = ImC , (15.36b)

where C is the complex Weyl scalar

C = − 1

(r − ia cos θ)3

(

M − Q2

r + ia cos θ

)

. (15.37)

In the Boyer-Lindquist tetrad, the photon 4-velocity vm on the principal null congruences is radial,

vt = ± ρ√∆, vr = ± ρ√

∆, vθ = 0 , vφ = 0 . (15.38)

Exercise 15.3 Dragging of inertial frames around a Kerr-Newman black hole. What is the

coordinate-frame 4-velocity uµ of the Boyer-Lindquist tetrad through the Boyer-Lindquist coordinates?

15.4 Doran waterfall

The picture of space falling into a black hole like a river or waterfall works also for rotating black holes. For

Kerr-Newman rotating black holes, the counterpart of the Gullstrand-Painleve metric is the Doran (2000)

metric.

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260 Black hole waterfalls

The space river that falls into a rotating black hole has a twist. One might have expected that the rotation

of the black hole would be manifested by a velocity that spirals inward, but this is not the case. Instead,

the river is characterized not merely by a velocity but also by a twist. The velocity and the twist together

comprise a 6-dimensional river bivector ωkm, equation (15.52) below, whose electric part is the velocity,

and whose magnetic part is the twist. Recall that the 6-dimensional group of Lorentz transformations is

generated by a combination of 3-dimensional Lorentz boosts and 3-dimensional spatial rotations. A fishy

that swims through the river is Lorentz boosted by tidal changes in the velocity, and rotated by tidal changes

in the twist, equation (15.61).

Thanks to the twist, unlike the Gullstrand-Painleve metric, the Doran metric is not spatially flat at

constant free-fall time tff . Rather, the spatial metric is sheared in the azimuthal direction. Just as the

velocity produces a Lorentz boost that makes the metric non-flat with respect to the time components, so

also the twist produces a rotation that makes the metric non-flat with respect to the spatial components.

15.4.1 Doran-Cartesian coordinates

In place of the polar coordinates r, θ, φff of the Doran metric, introduce corresponding Doran-Cartesian

coordinates x, y, z with z taken along the rotation axis of the black hole (the black hole rotates right-

handedly about z, for positive spin parameter a)

x ≡ R sin θ cosφff , y ≡ R sin θ sinφff , z ≡ r cos θ . (15.39)

The metric in Doran-Cartesian coordinates xµ ≡ tff , xi ≡ tff , x, y, z, is

ds2 = − dt2ff + δij(

dxi − βiακdxκ) (

dxj − βjαλdxλ)

(15.40)

where αµ is the rotational velocity vector

αµ =

1,ay

R2, − ax

R2, 0

, (15.41)

and βµ is the velocity vector

βµ =βR

ρ

0,xr

Rρ,yr

Rρ,zR

. (15.42)

The rotational velocity and radial velocity vectors are orthogonal

αµβµ = 0 . (15.43)

For the Kerr-Newman metric, the radial velocity β is

β = ∓√

2Mr −Q2

R(15.44)

with − for black hole (infalling), + for white hole (outfalling) solutions. Horizons occur where

β = ∓1 , (15.45)

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15.4 Doran waterfall 261

with β = −1 for black hole horizons, and β = 1 for white hole horizons. Note that the squared magnitude

βµβµ of the velocity vector is not β2, but rather differs from β2 by a factor of R2/ρ2:

βµβµ = βmβ

m =β2R2

ρ2. (15.46)

The point of the convention adopted here is that β(r) is any and only a function of r, rather than depending

also on θ through ρ. Moreover, with the convention here, β is ∓1 at horizons, equation (15.45). Finally, the

4-velocity βµ is simply related to β by βµ = (β/r) ∂r/∂xµ.

The Doran-Cartesian metric (15.40) encodes a vierbein emµ and inverse vierbein em

µ

emµ = δµ

m + αmβµ , em

µ = δmµ − αµβ

m . (15.47)

Here the tetrad-frame components αm of the rotational velocity vector and βm of the radial velocity vector

are

αm = emµαµ = δµ

mαµ , βm = emµβ

µ = δmµ β

µ , (15.48)

which works thanks to the orthogonality (15.43) of αµ and βµ. Equation (15.48) says that the covariant

tetrad-frame components of the rotational velocity vector α are the same as its covariant coordinate-frame

components in the Doran-Cartesian coordinate system, αm = αµ, and likewise the contravariant tetrad-frame

components of the radial velocity vector β are the same as its contravariant coordinate-frame components,

βm = βµ.

15.4.2 Doran-Cartesian tetrad

Like the Gullstrand-Painleve tetrad, the Doran-Cartesian tetrad γγm ≡ γγ0,γγ1,γγ2,γγ3 is aligned with the

Cartesian rest frame gµ ≡ gtff , gx, gy, gz at infinity, and is parallel-transported, without precessing, by

observers who free-fall from zero velocity and zero angular momentum at infinity, as can be confirmed by

checking that the tetrad connections with final index 0 all vanish, Γnm0 = 0, equation (15.14).

Let ‖ and ⊥ subscripts denote horizontal radial and azimuthal directions respectively, so that

γγ‖ ≡ cosφff γγ1 + sinφff γγ2 , γγ⊥ ≡ − sinφff γγ1 + cosφff γγ2 ,

g‖ ≡ cosφff gx + sinφff gy , g⊥ ≡ − sinφff gx + cosφff gy .(15.49)

Then the relation between Doran-Cartesian tetrad axes γγm and the tangent axes gµ of the Doran-Cartesian

metric (15.40) is

γγ0 = gtff + βigi , (15.50a)

γγ‖ = g‖ , (15.50b)

γγ⊥ = g⊥ −a sin θ

Rβigi , (15.50c)

γγ3 = gz . (15.50d)

The relations (15.50) resemble those (15.19) of the Gullstrand-Painleve tetrad, except that the azimuthal

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262 Black hole waterfalls

tetrad axis γγ⊥ is shifted radially relative to the azimuthal tangent axis g⊥. This shift reflects the fact that,

unlike the Gullstrand-Painleve metric, the Doran metric is not spatially flat at constant free-fall time, but

rather is sheared azimuthally.

15.4.3 Doran fishies

The tetrad-frame connections equal the ordinary coordinate partial derivatives in Doran-Cartesian coordi-

nates of a bivector (antisymmetric tensor) ωkm

Γkmn = − ∂ωkm

∂xn, (15.51)

which I call the river field because it encapsulates all the properties of the infalling river of space. The

bivector river field ωkm is

ωkm = αkβm − αmβk − ε0kmi ζi , (15.52)

where βm = ηmnβm, the totally antisymmetric tensor εklmn is normalized so that ε0123 = −1, and the vector

ζi points vertically upward along the rotation axis of the black hole

ζi ≡ 0, 0, 0, ζ , ζ ≡ a∫ r

β dr

R2. (15.53)

The electric part of ωkm, where one of the indices is time 0, constitutes the velocity vector βi

ω0i = βi (15.54)

while the magnetic part of ωkm, where both indices are spatial, constitutes the twist vector µi defined by

µi ≡ 12 ε

0ikmωkm = ε0ikmαkβm + ζi . (15.55)

The sense of the twist is that induces a right-handed rotation about an axis equal to the direction of µi by

an angle equal to the magnitude of µi. In 3-vector notation, with µ ≡ µi, α ≡ αi, β ≡ βi, ζ ≡ ζi,

µ ≡ α× β + ζ . (15.56)

In terms of the velocity and twist vectors, the river field ωkm is

ωkm =

0 βx βy βz

−βx 0 µz −µy

−βy −µz 0 µx

−βz µy −µx 0

. (15.57)

Note that the sign of the magnetic part β of ωkm is opposite to the sign of the analogous magnetic field

B associated with an electromagnetic field Fkm; but the adopted signs are natural in that the river field

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15.4 Doran waterfall 263

Rot

atio

nax

is

Inner horizon

Outer horizon

360° Rot

atio

nax

is

Inner horizon

Outer horizon

Figure 15.3 (Upper panel) velocity βi and (lower panel) twist µi vector fields for a Kerr black hole with spinparameter a = 0.96. Both vectors lie, as shown, in the plane of constant free-fall azimuthal angle φff . Thevertical bar in the lower panel shows the length of a twist vector corresponding to a full rotation of 360.

induces boosts in the direction of the velocity βi, and right-handed rotations about the twist µi. Like a

static electric field, the velocity vector βi is the gradient of a potential

βi =∂

∂xi

∫ r

β dr , (15.58)

but unlike a magnetic field the twist vector µi is not pure curl: rather, it is µi + ζi that is pure curl.

Figure 15.3 illustrates the velocity and twist fields in a Kerr black hole.

With the tetrad connection coefficients given by equation (15.51), the equation of motion (15.21) for a

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264 Black hole waterfalls

4-vector pk attached to a fishy following a geodesic in the Doran river translates to

dpk

dτ=∂ωk

m

∂xnunpm . (15.59)

In a proper time δτ , the fishy moves a proper distance δξm ≡ umδτ relative to the background Doran-

Cartesian coordinates. As a result, the fishy sees a tidal change δωkm in the river field

δωkm = δξn ∂ω

km

∂xn. (15.60)

Consequently the 4-vector pk is changed by

pk → pk + δωkm pm . (15.61)

But equation (15.61) corresponds to an infinitesimal Lorentz transformation by δωkm, equivalent to a Lorentz

boost by δβi and a rotation by δµi.

As discussed previously with regard to the Gullstrand-Painleve river, §15.2.3, the tidal change δωkm,

equation (15.60), in the river field seen by a fishy is not the full change δxν ∂ωkm/∂x

ν relative to the

background coordinates, but rather the change relative to the river

δωkm = (δxν − δxν

river)∂ωk

m

∂xν=[

δxν − βν(δtff − a sin2θ δφff)] ∂ωk

m

∂xν, (15.62)

with the change in the velocity and twist of the river itself subtracted off.

That there exists a tetrad (the Doran-Cartesian tetrad) where the tetrad-frame connections are a coor-

dinate gradient of a bivector, equation (15.51), is a peculiar feature of ideal black holes. It is an intriguing

thought that perhaps the 6 physical degrees of freedom of a general spacetime might always be encoded in

the 6 degrees of freedom of a bivector, but I suspect that that is not true.

Exercise 15.4 River model of the Friedmann-Robertson-Walker metric. Show that the flat FRW

line-element

ds2 = − dt2 + a2(dx2 + x2do2) (15.63)

can be re-expressed as

ds2 = − dt2 + (dr −Hr dt)2 + r2do2 , (15.64)

where r ≡ ax is the proper radial distance, and H ≡ a/a is the Hubble parameter. Interpret the line-

element (15.64). What is the generalization to a non-flat FRW universe?

Exercise 15.5 Program geodesics in a rotating black hole. Write a graphics (Java?) program

that uses the prescription (15.60) to draw geodesics of test particles in an ideal (Kerr-Newman) black

hole, expressed in Doran-Cartesian coordinates. Attach 3D bodies to your test particles, and use the same

prescription (15.60) to rotate the bodies. Implement an option to translate to Boyer-Lindquist coordinates.

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16

General spherically symmetric spacetime

16.1 Spherical spacetime

The most important equations in this chapter are the two Einstein equations (16.52). Spherical spacetimes

have 2 physical degrees of freedom. Spherical symmetry eliminates any angular degrees of freedom, leaving

4 adjustable metric coefficients gtt, gtr, grr, and gθθ. But coordinate transformations of the time t and radial

r coordinates remove 2 degrees of freedom, leaving a spherical spacetime with a net 2 physical degrees of

freedom. Spherical spacetimes have 4 distinct Einstein equations (16.30). But 2 of the Einstein equations

serve to enforce energy-momentum conservation, so the evolution of the spacetime is governed by 2 Einstein

equations, in agreement with the number of physical degrees of freedom of spherical spacetime.

The 2 degrees of freedom mean that spherical spacetimes in general relativity have a richer structure than

in Newtonian gravity, which has only degree of freedom, the Newtonian potential Φ. The richer structure

is most striking in the case of the mass inflation instability, Chapter 17, which is an intrinsically general

relativistic instability, with no Newtonian analogue.

16.1.1 Spherical line-element

The spherical line-element adopted in this chapter is, in spherical polar coordinates xµ ≡ t, r, θ, φ,

ds2 = − dt2

α2+

1

β2r

(

dr − βtdt

α

)2

+ r2do2 . (16.1)

Here r is the circumferential radius, defined such that the circumference around any great circle is 2πr. The

line-element (16.1) is somewhat unconventional in that it is not diagonal: gtr does not vanish. There are

two good reasons to consider a non-diagonal metric. Firstly, as discussed in §16.1.11, Einstein’s equations

take a more insightful form when expressed in a non-diagonal frame where βt does not vanish. Secondly, if a

horizon is present, as in the case of black holes, and if the radial coordinate is taken to be the circumferential

radius r, then a diagonal metric will have a coordinate singularity at the horizon, which is not ideal.

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266 General spherically symmetric spacetime

The vierbein emµ and inverse vierbein em

µ corresponding to the spherical line-element (16.1) are

emµ =

α βt 0 0

0 βr 0 0

0 0 1/r 0

0 0 0 1/(r sin θ)

, emµ =

1/α 0 0 0

− βt/(αβr) 1/βr 0 0

0 0 r 0

0 0 0 r sin θ

. (16.2)

As in the ADM formalism, Chapter 13, the tetrad time axis γγt is chosen to be orthogonal to hypersurfaces of

constant time t. However, the convention here for the vierbein coefficients differs from the ADM convention:

here 1/α is the ADM lapse, while βt/α is the ADM shift. The directed derivatives ∂t and ∂r along the time

and radial tetrad axes γγt and γγr are

∂t = etµ ∂

∂xµ= α

∂t+ βt

∂r, ∂r = er

µ ∂

∂xµ= βr

∂r. (16.3)

The tetrad-frame 4-velocity um of a person at rest in the tetrad frame is by definition um = 1, 0, 0, 0. It

follows that the coordinate 4-velocity uµ of such a person is

uµ = emµum = et

µ = α, βt, 0, 0 . (16.4)

A person instantaneously at rest in the tetrad frame satisfies dr/dt = βt/α according to equation (16.4),

so it follows from the line-element (16.1) that the proper time τ of a person at rest in the tetrad frame is

related to the coordinate time t by

dτ =dt

αin tetrad rest frame . (16.5)

The directed time derivative ∂t is just the proper time derivative along the worldline of a person continuously

at rest in the tetrad frame (and who is therefore not in free-fall, but accelerating with the tetrad frame),

which follows from

d

dτ=dxµ

∂xµ= uµ ∂

∂xµ= um∂m = ∂t . (16.6)

By contrast, the proper time derivative measured by a person who is instantaneously at rest in the tetrad

frame, but is in free-fall, is the covariant time derivative

D

Dτ=dxµ

dτDµ = uµDµ = umDm = Dt . (16.7)

Since the coordinate radius r has been defined to be the circumferential radius, a gauge-invariant definition,

it follows that the tetrad-frame gradient ∂m of the coordinate radius r is a tetrad-frame 4-vector (a coordinate

gauge-invariant object)

∂mr = emµ ∂r

∂xµ= em

r = βm = βt, βr, 0, 0 is a tetrad 4-vector . (16.8)

This accounts for the notation βt and βr introduced above. Since βm is a tetrad 4-vector, its scalar product

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16.1 Spherical spacetime 267

with itself must be a scalar. This scalar defines the interior mass M(t, r), also called the Misner-Sharp

mass, by

1− 2M

r≡ − βmβ

m = − β2t + β2

r is a coordinate and tetrad scalar . (16.9)

The interpretation of M as the interior mass will become evident below, §16.1.8.

16.1.2 Rest diagonal line-element

Although this is not the choice adopted here, the line-element (16.1) can always be brought to diagonal form

by a coordinate transformation t→ t× (subscripted × for diagonal) of the time coordinate. The t–r part of

the metric is

gtt dt2 + 2 gtr dt dr + grr dr

2 =1

gtt

[

(gtt dt+ gtr dr)2 + (gttgrr − g2

tr) dr2]

. (16.10)

This can be diagonalized by choosing the time coordinate t× such that

f dt× = gtt dt+ gtr dr (16.11)

for some integrating factor f(t, r). Equation (16.11) can be solved by choosing t× to be constant along

integral curves

dr

dt= − gtt

gtr. (16.12)

The resulting diagonal rest line-element is

ds2 = − dt2×α2×

+dr2

1− 2M/r+ r2do2 . (16.13)

The line-element (16.13) corresponds physically to the case where the tetrad frame is taken to be at rest in

the spatial coordinates, βt = 0, as can be seen by comparing it to the earlier line-element (16.1). In changing

the tetrad frame from one moving at dr/dt = βt/α to one that is at rest (at constant circumferential radius

r), a tetrad transformation has in effect been done at the same time as the coordinate transformation (16.11),

the tetrad transformation being precisely that needed to make the line-element (16.13) diagonal. The metric

coefficient grr in the line-element (16.13) follows from the fact that β2r = 1 − 2M/r when βt = 0, equa-

tion (16.9). The transformed time coordinate t× is unspecified up to a transformation t× → f(t×). If the

spacetime is asymptotically flat at infinity, then a natural way to fix the transformation is to choose t× to

be the proper time at rest at infinity.

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268 General spherically symmetric spacetime

16.1.3 Comoving diagonal line-element

Although once again this is not the path followed here, the line-element (16.1) can also be brought to diagonal

form by a coordinate transformation r → r×, where, analogously to equation (16.11), r× is chosen to satisfy

f dr× = gtr dt+ grr dr ≡1

βr

(

dr − βtdt

α

)

(16.14)

for some integrating factor f(t, r). The new coordinate r× is constant along the worldline of an object at

rest in the tetrad frame, with dr/dt = βt/α, equation (16.4), so r× can be regarded as a comoving radial

coordinate. The comoving radial coordinate r× could for example be chosen to equal the circumferential

radius r at some fixed instant of coordinate time t (say t = 0). The diagonal comoving line-element in

this comoving coordinate system takes the form

ds2 = − dt2

α2+dr2×λ2

+ r2do2 , (16.15)

where the circumferential radius r(t, r×) is considered to be an implicit function of time t and the comoving

radial coordinate r×. Whereas in the rest line-element (16.13) the tetrad was changed from one that was

moving at dr/dt = βt/α to one that was at rest, here the transformation keeps the tetrad unchanged. In

both the rest and comoving diagonal line-elements (16.13) and (16.15) the tetrad is at rest relative to the

respective radial coordinate r or r×; but whereas in the rest line-element (16.13) the radial coordinate was

fixed to be the circumferential radius r, in the comoving line-element (16.15) the comoving radial coordinate

r× is a label that follows the tetrad. Because the tetrad is unchanged by the transformation to the comoving

radial coordinate r×, the directed time and radial derivatives are unchanged:

∂t = α∂

∂t

= α∂

∂t

r

+ βt∂

∂r

t

, ∂r = λ∂

∂r×

t

= βr∂

∂r

t

. (16.16)

16.1.4 Tetrad connections

Now turn the handle to proceed towards the Einstein equations. The tetrad connections coefficients Γkmn

corresponding to the spherical line-element (16.1) are

Γrtt = ht , (16.17a)

Γrtr = hr , (16.17b)

Γθtθ = Γφtφ =βt

r, (16.17c)

Γθrθ = Γφrφ =βr

r, (16.17d)

Γφθφ =cot θ

r, (16.17e)

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16.1 Spherical spacetime 269

where ht is the proper radial acceleration (minus the gravitational force) experienced by a person at rest in

the tetrad frame

ht ≡ − ∂r lnα , (16.18)

and hr is the “Hubble parameter” of the radial flow, as measured in the tetrad rest frame, defined by

hr ≡ − βt∂ lnα

∂r+∂βt

∂r− ∂t lnβr . (16.19)

The interpretation of ht as a proper acceleration and hr as a radial Hubble parameter goes as follows. The

tetrad-frame 4-velocity um of a person at rest in the tetrad frame is by definition um = 1, 0, 0, 0. If the

person at rest were in free fall, then the proper acceleration would be zero, but because this is a general

spherical spacetime, the tetrad frame is not necessarily in free fall. The proper acceleration experienced by

a person continuously at rest in the tetrad frame is the proper time derivative Dum/Dτ of the 4-velocity,

which is

Dum

Dτ= Dtu

m = ∂tum + Γm

ttut = Γm

tt = 0,Γrtt, 0, 0 = 0, ht, 0, 0 , (16.20)

the first step of which follows from equation (16.7). Similarly, a person at rest in the tetrad frame will

measure the 4-velocity of an adjacent person at rest in the tetrad frame a small proper radial distance δξr

away to differ by δξrDrum. The Hubble parameter of the radial flow is thus the covariant radial derivative

Drum, which is

Drum = ∂ru

m + Γmtru

t = Γmtr = 0,Γr

tr, 0, 0 = 0, hr, 0, 0 . (16.21)

Confined to the t–r-plane (that is, considering only Lorentz transformations in the t–r-plane, which is to

say radial Lorentz boosts), the acceleration ht and Hubble parameter hr constitute the components of a

tetrad-frame 2-vector hn = ht, hr:

hn = Γrtn . (16.22)

The Riemann tensor, equations (16.24) below, involves covariant derivatives Dmhn of hn, which coincide

with the covariant derivatives D(2)hn confined to t–r-plane.

Since hr is a kind of radial Hubble parameter, it can be useful to define a corresponding radial scale factor

λ by

hr ≡ −∂t lnλ . (16.23)

The scale factor λ is the same as the λ in the comoving line-element of equation (16.15). This is true because

hr is a tetrad connection and therefore coordinate gauge-invariant, and the line-element (16.15) is related

to the line-element (16.1) being considered by a coordinate transformation r → r× that leaves the tetrad

unchanged.

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270 General spherically symmetric spacetime

16.1.5 Riemann, Einstein, and Weyl tensors

The non-vanishing components of the tetrad-frame Riemann tensor Rklmn corresponding to the spherical

line-element (16.1) are

Rtrtr = Drht −Dthr , (16.24a)

Rtθtθ = Rtφtφ = − 1

rDtβt , (16.24b)

Rrθrθ = Rrφrφ = − 1

rDrβr , (16.24c)

Rtθrθ = Rtφrφ = − 1

rDtβr = − 1

rDrβt , (16.24d)

Rθφθφ =2M

r3, (16.24e)

where Dm denotes the covariant derivative as usual.

The non-vanishing components of the tetrad-frame Einstein tensor Gkm are

Gtt = 2Rrθrθ +Rθφθφ , (16.25a)

Grr = 2Rtθtθ −Rθφθφ , (16.25b)

Gtr = − 2Rtθrθ , (16.25c)

Gθθ = Gφφ = Rtrtr +Rtθtθ −Rrθrθ , (16.25d)

whence

Gtt =2

r

(

−Drβr +M

r2

)

, (16.26a)

Grr =2

r

(

−Dtβt −M

r2

)

, (16.26b)

Gtr =2

rDtβr =

2

rDrβt , (16.26c)

Gθθ = Gφφ = Drht −Dthr +1

r(Drβr −Dtβt) . (16.26d)

The non-vanishing components of the tetrad-frame Weyl tensor Cklmn are

12 Ctrtr = −Ctθtθ = −Ctφtφ = Crθrθ = Crφrφ = − 1

2 Cθφθφ = C , (16.27)

where C is the Weyl scalar (the spin-0 component of the Weyl tensor),

C ≡ 1

6(Rtrtr −Rtθtθ +Rrθrθ −Rθφθφ) =

1

6

(

Gtt −Grr +Gθθ)

− M

r3. (16.28)

16.1.6 Einstein equations

The tetrad-frame Einstein equations

Gkm = 8πT km (16.29)

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16.1 Spherical spacetime 271

imply that

Gtt Gtr 0 0

Gtr Grr 0 0

0 0 Gθθ 0

0 0 0 Gφφ

= 8πT km = 8π

ρ f 0 0

f p 0 0

0 0 p⊥ 0

0 0 0 p⊥

(16.30)

where ρ ≡ T tt is the proper energy density, f ≡ T tr is the proper radial energy flux, p ≡ T rr is the proper

radial pressure, and p⊥ ≡ T θθ = T φφ is the proper transverse pressure.

16.1.7 Choose your frame

So far the radial motion of the tetrad frame has been left unspecified. Any arbitrary choice can be made.

For example, the tetrad frame could be chosen to be at rest,

βt = 0 , (16.31)

as in the Schwarzschild or Reissner-Nordstrom line-elements. Alternatively, the tetrad frame could be chosen

to be in free-fall,

ht = 0 , (16.32)

as in the Gullstrand-Painleve line-element. For situations where the spacetime contains matter, one natural

choice is the center-of-mass frame, defined to be the frame in which the energy flux f is zero

Gtr = 8πf = 0 . (16.33)

Whatever the choice of radial tetrad frame, tetrad-frame quantities in different radial tetrad frames are

related to each other by a radial Lorentz boost.

16.1.8 Interior mass

Equations (16.26a) with the middle expression of (16.26c), and (16.26b) with the final expression of (16.26c),

respectively, along with the definition (16.9) of the interior mass M , and the Einstein equations (16.30),

imply

p =1

βt

(

− 1

4πr2∂tM − βrf

)

, (16.34a)

ρ =1

βr

(

1

4πr2∂rM − βtf

)

. (16.34b)

In the center-of-mass frame, f = 0, these equations reduce to

∂tM = − 4πr2βt p , (16.35a)

∂rM = 4πr2βr ρ . (16.35b)

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272 General spherically symmetric spacetime

Equations (16.35) amply justify the interpretation of M as the interior mass. The first equation (16.35a)

can be written

∂tM + p 4πr2∂tr = 0 , (16.36)

which can be recognized as an expression of the first law of thermodynamics,

dE + p dV = 0 , (16.37)

with mass-energy E equal to M . The second equation (16.35b) can be written, since ∂r = βr ∂/∂r, equa-

tion (16.3),

∂M

∂r= 4πr2ρ , (16.38)

which looks exactly like the Newtonian relation between interior mass M and density ρ. Actually, this

apparently Newtonian equation (16.38) is deceiving. The proper 3-volume element d3r in the center-of-mass

tetrad frame is given by

d3r γγr ∧ γγθ ∧ γγφ = gr dr ∧ gθ dθ ∧ gφ dφ =r2 sin θ dr dθ dφ

βrγγr ∧ γγθ ∧ γγφ , (16.39)

so that the proper 3-volume element dV ≡ d3r of a radial shell of width dr is

dV =4πr2dr

βr. (16.40)

Thus the “true” mass-energy dMm associated with the proper density ρ in a proper radial volume element

dV might be expected to be

dMm = ρ dV =4πr2dr

βr, (16.41)

whereas equation (16.38) indicates that the actual mass-energy is

dM = ρ 4πr2dr = βr ρ dV . (16.42)

A person in the center-of-mass frame might perhaps, although there is really no formal justification for doing

so, interpret the balance of the mass-energy as gravitational mass-energy Mg

dMg = (βr − 1)ρ dV . (16.43)

Whatever the case, the moral of this is that you should beware of interpreting the interior mass M too

literally as palpable mass-energy.

16.1.9 Energy-momentum conservation

Covariant conservation of the Einstein tensor DmGmn = 0 implies conservation of energy-momentum

DmTmn = 0. The two non-vanishing equations represent conservation of energy and of radial momentum,

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16.1 Spherical spacetime 273

and are

DmTmt = ∂tρ+

2βt

r(ρ+ p⊥) + hr (ρ+ p) +

(

∂r +2βr

r+ 2 ht

)

f = 0 , (16.44a)

DmTmr = ∂rp+

2βr

r(p− p⊥) + ht (ρ+ p) +

(

∂t +2βt

r+ 2 hr

)

f = 0 . (16.44b)

In the center-of-mass frame, f = 0, these energy-momentum conservation equations reduce to

∂tρ+2βt

r(ρ+ p⊥) + hr (ρ+ p) = 0 , (16.45a)

∂rp+2βr

r(p− p⊥) + ht (ρ+ p) = 0 . (16.45b)

In a general situation where the mass-energy is the sum over several individual components a,

Tmn =∑

species a

Tmna , (16.46)

the individual mass-energy components a of the system each satisfy an energy-momentum conservation

equation of the form

DmTmna = Fn

a , (16.47)

where Fna is the flux of energy into component a. Einstein’s equations enforce energy-momentum conservation

of the system as a whole, so the sum of the energy fluxes must be zero∑

species a

Fna = 0 . (16.48)

16.1.10 First law of thermodynamics

For an individual species a, the energy conservation equation (16.44a) in the center-of-mass frame of the

species, fa = 0, can be written

DmTmta = ∂tρa + (ρa + p⊥a)∂t ln r2 + (ρa + pa)∂t lnλa = F t

a , (16.49)

where λa is the radial “scale factor,” equation (16.23), in the center-of-mass frame of the species (the scale

factor is different in different frames). Equation (16.49) can be recognized as an expression of the first law

of thermodynamics for a volume element V of species a, in the form

V −1[

∂t(ρaV ) + p⊥a Vr ∂tV⊥ + pa V⊥ ∂tVr

]

= F ta , (16.50)

with transverse volume (area) V⊥ ∝ r2, radial volume (width) Vr ∝ λa, and total volume V ∝ V⊥Vr . The

flux F ta on the right hand side is the heat per unit volume per unit time going into species a. If the pressure

of species a is isotropic, p⊥a = pa, then equation (16.50) simplifies to

V −1[

∂t(ρaV ) + pa ∂tV]

= F ta , (16.51)

with volume V ∝ r2λa.

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274 General spherically symmetric spacetime

16.1.11 Structure of the Einstein equations

The spherically symmetric spacetime under consideration is described by 3 vierbein coefficients, α, βt, and βr.

However, some combination of the 3 coefficients represents a gauge freedom, since the spherically symmetric

spacetime has only two physical degrees of freedom. As commented in §16.1.7, various gauge-fixing choices

can be made, such as choosing to work in the center-of-mass frame, f = 0.

Equations (16.26) give 4 equations for the 4 non-vanishing components of the Einstein tensor. The two

expressions for Gtr are identical when expressed in terms of the vierbein and vierbein derivatives, so are not

distinct equations. Conservation of energy-momentum of the system as a whole is built in to the Einstein

equations, a consequence of the Bianchi identities, so 2 of the Einstein equations are effectively equivalent to

the energy-momentum conservation equations (16.44). In the general case where the matter contains multiple

components, it is usually a good idea to include the equations describing the conservation or exchange of

energy-momentum separately for each component, so that global conservation of energy-momentum is then

satisfied as a consequence of the matter equations.

This leaves 2 independent Einstein equations to describe the 2 physical degrees of freedom of the spacetime.

The 2 equations may be taken to be the evolution equations (16.26c) and (16.26b) for βt and βr,

Dtβt = −Mr2− 4πrp , (16.52a)

Dtβr = 4πrf , (16.52b)

which are valid for any choice of tetrad frame, not just the center-of-mass frame.

Equations (16.52) are the most important of the general relativistic equations governing spherically sym-

metric spacetimes. It is these equations that are responsible (to the extent that equations may be con-

sidered responsible) for the strange internal structure of Reissner-Nordstrom black holes, and for mass

inflation. The coefficient βt equals the coordinate radial 4-velocity dr/dτ = ∂tr = βt of the tetrad

frame, equation (16.4), and thus equation (16.52a) can be regarded as giving the proper radial acceleration

D2r/Dτ2 = Dβt/Dτ = Dtβt of the tetrad frame as measured by a person who is in free-fall and instanta-

neously at rest in the tetrad frame. If the acceleration is measured by an observer who is continuously at

rest in the tetrad frame (as opposed to being in free-fall), then the proper acceleration is ∂tβt = Dtβt +βrht.

The presence of the extra term βrht, proportional to the proper acceleration ht actually experienced by

the observer continuously at rest in the tetrad frame, reflects the principle of equivalence of gravity and

acceleration.

The right hand side of equation (16.52a) can be interpreted as the radial gravitational force, which consists

of two terms. The first term, −M/r2, looks like the familiar Newtonian gravitational force, which is attractive

(negative, inward) in the usual case of positive mass M . The second term, −4πrp, proportional to the radial

pressure p, is what makes spherical spacetimes in general relativity interesting. In a Reissner-Nordstrom black

hole, the negative radial pressure produced by the radial electric field produces a radial gravitational repulsion

(positive, outward), according to equation (16.52a), and this repulsion dominates the gravitational force at

small radii, producing an inner horizon. In mass inflation, the (positive) radial pressure of relativistically

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16.1 Spherical spacetime 275

counter-streaming ingoing and outgoing streams just above the inner horizon dominates the gravitational

force (inward), and it is this that drives mass inflation.

Like the second half of a vaudeville act, the second Einstein equation (16.52b) also plays an indispensible

role. The quantity βr ≡ ∂rr on the left hand side is the proper radial gradient of the circumferential radius

r measured by a person at rest in the tetrad frame. The sign of βr determines which way an observer at

rest in the tetrad frame thinks is “outwards”, the direction of larger circumferential radius r. A positive

βr means that the observer thinks the outward direction points away from the black hole, while a negative

βr means that the observer thinks the outward direction points towards from the black hole. Outside the

outer horizon βr is necessarily positive, because βm must be spacelike there. But inside the horizon βr may

be either positive or negative. A tetrad frame can be defined as “ingoing” if the proper radial gradient βr

is positive, and “outgoing” if βr is negative. In the Reissner-Nordstrom geometry, ingoing geodesics have

positive energy, and outgoing geodesics have negative energy. However, the present definition of ingoing or

outgoing based on the sign of βr is general – there is no need for a timelike Killing vector such as would be

necessary to define the (conserved) energy of a geodesic.

Equation (16.52b) shows that the proper rate of change Dtβr in the radial gradient βr measured by an

observer who is in free-fall and instantaneously at rest in the tetrad frame is proportional to the radial energy

flux f in that frame. But ingoing observers tend to see energy flux pointing away from the black hole, while

outgoing observers tend to see energy flux pointing towards the black hole. Thus the change in βr tends to

be in the same direction as βr, amplifying βr whatever its sign.

Exercise 16.1 Birkhoff’s theorem. Prove Birkhoff’s theorem from equations (16.52).

16.1.12 Comment on the vierbein coefficient α

Whereas the Einstein equations (16.52) give evolution equations for the vierbein coefficients βt and βr, there

is no evolution equation for the vierbein coefficient α. Indeed, the Einstein equations involve the vierbein

coefficient α only in the combination ht ≡ −∂r lnα. This reflects the fact that, even after the tetrad frame

is fixed, there is still a coordinate freedom t → t′(t) in the choice of coordinate time t. Under such a gauge

transformation, α transforms as α→ α′ = f(t)α where f(t) = ∂t′/∂t is an arbitrary function of coordinate

time t. Only ht ≡ −∂r lnα is independent of this coordinate gauge freedom, and thus only ht, not α itself,

appears in the tetrad-frame Einstein equations.

Since α is needed to propagate the equations from one coordinate time to the next (because ∂t = α∂/∂t+

βt ∂/∂r), it is necessary to construct α by integrating ht ≡ −βr ∂ lnα/∂r along the radial direction r at each

time step. The arbitrary normalization of α at each step might be fixed by choosing α to be unity at infinity,

which corresponds to fixing the time coordinate t to equal the proper time at infinity.

In the particular case that the tetrad frame is taken to be in free-fall everywhere, ht = 0, as in the

Gullstrand-Painleve line-element, then α is constant at fixed t, and without loss of generality it can be fixed

equal to unity everywhere, α = 1. I like to think of a free-fall frame as being realized physically by tracer

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276 General spherically symmetric spacetime

“dark matter” particles that free-fall radially (from zero velocity, typically) at infinity, and stream freely,

without interacting, through any actual matter that may be present.

16.2 Spherical electromagnetic field

The internal structure of a charged black hole resembles that of a rotating black hole because the negative

pressure (tension) of the radial electric field produces a gravitational repulsion analogous to the centrifugal

repulsion in a rotating black hole. Since it is much easier to deal with spherical than rotating black holes, it

is common to use charge as a surrogate for rotation in exploring black holes.

16.2.1 Electromagnetic field

The assumption of spherical symmetry means that any electromagnetic field can consist only of a radial elec-

tric field (in the absence of magnetic monopoles). The only non-vanishing components of the electromagnetic

field Fmn are then

Ftr = −Frt = E =Q

r2, (16.53)

where E is the radial electric field, and Q(t, r) is the interior electric charge. Equation (16.53) can be

regarded as defining what is meant by the electric charge Q interior to radius r at time t.

16.2.2 Maxwell’s equations

A radial electric field automatically satisfies two of Maxwell’s equations, the source-free ones (11.64). For

the radial electric field (16.53), the other two Maxwell’s equations, the sourced ones (11.65), are

∂rQ = 4πr2q , (16.54a)

∂tQ = −4πr2j , (16.54b)

where q ≡ jt is the proper electric charge density and j ≡ jr is the proper radial electric current density in

the tetrad frame.

16.2.3 Electromagnetic energy-momentum tensor

For the radial electric field (16.53), the electromagnetic energy-momentum tensor (11.70) in the tetrad frame

is the diagonal tensor

Tmne =

Q2

8πr4

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

. (16.55)

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16.3 General relativistic stellar structure 277

The radial electric energy-momentum tensor is independent of the radial motion of the tetrad frame, which

reflects the fact that the electric field is invariant under a radial Lorentz boost. The energy density ρe and

radial and transverse pressures pe and p⊥e of the electromagnetic field are the same as those from a spherical

charge distribution with interior electric charge Q in flat space

ρe = −pe = p⊥e =Q2

8πr4=E2

8π. (16.56)

The non-vanishing components of the covariant derivative DmTmne of the electromagnetic energy-mom-

entum (16.55) are

DmTmte = ∂tρe +

4βt

rρe =

Q

4πr4∂tQ = − jQ

r2= − jE , (16.57a)

DmTmre = ∂rpe +

4βr

rpe = − Q

4πr4∂rQ = − qQ

r2= − qE . (16.57b)

The first expression (16.57a), which gives the rate of energy transfer out of the electromagnetic field as the

current density j times the electric field E, is the same as in flat space. The second expression (16.57b),

which gives the rate of transfer of radial momentum out of the electromagnetic field as the charge density q

times the electric field E, is the Lorentz force on a charge density q, and again is the same as in flat space.

16.3 General relativistic stellar structure

A star can be well approximated as static as well as spherically symmetric. In this case all time derivatives

can be taken to vanish, ∂/∂t = 0, and, since the center-of-mass frame coincides with the rest frame, it is

natural to choose the tetrad frame to be at rest, βt = 0. The Einstein equation (16.52b) then vanishes

identically, while the Einstein equation (16.52a) becomes

βrht =M

r2+ 4πrp , (16.58)

which expresses the proper acceleration ht in the rest frame in terms of the familiar Newtonian gravitational

force M/r2 plus a term 4πrp proportional to the radial pressure. The radial pressure p, if positive as is the

usual case for a star, enhances the inward gravitational force, helping to destabilize the star. Because βt is

zero, the interior mass M given by equation (16.9) reduces to

1− 2M/r = β2r . (16.59)

When equations (16.58) and (16.59) are substituted into the momentum equation (16.44b), and if the pressure

is taken to be isotropic, so p⊥ = p, the result is the Oppenheimer-Volkov equation for general relativistic

hydrostatic equilibrium

∂p

∂r= − (ρ+ p)(M + 4πr3p)

r2(1− 2M/r). (16.60)

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278 General spherically symmetric spacetime

In the Newtonian limit p≪ ρ and M ≪ r this goes over to (with units restored)

∂p

∂r= − ρ GM

r2, (16.61)

which is the usual Newtonian equation of spherically symmetric hydrostatic equilibrium.

16.4 Self-similar spherically symmetric spacetime

Even with the assumption of spherical symmetry, it is by no means easy to solve the system of partial

differential equations that comprise the Einstein equations coupled to mass-energy of various kinds. One

way to simplify the system of equations, transforming them into ordinary differential equations, is to consider

self-similar solutions.

16.4.1 Self-similarity

The assumption of self-similarity (also known as homothety, if you can pronounce it) is the assumption

that the system possesses conformal time translation invariance. This implies that there exists a conformal

time coordinate η such that the geometry at any one time is conformally related to the geometry at any

other time

ds2 = a(η)2[

g(c)ηη (x) dη2 + 2 g(c)

ηx (x) dη dx+ g(c)xx (x) dx2 + e2x do2

]

. (16.62)

Here the conformal metric coefficients g(c)µν (x) are functions only of conformal radius x, not of conformal time

η. The choice e2x of coefficient of do2 is a gauge choice of the conformal radius x, carefully chosen here so as

to bring the self-similar line-element into a form (16.66) below that resembles as far as possible the spherical

line-element (16.1). In place of the conformal factor a(η) it is convenient to work with the circumferential

radius r

r ≡ a(η)ex (16.63)

which is to be considered as a function r(η, x) of the coordinates η and x. The circumferential radius r has

a gauge-invariant meaning, whereas neither a(η) nor x are independently gauge-invariant. The conformal

factor r has the dimensions of length. In self-similar solutions, all quantities are proportional to some power of

r, and that power can be determined by dimensional analysis. Quantites that depend only on the conformal

radial coordinate x, independent of the circumferential radius r, are called dimensionless.

The fact that dimensionless quantities such as the conformal metric coefficients g(c)µν (x) are independent of

conformal time η implies that the tangent vector gη, which by definition satisfies

∂η= gη · ∂ , (16.64)

is a conformal Killing vector, also known as the homothetic vector. The tetrad-frame components of the

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16.4 Self-similar spherically symmetric spacetime 279

conformal Killing vector gη defines the tetrad-frame conformal Killing 4-vector ξm

∂η≡ r ξm∂m , (16.65)

in which the factor r is introduced so as to make ξm dimensionless. The conformal Killing vector gη is

the generator of the conformal time translation symmetry, and as such it is gauge-invariant (up to a global

rescaling of conformal time, η → bη for some constant b). It follows that its dimensionless tetrad-frame

components ξm constitute a tetrad 4-vector (again, up to global rescaling of conformal time).

16.4.2 Self-similar line-element

The self-similar line-element can be taken to have the same form as the spherical line-element (16.1), but

with the dependence on the dimensionless conformal Killing vector ξm made manifest:

ds2 = − r2[

(ξη dη)2 +1

β2x

(dx + βx ξxdη)

2+ do2

]

. (16.66)

The vierbein emµ and inverse vierbein em

µ corresponding to the self-similar line-element (16.66) are

emµ =

1

r

1/ξη − βx ξx/ξη 0 0

0 βx 0 0

0 0 1 0

0 0 0 1/ sin θ

, emµ = r

ξη 0 0 0

ξx 1/βx 0 0

0 0 1 0

0 0 0 sin θ

. (16.67)

It is straightforward to see that the coordinate time components of the inverse vierbein must be emη = r ξm,

since ∂/∂η = emη ∂m equals r ξm∂m, equation (16.65).

16.4.3 Tetrad-frame scalars and vectors

Since the conformal factor r is gauge-invariant, the directed gradient ∂mr constitutes a tetrad-frame 4-vector

βm (which unlike ξm is independent of any global rescaling of conformal time)

βm ≡ ∂mr . (16.68)

It is straightforward to check that βx defined by equation (16.68) is consistent with its appearance in the

vierbein (16.67) provided that r ∝ ex as earlier assumed, equation (16.63).

With two distinct dimensionless tetrad 4-vectors in hand, βm and the conformal Killing vector ξm, three

gauge-invariant dimensionless scalars can be constructed, βmβm, ξmβm, and ξmξm,

1− 2M

r= − βmβm = − β2

η + β2x , (16.69)

v ≡ ξmβm =1

r

∂r

∂η=

1

a

∂a

∂η, (16.70)

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280 General spherically symmetric spacetime

∆ ≡ − ξmξm = (ξη)2 − (ξx)2 . (16.71)

Equation (16.69) is essentially the same as equation (16.9). The dimensionless quantity v, equation (16.70),

may be interpreted as a measure of the expansion velocity of the self-similar spacetime. Equation (16.70)

shows that v is a function only of η (since a(η) is a function only of η), and it therefore follows that v must

be constant (since being dimensionless means that v must be a function only of x). Equation (16.70) then

also implies that the conformal factor a(η) must take the form

a(η) = evη . (16.72)

Because of the freedom of a global rescaling of conformal time, it is possible to set v = 1 without loss of

generality, but in practice it is convenient to keep v, because it is then transparent how to take the static

limit v → 0. Equation (16.72) along with (16.63) shows that the circumferential radius r is related to the

conformal coordinates η and x by

r = evη+x . (16.73)

The dimensionsless quantity ∆, equation (16.71), is the dimensionless horizon function: horizons occur where

the horizon function vanishes

∆ = 0 at horizons . (16.74)

16.4.4 Self-similar diagonal line-element

The self-similar line-element (16.66) can be brought to diagonal form by a coordinate transformation to

diagonal conformal coordinates η×, x× (subscripted × for diagonal)

η → η× = η + f(x) , x→ x× = x− vf(x) , (16.75)

which leaves unchanged the conformal factor r, equation (16.73). The resulting diagonal metric is

ds2 = r2(

−∆ dη2× +

dx2×

1− 2M/r + v2/∆+ do2

)

. (16.76)

The diagonal line-element (16.76) corresponds physically to the case where the tetrad frame is at rest in

the similarity frame, ξx = 0, as can be seen by comparing it to the line-element (16.66). The frame can be

called the similarity frame. The form of the metric coefficients follows from the line-element (16.66) and

the gauge-invariant scalars (16.69)–(16.71).

The conformal Killing vector in the similarity frame is ξm = ∆1/2, 0, 0, 0, and the 4-velocity of the

similarity frame in its own frame is um = 1, 0, 0, 0. Since both are tetrad 4-vectors, it follows that with

respect to a general tetrad frame

ξm = um ∆1/2 (16.77)

where um is the 4-velocity of the similarity frame with respect to the general frame. This shows that the

conformal Killing vector ξm in a general tetrad frame is proportional to the 4-velocity of the similarity frame

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16.4 Self-similar spherically symmetric spacetime 281

through the tetrad frame. In particular, the proper 3-velocity of the similarity frame through the tetrad

frame is

proper 3-velocity of similarity frame through tetrad frame =ξx

ξη. (16.78)

16.4.5 Ray-tracing line-element

It proves useful to introduce a “ray-tracing” conformal radial coordinate X related to the coordinate x× of

the diagonal line-element (16.76) by

dX ≡ ∆ dx×

[(1− 2M/r)∆ + v2]1/2

. (16.79)

In terms of the ray-tracing coordinate X , the diagonal metric is

ds2 = r2(

−∆ dη2× +

dX2

∆+ do2

)

. (16.80)

For the Reissner-Nordstrom geometry, ∆ = (1 − 2M/r)/r2, η× = t, and X = −1/r.

16.4.6 Geodesics

Spherical symmetry and conformal time translation symmetry imply that geodesic motion in spherically

symmetric self-similar spacetimes is described by a complete set of integrals of motion.

The integral of motion associated with conformal time translation symmetry can be obtained from La-

grange’s equations of motion

d

∂L

∂uη=∂L

∂η, (16.81)

with effective Lagrangian L = gµνuµuν for a particle with 4-velocity uµ. The self-similar metric depends on

the conformal time η only through the overall conformal factor gµν ∝ a(η)2. The derivative of the conformal

factor is given by ∂ ln a/∂η = v, equation (16.70), so it follows that ∂L/∂η = 2 vL. For a massive particle,

for which conservation of rest mass implies gµνuµuν = −1, Lagrange’s equations (16.81) thus yield

duη

dτ= − v . (16.82)

In the limit of zero accretion rate, v → 0, equation (16.82) would integrate to give uη as a constant, the

energy per unit mass of the geodesic. But here there is conformal time translation symmetry in place of

time translation symmetry, and equation (16.82) integrates to

uη = − v τ , (16.83)

in which an arbitrary constant of integration has been absorbed into a shift in the zero point of the proper

time τ . Although the above derivation was for a massive particle, it holds also for a massless particle, with the

understanding that the proper time τ is constant along a null geodesic. The quantity uη in equation (16.83)

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282 General spherically symmetric spacetime

is the covariant time component of the coordinate-frame 4-velocity uµ of the particle; it is related to the

covariant components um of the tetrad-frame 4-velocity of the particle by

uη = emη um = r ξmum . (16.84)

Without loss of generality, geodesic motion can be taken to lie in the equatorial plane θ = π/2 of the

spherical spacetime. The integrals of motion associated with conformal time translation symmetry, rotational

symmetry about the polar axis, and conservation of rest mass, are, for a massive particle

uη = − v τ , uφ = Lz , uµuµ = −1 , (16.85)

where Lz is the orbital angular momentum per unit rest mass of the particle. The coordinate 4-velocity

uµ ≡ dxµ/dτ that follows from equations (16.85) takes its simplest form in the conformal coordinates

η×, X, θ, φ of the ray-tracing metric (16.80)

uη× =v τ

r2∆, uX = ± 1

r2[

v2τ2 − (r2 + L2z)∆

]1/2, uφ =

Lz

r2. (16.86)

16.4.7 Null geodesics

The important case of a massless particle follows from taking the limit of a massive particle with infinite

energy and angular momentum, v τ →∞ and Lz → ∞. To obtain finite results, define an affine parameter

λ by CHECK dλ ≡ v τ dτ , and a 4-velocity in terms of it by vµ ≡ dxµ/dλ. The integrals of motion (16.85)

then become, for a null geodesic,

vη×= −1 , vφ = Jz , vµv

µ = 0 , (16.87)

where Jz ≡ Lz/(v τ) is the (dimensionless) conformal angular momentum of the particle. The 4-velocity vµ

along the null geodesic is then, in terms of the coordinates of the ray-tracing metric (16.80),

vη =1

r2∆, vX = ± 1

r2(

1− J2z ∆)1/2

, vφ =Jz

r2. (16.88)

Equations (16.88) yield the shape of a null geodesic by quadrature

φ =

Jz dX

(1− J2z ∆)1/2

. (16.89)

Equation (16.89) shows that the shape of null geodesics in spherically symmetric self-similar spacetimes

hinges on the behavior of the dimensionless horizon function ∆(X) as a function of the dimensionless ray-

tracing variable X . Null geodesics go through periapsis or apoapsis in the self-similar frame where the

denominator of the integrand of (16.89) is zero, corresponding to vX = 0.

In the Reissner-Nordstrom geometry there is a radius, the photon sphere, where photons can orbit in

circles for ever. In non-stationary self-similar solutions there is no conformal radius where photons can orbit

for ever (to remain at fixed conformal radius, the photon angular momentum would have to increase in

proportion to the conformal factor r). There is however a separatrix between null geodesics that do or do

not fall into the black hole, and the conformal radius where this occurs can be called the photon sphere

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16.4 Self-similar spherically symmetric spacetime 283

equivalent. The photon sphere equivalent occurs where the denominator of the integrand of equation (16.89)

not only vanishes, vX = 0, but is an extremum, which happens where the horizon function ∆ is an extremum,

d∆

dX= 0 at photon sphere equivalent . (16.90)

16.4.8 Dimensional analysis

Dimensional analysis shows that the conformal coordinates xµ ≡ η, x, θ, φ and the tetrad metric γmn are

dimensionless, while the coordinate metric gµν scales as r2,

xµ ∝ r0 , γmn ∝ r0 , gµν ∝ r2 . (16.91)

The vierbein emµ and inverse vierbein em

µ, equations (16.67), scale as

emµ ∝ r−1 , em

µ ∝ r . (16.92)

Coordinate derivatives ∂/∂xµ are dimensionless, while directed derivatives ∂m scale as 1/r,

∂xµ∝ r0 , ∂m ∝ r−1 . (16.93)

The tetrad connections Γkmn and the tetrad-frame Riemann tensor Rklmn scale as

Γkmn ∝ r−1 , Rklmn ∝ r−2 . (16.94)

16.4.9 Variety of self-similar solutions

Self-similar solutions exist provided that the properties of the energy-momentum introduce no additional

dimensional parameters. For example, the pressure-to-density ratio w ≡ p/ρ of any species is dimensionless,

and since the ratio can depend only on the nature of the species itself, not for example on where it happens

to be located in the spacetime, it follows that the ratio w must be a constant. It is legitimate for the

pressure-to-density ratio to be different in the radial and transverse directions (as it is for a radial electric

field), but otherwise self-similarity requires that

w ≡ p/ρ , w⊥ ≡ p⊥/ρ , (16.95)

be constants for each species. For example, w = 1 for an ultrahard fluid (which can mimic the behaviour

of a massless scalar field: E. Babichev, S. Chernov, V. Dokuchaev, Yu. Eroshenko, 2008, “Ultra-hard fluid

and scalar field in the Kerr-Newman metric,” Phys. Rev. D 78, 104027, arXiv:0807.0449), w = 1/3 for a

relativistic fluid, w = 0 for pressureless cold dark matter, w = −1 for vacuum energy, and w = −1 with

w⊥ = 1 for a radial electric field.

Self-similarity allows that the energy-momentum may consist of several distinct components, such as

a relativistic fluid, plus dark matter, plus an electric field. The components may interact with each other

provided that the properties of the interaction introduce no additional dimensional parameters. For example,

the relativistic fluid (and the dark matter) may be charged, and if so then the charged fluid will experience

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284 General spherically symmetric spacetime

a Lorentz force from the electric field, and will therefore exchange momentum with the electric field. If

the fluid is non-conducting, then there is no dissipation, and the interaction between the charged fluid and

electric field automatically introduces no additional dimensional parameters.

However, if the charged fluid is electrically conducting, then the electrical conductivity could potentially

introduce an additional dimensional parameter, and this must not be allowed if self-similarity is to be

maintained. In diffusive electrical conduction in a fluid of conductivity σ, an electric field E gives rise to a

current

j = σE , (16.96)

which is just Ohm’s law. Dimensional analysis shows that j ∝ r−2 and E ∝ r−1, so the conductivity must

scale as σ ∝ r−1. The conductivity can depend only on the intrinsic properties of the conducting fluid, and

the only intrinsic property available is its density, which scales as ρ ∝ r−2. If follows that the conductivity

must be proportional to the square root of the density ρ of the conducting fluid

σ = κ ρ1/2 , (16.97)

where κ is a dimensionless conductivity constant. The form (16.97) is required by self-similarity, and is

not necessarily realistic (although it is realistic that the conductivity increases with density). However, the

conductivity (16.97) is adequate for the purpose of exploring the consequences of dissipation in simple models

of black holes.

16.4.10 Tetrad connections

The expressions for the tetrad connections for the self-similar spacetime are the same as those (16.17) for

a general spherically symmetric spacetime, with just a relabelling of the time and radial coordinates into

conformal coordinates

t→ η , r → x . (16.98)

Specifically, equations (16.17) for the tetrad connections become become

Γηxη = hη , Γηxx = hx , Γηθθ = Γηφφ =βη

r, Γxθθ = Γxφφ =

βx

r, Γθφφ =

cot θ

r, (16.99)

in which hη and hx have the same physical interpretation discussed in §16.1.4 for the general spherically sym-

metric case: hη is the proper radial acceleration, and hx is the radial Hubble parameter. Expressions (16.18)

and (16.19) for hη and hx translate in the self-similar spacetime to

hη ≡ ∂x ln(r ξη) , hx ≡ ∂η ln(r ξx) . (16.100)

Comparing equations (16.100) to equations (16.18) and (16.23) shows that the vierbein coefficient α and

scale factor λ translate in the self-similar spacetime to

α =1

rξη, λ =

1

rξx. (16.101)

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16.4 Self-similar spherically symmetric spacetime 285

16.4.11 Spherical equations carry over to the self-similar case

The tetrad-frame Riemann, Weyl, and Einstein tensors in the self-similar spacetime take the same form

as in the general spherical case, equations (16.24)–(16.26), with just a relabelling (16.98) into conformal

coordinates.

Likewise, the equations for the interior mass in §16.1.8, for energy-momentum conservation in §16.1.9, for

the first law in §16.1.10, and the various equations for the electromagnetic field in §16.2, all carry through

unchanged except for a relabelling (16.98) of coordinates.

16.4.12 From partial to ordinary differential equations

The central simplifying feature of self-similar solutions is that they turn a system of partial differential

equations into a system of ordinary differential equations.

By definition, a dimensionless quantity F (x) is independent of conformal time η. It follows that the partial

derivative of any dimensionless quantity F (x) with respect to conformal time η vanishes

0 =∂F (x)

∂η= ξm∂mF (x) = (ξη∂η + ξx∂x)F (x) . (16.102)

Consequently the directed radial derivative ∂xF of a dimensionless quantity F (x) is related to its directed

time derivative ∂ηF by

∂xF (x) = − ξx

ξη∂ηF (x) . (16.103)

Equation (16.103) allows radial derivatives to be converted to time derivatives.

16.4.13 Integrals of motion

As remarked above, equation (16.102), in self-similar solutions ξm∂mF (x) = 0 for any dimensionless function

F (x). If both the directed derivatives ∂ηF (x) and ∂xF (x) are known from the Einstein equations or elsewhere,

then the result will be an integral of motion.

The spherically symmetric, self-similar Einstein equations admit two integrals of motion

0 = r ξm∂mβη =r βx(ξηhη + ξxhx)− ξη

(

M

r+ 4πr2p

)

+ ξx4πr2f , (16.104a)

0 = r ξm∂mβx =r βη(ξηhη + ξxhx) + ξx

(

M

r− 4πr2ρ

)

+ ξη4πr2f . (16.104b)

Taking ξx times (16.104a) plus ξη times (16.104b), and then βη times (16.104a) minus βx times (16.104b),

gives

0 = r v (ξηhη + ξxhx)− 4πr2[

ξηξx(ρ+ p)−(

(ξη)2 + (ξx)2)

f]

, (16.105a)

0 = r ξm∂mM

r= − v

M

r+ 4πr2 [βxξ

xρ− βηξηp+ (βηξ

x − βxξη)f ] . (16.105b)

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286 General spherically symmetric spacetime

The quantities in square brackets on the right hand sides of equations (16.105) are scalars for each species

a, so equations (16.105) can also be written

r v (ξηhη + ξxhx) = 4πr2∑

species a

ξηaξ

xa (ρa + pa) , (16.106a)

vM

r= 4πr2

species a

(βa,xξxaρa − βa,ηξ

ηapa) , (16.106b)

where the sum is over all species a, and βa,m and ξma are the 4-vectors βm and ξm expressed in the rest frame

of species a.

For electrically charged solutions, a third integral of motion comes from

0 = r ξm∂mQ

r= − v

Q

r+ 4πr2 (ξxq − ξηj) (16.107)

which is valid in any radial tetrad frame, not just the center-of-mass frame.

For a fluid with equation of state p = wρ, a fourth integral comes from considering

0 = r ξm∂m(r2p) = r[

w ξη∂η(r2ρ) + ξx∂x(r2p)]

(16.108)

and simplifying using the energy conservation equation for ∂ηρ and the momentum conservation equation

for ∂xp.

16.4.14 Integration variable

It is desirable to choose an integration variable that varies monotonically. A natural choice is the proper

time τ in the tetrad frame, since this is guaranteed to increase monotonically. Since the 4-velocity at rest

in the tetrad frame is by definition um = 1, 0, 0, 0, the proper time derivative is related to the directed

conformal time derivative in the tetrad frame by d/dτ = um∂m = ∂η.

However, there is another choice of integration variable, the ray-tracing variable X defined by equa-

tion (16.79), that is not specifically tied to any tetrad frame, and that has a desirable (tetrad and coordinate)

gauge-invariant meaning. The proper time derivative of any dimensionless function F (x) in the tetrad frame

is related to its derivative dF/dX with respect to the ray-tracing variable X by

∂ηF = um∂mF = uX∂XF = − ξx

r

dF

dX. (16.109)

In the third expression, uX∂XF is um∂mF expressed in the similarity frame of §16.4.4, the time contribu-

tion uη×∂η×F vanishing in the similarity frame because it is proportional to the conformal time derivative

∂F/∂η× = 0. In the last expression of (16.109), uX has been replaced by −ux = −ξx/∆1/2 in view of equa-

tion (16.77), the minus sign coming from the fact that uX is the radial component of the tetrad 4-velocity of

the tetrad frame relative to the similarity frame, while ux in equation (16.77) is the radial component of the

tetrad 4-velocity of the similarity frame relative to the tetrad frame. Also in the last expression of (16.109),

the directed derivative ∂X with respect to the ray-tracing variable X has been translated into its coordinate

partial derivative, ∂X = (∆1/2/r) ∂/∂X , which follows from the metric (16.80).

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16.4 Self-similar spherically symmetric spacetime 287

In summary, the chosen integration variable is the dimensionless ray-tracing variable −X (with a minus

because −X is monotonically increasing), the derivative with respect to which, acting on any dimensionless

function, is related to the proper time derivative in any tetrad frame (not just the baryonic frame) by

− d

dX=

r

ξx∂η . (16.110)

Equation (16.110) involves ξx, which is proportional to the proper velocity of the tetrad frame through the

similarity frame, equation (16.78), and which therefore, being initially positive, must always remain positive

in any tetrad frame attached to a fluid, as long as the fluid does not turn back on itself, as must be true for

the self-similar solution to be consistent.

16.4.15 Summary of equations for a single charged fluid

For reference, it is helpful to collect here the full set of equations governing self-similar spherically symmetric

evolution in the case of a single charged fluid with isotropic equation of state

p = p⊥ = w ρ , (16.111)

and conductivity

σ = κ ρ1/2 . (16.112)

In accordance with the arguments in §16.4.9, equations (16.95) and (16.97), self-similarity requires that the

pressure-to-density ratio w and the conductivity coefficient κ both be (dimensionless) constants.

It is natural to work in the center-of-mass frame of the fluid, which also coincides with the center-of-mass

frame of the fluid plus electric field (the electric field, being invariant under Lorentz boosts, does not pick

out any particular radial frame).

The proper time τ in the fluid frame evolves as

− dτ

dX=

r

ξx, (16.113)

which follows from equation (16.110) and the fact that ∂ητ = 1. The circumferential radius r evolves along

the path of the fluid as

− d ln r

dX=βη

ξx. (16.114)

Although it is straightforward to write down the equations governing how the tetrad frame moves through the

conformal coordinates η and x, there is not much to be gained from this because the conformal coordinates

have no fundamental physical significance.

Next, the defining equations (16.100) for the proper acceleration hη and Hubble parameter hx yield

equations for the evolution of the time and radial components of the conformal Killing vector ξm

− dξη

dX= βx − rhη , (16.115a)

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288 General spherically symmetric spacetime

− dξx

dX= − βη + rhx , (16.115b)

in which, in the formula for hη, equation (16.103) has been used to convert the conformal radial derivative

∂x to the conformal time derivative ∂η, and thence to −d/dX by equation (16.110).

Next, the Einstein equations (16.26c) and (16.26c) (with coordinates relabeled per (16.98) in the center-of-

mass frame (16.33) yield evolution equations for the time and radial components of the vierbein coefficients

βm

− dβη

dX= − βx

ξηrhx , (16.116a)

− dβx

dX=βη

ξxrhη , (16.116b)

where again, in the formula for βη, equation (16.103) has been used to convert the conformal radial derivative

∂x to the conformal time derivative ∂η. The 4 evolution equations (16.115) and (16.116) for ξm and βm are

not independent: they are related by ξmβm = v, a constant, equation (16.70). To maintain numerical

precision, it is important to avoid expressing small quantities as differences of large quantities. In practice,

a suitable choice of variables to integrate proves to be ξη + ξx, βη − βx, and βx, each of which can be

tiny in some circumstances. Starting from these variables, the following equations yield ξη − ξx, along with

the interior mass M and the horizon function ∆, equations (16.69) and (16.71), in a fashion that ensures

numerical stability:

ξη − ξx =2v− (ξη + ξx)(βη + βx)

βη − βx, (16.117a)

2M

r= 1 + (βη + βx)(βη − βx) , (16.117b)

∆ = (ξη + ξx)(ξη − ξx) . (16.117c)

The evolution equations (16.115) and (16.116) involve hη and hx. The integrals of motion considered in

§16.4.13 yield explicit expressions for hη and hx not involving any derivatives. For the Hubble parameter

hx, equation (16.105a) gives

rhx = − ξη

ξxrhη +

ξη

v4πε , (16.118)

where ε is the dimensionless enthalpy

ε ≡ r2(1 + w)ρ . (16.119)

For the proper acceleration hη, a somewhat lengthy calculation starting from the integral of motion (16.108),

and simplifying using the integral of motion (16.107) for Q, the expression (16.118) for hx, Maxwell’s equa-

tion (16.54b) [with the relabelling (16.98)], and the conductivity (16.112) in Ohm’s law (16.96), gives

rhη =ξx

8πw⊥(βxξx − wβηξ

η)r2ρ+ [v + (1 + w)4πrσξη ]Q2/r2 − w(4πξηε)2/v

4πε [(ξx)2 − w(ξη)2]. (16.120)

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16.4 Self-similar spherically symmetric spacetime 289

Two more equations complete the suite. The first, which represents energy conservation for the fluid, can

be written as an equation governing the entropy S of the fluid

− d lnS

dX=

σQ2

r(1 + w)ρξx, (16.121)

in which the S is (up to an arbitrary constant) the entropy of a comoving volume element V ∝ r3ξx of the

fluid

S ≡ r3ξxρ1/(1+w) . (16.122)

That equation (16.121) really is an entropy equation can be confirmed by rewriting it as

1

V

(

dρV

dτ+ p

dV

)

= jE =σQ2

r4, (16.123)

in which jE is recognized as the Ohmic dissipation, the rate per unit volume at which the volume element

V is being heated.

The final equation represents electromagnetic energy conservation, equation (16.57a), which can be written

− d lnQ

dX= − 4πrσ

ξx. (16.124)

The (heat) energy going into the fluid is balanced by the (free) energy coming out of the electromagnetic

field.

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17

The interiors of spherical black holes

As discussed in Chapter 8, the Reissner-Nordstrom geometry for an ideal charged spherical black hole

contains mathematical wormhole and white hole extensions to other universes. In reality, these extensions

are not expected to occur, thanks to the mass inflation instability discovered by Poisson & Israel (1990).

17.1 The mechanism of mass inflation

17.1.1 Reissner-Nordstrom phase

Figure 17.1 illustrates how the two Einstein equations (16.52) produce the three phases of mass inflation

inside a charged spherical black hole.

During the initial phase, illustrated in the top panel of Figure 17.1, the spacetime geometry is well-

approximated by the vacuum, Reissner-Nordstrom geometry. During this phase the radial energy flux f is

effectively zero, so βr remains constant, according to equation (16.52b). The change in the radial velocity βt,

equation (16.52a), depends on the competition between the Newtonian gravitational force −M/r2, which is

always attractive (tending to make the radial velocity βt more negative), and the gravitational force −4πrp

sourced by the radial pressure p. In the Reissner-Nordstrom geometry, the static electric field produces a

negative radial pressure, or tension, p = −Q2/(8πr4), which produces a gravitational repulsion −4πrp =

Q2/(2r3). At some point (depending on the charge-to-mass ratio) inside the outer horizon, the gravitational

repulsion produced by the tension of the electric field exceeds the attraction produced by the interior mass

M , so that the radial velocity βt slows down. This regime, where the (negative) radial velocity βt is slowing

down (becoming less negative), while βr remains constant, is illustrated in the top panel of Figure 17.1.

If the initial Reissner-Nordstrom phase were to continue, then the radial 4-gradient βm would become

lightlike. In the Reissner-Nordstrom geometry this does in fact happen, and where it happens defines the

inner horizon. The problem with this is that the lightlike 4-vector βm points in one direction for ingoing

frames, and in the opposite direction for outgoing frames. If βm becomes lightlike, then ingoing and outgoing

frames are streaming through each other at the speed of light. This is the infinite blueshift at the inner

horizon first pointed out by Penrose (1968).

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17.1 The mechanism of mass inflation 291

βt

βr

outg

oing

ingoing

1. RN

βr

outg

oing

ingoing

2. Inflation

βr

outg

oing

ingoing

3. Collapse

Figure 17.1 Spacetime diagrams of the tetrad-frame 4-vector βm, equation (16.8), illustrating qualitativelythe three successive phases of mass inflation: 1. (top) the Reissner-Nordstrom phase, where inflation ignites;2. (middle) the inflationary phase itself; and 3. (bottom) the collapse phase, where inflation comes to an end.In each diagram, the arrowed lines labeled ingoing and outgoing illustrate two representative examples ofthe 4-vector βt, βr, while the double-arrowed lines illustrate the rate of change of these 4-vectors impliedby Einstein’s equations (16.52). Inside the horizon of a black hole, all locally inertial frames necessarily fallinward, so the radial velocity βt ≡ ∂tr is always negative. A locally inertial frame is ingoing or outgoingdepending on whether the proper radial gradient βr ≡ ∂rr measured in that frame is positive or negative.

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292 The interiors of spherical black holes

If there were no matter present, or if there were only one stream of matter, either ingoing or outgoing but

not both, then βm could indeed become lightlike. But if both ingoing and outgoing matter are present, even

in the tiniest amount, then it is physically impossible for the ingoing and outgoing frames to stream through

each other at the speed of light.

If both ingoing and outgoing streams are present, then as they race through each other ever faster, they

generate a radial pressure p, and an energy flux f , which begin to take over as the main source on the right

hand side of the Einstein equations (16.52). This is how mass inflation is ignited.

17.1.2 Inflationary phase

The infalling matter now enters the second, mass inflationary phase, illustrated in the middle panel of

Figure 17.1.

During this phase, the gravitational force on the right hand side of the Einstein equation (16.52a) is

dominated by the pressure p produced by the counter-streaming ingoing and outgoing matter. The mass M

is completely sub-dominant during this phase (in this respect, the designation “mass inflation” is misleading,

since although the mass inflates, it does not drive inflation). The counter-streaming pressure p is positive,

and so accelerates the radial velocity βt (makes it more negative). At the same time, the radial gradient

βr is being driven by the energy flux f , equation (16.52b). For typically low accretion rates, the streams

are cold, in the sense that the streaming energy density greatly exceeds the thermal energy density, even if

the accreted material is at relativistic temperatures. This follows from the fact that for mass inflation to

begin, the gravitational force produced by the counter-streaming pressure p must become comparable to that

produced by the mass M , which for streams of low proper density requires a hyper-relativistic streaming

velocity. For a cold stream of proper density ρ moving at 4-velocity um ≡ ut, ur, 0, 0, the streaming energy

flux would be f ∼ ρutur, while the streaming pressure would be p ∼ ρ(ur)2. Thus their ratio f/p ∼ ut/ur

is slightly greater than one. It follows that, as illustrated in the middle panel of Figure 17.1, the change in

βr slightly exceeds the change in βt, which drives the 4-vector βm, already nearly lightlike, to be even more

nearly lightlike. This is mass inflation.

Inflation feeds on itself. The radial pressure p and energy flux f generated by the counter-streaming ingoing

and outgoing streams increase the gravitational force. But, as illustrated in the middle panel of Figure 17.1,

the gravitational force acts in opposite directions for ingoing and outgoing streams, tending to accelerate

the streams faster through each other. An intuitive way to understand this is that the gravitational force

is always inwards, meaning in the direction of smaller radius, but the inward direction is towards the black

hole for ingoing streams, and away from the black hole for outgoing streams.

The feedback loop in which the streaming pressure and flux increase the gravitational force, which ac-

celerates the streams faster through each other, which increases the streaming pressure and flux, is what

drives mass inflation. Inflation produces an exponential growth in the streaming energy, and along with it

the interior mass, and the Weyl curvature.

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17.2 The far future? 293

17.1.3 Collapse phase

It might seem that inflation is locked into an exponential growth from which there is no exit. But the

Einstein equations (16.52) have one more trick up their sleave.

For the counter-streaming velocity to continue to increase requires that the change in βr from equa-

tion (16.52b) continues to exceed the change in βt from equation (16.52a). This remains true as long as the

counter-streaming pressure p and energy flux f continue to dominate the source on the right hand side of

the equations. But the mass term −M/r2 also makes a contribution to the change in βt, equation (16.52a).

As will be seen in the examples of the next two sections, §§?? and ??, at least in the case of pressureless

streams the mass term exponentiates slightly faster than the pressure term. At a certain point, the additional

acceleration produced by the mass means that the combined gravitational force M/r2 + 4πrp exceeds 4πrf .

Once this happens, the 4-vector βm, instead of being driven to becoming more lightlike, starts to become

less lightlike. That is, the counter-streaming velocity starts to slow. At that point inflation ceases, and the

streams quickly collapse to zero radius.

It is ironic that it is the increase of mass that brings mass inflation to an end. Not only does mass not

drive mass inflation, but as soon as mass begins to contribute significantly to the gravitational force, it brings

mass inflation to an end.

17.2 The far future?

The Penrose diagram of a Reissner-Nordstrom or Kerr-Newman black hole indicates that an observer who

passes through the outgoing inner horizon sees the entire future of the outside universe go by. In a sense,

this is “why” the outside universe appears infinitely blueshifted.

This raises the question of whether what happens at the outgoing inner horizon of a real black hole indeed

depends on what happens in the far future. If it did, then the conclusions of previous sections, which are

based in part on the proposition that the accretion rate is approximately constant, would be suspect. A

lot can happen in the far future, such as black hole mergers, the Universe ending in a big crunch, Hawking

evaporation, or something else beyond our current ken.

Ingoing and outgoing observers both see each other highly blueshifted near the inner horizon. An outgoing

observer sees ingoing observers from the future, while an ingoing observer sees outgoing observers from the

past. If the streaming 4-velocity between ingoing and outgoing streams is umba, equation (??), then the proper

time dτb that elapses on stream b observed by stream a equals the blueshift factor utba times the proper time

dτa experienced by stream a,

dτb = utba dτa . (17.1)

17.2.1 Inflationary phase

A physically relevant timescale for the observing stream a is how long it takes for the blueshift to increase

by one e-fold, which is dτa/d lnutba. During the inflationary phase, stream a sees the amount of time elapsed

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294 The interiors of spherical black holes

on stream b through one e-fold of blueshift to be

utba

dτad lnut

ba

=2Cbr−λ

. (17.2)

The right hand side of equation (17.2) is derived from utba ≈ 2|uaub|, |dra/dτa| = |βa,t| ≈ βua, ra ≈ r−,

and the approximations (??) and (??) valid during the inflationary phase. The constants Cb and λ on the

right hand side of equation (17.2) are typically of order unity, while r− is the radius of the inner horizon

where mass inflation takes place. Thus the right hand side of equation (17.2) is of the order of one black hole

crossing time. In other words, stream a sees approximately one black hole crossing time elapse on stream b

for each e-fold of blueshift.

For astronomically realistic black holes, exponentiating the Weyl curvature up to the Planck scale will

take typically a few hundred e-folds of blueshift, as illustrated for example in Figure 17.10. Thus what

happens at the inner horizon of a realistic black hole before quantum gravity intervenes depends only on the

immediate past and future of the black hole – a few hundred black hole crossing times – not on the distant

future or past. This conclusion holds even if the accretion rate of one of the ingoing or outgoing streams is

tiny compared to the other, as considered in §??.

From a stream’s own point of view, the entire inflationary episode goes by in a flash. At the onset of

inflation, where β goes through its minimum, at µau2a = µbu

2b = λ according to equation (??), the blueshift

utba is already large

utba =

2λ√µaµb

, (17.3)

thanks to the small accretion rates µa and µb. During the first e-fold of blueshift, each stream experiences a

proper time of order√µaµb times the black hole crossing time, which is tiny. Subsequent e-folds of blueshift

race by in proportionately shorter proper times.

17.2.2 Collapse phase

The time to reach the collapse phase is another matter. According to the estimates in §§?? and ??, reaching

the collapse phase takes of order ∼ 1/µa e-folds of blueshift, where µa is the larger of the accretion rates

of the ingoing and outgoing streams, equation (??). Thus in reaching the collapse phase, each stream has

seen approximately 1/µa black hole crossing times elapse on the other stream. But 1/µa black hole crossing

times is just the accretion time – essentially, how long the black hole has existed. This timescale, the age of

the black hole, is not infinite, but it can hardly be expected that the accretion rate of a black hole would be

constant over its lifetime.

If the accretion rate were in fact constant, and if quantum gravity did not intervene and the streams

remained non-interacting, then indeed streams inside the black hole would reach the collapse phase, where-

upon they would plunge to a spacelike singularity at zero radius. For example, in the self-similar models

illustrated in Figures ?? and 17.10, outgoing baryons hitting the central singularity see ingoing dark matter

accreted a factor of two into the future (specifically, for ingoing baryons and outgoing dark matter hitting the

singularity, the radius of the outer horizon when the dark matter is accreted is twice that when the baryons

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17.3 Self-similar models of the interior structure of black holes 295

were accreted; the numbers are 2.11 for M• = 0.03, 1.97 for M• = 0.01, and unknown for M• = 0.003 because

in that case the numerics overflow before the central singularity is reached). The same conclusion applies to

the ultra-hard fluid models illustrated in Figure 17.8: if the accretion rate is constant, then outgoing streams

see only a factor of order unity or a few into the future before hitting the central singularity.

If on the other hard the accretion rate decreases sufficiently rapidly with time, then it is possible that

an outgoing stream never reaches the collapse phase, because the number of e-folds to reach the collapse

phase just keeps increasing as the accretion rate decreases. By contrast, an ingoing stream is always liable to

reach the collapse phase, if quantum gravity does not intervene, because an ingoing stream sees the outgoing

stream from the past, when the accretion rate was liable to have been larger.

However, as already remarked in §??, in astronomically realistic black holes, it is only for large accretion

rates, such as may occur when the black hole first collapses (M• & 0.01 for the models illustrated in

Fig. 17.10), that the collapse phase has a chance of being reached before the Weyl curvature exceeds the

Planck scale.

To summarize, it is only streams accreted during the first few hundred or so black hole crossing times

since a black hole’s formation that have a chance of hitting a central spacelike singularity. Streams accreted

at later times, whether ingoing or outgoing, are likely to meet their fate in the inflationary zone at the inner

horizon, where the Weyl curvature exponentiates to the Planck scale and beyond.

17.3 Self-similar models of the interior structure of black holes

The apparatus is now in hand actually to do some real calculations of the interior structure of black holes.

All the models presented in this section are spherical and self-similar. See Hamilton & Pollack (2005, PRD

71, 084031 & 084031) and Wallace, Hamilton & Polhemus (2008, arXiv:0801.4415) for more.

17.3.1 Boundary conditions at an outer sonic point

Because information can propagate only inward inside the horizon of a black hole, it is natural to set the

boundary conditions outside the horizon. The policy adopted here is to set them at a sonic point, where

the infalling fluid accelerates from subsonic to supersonic. The proper 3-velocity of the fluid through the

self-similar frame is ξx/ξη, equation (16.78) (the velocity ξx/ξη is positive falling inward), and the sound

speed is

sound speed =

pb

ρb=√wb , (17.4)

and sonic points occur where the velocity equals the sound speed

ξx

ξη= ±√wb at sonic points . (17.5)

The denominator of the expression (16.120) for the proper acceleration g is zero at sonic points, indicating

that the acceleration will diverge unless the numerator is also zero. What happens at a sonic point depends

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296 The interiors of spherical black holes

on whether the fluid transitions from subsonic upstream to supersonic downstream (as here) or vice versa.

If (as here) the fluid transitions from subsonic to supersonic, then sound waves generated by discontinuities

near the sonic point can propagate upstream, plausibly modifying the flow so as to ensure a smooth transition

through the sonic point, effectively forcing the numerator, like the denominator, of the expression (16.120)

to pass through zero at the sonic point. Conversely, if the fluid transitions from supersonic to subsonic, then

sound waves cannot propagate upstream to warn the incoming fluid that a divergent acceleration is coming,

and the result is a shock wave, where the fluid accelerates discontinuously, is heated, and thereby passes

from supersonic to subsonic.

The solutions considered here assume that the acceleration g at the sonic point is not only continuous [so

the numerator of (16.120) is zero] but also differentiable. Such a sonic point is said to be regular, and the

assumption imposes two boundary conditions at the sonic point.

The accretion in real black holes is likely to be much more complicated, but the assumption of a regular

sonic point is the simplest physically reasonable one.

17.3.2 Mass and charge of the black hole

The mass M• and charge Q• of the black hole at any instant can be defined to be those that would be

measured by a distant observer if there were no mass or charge outside the sonic point

M• = M +Q2

2r, Q• = Q at the sonic point . (17.6)

The mass M• in equation (17.6) includes the mass-energy Q2/2r that would be in the electric field outside

the sonic point if there were no charge outside the sonic point, but it does not include mass-energy from any

additional mass or charge that might be outside the sonic point.

In self-similar evolution, the black hole mass increases linearly with time, M• ∝ t, where t is the proper

time at rest far from the black hole. As discussed in §??, this time t equals the proper time τd = rξηd/v

recorded by dark matter clocks that free-fall radially from zero velocity at infinity. Thus the mass accretion

rate M• is

M• ≡dM•dt

=M•τd

=vM•rξη

d

at the sonic point . (17.7)

If there is no mass outside the sonic point (apart from the mass-energy in the electric field), then a freely-

falling dark matter particle will have

βd,x = 1 at the sonic point , (17.8)

which can be taken as the boundary condition on the dark matter at the sonic point, for either massive or

massless dark matter. Equation (17.8) follows from the facts that the geodesic equations in empty space

around a charged black hole (Reissner-Nordstrom metric) imply that βd,x = constant for a radially free-

falling particle (the same conclusion can drawn from the Einstein equation (16.26c)), and that a particle at

rest at infinity satisfies βd,η = ∂d,ηr = 0, and consequently βd,x = 1 from equation (16.69) with r →∞.

As remarked following equation (16.72), the residual gauge freedom in the global rescaling of conformal

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17.3 Self-similar models of the interior structure of black holes 297

time η allows the expansion velocity v to be adjusted at will. One choice suggested by equation (17.7) is to

set (but one could equally well set v to the expansion velocity of the horizon, v = r+, for example)

v = M• , (17.9)

which is equivalent to setting

ξηd =

M•r

at the sonic point . (17.10)

Equation (17.10) is not a boundary condition: it is just a choice of units of conformal time η. Equation (17.10)

and the boundary condition (17.8) coupled with the scalar relations (16.69) and (16.70) fully determine the

dark matter 4-vectors βd,m and ξmd at the sonic point.

17.3.3 Equation of state

The density ρb and temperature Tb of an ideal relativistic baryonic fluid in thermodynamic equilibrium are

related by

ρb =π2g

30T 4

b , (17.11)

where

g = gB +7

8gF (17.12)

is the effective number of relativistic particle species, with gB and gF being the number of bosonic and

fermionic species. If the expected increase in g with temperature T is modeled (so as not to spoil self-

similarity) as a weak power law g/gp = T ǫ, with gp the effective number of relativistic species at the Planck

temperature, then the relation between density ρb and temperature Tb is

ρb =π2gp30

T(1+w)/wb , (17.13)

with equation of state parameter wb = 1/(3 + ǫ) slightly less than the standard relativistic value w = 1/3.

In the models considered here, the baryonic equation of state is taken to be

wb = 0.32 . (17.14)

The effective number gp is fixed by setting the number of relativistic particles species to g = 5.5 at T =

10 MeV, corresponding to a plasma of relativistic photons, electrons, and positrons. This corresponds to

choosing the effective number of relativistic species at the Planck temperature to be gp ≈ 2,400, which is

not unreasonable.

The chemical potential of the relativistic baryonic fluid is likely to be close to zero, corresponding to equal

numbers of particles and anti-particles. The entropy Sb of a proper Lagrangian volume element V of the

fluid is then

Sb =(ρb + pb)V

Tb, (17.15)

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298 The interiors of spherical black holes

which agrees with the earlier expression (16.122), but now has the correct normalization.

17.3.4 Entropy creation

One fundamentally interesting question about black hole interiors is how much entropy might be created

inside the horizon. Bekenstein first argued that a black hole should have a quantum entropy proportional to

its horizon area A, and Hawking (1974) supplied the constant of proportionality 1/4 in Planck units. The

Bekenstein-Hawking entropy SBH is, in Planck units c = G = ~ = 1,

SBH =A

4. (17.16)

For a spherical black hole of horizon radius r+, the area is A = 4πr2+. Hawking showed that a black hole

has a temperature TH equal to 1/(2π) times the surface gravity g+ at its horizon, again in Planck units,

TH =g+2π

. (17.17)

The surface gravity is defined to be the proper radial acceleration measured by a person in free-fall at the

horizon. For a spherical black hole, the surface gravity is g+ = −Dtβt = M/r2 + 4πrp evaluated at the

horizon, equation (16.52a).

The proper velocity of the baryonic fluid through the sonic point equals ξx/ξη, equation (16.78). Thus

the entropy Sb accreted through the sonic point per unit proper time of the fluid is

dSb

dτ=

(1 + wb)ρb

Tb

4πr2ξx

ξη. (17.18)

The horizon radius r+, which is at fixed conformal radius x, expands in proportion to the conformal factor,

r+ ∝ a, and the conformal factor a increases as d ln a/dτ = ∂η ln a = v/(rξη), so the Bekenstein-Hawking

entropy SBH = πr2+ increases as

dSBH

dτ=

2πr2+v

rξη. (17.19)

Thus the entropy Sb accreted through the sonic point per unit increase of the Bekenstein-Hawking entropy

SBH is

dSb

dSBH=

(1 + wb)ρb4πr3ξx

2πr2+vTb

r=rs

. (17.20)

Inside the sonic point, dissipation increases the entropy according to equation (16.121). Since the entropy

can diverge at a central singularity where the density diverges, and quantum gravity presumably intervenes

at some point, it makes sense to truncate the production of entropy at a “splat” point where the density ρb

hits a prescribed splat density ρ#

ρb = ρ# . (17.21)

Integrating equation (16.121) from the sonic point to the splat point yields the ratio of the entropies at the

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17.3 Self-similar models of the interior structure of black holes 299

sonic and splat points. Multiplying the accreted entropy, equation (17.20), by this ratio yields the rate of

increase of the entropy of the black hole, truncated at the splat point, per unit increase of its Bekenstein-

Hawking entropy

dSb

dSBH=

(1 + wb)ρb4πr3ξx

2πr2+vTb

ρb=ρ#

. (17.22)

17.3.5 Holography

The idea that the entropy of a black hole cannot exceed its Bekenstein-Hawking entropy has motivated

holographic conjectures that the degrees of freedom of a volume are somehow encoded on its boundary,

and consequently that the entropy of a volume is bounded by those degrees of freedom. Various counter-

examples dispose of most simple-minded versions of holographic entropy bounds. The most successful entropy

bound, with no known counter-examples, is Bousso’s covariant entropy bound (Bousso 2002, Rev. Mod.

Phys. 74, 825). The covariant entropy bound concerns not just any old 3-dimensional volume, but rather the

3-dimensional volume formed by a null hypersurface, a lightsheet. For example, the horizon of a black hole is

a null hypersurface, a lightsheet. The covariant entropy bound asserts that the entropy that passes (inward

or outward) through a lightsheet that is everywhere converging cannot exceed 1/4 of the 2-dimensional area

of the boundary of the lightsheet.

In the self-similar black holes under consideration, the horizon is expanding, and outgoing lightrays that

sit on the horizon do not constitute a converging lightsheet. However, a spherical shell of ingoing lightrays

that starts on the horizon falls inwards and therefore does form a converging lightsheet, and a spherical shell

of outgoing lightrays that starts just slightly inside the horizon also falls inward and forms a converging

lightsheet. The rate at which entropy Sb passes through such ingoing or outgoing spherical lightsheets per

unit decrease in the area Scov ≡ πr2 of the lightsheet is∣

dSb

dScov

=dSb

dSBH

r2+r2

v

ξx|βη ∓ βx|, (17.23)

in which the ∓ sign is − for ingoing, + for outgoing lightsheets. A sufficient condition for Bousso’s covariant

entropy bound to be satisfied is

|dSb/dScov| ≤ 1 . (17.24)

17.3.6 Black hole accreting a neutral relativistic plasma

The simplest case to consider is that of a black hole accreting a neutral relativistic plasma. In the self-similar

solutions, the charge of the black hole is produced self-consistently by the accreted charge of the baryonic

fluid, so a neutral fluid produces an uncharged black hole.

Figure 17.2 shows the baryonic density ρb and Weyl curvature C inside the uncharged black hole. The

mass and accretion rate have been taken to be

M• = 4× 106 M⊙ , M• = 10−16 , (17.25)

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300 The interiors of spherical black holes

hori

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Plan

cksc

ale

ρb

−C

dSb/dSBH

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.2 An uncharged baryonic plasma falls into an uncharged spherical black hole. The plot shows inPlanck units, as a function of radius, the plasma density ρb, the Weyl curvature scalar C (which is negative),and the rate dSb/dSBH of increase of the plasma entropy per unit increase in the Bekenstein-Hawking entropy

of the black hole. The mass is M• = 4×106 M⊙, the accretion rate is M• = 10−16, and the equation of stateis wb = 0.32.

which are motivated by the fact that the mass of the supermassive black hole at the center of the Milky Way

is 4× 106 M⊙, and its accretion rate is

Mass of MW black hole

age of Universe≈ 4× 106 M⊙

1010 yr≈ 6× 1060 Planck units

4× 1044 Planck units≈ 10−16 . (17.26)

Figure 17.2 shows that the baryonic plasma plunges uneventfully to a central singularity, just as in the

Schwarzschild solution. The Weyl curvature scalar hits the Planck scale, |C| = 1, while the baryonic proper

density ρb is still well below the Planck density, so this singularity is curvature-dominated.

17.3.7 Black hole accreting a non-conducting charged relativistic plasma

The next simplest case is that of a black hole accreting a charged but non-conducting relativistic plasma.

Figure 17.3 shows a black hole with charge-to-mass Q•/M• = 10−5, but otherwise the same parameters

as in the uncharged black hole of §17.3.6: M• = 4 × 106 M⊙, M• = 10−16, and wb = 0.32. Inside the outer

horizon, the baryonic plasma, repelled by the electric charge of the black hole self-consistently generated by

the accretion of the charged baryons, becomes outgoing. Like the Reissner-Nordstrom geometry, the black

hole has an (outgoing) inner horizon. The baryons drop through the inner horizon, shortly after which the

self-similar solution terminates at an irregular sonic point, where the proper acceleration diverges. Normally

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17.3 Self-similar models of the interior structure of black holes 301

hori

zon

inne

rho

rizo

n

ρb

ρe

|C|

dSb/dSBH

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.3 A plasma that is charged but non-conducting. The black hole has an inner horizon like theReissner-Nordstrom geometry. The self-similar solution terminates at an irregular sonic point just beneaththe inner horizon. The mass is M• = 4 × 106 M⊙, accretion rate M• = 10−16, equation of state wb = 0.32,and black hole charge-to-mass Q•/M• = 10−5.

this is a signal that a shock must form, but even if a shock is introduced, the plasma still terminates at an

irregular sonic point shortly downstream of the shock. The failure of the self-similar to continue does not

invalidate the solution, because the failure is hidden beneath the inner horizon, and cannot be communicated

to infalling matter above it.

The solution is nevertheless not realistic, because it assumes that there is no ingoing matter, such as

would inevitably be produced for example by infalling neutral dark matter. Such ingoing matter would

appear infinitely blueshifted to the outgoing baryons falling through the inner horizon, which would produce

mass inflation, as in §17.3.10.

17.3.8 Black hole accreting a conducting relativistic plasma

What happens if the baryonic plasma is not only electrically charged but also electrically conducting? If the

conductivity is small, then the solutions resemble the non-conducting solutions of the previous subsection,

§17.3.7. But if the conductivity is large enough effectively to neutralize the plasma as it approaches the

center, then the plasma can plunge all the way to the central singularity, as in the uncharged case in §17.3.6.

Figure 17.4 shows a case in which the conductivity has been tuned to equal, within numerical accuracy, the

critical conductivity κb = 0.35 above which the plasma collapses to a central singularity. The parameters are

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302 The interiors of spherical black holes

hori

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Plan

cksc

ale

ρb

ρe

|C|

|dSb/dScov|

dSb/dSBH

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.4 Here the baryonic plasma is charged, and electrically conducting. The conductivity is at (withinnumerical accuracy) the threshold above which the plasma plunges to a central singularity. The mass is

M• = 4 × 106 M⊙, the accretion rate M• = 10−16, the equation of state wb = 0.32, the charge-to-massQ•/M• = 10−5, and the conductivity parameter κb = 0.35. Arrows show how quantities vary a factor of 10into the past and future.

otherwise the same as in previous subsections: a mass of M• = 4× 106 M⊙, an accretion rate M• = 10−16,

an equation of state wb = 0.32, and a black hole charge-to-mass of Q•/M• = 10−5.

The solution at the critical conductivity exhibits the periodic self-similar behavior first discovered in

numerical simulations by Choptuik (1993, PRL 70, 9), and known as “critical collapse” because it happens

at the borderline between solutions that do and do not collapse to a black hole. The ringing of curves in

Figure 17.4 is a manifestation of the self-similar periodicity, not a numerical error.

These solutions are not subject to the mass inflation instability, and they could therefore be prototypical

of the behavior inside realistic rotating black holes. For this to work, the outward transport of angular

momentum inside a rotating black hole must be large enough effectively to produce zero angular momentum

at the center. My instinct is that angular momentum transport is probably not strong enough, but I do not

know this for sure. If angular momentum transport is not strong enough, then mass inflation will take place.

Figure 17.4 shows that the entropy produced by Ohmic dissipation inside the black hole can potentially

exceed the Bekenstein-Hawking entropy of the black hole by a large factor. The Figure shows the rate

dSb/dSBH of increase of entropy per unit increase in its Bekenstein-Hawking entropy, as a function of the

hypothetical splat point above which entropy production is truncated. The rate is almost independent of

the black hole mass M• at fixed splat density ρ#, so it is legitimate to interpret dSb/dSBH as the cumulative

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17.3 Self-similar models of the interior structure of black holes 303

hori

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Plan

cksc

ale

ρb

ρe|C|

|dSb/dScov|

dSb/dSBH

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.5 This black hole creates a lot of entropy by having a large charge-to-mass Q•/M• = 0.8 and a

low accretion rate M• = 10−28. The conductivity parameter κb = 0.35 is again at the threshold above whichthe plasma plunges to a central singularity. The equation of state is wb = 0.32.

entropy created inside the black hole relative to the Bekenstein-Hawking entropy. Truncated at the Planck

scale, |C| = 1, the entropy relative to Bekenstein-Hawking is dSb/dSBH ≈ 1010.

Generally, the smaller the accretion rate M•, the more entropy is produced. If moreover the charge-to-

mass Q•/M• is large, then the entropy can be produced closer to the outer horizon. Figure 17.5 shows a

model with a relatively large charge-to-mass Q•/M• = 0.8, and a low accretion rate M• = 10−28. The large

charge-to-mass ratio in spite of the relatively high conductivity requires force-feeding the black hole: the

sonic point must be pushed to just above the horizon. The large charge and high conductivity leads to a

burst of entropy production just beneath the horizon.

If the entropy created inside a black hole exceeds the Bekenstein-Hawking entropy, and the black hole later

evaporates radiating only the Bekenstein-Hawking entropy, then entropy is destroyed, violating the second

law of thermodynamics.

This startling conclusion is premised on the assumption that entropy created inside a black hole accumu-

lates additively, which in turn derives from the assumption that the Hilbert space of states is multiplicative

over spacelike-separated regions. This assumption, called locality, derives from the fundamental proposi-

tion of quantum field theory in flat space that field operators at spacelike-separated points commute. This

reasoning is essentially the same as originally led Hawking (1976) to conclude that black holes must destroy

information.

The same ideas that motivate holography also rescue the second law. If the future lightcones of spacelike-

separated points do not intersect, then the points are permanently out of communication, and can behave

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304 The interiors of spherical black holes

matterInfalling

Sb >> SBH

Sb = SBH

Sb <

SBH

Figure 17.6 Partial Penrose diagram of the black hole. The entropy passing through the spacelike slice beforethe black hole evaporates exceeds that passing through the spacelike slice after the black hole evaporates,apparently violating the second law of thermodynamics. However, the entropy passing through any null slicerespects the second law.

like alternate quantum realities, like Schrodinger’s dead-and-alive quantum cat. Just as it is not legitimate

to the add the entropies of the dead cat and the live cat, so also it is not legitimate to add the entropies of

regions inside a black hole whose future lightcones do not intersect. The states of such separated regions,

instead of being distinct, are quantum entangled with each other.

Figures 17.4 and 17.5 show that the rate |dSb/dScov| at which entropy passes through ingoing or outgoing

spherical lightsheets is less than one at all scales below the Planck scale. This shows not only that the black

holes obey Bousso’s covariant entropy bound, but also that no individual observer inside the black hole sees

more than the Bekenstein-Hawking entropy on their lightcone. No observer actually witnesses a violation of

the second law.

17.3.9 Black hole accreting a charged massless scalar field

The charged, non-conducting plasma considered in §17.3.7 fell through an (outgoing) inner horizon without

undergoing mass inflation. This can be attributed to the fact that relativistic counter-streaming could not

occur: there was only a single (outgoing) fluid, and the speed of sound in the fluid was less than the speed

of light.

In reality, unless dissipation destroys the inner horizon as in §17.3.8, then relativistic counter-streaming

between ingoing and outgoing fluids will undoubtedly take place, through gravitational waves if nothing else.

One way to allow relativistic counter-streaming is to let the speed of sound be the speed of light. This is

true in a massless scalar (= spin-0) field φ, which has an equation of state wφ = 1. Figure 17.7 shows a black

hole that accretes a charged, non-conducting fluid with this equation of state. The parameters are otherwise

the same as as in previous subsections: a mass of M• = 4 × 106 M⊙, an accretion rate of M• = 10−16, and

a black hole charge-to-mass of Q•/M• = 10−5. As the Figure shows, mass inflation takes place just above

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17.3 Self-similar models of the interior structure of black holes 305

hori

zonm

ass

infl

atio

n

ρφ

ρe

|C|

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.7 Instead of a relativistic plasma, this shows a charged scalar field φ whose equation of statewφ = 1 means that the speed of sound equals the speed of light. The scalar field therefore supports relativisticcounter-streaming, as a result of which mass inflation occurs just above the erstwhile inner horizon. Themass is M• = 4 × 106 M⊙, the accretion rate M• = 10−16, the charge-to-mass Q•/M• = 10−5, and theconductivity is zero.

the place where the inner horizon would be. During mass inflation, the density ρφ and the Weyl scalar C

rapidly exponentiate up to the Planck scale and beyond.

One of the remarkable features of the mass inflation instability is that the smaller the accretion rate, the

more violent the instability. Figure 17.8 shows mass inflation in a black hole of charge-to-mass Q•/M• = 0.8

accreting a massless scalar field at rates M• = 0.01, 0.003, and 0.001. The charge-to-mass has been chosen

to be largish so that the inner horizon is not too far below the outer horizon, and the accretion rates have

been chosen to be large because otherwise the inflationary growth rate is too rapid to be discerned easily on

the graph. The density ρφ and Weyl scalar C exponentiate along with, and in proportion to, the interior

mass M , which increases as the radius r decreases as

M ∝ exp(− ln r/M•) . (17.27)

Physically, the scale of length of inflation is set by how close to the inner horizon infalling material approaches

before mass inflation begins. The smaller the accretion rate, the closer the approach, and consequently the

shorter the length scale of inflation.

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306 The interiors of spherical black holes

hori

zon

innerhorizon

0.01 hori

zon

innerhorizon

0.003

hori

zon

innerhorizon

ρφ

0.001

ρe

|C|

.1 .2 .5 1 210−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (geometric units)

(Pla

nck

unit

s)

Figure 17.8 The density ρφ and Weyl curvature scalar |C| inside a black hole accreting a massless scalar field.

The graph shows three cases, with mass accretion rates M• = 0.01, 0.003, and 0.001. The graph illustratesthat the smaller the accretion rate, the faster the density and curvature inflate. Mass inflation destroys theinner horizon: the dashed vertical line labeled “inner horizon” shows the position that the inner horizonwould have if mass inflation did not occur. The black hole mass is M• = 4 × 106 M⊙, the charge-to-mass isQ•/M• = 0.8, and the conductivity is zero.

17.3.10 Black hole accreting charged baryons and dark matter

No scalar field (massless or otherwise) has yet been observed in nature, although it is supposed that the

Higgs field is a scalar field, and it is likely that cosmological inflation was driven by a scalar field. Another

way to allow mass inflation in simple models is to admit not one but two fluids that can counter-stream

relativistically through each other. A natural possibility is to feed the black hole not only with a charged

relativistic fluid of baryons but also with neutral pressureless dark matter that streams freely through the

baryons. The charged baryons, being repelled by the electric charge of the black hole, become outgoing,

while the neutral dark matter remains ingoing.

Figure 17.9 shows that relativistic counter-streaming between the baryons and the dark matter causes

the center-of-mass density ρ and the Weyl curvature scalar C to inflate quickly up to the Planck scale and

beyond. The ratio of dark matter to baryonic density at the sonic point is ρd/ρb = 0.1, but otherwise the

parameters are the generic parameters of previous subsections: M• = 4 × 106 M⊙, M• = 10−16, wb = 0.32,

Q•/M• = 10−5, and zero conductivity. Almost all the center-of-mass energy ρ is in the counter-streaming

energy between the outgoing baryonic and ingoing dark matter. The individual densities ρb of baryons and

ρd of dark matter (and ρe of electromagnetic energy) increase only modestly.

As in the case of the massless scalar field considered in the previous subsection, §17.3.9, the smaller the

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17.3 Self-similar models of the interior structure of black holes 307

hori

zonm

ass

infl

atio

n

ρd

ρbρe

|C|

ρ

dSb/dSBH

1010 1020 1030 1040 105010−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (Planck units)

(Pla

nck

unit

s)

Figure 17.9 Back to the relativistic charged baryonic plasma, but now in addition the black hole accretesneutral pressureless uncharged dark matter, which streams freely through the baryonic plasma. The rel-ativistic counter-streaming produces mass inflation just above the erstwhile inner horizon. The mass isM• = 4 × 106 M⊙, the accretion rate M• = 10−16, the baryonic equation of state wb = 0.32, the charge-to-mass Q•/M• = 10−5, the conductivity is zero, and the ratio of dark matter to baryonic density at the outersonic point is ρd/ρb = 0.1.

accretion rate, the shorter the length scale of inflation. Not only that, but the smaller one of the ingoing or

outgoing streams is relative to the other, the shorter the length scale of inflation. Figure 17.10 shows a black

hole with three different ratios of the dark-matter-to-baryon density ratio at the sonic point, ρd/ρb = 0.3,

0.1, and 0.03, all with the same total accretion rate M• = 10−2. The smaller the dark matter stream, the

faster is inflation. The accretion rate M• and the dark-matter-to-baryon ratio ρd/ρb have been chosen to be

relatively large so that the inflationary growth rate is discernable easily on the graph.

Figure 17.10 shows that, as in Figure 17.9, almost all the center-of-mass energy is in the streaming energy

between the baryons and the dark matter. For one case, ρd/ρb = 0.3, Figure 17.10 shows the individual

densities ρb of baryons, ρd of dark matter, and ρe of electromagnetic energy, all of which remain tiny compared

to the streaming energy.

17.3.11 The black hole particle accelerator

The previous subsection, §17.3.10, showed that almost all the center-of-mass energy during mass inflation

is in the energy of counter-streaming. Thus the black hole acts like an extravagantly powerful particle

accelerator.

Mass inflation is an exponential instability. The nature of the black hole particle accelerator is that an

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308 The interiors of spherical black holes

hori

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innerhorizon

ρd ρb

ρe |C|ρ

0.3

hori

zon

innerhorizon

ρe |C|ρ

0.1

hori

zon

innerhorizon

ρe |C|ρ

0.03.1 .2 .5 1 2

10−12010−11010−10010−9010−8010−7010−6010−5010−4010−3010−2010−10

10010101020

Radius r (geometric units)

(Pla

nck

unit

s)

Figure 17.10 The center-of-mass density ρ and Weyl curvature |C| inside a black hole accreting baryons

and dark matter at rate M• = 0.01. The graph shows three cases, with dark-matter-to-baryon ratio at thesonic point of ρd/ρb = 0.3, 0.1, and 0.03. The smaller the ratio of dark matter to baryons, the faster thecenter-of-mass density ρ and curvature C inflate. For the largest ratio, ρd/ρb = 0.3 (to avoid confusion,only this case is plotted), the graph also shows the individual proper densities ρb of baryons, ρd of darkmatter, and ρe of electromagnetic energy. During mass inflation, almost all the center-of-mass energy ρ isin the streaming energy: the proper densities of individual components remain small. The black hole massis M• = 4 × 106 M⊙, the baryonic equation of state is wb = 0.32, the charge-to-mass is Q•/M• = 0.8, andthe conductivity is zero.

individual particle spends approximately an equal interval of proper time being accelerated through each

decade of collision energy.

Each baryon in the black hole collider sees a flux ndur of dark matter particles per unit area per unit

time, where nd = ρd/md is the proper number density of dark matter particles in their own frame, and ur is

the radial component of the proper 4-velocity, the γv, of the dark matter through the baryons. The γ factor

in ur is the relavistic beaming factor: all frequencies, including the collision frequency, are speeded up by

the relativistic beaming factor γ. As the baryons accelerate through the collider, they spend a proper time

interval dτ/d lnur in each e-fold of Lorentz factor ur. The number of collisions per baryon per e-fold of ur is

the dark matter flux (ρd/md)ur, multiplied by the time dτ/d ln ur, multiplied by the collision cross-section

σ. The total cumulative number of collisions that have happened in the black hole particle collider equals

this multiplied by the total number of baryons that have fallen into the black hole, which is approximately

equal to the black hole mass M• divided by the mass mb per baryon. Thus the total cumulative number of

Page 321: General Relativity, Black Holes, And Cosmology

17.3 Self-similar models of the interior structure of black holes 309

100 1050 10100

10−2

10−1

1

10

Velocity u

Col

lisi

onra

teρ d

udτ

/dln

u(M

•/M

•)

0.03

0.01

0.003

10−16

Figure 17.11 Collision rate of the black hole particle accelerator per e-fold of velocity u (meaning γv),

expressed in units of the inverse black hole accretion time M•/M•. The models illustrated are the same as

those in Figure 17.10. The curves are labeled with their mass accretion rates: M• = 0.03, 0.01, 0.003, and10−16. Stars mark where the center-of-mass energy of colliding baryons and dark matter particles exceedsthe Planck energy, while disks show where the Weyl curvature scalar C exceeds the Planck scale.

collisions in the black hole collider is

number of collisions

e-fold of ur =M•mb

ρd

mdσur dτ

d lnur. (17.28)

Figure 17.11 shows, for several different accretion rates M•, the collision rate M•ρdurdτ/d lnur of the black

hole collider, expressed in units of the black hole accretion rate M•. This collision rate, multiplied by

M•σ/(mdmb), gives the number of collisions (17.28) in the black hole. In the units c = G = 1 being used

here, the mass of a baryon (proton) is 1 GeV ≈ 10−54 m. If the cross-section σ is expressed in canonical

accelerator units of femtobarns (1 fb = 10−43 m2) then the number of collisions (17.28) is

number of collisions

e-fold of ur = 1045( σ

1 fb

)

(

300 GeV2

mbmd

)

(

M•10−16

)

(

M•ρdurdτ/d lnur

0.03 M•

)

. (17.29)

Particle accelerators measure their cumulative luminosities in inverse femtobarns. Equation (17.29) shows

that the black hole accelerator delivers about 1045 femtobarns, and it does so in each e-fold of collision energy

up to the Planck energy and beyond.

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310 The interiors of spherical black holes

17.4 Instability at outer horizon?

A number of papers have suggested that a magical phase transition at, or just outside, the outer horizon

prevents any horizon from forming. Is it true? For example, is there a mass inflation instability at the outer

horizon?

If the were a White Hole on the other side of the outer horizon, then indeed an object entering the outer

horizon would encounter an inflationary instability. But otherwise, no.

Page 323: General Relativity, Black Holes, And Cosmology

PART SEVEN

GENERAL RELATIVISTIC PERTURBATION THEORY

Page 324: General Relativity, Black Holes, And Cosmology
Page 325: General Relativity, Black Holes, And Cosmology

Concept Questions

1. Why do general relativistic perturbation theory using the tetrad formalism as opposed to the coordinate

approach?

2. Why is the tetrad metric γmn assumed fixed in the presence of perturbations?

3. Are the tetrad axes γγm fixed under a perturbation?

4. Is it true that the tetrad components ϕmn of a perturbation are (anti-)symmetric in m↔ n if and only if

its coordinate components ϕµν are (anti-)symmetric in µ↔ ν?

5. Does an unperturbed quantity, such as the unperturbed metric0

gµν , change under an infinitesimal coordi-

nate gauge transformation?

6. How can the vierbein perturbation ϕmn be considered a tetrad tensor field if it changes under an infinites-

imal coordinate gauge transformations?

7. What properties of the unperturbed spacetime allow decomposition of perturbations into independently

evolving Fourier modes?

8. What properties of the unperturbed spacetime allow decomposition of perturbations into independently

evolving scalar, vector, and tensor modes?

9. In what sense do scalar, vector, and tensor modes have spin 0, 1, and 2 respectively?

10. Tensor modes represent gravitational waves that, in vacuo, propagate at the speed of light. Do scalar

and vector modes also propagate at the speed of light in vacuo? If so, do scalar and vector modes also

constitute gravitational waves?

11. If scalar, vector, and tensor modes evolve independently, does that mean that scalar modes can exist and

evolve in the complete absence of tensor modes? If so, does it mean that scalar modes can propagate

causally, in vacuo at the speed of light, without any tensor modes being present?

12. Equation (20.74) defines the mass M of a body as what a distant observer would measure from its

gravitational potential. Similarly equation (20.82) defines the angular momentum L of a body as what a

distant observer would measure from the dragging of inertial frames. In what sense are these definitions

legitimate?

13. Can an observer far from a body detect the difference between the scalar potentials Ψ and Φ produced by

the body?

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314 Concept Questions

14. If a gravitational wave is a wave of spacetime itself, distorting the very rulers and clocks that measure

spacetime, how is it possible to measure gravitational waves at all?

15. Have gravitational waves been detected?

16. If gravitational waves carry energy-momentum, then can gravitational waves be present in a region of

spacetime with vanishing energy-momentum tensor, Tmn = 0?

Page 327: General Relativity, Black Holes, And Cosmology

What’s important?

1. Getting your brain around coordinate and tetrad gauge transformations.

2. A central aim of general relativistic perturbation theory is to identify the coordinate and tetrad gauge-

invariant perturbations, since only these have physical meaning.

3. A second central aim is to classify perturbations into independently evolving modes, to the extent that

this is possible.

4. In background spacetimes with spatial translation and rotation symmetry, which includes Minkowski space

and the Friedmann-Roberston-Walker metric of cosmology, modes decompose into independently evolving

scalar (spin-0), vector (spin-1), and tensor (spin-2) modes. In background spacetimes without spatial

translation and rotation symmetry, such as black holes, scalar, vector, and tensor modes scatter off the

curvature of space, and therefore mix with each other.

5. In background spacetimes with spatial translation and rotation symmetry, there are 6 algebraic combina-

tions of metric coefficients that are coordinate and tetrad gauge-invariant, and therefore represent physical

perturbations. There are 2 scalar modes, 2 vector modes, and 2 tensor modes. A spin-m mode varies as

eimχ where χ is the rotational angle about the spatial wavevector k of the mode.

6. In background spacetimes without spatial translation and rotation symmetry, the coordinate and tetrad

gauge-invariant perturbations are not algebraic combinations of the metric coefficients, but rather com-

binations that involve first and second derivatives of the metric coefficients. Gravitational waves are

described by the Weyl tensor, which can be decomposed into 5 complex components, with spin 0, ±1,

and ±2. The spin-±2 components describe propagating gravitational waves, while the spin-0 and spin-±1

components describe the non-propagating gravitational field near a source.

7. The preeminent application of general relativistic perturbation theory is to cosmology. Coupled with

physics that is either well understood (such as photon-electron scattering) or straightforward to model

even without a deep understanding (such as the dynamical behavior of non-baryonic dark matter and

dark energy), the theory has yielded predictions that are in spectacular agreement with observations of

fluctuations in the CMB and in the large scale distribution of galaxies and other tracers of the distribution

of matter in the Universe.

Page 328: General Relativity, Black Holes, And Cosmology

18

Perturbations and gauge transformations

This chapter sets up the basics equations that define perturbations to an arbitrary spacetime in the tetrad

formalism of general relativity, and it examines the effect of tetrad and coordinate gauge transformations

on those perturbations. The perturbations are supposed to be small, in the sense that quantities quadratic

in the perturbations can be neglected. The formalism set up in this chapter provides a foundation used in

subsequent chapters.

18.1 Notation for perturbations

A 0 (zero) overscript signifies an unperturbed quantity, while a 1 (one) overscript signifies a perturbation.

No overscript means the full quantity, including both unperturbed and perturbed parts. An overscript is

attached only where necessary. Thus if the unperturbed part of a quantity is zero, then no overscript is

needed, and none is attached.

18.2 Vierbein perturbation

Let the vierbein perturbation ϕmn be defined so that the perturbed vierbein is

emµ = (δn

m + ϕmn)

0

enµ , (18.1)

with corresponding inverse

emµ = (δm

n − ϕnm)

0

enµ . (18.2)

Since the perturbation ϕmn is already of linear order, to linear order its indices can be raised and lowered

with the unperturbed metric, and transformed between tetrad and coordinate frames with the unperturbed

vierbein. In practice it proves convenient to work with the covariant tetrad-frame components ϕmn of the

vierbein pertubation

ϕmn = γnlφml . (18.3)

Page 329: General Relativity, Black Holes, And Cosmology

18.3 Gauge transformations 317

The perturbation ϕmn can be regarded as a tetrad tensor field defined on the unperturbed background.

18.3 Gauge transformations

The vierbein perturbation ϕmn has 16 degrees of freedom, but only 6 of these degrees of freedom correspond

to real physical perturbations, since 6 degrees of freedom are associated with arbitrary infinitesimal changes

in the choice of tetrad, which is to say arbitrary infinitesimal Lorentz transformations, and a further 4 degrees

of freedom are associated with arbitrary infinitesimal changes in the coordinates.

In the context of perturbation theory, these infinitesimal tetrad and coordinate transformations are called

gauge transformations. Real physical perturbations are perturbations that are gauge-invariant under

both tetrad and coordinate gauge transformations.

18.4 Tetrad metric assumed constant

In the tetrad formalism, tetrad axes γγm are introduced as locally inertial (or other physically motivated)

axes attached to an observer. The axes enable quantities to be projected into the frame of the observer.

In a spacetime buffeted by perturbations, it is natural for an observer to cling to the rock provided by the

locally inertial (or other) axes, as opposed to allowing the axes to bend with the wind. For example, when

a gravitational wave goes by, the tidal compression and rarefaction causes the proper distance between two

freely falling test masses to oscillate. It is natural to choose the tetrad so that it continues to measure proper

times and distances in the perturbed spacetime.

In these notes on general relativistic perturbation theory, the tetrad metric will be taken to be constant

everywhere, and unchanged by a perturbation

γmn =0

γmn = constant . (18.4)

For example, if the tetrad is orthonormal, then the tetrad metric is constant, the Minkowski metric ηmn.

However, the tetrad could also be some other tetrad for which the tetrad metric is constant, such as a spinor

tetrad (§12.1.1), or a Newman-Penrose tetrad (§12.2.1).

18.5 Perturbed coordinate metric

The perturbed coordinate metric is

gµν = γmn em

µen

ν

= γkl(δkm − ϕm

k)0

emµ(δl

n − ϕnl)

0

enν

=0

gµν − (ϕµν + ϕνµ) . (18.5)

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318 Perturbations and gauge transformations

Thus the perturbation of the coordinate metric depends only on the symmetric part of the vierbein pertur-

bation ϕmn, not the antisymmetric part

1

gµν = − (ϕµν + ϕνµ) . (18.6)

18.6 Tetrad gauge transformations

Under an infinitesimal tetrad transformation, the covariant vierbein perturbations ϕmn transform as

ϕmn → ϕmn + ǫmn , (18.7)

where ǫmn is the generator of a Lorentz transformation, which is to say an arbitrary antisymmetric tensor

(Exercise 11.2). Thus the antisymmetric part ϕmn − ϕnm of the covariant perturbation ϕmn is arbitrarily

adjustable through an infinitesimal tetrad transformation, while the symmetric part ϕmn + ϕnm is tetrad

gauge-invariant.

It is easy to see when a quantity is tetrad gauge-invariant: it is tetrad gauge-invariant if and only if it

depends only on the symmetric part of the vierbein perturbation, not on the antisymmetric part. Evidently

the perturbation (18.6) to the coordinate metric gµν is tetrad gauge-invariant. This is as it should be, since

the coordinate metric gµν is a coordinate-frame quantity, independent of the choice of tetrad frame.

If only tetrad gauge-invariant perturbations are physical, why not just discard tetrad perturbations (the

antisymmetric part of ϕmn) altogether, and work only with the tetrad gauge-invariant part (the symmetric

part of ϕmn)? The answer is that tetrad-frame quantities such as the tetrad-frame Einstein tensor do change

under tetrad gauge transformations (infinitesimal Lorentz transformations of the tetrad). It is true that

the only physical perturbations of the Einstein tensor are those combinations of it that are tetrad gauge-

invariant. But in order to identify these tetrad gauge-invariant combinations, it is necessary to carry through

the dependence on the non-tetrad-gauge-invariant part, the antisymmetric part of ϕmn.

Much of the professional literature on general relativistic perturbation theory works with the traditional

coordinate formalism, as opposed to the tetrad formalism. The term “gauge-invariant” then means coor-

dinate gauge-invariant, as opposed to both coordinate and tetrad gauge-invariant. This is fine as far as it

goes: the coordinate approach is perfectly able to identify physical perturbations versus gauge perturbations.

However, there still remains the problem of projecting the perturbations into the frame of an observer, so

ultimately the issue of perturbations of the observer’s frame, tetrad perturbations, must be faced.

Concept question 18.1 In perturbation theory, can tetrad gauge transformations be non-infinitesimal?

Page 331: General Relativity, Black Holes, And Cosmology

18.7 Coordinate gauge transformations 319

18.7 Coordinate gauge transformations

A coordinate gauge transformation is a transformation of the coordinates xµ by an infinitesimal shift ǫµ

xµ → x′µ = xµ + ǫµ . (18.8)

You should not think of this as shifting the underlying spacetime around; rather, it is just a change of the

coordinate system, which leaves the underlying spacetime unchanged. Because the shift ǫµ is, like the vierbein

perturbations ϕmn, already of linear order, its indices can be raised and lowered with the unperturbed metric,

and transformed between coordinate and tetrad frames with the unperturbed vierbein. Thus the shift ǫµ

can be regarded as a vector field defined on the unperturbed background. The tetrad components ǫm of the

shift ǫµ are

ǫm =0

emµ ǫ

µ . (18.9)

Physically, the tetrad-frame shift ǫm is the shift measured in locally inertial coordinates

ξm → ξ′m = ξm + ǫm . (18.10)

18.8 Coordinate gauge transformation of a coordinate scalar

Under a coordinate transformation (18.8), a coordinate-frame scalar Φ(x) remains unchanged

Φ(x)→ Φ′(x′) = Φ(x) . (18.11)

Here the scalar Φ′(x′) is evaluated at position x′, which is the same as the original physical position x since

all that has changed is the coordinates, not the physical position. However, in perturbation theory, quantities

are evaluated at coordinate position x, not x′. The value of Φ at x is related to that at x′ by

Φ′(x) = Φ′(x′ − ǫ) = Φ′(x′)− ǫκ ∂Φ′

∂xκ. (18.12)

Since ǫκ is a small quantity, and Φ′ differs from Φ by a small quantity, the last term ǫκ∂Φ′/∂xκ in equa-

tion (18.16) can be replaced by ǫκ∂Φ/∂xκ to linear order. Putting equations (18.11) and (18.12) together

shows that the scalar Φ changes under a coordinate gauge transformation (18.8) as

Φ(x)→ Φ′(x) = Φ(x)− ǫκ ∂Φ

∂xκ. (18.13)

The transformation (18.13) can also be written

Φ(x)→ Φ′(x) = Φ(x) + LǫΦ , (18.14)

where Lǫ is the Lie derivative, §18.13.

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320 Perturbations and gauge transformations

18.9 Coordinate gauge transformation of a coordinate vector or tensor

A similar argument applies to coordinate vectors and tensors. Under a coordinate transformation (18.8), a

coordinate-frame 4-vector Aµ(x) transforms in the usual way as

Aµ(x)→ A′µ(x′) = Aκ(x)∂x′µ

∂xκ= Aµ(x) +Aκ(x)

∂ǫµ

∂xκ. (18.15)

As in the scalar case, the vector A′µ(x′) is evaluated at position x′, which is the same as the original physical

position x since all that has changed is the coordinates, not the physical position. Again, in perturbation

theory, quantities are evaluated at coordinate position x, not x′. The value of A′µ at x is related to that at

x′ by

A′µ(x) = A′µ(x′ − ǫ) = A′µ(x′)− ǫκ ∂A′µ

∂xκ. (18.16)

The last term ǫκ∂A′µ/∂xκ in equation (18.16) can be replaced by ǫκ∂Aµ/∂xκ to linear order. Putting

equations (18.15) and (18.16) together shows that the 4-vector Aµ changes under a coordinate gauge trans-

formation (18.8) as

Aµ(x)→ A′µ(x) = Aµ(x) +Aκ ∂ǫµ

∂xκ− ǫκ ∂A

µ

∂xκ. (18.17)

The transformation (18.17) can also be written

Aµ(x)→ A′µ(x) = Aµ(x) + LǫAµ , (18.18)

where Lǫ is the Lie derivative, §18.13.

More generally, under a coordinate gauge transformation (18.8), a coordinate tensor Aκλ...µν... transforms as,

equation (18.32),

Aκλ...µν...(x)→ A′κλ...

µν...(x) = Aκλ...µν...(x) + LǫA

κλ...µν... . (18.19)

18.10 Coordinate gauge transformation of a tetrad vector

A tetrad-frame 4-vector Am is a coordinate-invariant quantity, and therefore acts like a coordinate scalar,

equation (18.13), under a coordinate gauge transformation (18.8)

Am(x)→ A′m(x) = Am(x)− ǫκ ∂Am

∂xκ= Am(x) − ǫk∂kA

m . (18.20)

The change −ǫk∂kAm is a coordinate tensor (specifically, a coordinate scalar), but it is not a tetrad tensor.

More generally, a tetrad-frame tensor Akl...mn... transforms under a coordinate gauge transformation (18.8)

as

Akl...mn...(x)→ A′kl...

mn...(x) = Akl...mn...(x)− ǫa∂aA

kl...mn... . (18.21)

Again, the change −ǫa∂aAkl...mn... is a coordinate tensor (a coordinate scalar), but not a tetrad tensor.

Page 333: General Relativity, Black Holes, And Cosmology

18.11 Coordinate gauge transformation of the vierbein 321

18.11 Coordinate gauge transformation of the vierbein

The inverse vierbein emµ equals the scalar product of the tetrad and coordinate axes, em

µ = γγm · gµ.

Therefore the transformation of the vierbein under a coordinate gauge transformation (18.8) follows from

the transformations of γγm and gµ. The tetrad axes γγm transform in accordance with (18.20) as

γγm → γγ′m = γγm − ǫk∂kγγm

= γγm + ǫkΓmnkγγn . (18.22)

The coordinate axes gµ transform in accordance with (18.32) and (18.34) as (including torsion Sνµκ)

gµ → g′µ = gµ + Lǫgµ

= gµ − gνDµǫν − ǫνDνgµ − gνS

νµκǫ

κ

= gµ −Dµ (gνǫν)− gνS

νµκǫ

κ

= gµ − γγn

(

Dmǫn + Sn

mkǫk) 0

emµ , (18.23)

where the third line follows from the second because the axes gν are by definition covariantly constant,

Dµgν = 0. It follows from (18.22) and (18.23) that the inverse vierbein emµ transforms under an infinitesimal

coordinate gauge transformation (18.8) as

emµ → e′mµ = γγ′m · g′

µ

= emµ +

(

−Dnǫm − Sm

nkǫk + Γm

nkǫk) 0

enµ . (18.24)

From equation (18.24) and the definition (18.2) of the vierbein perturbations ϕmn, it follows that the vierbein

perturbations transform under a coordinate gauge transformation (18.8) as

ϕmn → ϕ′mn = ϕmn + ∂mǫn − (Γknm + Γnmk − Snmk) ǫk , (18.25)

in which ǫk are the tetrad components of the coordinate shift, and Γkmn are tetrad connection coefficients.

If torsion vanishes, Snmk = 0, as general relativity assumes, then the transfomation (18.25) of the vierbein

perturbations under a coordinate gauge transformation reduces to

ϕmn → ϕ′mn = ϕmn + ∂mǫn − (Γknm + Γnmk) ǫk . (18.26)

18.12 Coordinate gauge transformation of the metric

The tetrad metric γmn transforms under an infinitesimal coordinate gauge transformation (18.8) as

γmn → γ′mn = γmn − (Γmnk + Γnmk)ǫk = γmn , (18.27)

where the last expression is true because the tetrad metric γmn is being assumed constant, equation (18.4),

in which case Γmnk + Γnmk = ∂kγnm = 0.

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322 Perturbations and gauge transformations

The coordinate metric gµν transforms under an infinitesimal coordinate gauge transformation (18.8) as

gµν → g′µν = gµν + Lǫgµν = gµν − (Dµǫν +Dνǫµ)− (Sµνκ + Sνµκ)ǫκ . (18.28)

18.13 Lie derivative

The change in the coordinate 4-vector Aµ on the right hand side of equation (18.17) is called the Lie

derivative of Aµ along the direction ǫκ, and it is designated by the operator Lǫ

LǫAµ ≡ Aκ ∂ǫ

µ

∂xκ− ǫκ∂A

µ

∂xκ. (18.29)

The Lie derivative has the important property of being a tensor, which is one reason that it merits a special

name. As its name suggests, the Lie derivative acts like a derivative: it is linear, and it satisfies the Leibniz

rule. Translating from ordinary partial derivatives to covariant derivatives yields the following expression

for the Lie derivative in covariant form

LǫAµ = AκDκǫ

µ − ǫκDκAµ +AκǫλSµ

κλ is a coordinate vector , (18.30)

where Sµκλ is the torsion. If torsion vanishes, as GR assumes, then the Lie derivative of a 4-vector is

LǫAµ = AκDκǫ

µ − ǫκDκAµ is a coordinate vector . (18.31)

More generally, under a coordinate gauge transformation (18.8), a coordinate tensor Aκλ...µν... transforms as

Aκλ...µν...(x)→ A′κλ...

µν...(x) = Aκλ...µν...(x) + LǫA

κλ...µν... (18.32)

where the Lie derivative is defined by

LǫAκλ...µν... ≡ Aαλ...

µν...

∂ǫκ

∂xα+Aκα...

µν...

∂ǫλ

∂xα... −Aκλ...

αν...

∂ǫα

∂xµ−Aκλ...

µα...

∂ǫα

∂xν− ǫα ∂A

κλ...µν...

∂xα. (18.33)

In covariant form, equation (18.33) is

LǫAκλ...µν... = Aαλ...

µν...Dαǫκ +Aκα...

µν...Dαǫλ ... −Aκλ...

αν...Dµǫα −Aκλ...

µα...Dνǫα ... − ǫαDαA

κλ...µν... (18.34)

+(

Aαλ...µν...S

καβ +Aκα...

µν...Sλαβ ... −Aκλ...

αν...Sαµβ −Aκλ...

µα...Sανβ

)

ǫβ is a coordinate tensor ,

which in the case of vanishing torsion, as GR assumes, reduces to

LǫAκλ...µν... = Aαλ...

µν...Dαǫκ +Aκα...

µν...Dαǫλ ... −Aκλ...

αν...Dµǫα −Aκλ...

µα...Dνǫα ... − ǫαDαA

κλ...µν... is a coordinate tensor .

(18.35)

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19

Scalar, vector, tensor decomposition

In the particular case that the unperturbed spacetime is spatially homogeneous and isotropic, which includes

not only Minkoswki space but also the important case of the cosmological Friedmann-Robertson-Walker

metric, perturbations decompose into independently evolving scalar (spin-0), vector (spin-1), and tensor

(spin-2) modes.

Similarly to Fourier decomposition, decomposition into scalar, vector, and tensor modes is non-local, in

principle requiring knowledge of perturbation amplitudes simultaneously throughout all of space. In practical

problems however, an adequate decomposition is possible as long as the scales probed are sufficiently larger

than the wavelengths of the modes probed. Ultimately, the fact that an adequate decomposition is possible

is a consequence of the fact that gravitational fluctuations in the real Universe appear to converge at the

cosmological horizon, so that what happens locally is largely independent of what is happening far away.

19.1 Decomposition of a vector in flat 3D space

Theorem: In flat 3-dimensional space, a 3-vector field w(x) can be decomposed uniquely (subject to the

boundary condition that w vanishes sufficiently rapidly at infinity) into a sum of scalar and vector parts

w = ∇w‖scalar

+ w⊥vector

. (19.1)

In this context, the term vector signifies a 3-vector w⊥ that is transverse, that is to say, it has vanishing

divergence,

∇ ·w⊥ = 0 . (19.2)

Here ∇ ≡ ∂/∂x ≡ ∇i ≡ ∂/∂xi is the gradient in flat 3D space. The scalar and vector parts are also known

as spin-0 and spin-1, or gradient and curl, or longitudinal and transverse. The scalar part ∇w‖ contains 1

degree of freedom, while the vector part w⊥ contains 2 degrees of freedom. Together they account for the 3

degrees of freedom of the vector w.

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324 Scalar, vector, tensor decomposition

Proof: Take the divergence of equation (19.1)

∇ ·w = ∇2w‖ . (19.3)

The operator∇2 on the right hand side of equation (19.3) is the 3D Laplacian. The solution of equation (19.3)

is

w‖(x) = −∫

∇′ ·w(x′)

|x′ − x|d3x′

4π. (19.4)

The solution (19.4) is valid subject to boundary conditions that the vector w vanish sufficiently rapidly at

infinity. In cosmology, the required boundary conditions, which are set at the Big Bang, are apparently

satisfied because fluctuations at the Big Bang were small. Equation (19.1) then immediately implies that

the vector part is w⊥ = w −∇w‖.

19.2 Fourier version of the decomposition of a vector in flat 3D space

When the background has some symmetry, it is natural to expand perturbations in eigenmodes of the

symmetry. If the background space is flat, then it is translation symmetric. Eigenmodes of the translation

operator ∇ are Fourier modes.

A function a(x) in flat 3D space and its Fourier transform a(k) are related by (the disposition of factors

of 2π in the following definition follows the convention most commonly adopted by cosmologists)

a(k) =

a(x)eik·x d3x , a(x) =

a(k)e−ik·x d3k

(2π)3. (19.5)

You may not be familiar with the practice of using the same symbol a in both real and Fourier space; but

a is the same vector in Hilbert space, with components ax = a(x) in real space, and ak = a(k) in Fourier

space.

Taking the gradient ∇ in real space is equivalent to multiplying by −ik in Fourier space

∇→ −ik . (19.6)

Thus the decomposition (19.1) of the 3D vector w translates into Fourier space as

w = −ikw‖scalar

+ w⊥vector

, (19.7)

where the vector part w⊥ satisfies

k ·w⊥ = 0 . (19.8)

In other words, in Fourier space the scalar part ∇w‖ of the vector w is the part parallel (longitudinal) to

the wavevector k, while the vector part w⊥ is the part perpendicular (transverse) to the wavevector k.

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19.3 Decomposition of a tensor in flat 3D space 325

19.3 Decomposition of a tensor in flat 3D space

Similarly, the 9 components of a 3× 3 spatial matrix hij can be decomposed into 3 scalars, 2 vectors, and 1

tensor:

hij = δij φscalar

+∇i∇jhscalar

+ εijk∇khscalar

+∇ihjvector

+∇j hivector

+ hTij

tensor

. (19.9)

In this context, the term tensor signifies a 3× 3 matrix hTij that is traceless, symmetric, and transverse:

hT ii = 0 , hT

ij = hTji , ∇ih

Tij = 0 . (19.10)

The transverse-traceless-symmetric matrix hTij has two degrees of freedom. The vector components hi and

hi are by definition transverse,

∇ihi = ∇ihi = 0 . (19.11)

The tildes on h and hi simply distinguish those symbols; the tildes have no other significance. The trace of

the 3× 3 matrix hij is

hii = 3φ+∇2h . (19.12)

Spinor decomposition.

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20

Flat space background

General relativistic perturbation theory is simplest in the case that the unperturbed background space

is Minkowski space. In Cartesian coordinates xµ ≡ t, x, y, z, the unperturbed coordinate metric is the

Minkowski metric0

gµν = ηµν . (20.1)

In this chapter the tetrad is taken to be orthonormal, and aligned with the unperturbed coordinate axes, so

that the unperturbed vierbein is the unit matrix

0

emµ = δµ

m . (20.2)

Let overdot denote partial differentiation with respect to time t,

overdot ≡ ∂

∂t, (20.3)

and let ∇ denote the spatial gradient

∇ ≡ ∂

∂x≡ ∇i ≡

∂xi. (20.4)

Sometimes it will also be convenient to use ∇m to denote the 4-dimensional spacetime derivative

∇m ≡ ∂

∂t,∇

. (20.5)

20.1 Classification of vierbein perturbations

The aims of this section are two-fold. First, decompose perturbations into scalar, vector, and tensor parts.

Second, identify the coordinate and tetrad gauge-invariant perturbations. It will be found, equations (20.13),

that there are 6 coordinate and tetrad gauge-invariant perturbations, comprising 2 scalars Ψ and Φ, 1 vector

Wi containing 2 degrees of freedom, and 1 tensor hij containing 2 degrees of freedom.

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20.1 Classification of vierbein perturbations 327

The vierbein perturbations ϕmn decompose into 6 scalars, 4 vectors, and 1 tensor

ϕtt = ψscalar

, (20.6a)

ϕti = ∇iwscalar

+ wivector

, (20.6b)

ϕit = ∇iwscalar

+ wivector

, (20.6c)

ϕij = δij Φscalar

+∇i∇jhscalar

+ εijk∇khscalar

+∇ihjvector

+∇j hivector

+ hijtensor

. (20.6d)

The tildes on w and h simply distinguish those symbols; the tildes have no other significance. The vector

components are by definition transverse (have vanishing divergence), while the tensor component hij is by

definition traceless, symmetric, and transverse. For a single Fourier mode whose wavevector k is taken

without loss of generality to lie in the z-direction, equations (20.6) are

ϕmn =

ψ wx wy ∇zw

wx Φ + hxx hxy +∇zh ∇zhx

wy hxy −∇zh Φ− hxx ∇zhy

∇zw ∇zhx ∇zhy Φ +∇2zh

. (20.7)

To identify coordinate gauge-invariant quantities, it is necessary to consider infinitesimal coordinate gauge

transformations (18.8). The tetrad-frame components ǫm of the coordinate shift of the coordinate gauge

transformation decompose into 2 scalars and 1 vector

ǫm = ǫtscalar

, ∇iǫscalar

+ ǫivector

. (20.8)

In the flat space background space being considered the coordinate gauge transformation (18.26) of the

vierbein perturbation simplifies to

ϕmn → ϕ′mn = ϕmn +∇mǫn . (20.9)

In terms of the scalar, vector, and tensor potentials introduced in equations (20.6), the gauge transforma-

tions (20.9) are

ϕtt → ψ + ǫtscalar

, (20.10a)

ϕti → ∇i(w + ǫ)scalar

+ (wi + ǫi)vector

, (20.10b)

ϕit → ∇i(w + ǫt)scalar

+ wivector

, (20.10c)

ϕij → δij Φscalar

+∇i∇j(h+ ǫ)scalar

+ εijk∇khscalar

+∇i(hj + ǫj)vector

+∇j hivector

+ hijtensor

. (20.10d)

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328 Flat space background

Equations (20.10a) imply that under an infinitesimal coordinate gauge transformation

ψ → ψ + ǫt , (20.11a)

w → w + ǫ , wi → wi + ǫi , (20.11b)

w → w + ǫt , wi → wi , (20.11c)

Φ→ Φ , h→ h+ ǫ , h→ h , hi → hi + ǫi , hi → hi , hij → hij . (20.11d)

Eliminating the coordinate shift ǫm from the transformations (20.11) yields 12 coordinate gauge-invariant

combinations of the potentials

ψ − ˙w , w − h , wi − hi , wi , Φ , h , hi , hij . (20.12)

Physical perturbations are not only coordinate but also tetrad gauge-invariant. A quantity is tetrad gauge-

invariant if and only if it depends only on the symmetric part of the vierbein pertubations, not on the

antisymmetric part, §18.6. There are 6 combinations of the coordinate gauge-invariant perturbations (20.12)

that are symmetric, and therefore not only coordinate but also tetrad gauge-invariant. These 6 coordinate

and tetrad gauge-invariant perturbations comprise 2 scalars, 1 vector, and 1 tensor

Ψscalar

≡ ψ − w − ˙w + h , (20.13a)

Φscalar

, (20.13b)

Wivector

≡ wi + wi − hi − ˙hi , (20.13c)

hijtensor

. (20.13d)

Since only the 6 tetrad and coordinate gauge-invariant potentials Ψ, Φ, Wi, and hij have physical signifi-

cance, it is legitimate to choose a particular gauge, a set of conditions on the non-gauge-invariant potentials,

arranged to simplify the equations, or to bring out some physical aspect. Three gauges considered later are

harmonic gauge (§20.7), Newtonian gauge (§20.9), and synchronous gauge (§20.10). However, for the next

several sections, no gauge will be chosen: the exposition will continue to be completely general.

20.2 Metric, tetrad connections, and Einstein and Weyl tensors

This section gives expressions in a completely general gauge for perturbed quantities in flat background

Minkowski space.

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20.2 Metric, tetrad connections, and Einstein and Weyl tensors 329

The perturbed coordinate metric gµν , equation (18.5), is

gtt = −(1 + 2ψ) , (20.14a)

gti = −∇i(w + w)− (wi + wi) , (20.14b)

gij = δij (1− 2 Φ)− 2∇i∇jh−∇i(hj + hj)−∇j(hi + hi)− 2 hij . (20.14c)

The coordinate metric is tetrad gauge-invariant, but not coordinate gauge-invariant.

The perturbed tetrad connections Γkmn are

Γtit = −∇i(ψ − ˙w) + ˙wi , (20.15a)

Γtij = δij Φ−∇i∇j(w − h)− 12 (∇iWj +∇jWi) +∇jwi + hij , (20.15b)

Γijt = 12 (∇iWj −∇jWi)−

∂t(εijl∇lh−∇ihj +∇j hi) , (20.15c)

Γijk = (δjk∇i − δik∇j)Φ−∇k(εijl∇lh−∇ihj +∇j hi) +∇ihjk −∇jhik . (20.15d)

Being purely tetrad-frame quantities, the tetrad connections are automatically coordinate gauge-invariant.

However, they are not tetrad gauge-invariant, as is evident from the fact that they (all) depend on antisym-

metric parts of the vierbein perturbations ϕmn.

One of the advantages of working with tetrads is that tetrad-frame quantities such as the tetrad connec-

tions Γkmn and the tetrad-frame Riemann tensor Rklmn are by construction independent of the choice of

coordinates, and are therefore automatically coordinate gauge-invariant. In the tetrad formalism, you do

not have to work too hard (is that really ever true?) to construct coordinate gauge-invariant combinations

of the vierbein perturbations ϕmn: the tetrad-frame connections and Riemann tensor will automatically

give you the coordinate gauge-invariant combinations. You can check that in the present case the tetrad

connections (20.15) depend only on, and on all 12 of, the coordinate gauge-invariant combinations (20.12).

The perturbed tetrad-frame Einstein tensor Gmn is

Gtt = 2∇2Φscalar

, (20.16a)

Gti = 2∇iΦscalar

+ 12 ∇2Wivector

, (20.16b)

Gij = 2 δij Φscalar

− (∇i∇j − δij∇2)(Ψ− Φ)scalar

+ 12 (∇iWj +∇jWi)

vector

− hijtensor

, (20.16c)

where is the d’Alembertian, the 4-dimensional wave operator

≡ −∇m∇m =∂2

∂t2−∇2 . (20.17)

Being a tetrad-frame quantity, the tetrad-frame Einstein tensor is automatically coordinate gauge-invariant.

Equations (20.16) show that the tetrad-frame Einstein tensor Gmn is also tetrad gauge-invariant, since it

depends only on the tetrad-gauge invariant combinations (20.13) of the vierbein perturbations. The property

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330 Flat space background

that the Einstein tensor is tetrad as well as coordinate gauge-invariant is a feature of empty background

space, and does not persist to more general spacetimes, such as the Friedmann-Robertson-Walker spacetime.

In a frame with the wavector k taken along the z-axis, the perturbed Einstein tensor is

Gmn =

2∇2zΦ

12 ∇2

zWx12 ∇2

zWy 2∇zΦ

12 ∇2

zWx 2 Φ +∇2z(Ψ− Φ)−h+ −h×

12 ∇zWx

12 ∇2

zWy −h× 2 Φ +∇2z(Ψ− Φ) + h+

12 ∇zWy

2∇zΦ12 ∇zWx

12 ∇zWy 2 Φ

(20.18)

where h+ and h× are the two polarizations of gravitational waves, discussed further in §20.14,

h+ ≡ hxx = − hyy , h× ≡ hxy = hyx . (20.19)

The tetrad-frame complexified Weyl tensor is

Ctitj = 14 (∇i∇j − 1

3 δij∇2)(Ψ + Φ)scalar

+ 18

[

− (∇iWj +∇jWi) + i(εikl∇j + εjkl∇i)∇kWl

]

vector

+ 14

[

hij − εiklεjmn∇k∇mhln − i(εikl∇khjl + εjkl∇khil)]

tensor

. (20.20)

Like the tetrad-frame Einstein tensor, the tetrad-frame Weyl tensor is both coordinate and tetrad gauge-

invariant, depending only on the coordinate and tetrad gauge-invariant potentials Ψ, Φ, Wi, and hij .

20.3 Spinor components of the Einstein tensor

Scalar, vector, and tensor perturbations correspond respectively to perturbations of spin 0, 1, and 2. An

object has spin m if it is unchanged by a rotation of 2π/m about a prescribed direction. In perturbed

Minkowski space, the prescribed direction is the direction of the wavevector k in the Fourier decomposition

of the modes. The spin components may be projected out by working in a spinor tetrad, §12.1.1.

In a frame where the wavevector k is taken along the z-axis, the spinor components of the perturbed

Einstein tensor Gmn are (compare equations (20.16))

Gtt = 2∇2zΦ

spin-0

, Gtz = 2∇zΦspin-0

, Gzz = 2 Φspin-0

, (20.21a)

G+− −Gzz = ∇2z(Ψ− Φ)spin-0

, (20.21b)

Gt± = 12 ∇2

zW±

spin-±1

, Gz± = 12 ∇zW±

spin-±1

, (20.21c)

G±± = −h±±

spin-±2, (20.21d)

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20.4 Too many Einstein equations? 331

where W± are the spin-±1 components of the vector perturbation Wi

W± = 1√2

(Wx ± iWy) , (20.22)

and h±± are the spin-±2 components of the tensor perturbation hij

h±± = hxx ± i hxy = h+ ± i h× . (20.23)

The spin +2 and −2 components h++ and h−− of the tensor perturbation are called the right- and left-handed

circular polarizations. The spin +2 and −2 circular polarizations h++ and h−− have respective shapes ei2χ

and e−i2χ, under a right-handed rotation by angle χ about the z-axis, which may be compared to the cos 2χ

and sin 2χ shapes of the linear polarizations h+ and h×.

20.4 Too many Einstein equations?

The Einstein equations are as usual (units c = G = 1)

Gmn = 8πTmn . (20.24)

There are 10 Einstein equations, but the Einstein tensor (20.16) depends on only 6 independent potentials:

the two scalars Ψ and Φ, the vector Wi, and the tensor hij . The system of Einstein equations is thus over-

complete. Why? The answer is that 4 of the Einstein equations enforce conservation of energy-momentum,

and can therefore be considered as governing the evolution of the energy-momentum as opposed to being

equations for the gravitational potentials. For example, the form of equations (20.16a) and (20.16b) for Gtt

and Gti enforces conservation of energy

DmGmt = 0 , (20.25)

while the form of equations (20.16b) and (20.16c) for Gti and Gij enforces conservation of momentum

DmGmi = 0 . (20.26)

Normally, the equations governing the evolution of the energy-momentum Tmna of each species a of mass-

energy would be set up so as to ensure overall conservation of energy-momentum. If this is done, then

the conservation equations (20.25) and (20.26) can be regarded as redundant. Since equations (20.25) and

(20.26) are equations for the time evolution of Gtt and Gti, one might think that the Einstein equations

for Gtt and Gti would become redundant, but this is not quite true. In fact the Einstein equations for

Gtt and Gti impose constraints that must be satisfied on the initial spatial hypersurface. Conservation

of energy-momentum guarantees that those constraints will continue to be satisfied on subsequent spatial

hypersurfaces, but still the initial conditions must be arranged to satisfy the constraints. Because the Einstein

equations for Gtt and Gti must be satisfied as constraints on the initial conditions, but thereafter can be

ignored, the equations are called constraint equations. The Einstein equation for Gtt is called the energy

constraint, or Hamiltonian constraint. The Einstein equations for Gti are called the momentum constraints.

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332 Flat space background

Exercise 20.1 Energy and momentum constraints. Confirm the argument of this section. Suppose

that the spatial Einstein equations are true, Gij = 8πT ij. Show that if the time-time and time-space

Einstein equations Gtm = 8πT tm are initially true, then conservation of energy-momentum implies that

these equations must necessarily remain true at all times. [Hint: Conservation of energy-momentum requires

that DmTmn = 0, and the Bianchi identities require that the Einstein tensor satisfies DmG

mn = 0, so

Dm(Gmn − 8πTmn) = 0 . (20.27)

By expanding out these equations in full, or otherwise, show that the solution satisfying Gij − 8πT ij = 0 at

all times, and Gtm − 8πT tm = 0 initially, is Gtm − 8πT tm = 0 at all times.]

Concept question 20.2 Which Einstein equations are redundant? It has been argued in this

section that, if the energy-momentum tensor Tmn is arranged to satisfy energy-conservation DmTmn = 0 as

it should, then the time-time and time-space Einstein equations must be satisfied by the initial conditions,

but thereafter become redundant. Question: Can any 4 of the 10 Einstein equations be dropped, or just the

time-time and time-space Einstein equations?

20.5 Action at a distance?

The tensor component of the Einstein equations shows that, in a vacuum Tmn = 0, the tensor perturbations

hij propagate at the speed of light, satisfying the wave equation

hij = 0 . (20.28)

The tensor perturbations represent propagating gravitational waves.

It is to be expected that scalar and vector perturbations would also propagate at the speed of light, yet

this is not obvious from the form of the Einstein tensor (20.16). Specifically, there are 4 components of the

Einstein tensor (20.16) that apparently depend only on spatial derivatives, not on time derivatives. The 4

corresponding Einstein equations are

∇2Φ = 4πTttscalar

, (20.29a)

∇2Wi = 16πTtivector

, (20.29b)

∇2(Ψ− Φ) = − 8πQijTijscalar

, (20.29c)

where Qij in equation (20.29c) is the quadrupole operator defined below, equation (20.87). These condi-

tions must be satisfied everywhere at every instant of time, giving the impression that signals are traveling

instantaneously from place to place.

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20.6 Comparison to electromagnetism 333

20.6 Comparison to electromagnetism

The previous two sections §§20.4, 20.5 brought up two issues:

1. There are 10 Einstein equations, but only 6 independent gauge-invariant potentials Ψ, Φ, Wi, and hij .

The additional 4 Einstein equations serve to enforce conservation of energy-momentum.

2. Only 2 of the gauge-invariant potentials, the tensor potentials hij , satisfy causal wave equations. The

remaining 4 gauge-invariant potentials Ψ, Φ, and Wi, satisfy equations (20.29) that depend on the instan-

taneous distribution of energy-momentum throughout space, on the face of it violating causality.

These facts may seem surprising, but in fact the equations of electromagnetism have a similar structure, as

will now be shown. In this section, the spacetime is assumed to be flat Minkowski space. The discussion

in this section is based in part on the exposition by Bertschinger (1995 “Cosmological Dynamics,” 1993 Les

Houches Lectures, arXiv:astro-ph/9503125).

In accordance with the usual procedure, the electromagnetic field may be defined in terms of an elec-

tromagnetic 4-potential Am, whose time and spatial parts constitute the scalar potential φ and the vector

potential A:

Am ≡ φ,A . (20.30)

The electric and magnetic fields E and B may be defined in terms of the potentials φ and A by

E ≡ −∇φ− ∂A

∂t, (20.31a)

B ≡∇×A . (20.31b)

Given their definition (20.31), the electric and magnetic fields automatically satisfy the two source-free

Maxwell’s equations

∇ ·B = 0 , (20.32a)

∇×E +∂B

∂t= 0 . (20.32b)

The remaining two Maxwell’s equations, the sourced ones, are

∇ ·E = 4πq , (20.33a)

∇×B − ∂E

∂t= 4πj , (20.33b)

where q and j are the electric charge and current density, the time and space components of the electric

4-current density jm

jm ≡ q, j . (20.34)

The electromagnetic potentials φ and A are not unique, but rather are defined only up to a gauge transfor-

mation by some arbitrary gauge field χ

φ→ φ+∂χ

∂t, A→ A−∇χ . (20.35)

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334 Flat space background

The gauge transformation (20.35) evidently leaves the electric and magnetic fields E and B, equations (20.31),

invariant.

Following the path of previous sections, §20.1 and thereafter, decompose the vector potential A into its

scalar and vector parts

A = ∇A‖scalar

+ A⊥vector

, (20.36)

in which the vector part by definition satisfies the transversality condition ∇ · A⊥ = 0. Under a gauge

transformation (20.35), the potentials transform as

φ→ φ+∂χ

∂t, (20.37a)

A‖ → A‖ − χ , (20.37b)

A⊥ → A⊥ . (20.37c)

Eliminating the gauge field χ yields 3 gauge-invariant potentials, comprising 1 scalar Φ, and 1 vector A⊥containing 2 degrees of freedom:

Φscalar

≡ φ+∂A‖∂t

, (20.38a)

A⊥vector

. (20.38b)

This shows that the electromagnetic field contains 3 independent degrees of freedom, consisting of 1 scalar

and 1 vector.

Concept question 20.3 Are gauge-invariant potentials Lorentz invariant? The potentials Φ and

A⊥, equations (20.38), are by construction gauge-invariant, but is this construction Lorentz invariant? Do

Φ and A⊥ constitute the components of a 4-vector?

In terms of the gauge-invariant potentials Φ and A⊥, equations (20.38), the electric and magnetic fields

are

E = −∇Φ− ∂A⊥∂t

, (20.39a)

B = ∇×A⊥ . (20.39b)

The sourced Maxwell’s equations (20.33) thus become, in terms of Φ and A⊥,

−∇2Φscalar

= 4πqscalar

, (20.40a)

∇Φscalar

+ A⊥vector

= 4π∇j‖scalar

+ 4πj⊥vector

, (20.40b)

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20.6 Comparison to electromagnetism 335

where ∇j‖ and j⊥ are the scalar and vector parts of the current density j. Equations (20.40) bear a striking

similarity to the Einstein equations (20.16). Only the vector part A⊥ satisfies a wave equation,

A⊥ = 4πj⊥ , (20.41)

while the scalar part Φ satisfies an instantaneous equation (20.40a) that seemingly violates causality. And just

as Einstein’s equations (20.16) enforce conservation of energy-momentum, so also Maxwell’s equations (20.40)

enforce conservation of electric charge

∂q

∂t+ ∇ · j = 0 , (20.42)

or in 4-dimensional form

∇mjm = 0 . (20.43)

The fact that only the vector part A⊥ satisfies a wave equation (20.41) reflects physically the fact that

electromagnetic waves are transverse, and they contain only two propagating degrees of freedom, the vector,

or spin ±1, components.

Why do Maxwell’s equations (20.40) have this structure? Although equation (20.41) appears to be a local

wave equation for the vector part A⊥ of the potential sourced by the vector part j⊥ of the current, in fact

the wave equation is non-local because the decomposition of the potential and current into scalar and vector

parts is non-local (it involves the solution of a Laplacian equation, eq. (19.3)). It is only the sum j = ∇j‖+j⊥of the scalar and vector parts of the current density that is local. Therefore, the Maxwell’s equation (20.40b)

must have a scalar part to go along with the vector part, such that the source on the right hand side, the

current density j, is local. Given this Maxwell equation (20.40b), the Maxwell equation (20.40a) then serves

precisely to enforce conservation of electric charge, equation (20.42).

Just as it is possible to regard the Einstein equations (20.16a) and (20.16b) as constraint equations

whose continued satisfaction is guaranteed by conservation of energy-momentum, so also the Maxwell equa-

tion (20.40a) for Φ can be regarded as a constraint equation whose continued satisfaction is guaranteed

by conservation of electric charge. For charge conservation (20.42) coupled with the spatial Maxwell equa-

tion (20.40b) ensures that

∂t

(

4πq +∇2Φ)

= 0 , (20.44)

the solution of which, subject to the condition that 4πq +∇2Φ = 0 initially, is 4πq +∇2Φ = 0 at all times,

which is precisely the Maxwell equation (20.40a).

Exercise 20.4 Is it possible to discard the scalar part of the spatial Maxwell equation (20.40b),

rather than equation (20.40a) for Φ? Project out the scalar part of equation (20.40b) by taking its

divergence,

∇2(

4πj‖ − Φ)

= 0 . (20.45)

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336 Flat space background

Argue that the Maxwell equation (20.40a), coupled with charge conservation (20.42), ensures that equa-

tion (20.45) is true, subject to boundary condition that the current j vanish sufficiently rapidly at spatial

infinity, in accordance with the decomposition theorem of §19.1.

Since only gauge-invariant quantities have physical significance, it is legitimate to impose any condition

on the gauge field χ. A gauge in which the potentials φ and A individually satisfy wave equations is Lorenz

(not Lorentz!) gauge, which consists of the Lorentz-invariant condition

∇mAm = 0 . (20.46)

Under a gauge transformation (20.35), the left hand side of equation (20.46) transforms as

∇mAm → ∇mA

m + χ , (20.47)

and the Lorenz gauge condition (20.46) can be accomplished as a particular solution of the wave equation

for the gauge field χ. In terms of the potentials φ and A‖, the Lorenz gauge condition (20.46) is

∂φ

∂t+∇2A‖ = 0 . (20.48)

In Lorenz gauge, Maxwell’s equations (20.40) become

φ = 4πq , (20.49a)

A = 4πj , (20.49b)

which are manifestly wave equations for the potentials φ and A.

Does the fact that the potentials φ and A in one particular gauge, Lorenz gauge, satisfy wave equations

necessarily guarantee that the electric and magnetic fields E and B satisfy wave equations? Yes, because it

follows from the definitions (20.31) of E and B that if the potentials φ and A satisfy wave equations, then

so also must the fields E and B themselves; but the fields E and B are gauge-invariant, so if they satisfy

wave equations in one gauge, then they must satisfy the same wave equations in any gauge.

In electromagnetism, the most physical choice of gauge is one in which the potentials φ and A coincide

with the gauge-invariant potentials Φ and A⊥, equations (20.38). This gauge, known as Coulomb gauge,

is accomplished by setting

A‖ = 0 , (20.50)

or equivalently

∇ ·A = 0 . (20.51)

The gravitational analogue of this gauge is the Newtonian gauge discussed in the next section but one, §20.9.

Does the fact that in Lorenz gauge the potentials φ and A propagate at the speed of light (in the absence

of sources, jm = 0) imply that the gauge-invariant potentials Φ and A⊥ propagate at the speed of light?

No. The gauge-invariant potentials Φ and A⊥, equations (20.38), are related to the Lorenz gauge potentials

φ and A by a non-local decomposition.

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20.7 Harmonic gauge 337

20.7 Harmonic gauge

The fact that all locally measurable gravitational perturbations do propagate causally, at the speed of light

in the absence of sources, can be demonstrated by choosing a particular gauge, harmonic gauge, equa-

tion (20.52), which can be considered an analogue of the Lorenz gauge of electromagnetism, equation (20.46).

In harmonic gauge, all 10 of the tetrad gauge-variant (i.e. symmetric) combinations ϕmn+ϕnm of the vierbein

perturbations satisfy wave equations (20.56), and therefore propagate causally. This does not imply that

the scalar, vector, and tensor components of the vierbein perturbations individually propagate causally, be-

cause the decomposition into scalar, vector, and tensor modes is non-local. In particular, the coordinate and

tetrad-gauge invariant potentials Ψ, Φ, Wi, and hij defined by equations (20.13) do not propagate causally.

The situation is entirely analogous to that of electromagnetism, §20.6, where in Lorenz gauge the potentials

φ and A propagate causally, equations (20.49), yet the gauge-invariant potentials Φ and A⊥ defined by

equations (20.38) do not.

Harmonic gauge is the set of 4 coordinate conditions

∇m(ϕmn + ϕnm)−∇nϕmm = 0 . (20.52)

The conditions (20.52) are arranged in a form that is tetrad gauge-invariant (the conditions depend only on

the symmetric part of ϕmn). The quantities on the left hand side of equations (20.52) transform under a

coordinate gauge transformation, in accordance with (20.9), as

∇m(ϕmn + ϕnm)−∇nϕmm → ∇m(ϕmn + ϕnm)−∇nϕm

m + ǫn . (20.53)

The change ǫn resulting from the coordinate gauge transformation is the 4-dimensional wave operator

acting on the coordinate shift ǫn. Indeed, the harmonic gauge conditions (20.52) follow uniquely from the

requirements (a) that the change produced by a coordinate gauge transformation be ǫn, as suggested by

the analogous electromagnetic transformation (20.47), and (b) that the conditions be tetrad gauge-invariant.

The harmonic gauge conditions (20.52) can be accomplished as a particular solution of the wave equation for

the coordinate shift ǫn. In terms of the potentials defined by equations (20.6) and (20.13), the 4 harmonic

gauge conditions (20.52) are

Ψ + 3Φ + (w + w − h) = 0 , (20.54a)

Wi + (hi + hi) = 0 , (20.54b)

−Ψ + Φ + h = 0 , (20.54c)

or equivalently

− 4 Φ = (w + w) , (20.55a)

− Wi = (hi + hi) , (20.55b)

Ψ− Φ = h . (20.55c)

Substituting equations (20.55) into the Einstein tensor Gmn leads, after some calculation, to the result that

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338 Flat space background

in harmonic gauge

− 12 (ϕmn + ϕnm) = Rmn , (20.56)

where Rmn = Gmn− 12 ηmnG is the Ricci tensor. Equation (20.56) shows that in harmonic gauge, all tetrad

gauge-invariant (i.e. symmetric) combinations ϕmn+ϕnm of the vierbein potentials propagate causally, at

the speed of light in vacuo, Rmn = 0. Although the result (20.56) is true only in a particular gauge,

harmonic gauge, it follows that all quantities that are (coordinate and tetrad) gauge-invariant, and that can

be constructed from the vierbein potentials ϕmn and their derivatives (and are therefore local), must also

propagate at the speed of light.

20.8 What is the gravitational field?

In electromagnetism, the electromagnetic fields are the electric field E and the magnetic field B. These fields

have the property that they are gauge-invariant, and measurable locally. The electromagnetic potentials Φ

and A⊥, equations (20.38), are gauge-invariant, but they are not measurable locally.

What are the analogous gauge-invariant and locally measurable quantities for the gravitational field in

perturbed Minkowski space? The answer is, the Weyl tensor Cklmn, the trace-free or tidal part of the

Riemann tensor, the expression (20.20) for which depends only on the coordinate and tetrad gauge-invariant

potentials.

20.9 Newtonian (Copernican) gauge

If the unperturbed background is Minkowski space, then the most physical gauge is one in which the 6

perturbations retained coincide with the 6 coordinate and tetrad gauge-invariant perturbations (20.13).

This gauge is called Newtonian gauge. Because in Newtonian gauge the perturbations are precisely the

physical perturbations, if the perturbations are physically weak (small), then the perturbations in Newtonian

gauge will necessarily be small.

I think Newtonian gauge should be called Copernican gauge. Even though the solar system is a highly non-

linear system, from the perspective of general relativity it is a weakly perturbed gravitating system. Applied

to the solar system, Newtonian gauge effectively keeps the coordinates aligned with the classical Sun-centred

Copernican coordinate frame. By contrast, the coordinates of synchronous gauge (§20.10), which are chosen

to follow freely-falling bodies, would quickly collapse or get wound up by orbital motions if applied to the

solar system, and would cease to provide a useful description.

Newtonian gauge sets

w = w = wi = h = h = hi = hi = 0 , (20.57)

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20.10 Synchronous gauge 339

so that the retained perturbations are the 6 coordinate and tetrad gauge-invariant perturbations (20.13)

Ψscalar

= ψ , (20.58a)

Φscalar

, (20.58b)

Wivector

= wi , (20.58c)

hijtensor

. (20.58d)

The Newtonian line-element is, in a form that keeps the Newtonian tetrad manifest,

ds2 = −[

(1 + Ψ) dt]2

+ δij[

(1− Φ)dxi − hikdx

k −W idt][

(1− Φ)dxj − hjl dx

l −W jdt]

, (20.59)

which reduces to the Newtonian metric

ds2 = − (1 + 2 Ψ) dt2 − 2Wi dt dxi +[

δij(1 − 2 Φ)− 2 hij

]

dxi dxj . (20.60)

Since scalar, vector, and tensor perturbations evolve independently, it is legitimate to consider each in

isolation. For example, if one is interested only in scalar perturbations, then it is fine to keep only the

scalar potentials Ψ and Φ non-zero. Furthermore, as discussed in §20.13, since the difference Ψ − Φ in

scalar potentials is sourced by anisotropic relativistic pressure, which is typically small, it is often a good

approximation to set Ψ = Φ.

The tetrad-frame 4-velocity of a person at rest in the tetrad frame is by definition um = 1, 0, 0, 0, and

the corresponding coordinate 4-velocity uµ is, in Newtonian gauge,

uµ = etµ = 1−Ψ,Wi . (20.61)

This shows that Wi can be interpreted as a 3-velocity at which the tetrad frame is moving through the coor-

dinates. This is the “dragging of inertial frames” discussed in §20.12. The proper acceleration experienced

by a person at rest in the tetrad frame, with tetrad 4-velocity um = 1, 0, 0, 0, is

Dui

Dτ= utDtu

i = ut(

∂tui + Γi

ttut)

= Γitt = ∇iΨ . (20.62)

This shows that the “gravity,” or minus the proper acceleration, experienced by a person at rest in the tetrad

frame is minus the gradient of the potential Ψ.

Concept question 20.5 If the decomposition into scalar, vector, and tensor modes is non-local, how can

it be legimate to consider the evolution of the modes in isolation from each other?

20.10 Synchronous gauge

One of the earliest gauges used in general relativistic perturbation theory, and still (in its conformal version)

widely used in cosmology, is synchronous gauge. As will be seen below, equations (20.69) and (20.70),

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340 Flat space background

synchronous gauge effectively chooses a coordinate system and tetrad that is attached to the locally inertial

frames of freely falling observers. This is fine as long as the observers move only slightly from their initial

positions, but the coordinate system will fail when the system evolves too far, even if, as in the solar system,

the gravitational perturbations remain weak and therefore treatable in principle with perturbation theory.

Synchronous gauge sets the time components ϕmn with m = t or n = t of the vierbein perturbations to

zero

ψ = w = w = wi = wi = 0 , (20.63)

and makes the additional tetrad gauge choices

h = hi = 0 , (20.64)

with the result that the retained perturbations are the spatial perturbations

Φscalar

, hscalar

, hivector

, hijtensor

. (20.65)

In terms of these spatial perturbations, the gauge-invariant perturbations (20.13) are

Ψscalar

= h , (20.66a)

Φscalar

, (20.66b)

Wivector

= − hi , (20.66c)

hijtensor

. (20.66d)

The synchronous line-element is, in a form that keeps the synchronous tetrad manifest,

ds2 = − dt2 + δij[

(1−Φ)dxi − (∇k∇ih+∇khi + hi

k)dxk][

(1−Φ)dxj − (∇l∇jh+∇lhj + hj

l )dxl]

, (20.67)

which reduces to the synchronous metric

ds2 = − dt2 + [(1 − 2 Φ)δij − 2∇i∇jh−∇ihj −∇jhi − 2 hij ] dxi dxj . (20.68)

In synchronous gauge, a person at rest in the tetrad frame has coordinate 4-velocity

uµ = etµ = 1, 0, 0, 0 , (20.69)

so that the tetrad rest frame coincides with the coordinate rest frame. Moreover a person at rest in the

tetrad frame is freely falling, which follows from the fact that the acceleration experienced by a person at

rest in the tetrad frame is zero

Duk

Dτ= ut

(

∂tuk + Γk

ttut)

= Γktt = 0 , (20.70)

in which ∂tuk = 0 because the 4-velocity at rest in the tetrad frame is constant, uk = 1, 0, 0, 0, and Γk

tt = 0

from equations (20.15a) with the synchronous gauge choices (20.63) and (20.64).

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20.11 Newtonian potential 341

20.11 Newtonian potential

The next few sections examine the physical meaning of each of the gauge-invariant potentials Ψ, Φ, Wi, and

hij by looking at the potentials at large distances produced by a finite body containing energy-momentum,

such as the Sun.

Einstein’s equations Gmn = 8πTmn applied to the time-time component Gtt of the Einstein tensor, equa-

tion (20.16a), imply Poisson’s equation

∇2Φ = 4πρ , (20.71)

where ρ is the mass-energy density

ρ ≡ Ttt . (20.72)

The solution of Poisson’s equation (20.71) is

Φ(x) = −∫

ρ(x′) d3x′

|x′ − x| . (20.73)

Consider a finite body, for example the Sun, whose energy-momentum is confined within a certain region.

Define the mass M of the body to be the integral of the mass-energy density ρ,

M ≡∫

ρ(x′) d3x′ . (20.74)

Equation (20.74) agrees with what the definition of the mass M would be in the non-relativistic limit, and

as seen below, equation (20.77), it is what a distant observer would infer the mass of the body to be based

on its gravitational potential Φ far away. Thus equation (20.74) can be taken as the definition of the mass

of the body even when the energy-momentum is relativistic. Choose the origin of the coordinates to be at

the centre of mass, meaning that∫

x′ ρ(x′) d3x′ = 0 . (20.75)

Consider the potential Φ at a point x far outside the body. Expand the denominator of the integral on the

right hand side of equation (20.73) as a Taylor series in 1/x where x ≡ |x|

1

|x′ − x| =1

x

∞∑

ℓ=0

(

x′

x

)ℓ

Pℓ(x · x′) =1

x+

x · x′

x2+ ... (20.76)

where Pℓ(µ) are Legendre polynomials. Then

Φ(x) = − 1

x

ρ(x′) d3x′ − 1

x2x ·∫

x′ ρ(x′) d3x′ −O(x−3)

= − M

x−O(x−3) . (20.77)

Equation (20.77) shows that the potential far from a body goes as Φ = −M/x, reproducing the usual

Newtonian formula.

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342 Flat space background

20.12 Dragging of inertial frames

In Newtonian gauge, the vector potential W ≡Wi is the velocity at which the locally inertial tetrad frame

moves through the coordinates, equation (20.61). This is called the dragging of inertial frames. As shown

below, a body of angular momentum L drags frames around it with an angular velocity that goes to 2L/x3

at large distances x.

Einstein’s equations applied to the vector part of the time-space component Gti of the Einstein tensor,

equation (20.16b), imply

∇2W = − 16πf , (20.78)

where W ≡Wi is the gauge-invariant vector potenial, and f is the vector part of the energy flux T ti

f ≡ fi = f i ≡ T ti

vector= −Tti

vector. (20.79)

The solution of equation (20.78) is

W (x) = 4

f(x′) d3x′

|x′ − x| . (20.80)

As in the previous section, §20.11, consider a finite body, such as the Sun, whose energy-momentum is

confined within a certain region. Work in the rest frame of the body, defined to be the frame where the

energy flux f integrated over the body is zero,

f(x′) d3x′ = 0 . (20.81)

Define the angular momentum L of the body to be

L ≡∫

x′ × f(x′) d3x′ . (20.82)

Equation (20.82) agrees with what the definition of angular momentum would be in the non-relativistic limit,

where the mass-energy flux of a mass density ρ moving at velocity v is f = ρv. As will be seen below, the

angular momentum (20.82) is what a distant observer would infer the angular momentum of the body to be

based on the potential W far away, and equation (20.82) can be taken to be the definition of the angular

momentum of the body even when the energy-momentum is relativistic. As will be proven momentarily,

equation (20.83), the integral∫

x′ifj(x′) d3x′ is antisymmetric in ij. To show this, write fj = εjkl∇kφl for

some potential φl, which is valid because fj is the vector (curl) part of the energy flux. Then

x′ifj(x′) d3x′ =

x′iεjkl∇′kφl(x

′) d3x′ = −∫

εjklφl(x′)∇′

kx′i d

3x′ =

εijlφl(x′) d3x′ , (20.83)

where the third expression follows from the second by integration by parts, the surface term vanishing

because of the assumption that the energy-momentum of the body is confined within a certain region.

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20.13 Quadrupole pressure 343

Taylor expanding equation (20.80) using equation (20.76) gives

W (x) =4

x

f(x′) d3x′ +4

x2

(x · x′)f(x′) d3x+O(x−3)

=2

x2

[(x · x′)f(x′)− (x · f(x′))x′] d3x+O(x−3)

=2

x2L× x +O(x−3) , (20.84)

where the first integral on the right hand side of the first line of equation (20.84) vanishes because the frame

is the rest frame of the body, equation (20.81), and the second integral on the right hand side of the first line

equals the first integral on the second line thanks to the antisymmetry of∫

x′f(x′) d3x, equation (20.83).

The vector potential W ≡ Wi points in the direction of rotation, right-handedly about the axis of angular

momentum L. Equation (20.84) says that a body of angular momemtum L drags frames around it at angular

velocity Ω at large distances x

W = Ω× x , Ω =2L

x3. (20.85)

20.13 Quadrupole pressure

Einstein’s equations applied to the part of the Einstein tensor (20.16c) involving Ψ− Φ imply

∇2(Ψ− Φ) = − 8πQijTij , (20.86)

where Qij is the quadrupole operator (an integro-differential operator) defined by

Qij ≡ 32 ∇i∇j ∇−2 − 1

2 δij , (20.87)

with ∇−2 the inverse spatial Laplacian operator. In Fourier space, the quadrupole operator is

Qij = 32 kikj − 1

2 δij . (20.88)

The quadrupole operator Qij yields zero when acting on δij , and the Laplacian operator ∇2 when acting on

∇i∇j

Qijδij = 0 , Qij∇i∇j = ∇2 . (20.89)

The solution of equation (20.86) is

Ψ− Φ = −∫ [

3

2

(xi − x′i)(xj − x′j)|x− x′|2 − 1

2δij

]

Tij(x′) d3x′

|x− x′| . (20.90)

At large distance in the z-direction from a finite body

Ψ− Φ = − 1

x

[

Tzz − 12 (Txx + Tyy)

]

d3x′ +O(x−2) . (20.91)

Equation (20.86) shows that the source of the difference Ψ − Φ between the two scalar potentials is the

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344 Flat space background

quadrupole pressure. Since the quadrupole pressure is small if either there are no relativistic sources, or any

relativistic sources are isotropic, it is often a good approximation to set Ψ = Φ. An exception is where there

is a significant anisotropic relativistic component. For example, the energy-momentum tensor of a static

electric field is relativistic and anisotropic. However, this is still not enough to ensure that Ψ differs from Φ:

as found in Exercise 20.6, if the energy-momentum of a body is spherically symmetric, then Ψ−Φ vanishes

outside (but not inside) the body.

One situation where the difference between Ψ and Φ is appreciable is the case of freely-streaming neutrinos

at around the time of recombination in cosmology. The 2008 analysis of the CMB by the WMAP team claims

to detect a non-zero value of Ψ− Φ from a slight shift in the third acoustic peak.

Exercise 20.6 Argue that the traceless part of the energy-momentum tensor of a spherically symmetric

distribution must take the form

Tij(r) =(

rirj − 13 δij

)(

p(r) − p⊥(r))

, (20.92)

where p(r) and p⊥(r) are the radial and transverse pressures at radius r. From equation (20.90), show that

Ψ− Φ at radial distance x from the centre of a spherically symmetric distribution is

Ψ(x)− Φ(x) = −∫ ∞

x

(r2 − x2)(

p(r)− p⊥(r)) 4πdr

r. (20.93)

Notice that the integral is over r > x, that is, only energy-momentum outside radius x produces non-

vanishing Ψ− Φ. In particular, if the body has finite extent, then Ψ− Φ vanishes outside the body.

20.14 Gravitational waves

The tensor perturbations hij describe propagating gravitational waves. The two independent components of

the tensor perturbations describe two polarizations. The two components are commonly designated h+ and

h×, equations (20.19). Gravitational waves induce a quadrupole tidal oscillation transverse to the direction

of propagation, and the subscripts + and × represent the shape of the quadrupole oscillation, as illustrated

by Figure 20.1. The h+ polarization has a cos 2χ shape, while the h× polarization has a sin 2χ shape, where

χ is the azimuthal angle with respect to the y-axis about the direction x of propagation.

Einstein’s equations applied to the tensor component of the spatial Einstein tensor (20.16c) imply that

gravitational waves are sourced by the tensor component of the energy-momentum

hij = − 8π Tijtensor

. (20.94)

The solution of the wave equation (20.94) can be obtained from the Green’s function of the d’Alembertian

wave operator . The Green’s function is by definition the solution of the wave equation with a delta-

function source. There are retarded solutions, which propagate into the future along the future light cone,

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20.14 Gravitational waves 345

Figure 20.1 The two polarizations of gravitational waves. The (top) polarization h+ has a cos 2χ shapeabout the direction of propagation (into the paper), while the (bottom) polarization h× has a sin 2χ shape.A gravitational wave causes a system of freely falling test masses to oscillate relative to a grid of points afixed proper distance apart.

and advanced solutions, which propagate into the past along the past light cone. In the present case, the

solutions of interest are the retarded solutions, since these represent gravitational waves emitted by a source.

Because of the time and space translation symmetry of the d’Alembertian, the delta-function source of the

Green’s function can without loss of generality be taken at the origin t = x = 0. Thus the Green’s function

F is the solution of

F = δ4(x) , (20.95)

where δ4(x) ≡ δ(t)δ3(x) is the 4-dimensional Dirac delta-function. The solution of equation (20.95) subject

to retarded boundary conditions is (a standard exercise in mathematics) the retarded Green’s function

F =δ(x− t)Θ(t)

4πx, (20.96)

where x ≡ |x| and Θ(t) is the Heaviside function, Θ(t) = 0 for t < 0 and Θ(t) = 1 for t ≥ 0. The solution of

the sourced gravitational wave equation (20.94) is thus

hij(t,x) = − 2

∫ Tij(t′,x′)

tensor

d3x′

|x′ − x| , (20.97)

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346 Flat space background

where t′ is the retarded time

t′ ≡ t− |x′ − x| , (20.98)

which lies on the past light cone of the observer, and is the time at which the source emitted the signal. The

solution (20.97) resembles the solution of Poisson’s equation, except that the source is evaluated along the

past light cone of the observer.

As in §§20.11 and 20.12, consider a finite body, whose energy-momentum is confined within a certain region,

and which is a source of gravitational waves. The Hulse-Taylor binary pulsar is a fine example. Far from the

body, the leading order contribution to the tensor potential hij is, from the multipole expansion (20.76),

hij(t,x) = − 2

x

Tij(t′,x′)

tensor

d3x′ . (20.99)

The integral (20.99) is hard to solve in general, but there is a simple solution for gravitational waves whose

wavelengths are large compared to the size of the body. To obtain this solution, first consider that conser-

vation of energy-momentum implies that

∂2T tt

∂t2−∇i∇jT

ji =∂

∂t

(

∂T tt

∂t+∇iT

ti

)

−∇i

(

∂T ti

∂t+∇jT

ji

)

= 0 . (20.100)

Multiply by xixj and integrate∫

xixj ∂2T tt

∂t2d3x =

xixj ∇k∇lTkl d3x =

T kl∇k∇l(xixj) d3x = 2

T ij d3x , (20.101)

where the third expression follows from the second by a double integration by parts. For wavelengths that

are long compared to the size of the body, the first expression of equations (20.101) is∫

xixj∂2T tt

∂t2d3x ≈ ∂2

∂t2

xixj Ttt d3x =

∂2Iij∂t2

(20.102)

where Iij is the second moment of the mass

Iij ≡∫

xixj Ttt d3x . (20.103)

The tensor (spin-2) part of the energy-momentum is trace-free. The trace-free part –Iij of the second moment

Iij is the quadrupole moment of the mass distribution (this definition is conventional, but differs by a factor

of 2/3 from what is called the quadrupole moment in spherical harmonics)

–Iij ≡ Iij − 13 δij I

kk =

(xixj − 13 δij x

2)T tt d3x . (20.104)

Substituting the last expression of equations (20.101) into equation (20.99) gives the quadrupole formula for

gravitational radiation at wavelengths long compared to the size of the emitting body

hij(t,x) = − 1

x–Iij

tensor

. (20.105)

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20.15 Energy-momentum carried by gravitational waves 347

If the gravitational wave is moving in the z-direction, then the tensor components of the quadrupole moment–Iij are

–I+ = 12 (Ixx − Iyy) , –I× = 1

2 (Ixy + Iyx) . (20.106)

20.15 Energy-momentum carried by gravitational waves

The gravitational wave equation (20.28) in empty space appears to describe gravitational waves propagating

in a region where the energy-momentum tensor Tmn is zero. However, gravitational waves do carry energy-

momentum, just as do other kinds of waves, such as electromagnetic waves. The energy-momentum is

quadratic in the tensor perturbation hij , and so vanishes to linear order.

To determine the energy-momentum in gravitational waves, calculate the Einstein tensor Gmn to second

order, imposing the vacuum conditions that the unperturbed and linear parts of the Einstein tensor vanish

0

Gmn =1

Gmn = 0 . (20.107)

The parts of the second-order perturbation that depend on the tensor perturbation hij are, in a frame where

the wavevector k is along the z-axis,

2

Gtt = − (hij)(hij) +

1

4

( ∂2

∂t2+∇2

z

)

h2 , (20.108a)

2

Gtz = − (hij)(∇zhij) +

1

2

∂t∇zh

2 , (20.108b)

2

Gzz = − (∇zhij)(∇zhij) +

1

4

( ∂2

∂t2+∇2

z

)

h2 , (20.108c)

where

h2 ≡ hijhij = 2(h2

++ h2

×) = 2h++h−−. (20.109)

Being tetrad-frame quantities, the expressions (20.108) are automatically coordinate gauge-invariant, and

they are also tetrad gauge-invariant since they depend only on the (coordinate and) tetrad gauge-invariant

perturbation hij . The rightmost set of terms on the right hand side of each of equations (20.108) are total

derivatives (with respect to time t or space z). These terms yield surface terms when integrated over a

region, and tend to average to zero when integrated over a region much larger than a wavelength. On the

other hand, the leftmost set of terms on the right hand side of each of equations (20.108) do not average

to zero; for example, the terms for Gtt and Gzz are negative everywhere, being minus a sum of squares. A

negative energy density? The interpretation is that these terms are to be taken over to the right hand side

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348 Flat space background

of the Einstein equations, and re-interpreted as the energy-momentum T gwmn in gravitational waves

T gwtt ≡

1

[

(hij)(hij)− 1

4

( ∂2

∂t2+∇2

z

)

h2

]

, (20.110a)

T gwtz ≡

1

[

(hij)(∇zhij)− 1

2

∂t∇zh

2

]

, (20.110b)

T gwzz ≡

1

[

(∇zhij)(∇zhij)− 1

4

( ∂2

∂t2+∇2

z

)

h2

]

. (20.110c)

The terms involving total derivatives, although they vanish when averaged over a region larger than many

wavelengths, ensure that the energy-momentum T gwmn in gravitational waves satisfies conservation of energy-

momentum in the flat background space

∇mT gwmn = 0 . (20.111)

Averaged over a region larger than many wavelengths, the energy-momentum in gravitational waves is

〈T gwmn〉 =

1

8π(∇mhij)(∇nh

ij) . (20.112)

Equation (20.112) may also be written explicitly as a sum over the two linear or circular polarizations

〈T gwmn〉 =

1

4π[(∇mh+)(∇nh+) + (∇mh×)(∇nh×)]

=1

8π[(∇mh++)(∇nh−−) + (∇nh++)(∇mh−−)] . (20.113)

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PART EIGHT

COSMOLOGICAL PERTURBATIONS

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Concept Questions

1. Why do the wavelengths of perturbations in cosmology expand with the Universe, whereas perturbations

in Minkowski space do not expand?

2. What does power spectrum mean?

3. Why is the power spectrum a good way to characterize the amplitude of fluctuations?

4. Why is the power spectrum of fluctuations of the Cosmic Microwave Background (CMB) plotted as a

function of harmonic number?

5. What causes the acoustic peaks in the power spectrum of fluctuations of the CMB?

6. Are there acoustic peaks in the power spectrum of matter (galaxies) today?

7. What sets the scale of the first peak in the power spectrum of the CMB? [What sets the physical scale?

Then what sets the angular scale?]

8. The odd peaks (including the first peak) in the CMB power spectrum are compression peaks, while the

even peaks are rarefaction peaks. Why does a rarefaction produce a peak, not a trough?

9. Why is the first peak the most prominent? Why do higher peaks generally get progressively weaker?

10. The third peak is about as strong as the second peak? Why?

11. The matter power spectrum reaches a maximum at a scale that is slightly larger than the scale of the first

baryonic acoustic peak. Why?

12. The physical density of species x at the time of recombination is proportional to Ωxh2 where Ωx is the

ratio of the actual to critical density of species x at the present time, and h ≡ H0/100 kms−1 Mpc−1 is

the present-day Hubble constant. Explain.

13. How does changing the baryon density Ωbh2 affect the CMB power spectrum?

14. How does changing the non-baryonic cold dark matter density Ωch2, without changing the baryon density

Ωbh2, affect the CMB power spectrum?

15. What effects do neutrinos have on perturbations?

16. How does changing the curvature Ωk affect the CMB power spectrum?

17. How does changing the dark energy ΩΛ affect the CMB power spectrum?

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21

An overview of cosmological perturbations

Undoubtedly the preeminent application of general relativistic perturbation theory is to cosmology. Flucta-

tions in the temperature and polarization of the Cosmic Microwave Background (CMB) provide an observa-

tional window on the Universe at 400,000 years old that, coupled with other astronomical observations, has

yielded impressively precise measurements of cosmological parameters.

The theory of cosmological perturbations is based principally on general relativistic perturbation theory

coupled to the physics of 5 species of energy-momentum: photons, baryons, non-baryonic cold dark matter,

neutrinos, and dark energy.

Dark energy was not important at the time of recombination, where the CMB that we see comes from,

but it is important today. If dark energy has a vacuum equation of state, p = −ρ, then dark energy does

not cluster (vacuum energy density is a constant), but it affects the evolution of the cosmic scale factor,

and thereby does affect the clustering of baryons and dark matter today. Moreover the evolution of the

gravitational potential along the line-of-sight to the CMB does affect the observed power spectrum of the

CMB, the so-called integrated Sachs-Wolfe effect.

Unfortunately, it is beyond the scope of these notes to treat cosmological perturbations in full. For that,

consult Scott Dodelson’s incomparable text “Modern Cosmology”.

1. Inflationary initial conditions. The theory of inflation has been remarkably successful in accounting for

many aspects of observational cosmology, even though a fundamental understanding of the inflaton scalar

field that supposedly drove inflation is missing. The current paradigm holds that primordial fluctuations

were generated by vacuum quantum fluctuations in the inflaton field at the time of inflation. The theory

makes the generic predictions that the gravitational potentials generated by vacuum fluctuations were (a)

Gaussian, (b) adiabatic (meaning that all species of mass-energy fluctuated together, as opposed to in

opposition to each other), and (c) scale-free, or rather almost scale-free (the fact that inflation came to

an end modifies slightly the scale-free character). The three predictions fit the observed power spectrum

of the CMB astonishingly well.

2. Comoving Fourier modes. The spatial homogeneity of the Friedmann-Robertson-Walker background

spacetime means that its perturbations are characterized by Fourier modes of constant comoving wavevec-

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An overview of cosmological perturbations 353

tor. Each Fourier mode generated by inflation evolved independently, and its wavelength expanded with

the Universe.

3. Scalar, vector, tensor modes. Spatial isotropy on top of spatial homogenity means that the pertur-

bations comprised independently evolving scalar, vector, and tensor modes. Scalar modes dominate the

fluctuations of the CMB, and caused the clustering of matter today. Vector modes are usually assumed

to vanish, because there is no mechanism to generate the vorticity that sources vector modes, and the

expansion of the Universe tends to redshift away any vector modes that might have been present. Inflation

generates gravitational waves, which then propagate essentially freely to the present time. Gravitational

waves leave an observational imprint in the “B” (curl) mode of polarization of the CMB, whereas scalar

modes produce only an “E” (gradient) mode of polarization.

4. Power spectrum. The primary quantity measurable from observations is the power spectrum, which

is the variance of fluctuations of the CMB or of matter (as traced by galaxies, galaxy clusters, the Lyman

alpha forest, peculiar velocities, weak lensing, or 21 centimeter observations at high redshift). The statis-

tics of a Gaussian field are completely characterized by its mean and variance. The mean characterizes

the unperturbed background, while the variance characterizes the fluctuations. For a 3-dimensional sta-

tistically homogeneous and isotropic field, the variance of Fourier modes δk defines the power spectrum

P (k)

〈δkδk′〉 = 1kk′P (k) , (21.1)

where 1kk′ is the unit matrix in the Hilbert space of Fourier modes

1kk′ ≡ (2π)3δ3D(k + k′) . (21.2)

The “momentum-conserving” Dirac delta-function in equation (21.2) is a consequence of spatial translation

symmetry. Isotropy implies that the power spectrum P (k) is a function only of the absolute value k ≡ |k|of the wavevector. For a statistically rotation invariant field projected on the sky, such as the CMB, the

variance of spherical harmonic modes Θℓm ≡ δTℓm/T defines the power spectrum Cℓ

〈ΘℓmΘℓ′m′〉 = 1ℓm,ℓ′m′Cℓ (21.3)

where 1ℓm,ℓ′m′ is the unit matrix in the Hilbert space of spherical harmonics (distinguish the three usages

of δ in this paragraph: δ meaning fluctuation, δD meaning Dirac delta-function, and δ meaning Kronecker

delta, as in the following equation)

1ℓm,ℓ′m′ ≡ δℓℓ′δm,−m′ . (21.4)

Again, the “angular momentum-preserving” condition (21.4) that ℓ = ℓ′ and m+m′ = 0 is a consequence

of rotational symmetry. The same rotational symmetry implies that the power spectrum Cℓ is a function

only of the harmonic number ℓ, not of the directional harmonic number m.

5. Reheating. Early Universe inflation evidently came to an end. It is presumed that the vacuum energy

released by the decay of the inflaton field, an event called reheating, somehow efficiently produced the

matter and radiation fields that we see today. After reheating, the Universe was dominated by relativistic

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354 An overview of cosmological perturbations

fields, collectively called “radiation”. Reheating changed the evolution of the cosmic scale factor from

acceleration to deceleration, but is presumed not to have generated additional fluctuations.

6. Photon-baryon fluid and the sound horizon. Photon-electron (Thomson) scattering kept photons

and baryons tightly coupled to each other, so that they behaved like a relativistic fluid. As long as the

radiation density exceeded the baryon density, which remained true up to near the time of recombination,

the speed of sound in the photon-baryon fluid was√

p/ρ ≈√

13 of the speed of light. Fluctuations with

wavelengths outside the sound horizon grew by gravity. As time went by, the sound horizon expanded

in comoving radius, and fluctuations thereby came inside the sound horizon. Once inside the sound

horizon, sound waves could propagate, which tended to decrease the gravitational potential. However,

each individual sound wave itself continued to oscillate, its oscillation amplitude δT/T relative to the

background temperature T remaining approximately constant. The relativistic suppression of the potential

at small scales is responsible for the fact that the power spectrum of matter declines at small scales.

7. Acoustic peaks in the power spectrum. The oscillations of the photon-baryon fluid produced the

characteristic pattern of peaks and troughs in the CMB power spectrum observed today. The same

peaks and troughs occur in the matter power spectrum, but are much less prominent, at a level of about

10% as opposed to the order unity oscillations observed in the CMB power spectrum. For adiabatic

fluctuations, the amplitude of the temperature fluctuations follows a pattern ∼ − cos(kηs) where ηs is the

comoving sound horizon. The n’th peak occurs at a wavenumber k where kηs ≈ nπ. In the observed

CMB power spectrum, the relevant value of the sound horizon ηs is its value ηs,∗ at recombination.

Thus the wavenumber k of the first peak of the observed CMB power spectrum occurs where kηs,∗ ≈ π.

Two competing forces cause a mode to evolve: a gravitational force that amplifies compression, and

a restoring pressure force that counteracts compression. When a mode enters the sound horizon for

the first time, the compressing gravitational force beats the restoring pressure force, so the first thing

that happens is that the mode compresses further. Consequently the first peak is a compression peak.

This sets the subsequent pattern: odd peaks are compression peaks, while even peaks are rarefaction

peaks. The observed temperature fluctuations of the CMB are produced by a combination of intrinsic

temperature fluctuations, Doppler shifts, and gravitational redshifting out of potential wells. The Doppler

shift produced by the velocity of a perturbation is 90 out of phase with the temperature fluctuation, and

so tends to fill in the troughs in the power spectrum of the temperature fluctuation. This is the main

reason that the observed CMB power spectrum remains above zero at all scales.

8. Logarithmic growth of matter fluctuations. Non-baryonic cold dark matter interacts weakly except

by gravity, and is needed to explain the observed clustering of matter in the Universe today in spite of the

small amplitude of temperature fluctuations in the CMB. The adjective “cold” refers to the requirement

that the dark matter became non-relativistic (p = 0) at some early time. If the dark matter is both

non-baryonic and cold, then it did not participate in the oscillations of the photon-baryon fluid. During

the radiation-dominated phase prior to matter-radiation equality, dark matter matter fluctuations inside

the sound horizon grow logarithmically. The logarithmic growth translates into a logarithmic increase

in the amplitude of matter fluctuations at small scales, and is a characteristic signature of non-baryonic

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An overview of cosmological perturbations 355

cold dark matter. Unfortunately this signature is not readily discernible in the power spectrum of matter

today, because of nonlinear clustering.

9. Epoch of matter-radiation equality. The density of non-relativistic matter decreases more slowly than

the density of relativistic radiation. There came a point where the matter density equaled the radiation

density, an epoch called matter-radiation equality, after which the matter density exceeded the radiation

density. The observed ratio of the density of matter and radiation (CMB) today require that matter-

radiation equality occurred at a redshift of zeq ≈ 3200, a factor of 3 higher in redshift than recombination

at z∗ ≈ 1100. After matter-radiation equality, dark matter perturbations grew more rapidly, linearly

instead of just logarithmically with cosmic scale factor. A larger dark matter density causes matter-

radiation equality to occur earlier. The sound horizon at matter-radiation equality corresponds to a scale

roughly around the 2.5’th peak in the CMB power spectrum. For adiabatic fluctuations, the way that the

temperature and gravitational perturbations interact when a mode first enters the sound horizon means

that the temperature oscillation is 5 times larger for modes that enter the horizon well into the radiation-

dominated epoch versus well into the matter-dominated epoch. The effect enhances the amplitude of

observed CMB peaks higher than 2.5 relative to those lower than 2.5. The observed relative strengths

of the 3rd versus the 2nd peak of the CMB power spectrum provides a measurement of the redshift of

matter-radiation equality, and direct evidence for the presence of non-baryonic cold dark matter.

10. Sound speed. The density of baryons decreased more slowly than the density of radiation, so that at

around recombination the baryon density was becoming comparable to the radiation density. The sound

speed√

p/ρ depends on the ratio of pressure p, which was essentially entirely that of the photons, to the

density ρ, which was produced by both photons and baryons. The sound speed consequently decreased

below√

13 . Increasing the baryon-to-photon ratio at recombination has several observational effects on

the acoustic peaks of the CMB power spectrum, making it a prime measurable parameter from the CMB.

First, an increased baryon fraction increases the gravitational forcing (baryon loading), which enhances

the compression (odd) peaks while reducing the rarefaction (even) peaks. Second, increasing the baryon

fraction reduces the sound speed, which: (a) decreases the amplitude of the radiation dipole relative to the

radiation monopole, so increasing the prominence of the peaks; and (b) reduces the oscillation frequency

of the photon-baryon fluid, which shifts the peaks to larger scales. The reduced sound speed also causes

an adiabatic reduction of the amplitudes of all modes by the square root of the sound speed, but this effect

is degenerate with an overall reduction in the initial amplitudes of modes produced by inflation.

11. Recombination. As the temperature cooled below about 3,000 K, electrons combined with hydrogen

and helium nuclei into neutral atoms. This drastically reduced the amount of photon-electron scattering,

releasing the CMB to propagate almost freely. At the same time, the baryons were released from the

photons. Without radiation pressure to support them, fluctuations in the baryons began to grow like the

dark matter fluctuations.

12. Neutrinos. Probably all three species of neutrino have mass less than 0.3 eV and were therefore relativistic

up to and at the time of recombination. Each of the 3 species of neutrino had an abundance comparable

to that of photons, and therefore made an important contribution to the relativistic background and its

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356 An overview of cosmological perturbations

fluctuations. Unlike photons, neutrinos streamed freely, without scattering. The relativistic free-streaming

of neutrinos provided the main source of the quadrupole pressure that produces a non-vanishing difference

Ψ − Φ between the scalar potentials. However, the neutrino quadrupole pressure was still only ∼ 10% of

the neutrino monopole pressure. To the extent that the neutrino quadrupole pressure can be approximated

as negligible, the neutrinos and their fluctuations can be treated the same as photons.

13. CMB fluctuations. The CMB fluctuations seen on the sky today represent a projection of fluctuations

on a thin but finite shell at a redshift of about 1100, corresponding to an age of the Universe of about

400,000 yr. The temperature, and the degrees of polarization in two different directions, provide 3 inde-

pendent observables at each point on the sky. The isotropy of the unperturbed radiation means that it is

most natural to measure the fluctuations in spherical harmonics, which are the eigenmodes of the rotation

operator. Similarly, it is natural to measure the CMB polarization in spin harmonics.

14. Matter fluctuations. After recombination, perturbations in the non-baryonic and baryonic matter grew

by gravity, essentially unaffected any longer by photon pressure. If one or more of the neutrino types

had a mass small enough to be relativistic but large enough to contribute appreciable density, then its

relativistic streaming could have suppressed power in matter fluctuations at small scales, but observations

show no evidence of such suppression, which places an upper limit of about an eV on the mass of the

most massive neutrino. The matter power spectrum measured from the clustering of galaxies contains

acoustic oscillations like the CMB power spectrum, but because the non-baryonic dark matter dominates

the baryons, the oscillations are much smaller.

15. Integrated Sachs-Wolfe effect. Variations in the gravitational potential along the line-of-sight to the

CMB affect the CMB power spectrum at large scales. This is called the integrated Sachs-Wolfe (ISW)

effect. If matter dominates the background, then the gravitational potential Φ has the property that it

remains constant in time for (subhorizon) linear fluctuations, and there is no ISW effect. In practice,

ISW effects are produced by at least three distinct causes. First, an early-time ISW effect is produced

by the fact that the Universe at recombination still has an appreciable component of radiation, and is

not yet wholly matter-dominated. Second, a late-time ISW effect is produced either by curvature or by a

cosmological constant. Third, a non-linear ISW effect is produced by non-linear evolution of the potential.

Page 369: General Relativity, Black Holes, And Cosmology

22

∗Cosmological perturbations in a flatFriedmann-Robertson-Walker background

For simplicity, this book considers only a flat (not closed or open) Friedmann-Robertson-Walker background.

The comoving cosmological horizon size at recombination was much smaller than today, and consequently the

cosmological density Ω was much closer to 1 at recombination than it is today. Since observations indicate

that the Universe today is within 1% of being spatially flat, it is an excellent approximation to treat the

Universe at the time of recombination as being spatially flat.

With some modifications arising from cosmological expansion, perturbation theory on a flat FRW back-

ground is quite similar to perturbation theory in flat (Minkowski) space, Chapter 20.

The strategy is to start in a completely general gauge, and to discover how the conformal Newtonian

gauge, which is used in subsequent Chapters, emerges naturally as that gauge in which the perturbations

are precisely the physical perturbations.

22.1 Unperturbed line-element

It is convenient to choose the coordinate system xµ = η, xi to consist of conformal time η together with

Cartesian comoving coordinates x ≡ xi ≡ x, y, z. The coordinate metric of the unperturbed background

FRW geometry is then

ds2 = a(η)2(

− dη2 + dx2 + dy2 + dz2)

, (22.1)

where a(η) is the cosmic scale factor. The unperturbed coordinate metric is thus the conformal Minkowski

metric0

gµν = a(η)2ηµν . (22.2)

The tetrad is taken to be orthonormal, with the unperturbed tetrad axes γγm ≡ γγ0.γγ1,γγ2,γγ3 being aligned

with the unperturbed coordinate axes gµ ≡ gη, gx, gy, gz so that the unperturbed vierbein and inverse

vierbein are respectively 1/a and a times the unit matrix

0

emµ =

1

aδµm ,

0

em

µ = a δmµ . (22.3)

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358 ∗Cosmological perturbations in a flat Friedmann-Robertson-Walker background

Let overdot denote partial differentiation with respect to conformal time η,

overdot ≡ ∂

∂η, (22.4)

so that for example a ≡ da/dη. The coordinate time derivative ∂/∂η is to be distinguished from the directed

time derivative ∂0 ≡ e0µ ∂/∂xµ. Let ∇i denote the comoving gradient

∇i ≡∂

∂xi, (22.5)

which should be distinguished from the directed derivative ∂i ≡ eiµ ∂/∂xµ.

22.2 Comoving Fourier modes

Since the unperturbed Friedmann-Robertson-Walker spacetime is spatially homogeneous and isotropic, it

is natural to work in comoving Fourier modes. Comoving Fourier modes have the key property that they

evolve independently of each other, as long as perturbations remain linear. Equations in Fourier space are

obtained by replacing the comoving spatial gradient ∇i by −i times the comoving wavevector ki (the choice

of sign is the standard convention in cosmology)

∇i → −iki . (22.6)

By this means, the spatial derivatives become algebraic, so that the partial differential equations governing

the evolution of perturbations become ordinary differential equations.

In what follows, the comoving gradient ∇i will be used interchangeably with −iki, whichever is most

convenient.

22.3 Classification of vierbein perturbations

The tetrad-frame components ϕmn of the vierbein perturbation of the FRW geometry decompose in much

the same way as in flat Minkowski case into 6 scalars, 4 vectors (8 degrees of freedom), and 1 tensor (2

degrees of freedom) (the following equations are essentially the same as those (20.6) for the flat Minkowski

background),

ϕ00 = ψscalar

, (22.7a)

ϕ0i = ∇iwscalar

+ wivector

, (22.7b)

ϕi0 = ∇iwscalar

+ wivector

, (22.7c)

ϕij = δij φscalar

+∇i∇jhscalar

+ εijk∇khscalar

+∇ihjvector

+∇j hivector

+ hijtensor

. (22.7d)

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22.3 Classification of vierbein perturbations 359

The tetrad-frame components ǫm of the coordinate shift of the coordinate gauge transformation (18.8)

similarly decompose into 2 scalars and 1 vector (the following equation is essentially the same as that (20.8)

for the flat Minkowski background)

ǫm = ǫ0scalar

, ∇iǫscalar

+ ǫivector

. (22.8)

The vierbein perturbations ϕmn transform under a coordinate gauge transformation (18.8) as, equation (18.25),

ϕ00 → ψ +1

a

∂ǫ0∂η

scalar

, (22.9a)

ϕ0i → ∇i

(

w +1

a

( ∂

∂η− a

a

)

ǫ

)

scalar

+

(

wi +1

a

( ∂

∂η− a

a

)

ǫi

)

vector

, (22.9b)

ϕi0 → ∇i

(

w +1

aǫ0

)

scalar

+ wivector

, (22.9c)

ϕij → δij

(

φ− a

a2ǫ0

)

scalar

+∇i∇j

(

h+1

)

scalar

+ εijk∇khscalar

+∇i

(

hj +1

aǫj

)

vector

+∇j hivector

+ hijtensor

, (22.9d)

or equivalently

ψ → ψ +1

a

∂ǫ0∂η

, (22.10a)

w → w +1

a

( ∂

∂η− a

a

)

ǫ , wi → wi +1

a

( ∂

∂η− a

a

)

ǫi , (22.10b)

w → w +1

aǫ0 , wi → wi , (22.10c)

φ→ φ− a

a2ǫ0 , h→ h+

1

aǫ , h→ h , hi → hi +

1

aǫi , hi → hi , hij → hij . (22.10d)

Eliminating the coordinate shift ǫm from the transformations (22.10) yields 12 coordinate gauge-invariant

combinations of the perturbations

ψ −( ∂

∂η+a

a

)

w , w − h , wi − hi , wi , φ+a

aw , h , hi , hij . (22.11)

Six combinations of these coordinate gauge-invariant perturbations depend only on the symmetric part

ϕmn + ϕnm of the vierbein perturbations, and are therefore tetrad gauge-invariant as well as coordinate

gauge-invariant. These 6 coordinate and tetrad gauge-invariant perturbations comprise 2 scalars, 1 vector,

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360 ∗Cosmological perturbations in a flat Friedmann-Robertson-Walker background

and 1 tensor

Ψscalar

≡ ψ −( ∂

∂η+a

a

)

(w + w − h) , (22.12a)

Φscalar

≡ φ+a

a(w + w − h) , (22.12b)

Wivector

≡ wi + wi − hi − ˙hi , (22.12c)

hijtensor

. (22.12d)

22.4 Metric, tetrad connections, and Einstein tensor

This section gives expressions in a completely general gauge for perturbed quantities in the flat Friedmann-

Robertson-Walker background geometry.

The perturbed coordinate metric gµν is

gηη = −a2(1 + 2ψ) , (22.13a)

gηi = −a2[

∇i(w + w) + (wi + wi)]

, (22.13b)

gij = a2[

(1− 2φ)δij − 2∇i∇jh−∇i(hj + hj)−∇j(hi + hi)− 2 hij

]

. (22.13c)

The coordinate metric is tetrad gauge-invariant, but not coordinate gauge-invariant.

The perturbed tetrad connections Γkmn are

Γ0i0 =1

a

[

−∇i

(

ψ −( ∂

∂η+a

a

)

w

)

+( ∂

∂η+a

a

)

wi

]

, (22.14a)

Γ0ij =1

a

[

(

− aa

+ F)

δij −∇i∇j(w − h)−1

2(∇iWj +∇jWi) +∇jwi + hij

]

, (22.14b)

Γij0 =1

a

[

1

2(∇iWj −∇jWi)−

∂η(εijl∇lh−∇ihj +∇j hi)

]

, (22.14c)

Γijk =1

a

[

(δjk∇i − δik∇j)(

φ+a

aw)

− a

a(δikδjl − δjkδil)wl

−∇k(εijl∇lh−∇ihj +∇j hi) +∇ihjk −∇jhik

]

, (22.14d)

where F is defined by

F ≡ a

aψ + φ . (22.15)

Being purely tetrad-frame quantities, the tetrad connections are automatically coordinate gauge-invariant,

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22.4 Metric, tetrad connections, and Einstein tensor 361

but they are not tetrad gauge-invariant. The quantity F defined by equation (22.15) is not coordinate gauge-

invariant, but the combination − a/a2 + F/a that appears in the expression (22.14b) for Γ0ij is coordinate

and tetrad gauge-invariant.

Exercise 22.1 Coordinate gauge invariance of − a/a2 + F/a. Argue that under a coordinate gauge

transformation of the conformal time, η → η′ = η + ǫη, the cosmic scale factor a(η) and its derivative

a ≡ da/dη transforms as (see §18.8)

a→ a+ Lǫa = a− aǫη = a+a

aǫ0 , a→ a+ Lǫa = a− aǫη = a+

a

aǫ0 . (22.16)

Check that this behaviour is consistent with the gauge transformation (18.28) of gηη, equation (22.13a).

Hence show that − a/a2 + F/a is coordinate gauge invariant. THIS IS NOT WORKING.

The perturbed tetrad-frame Einstein tensor Gmn is

G00 =1

a2

3a2

a2− 6

a

aF + 2∇2Φ

scalar

, (22.17a)

G0i =1

a2

2∇i

(

F +( a

a− 2

a2

a2

)

w

)

scalar

+1

2∇2Wi + 2

( a

a− 2

a2

a2

)

wi

vector

, (22.17b)

Gij =1

a2

(

− 2a

a+a2

a2+ 2

( ∂

∂η+ 2

a

a

)

F + 2( a

a− 2

a2

a2

)

ψ

)

δij

scalar

− (∇i∇j − δij∇2)(Ψ − Φ)scalar

+1

2

( ∂

∂η+ 2

a

a

)

(∇iWj +∇jWi)

vector

−( ∂2

∂η2+ 2

a

a

∂η−∇2

)

hij

tensor

. (22.17c)

Being tetrad-frame quantities, all components of the tetrad-frame Einstein tensor are automatically coordi-

nate gauge-invariant. The time-time G00 and space-space Gij components are not only coordinate but also

tetrad gauge-invariant, as follows from the fact that these components depend only on symmetric combi-

nations of the vierbein potentials. Specifically, the quantities 3(a2/a4) − 6(a/a3)F on the right hand side

of equation (22.17a) for G00, and the coefficient of δij (including the overall 1/a2 factor) on the right hand

side of equation (22.17c) for Gij , are coordinate and tetrad gauge-invariant. However, the time-space com-

ponents G0i are not tetrad gauge-invariant, as is evident from the fact that equation (22.17b) involves the

non-tetrad-gauge-invariant perturbations w and wi. Physically, under a tetrad boost by a velocity v of linear

order, the time-space components G0i change by first order v, but G00 and Gij change only to second order

v2. Thus to linear order, only G0i changes under a tetrad boost. Note that G0i changes under a tetrad boost

(w and wi), but not under a tetrad rotation (h and hi).

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362 ∗Cosmological perturbations in a flat Friedmann-Robertson-Walker background

22.5 ADM gauge choices

The ADM (3+1) formalism, Chapter 13, chooses the tetrad time axis γγ0 to be orthogonal to hypersurfaces

of constant time, η = constant, equivalent to requiring that the tetrad time axis be orthogonal to each of

the spatial coordinate axes, γγ0 · gi = 0, equation (13.2). The ADM choice is equivalent to setting

w = wi = 0 . (22.18)

The ADM choice simplifies the tetrad-frame connections (22.14) and the time-space component G0i of the

tetrad-frame frame Einstein tensor, equation (22.17b).

Another gauge choice that significantly simplifies the tetrad connections (22.14), though does not affect

the Einstein tensor (22.17), is

h = hi = 0 . (22.19)

If the wavevector k is taken along the coordinate z-direction, then the gauge choice hi = 0 is equivalent to

choosing the tetrad 3-axis (z-axis) γγ3 to be orthogonal to the coordinate x and y-axes, γγ3 · gx = γγ3 · gy = 0.

The gauge choice h = 0 is equivalent to rotating the tetrad axes about the 3-axis (z-axis) so that γγ1 · gy =

γγ2 · gx.

22.6 Conformal Newtonian gauge

Conformal Newtonian gauge sets

w = w = wi = h = h = hi = hi = 0 , (22.20)

so that the retained perturbations are the 6 coordinate and tetrad gauge-invariant perturbations (22.12)

Ψscalar

= ψ , (22.21a)

Φscalar

= φ , (22.21b)

Wivector

= wi , (22.21c)

hijtensor

. (22.21d)

In conformal Newtonian gauge, the scalar perturbations of the Einstein equations are the energy density,

energy flux, monopole pressure, and quadrupole pressure equations

− 3a

aF − k2Φ = 4πGa2

1

T 00 , (22.22a)

ikF = 4πGa2 ki

1

T 0i , (22.22b)

F + 2a

aF +

( a

a− 2

a2

a2

)

Ψ− k2

3(Ψ − Φ) =

4

3Gπa2 δij

1

T ij , (22.22c)

k2(Ψ − Φ) = 8πGa2(

32 kikj − 1

2 δij

)

1

T ij , (22.22d)

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22.7 Synchronous gauge 363

where F is the coordinate and tetrad gauge-invariant quantity

F ≡ a

aΨ + Φ . (22.23)

All 4 of the scalar Einstein equations (22.22) are expressed in terms of gauge-invariant variables, and are

therefore fully gauge-invariant.

The energy and momentum equations (22.22a) and (22.22b) can be combined to eliminate F , yielding an

equation for Φ alone

−k2Φ = 4πa2

(

1

T 00 + 3a

a

ki

ik

1

T 0i

)

. (22.24)

The quantity in parentheses on the right hand side of equation (22.24) is the source for the scalar potential

Φ, and can be interpreted as a measure of the “true” energy fluctuation. This is however just a matter of

interpretation: the individual perturbations1

T 00 and1

T 01 are both individually gauge-invariant, and therefore

have physical meaning.

If the energy-momentum tensors of the various matter components are arranged so as to conserve overall

energy-momentum, as they should, then 2 of the 4 equations (22.22a)–(22.22d) are redundant, since they

serve simply to enforce conservation of energy and scalar momentum. Thus it suffices to take, as the equations

governing the potentials Φ and Ψ, any 2 of the equations (22.22a)–(22.22d). One is free to retain whichever 2

of the equations is convenient. Usually the 1st equation, the energy equation (22.22a), and the 4th equation,

the quadrupole pressure equation (22.22d), are most convenient to retain. But sometimes the 2nd equation,

the scalar momentum equation (22.22b), is more convenient in place of the energy equation (22.22a).

22.7 Synchronous gauge

One gauge that remains in common use in cosmology, but is not used here, is synchronous gauge, discussed

in the case of Minkowski background space in §20.10. The cosmological synchronous gauge choices are the

same as for the Minkowski background, equations (20.63) and (20.64):

ψ = w = w = wi = wi = h = hi = 0 . (22.25)

The gauge-invariant perturbations (22.12) in synchronous gauge are

Ψscalar

= h+a

ah , (22.26a)

Φscalar

= φ− a

ah , (22.26b)

Wivector

= − hi , (22.26c)

hijtensor

. (22.26d)

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23

Cosmological perturbations: a simplest set ofassumptions

1. Consider only scalar modes.

2. Consider explicitly only two species: non-baryonic cold dark matter, and radiation consisting of photons

and neutrinos lumped together. Neglect the contribution of baryons to the mass density.

3. Treat the radiation as almost isotropic, so it is dominated by its first two moments, the monopole and

dipole. In practice, electron-photon scattering keeps photons almost isotropic. Unlike photons, neutrinos

stream freely, but they inherit an approximately isotropic distribution from an early time when they were

in thermodynamic equilibrium.

4. Include damping from photon-electron (Thomson) scattering by allowing the radiation a small quadrupole

moment, the diffusion approximation.

23.1 Perturbed FRW line-element

Perturbed FRW line-element in conformal Newtonian gauge

ds2 = a2[

−(1 + 2Ψ)dη2 + δij(1− 2Φ)dxidxj]

, (23.1)

where a(η) is the cosmic scale factor, a function only of conformal time η.

23.2 Energy-momenta of ideal fluids

In the simplest approximation, matter, radiation, and dark energy can each be treated as ideal fluids. The

energy-momentum tensor of an ideal fluid with proper density ρ and isotropic pressure p in its own rest

frame, and moving with bulk 4-velocity um relative to the conformal Newtonian tetrad frame, is

Tmn = (ρ+ p)umun + p ηmn . (23.2)

In the situation under consideration, the fluids satisfy equations of state

p = wρ (23.3)

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23.2 Energy-momenta of ideal fluids 365

with w constant. Specifically, w = 0 for non-relativistic matter, w = 1/3 for relativistic radiation, and

w = −1 for dark energy with constant density. Furthermore, all the fluids are moving with non-relativistic

bulk velocities, including the radiation, which is almost isotropic, and therefore has a small bulk velocity

even though the individual particles of radiation move at the speed of light. The bulk tetrad-frame 4-velocity

um is thus, to linear order

um = 1, vi , (23.4)

where vi is the non-relativistic spatial bulk 3-velocity (the spatial tetrad metric is Euclidean, so vi = vi).

The density ρ can be written in terms of the unperturbed density ρ and a fluctuation δ defined by

ρ = ρ[1 + (1 + w)δ] . (23.5)

The factor 1+w is included in the definition of the fluctuation δ because it simplifies the resulting perturbation

equations (23.12). As you will discover in Exercise 23.1, the fluctuation δ can be interpreted physically as

the entropy fluctuation,

δ =1

1 + w

δρ

ρ=δs

s. (23.6)

For matter, w = 0, the entropy fluctuation coincides with the density fluctuation, δ = δρ/ρ. For dark

energy, w = −1, the density fluctuation is necessarily zero, δρ/ρ = 0. To linear order in the velocity vi, the

tetrad-frame energy-momentum tensor (23.2) of the ideal fluid is then

T 00 ≡ ρ[1 + (1 + w)δ] , (23.7a)

T 0i ≡ (1 + w)ρ vi , (23.7b)

T ij = wρ[1 + (1 + w)δ] δij . (23.7c)

To linear order in the fluctuation δ, velocity vi, and potentials Ψ and Φ, and in conformal Newtonian gauge,

conservation of energy and momentum requires

DmTm0 =

1−Ψ

a

[

∂ρ

∂η+ (1 + w)∇i(ρvi) + 3(1 + w)ρ

( a

a− Φ

)

]

= 0 , (23.8a)

DmTmi =

1

a

[

(1 + w)∂ρvi

∂η+ 4(1 + w)

a

aρvi + w∇iρ+ (1 + w)ρ∇iΨ

]

= 0 . (23.8b)

The energy conservation equation (23.8a) has an unperturbed part,

Dm

0

Tm0 =1

a

[

∂ρ

∂η+ 3(1 + w)ρ

a

a

]

= 0 , (23.9)

which implies the usual result that the mean density evolves as a power law with cosmic scale factor,

ρ ∝ a−3(1+w). Subtracting appropriate amounts of the unperturbed energy conservation equation (23.9)

from the perturbed energy-momentum conservation equations (23.8) yields equations for the fluctuation δ

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366 Cosmological perturbations: a simplest set of assumptions

and velocity vi:

δ +∇ivi = 3Φ , (23.10a)

vi + (1− 3w)a

avi + w∇iδ = −∇iΨ . (23.10b)

Now decompose the 3-velocity vi into its scalar v and vector v⊥,i parts. Up to this point, the scalar part of

a vector has been taken to be the gradient of a potential. But here it is advantageous to absorb a factor of k

into the definition of the scalar part v of the velocity, so that instead of vi = −ikiv + v⊥,i in Fourier space,

the velocity is given in Fourier space by

vi = −ikiv + v⊥,i . (23.11)

The advantage of this choice is that v is dimensionless, as are δ and Ψ and Φ. The scalar parts of the

perturbation equations (23.10) are then

δ − kv = 3Φ , (23.12a)

v + (1− 3w)a

av + wkδ = −kΨ . (23.12b)

Combining the two equations (23.12) for the fluctuation δ and velocity v yields a second-order differential

equation for δ − 3Φ,[

d2

dη2+ (1− 3w)

a

a

d

dη+ k2w

]

(δ − 3Φ) = −k2(Ψ + 3wΦ) . (23.13)

For positive w, equation (23.13) is a wave equation for a damped, forced oscillator with sound speed√w. The

resulting generic behaviour for the particular cases of matter (w = 0) and radiation (w = 13 ) is considered

in §23.6 and §23.7 below.

A more careful treatment, deferred to Chapter 24, accounts for the complete momentum distribution of

radiation by expanding the temperature perturbation Θ ≡ δT/T in multipole moments, equation (24.48).

The radiation fluctuation δr and scalar bulk velocity vr are related to the first two multipole moments of

the temperature perturbation, the monopole Θ0 and the dipole Θ1, by

δr = 3Θ0 , (23.14a)

vr = 3Θ1 . (23.14b)

The factor of 3 arises because the unperturbed radiation distribution is in thermodynamic equilibrium, for

which the entropy density is s ∝ T 3, so δr = 3δT/T .

The energy-momentum perturbation1

Tmn that goes into the Einstein equations (22.22) are, from equa-

tions (23.7) with the unperturbed part subtracted,

1

T 00 ≡ (1 + w)ρ δ , (23.15a)1

T 0i ≡ (1 + w)ρ vi , (23.15b)1

T ij = w(1 + w)ρ δ δij . (23.15c)

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23.3 Diffusive damping 367

Exercise 23.1 Entropy fluctuation. The purpose of this exercise is to discover that the fluctuation

δ defined by equation (23.5) can be interpreted as the entropy fluctuation. According to the first law of

thermodynamics, the entropy density s of a fluid of energy density ρ, pressure p, and temperature T in a

volume V satisfies

d(ρV ) + pdV = Td(sV ) . (23.16)

If the fluid is ideal, so that ρ, p, T , and s are independent of volume V , then integrating the first law (23.16)

implies that

ρV + pV = TsV . (23.17)

This implies that the entropy density s is related to the other variables by

s =ρ+ p

T. (23.18)

Show that, for ideal fluid with equation of state p/ρ = w = constant, the first law (23.16) together with the

expression (23.18) for entropy implies that

T ∝ ρw/(1+w) , (23.19)

and hence

s ∝ ρ1/(1+w) . (23.20)

Conclude that small variations of the entropy and density are related by

δs

s=

1

1 + w

δρ

ρ, (23.21)

confirming equation (23.6). [Hint: Do not confuse what is being asked here with adiabatic expansion. The

results (23.19) are properties of the fluid, independent of whether the fluid is changing adiabatically. For

adiabatic expansion, the fluid satisfies the additional condition sV = constant.]

23.3 Diffusive damping

The treatment of matter and radiation as ideal fluids misses a feature that has a major impact on observed

fluctuations in the CMB, namely the diffusive damping of sound waves that results from the finite mean

free path to electron-photon scattering. As recombination approaches, the scattering mean free path length-

ens, until at recombination photons are able to travel freely across the Universe, ready to be observed by

astronomers. The damping is greater at smaller scales, and is responsible for the systematic decrease in the

CMB power spectrum to smaller scales.

As expounded in Chapter 24, in §24.13 and following, the damping can be taken into account to lowest

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368 Cosmological perturbations: a simplest set of assumptions

order, the diffusion approximation, by admitting a small quadrupole moment Θ2 to the photon distribu-

tion. A detailed analysis of the collisional Boltzmann equation for photons, §24.12, reveals that the photon

quadrupole Θ2 is given in the diffusion approximation by equation (24.94). There is an additional source of

damping that arises from viscous baryon drag, §24.15, but this effect vanishes in the limit of small baryon

density, and is neglected in the present Chapter.

The diffusive damping resulting from a small photon quadrupole conserves the energy and momentum

of the photon fluid, so that covariant momentum conservation DmTmn = 0 continues to hold true within

the photon fluid. By contrast, viscous baryon drag, §24.15, neglected in this Chapter, transfers momentum

between photons and baryons.

Define the dimensionless quadrupole q by

T ijquadrupole = (1 + w)ρq

(

32 kikj − 1

2 δij

)

, (1 + w)ρq ≡(

kikj − 13 δij

)

T ij . (23.22)

For photons, the dimensionless quadrupole q is related to the photon quadrupole harmonic Θ2 by, equa-

tion (24.55d),

q = − 2Θ2 . (23.23)

In the presence of a quadrupole pressure, the energy conservation equation (23.8a) is unchanged, but the

momentum conservation equation (23.8b) is modified by the change Ψ→ Ψ + q:

DmTmi =

1

a

[

(1 + w)∂ρvi

∂η+ 4(1 + w)

a

aρvi + w∇iρ+ (1 + w)ρ∇i(Ψ + q)

]

= 0 . (23.24)

The consequent equations (23.10b), (23.12b), (23.13) are similarly modified by Ψ → Ψ + q. In particular,

the velocity equation (23.12b) is modified to

v + (1− 3w)a

av + wkδ = −k(Ψ + q) . (23.25)

23.4 Equations for the simplest set of assumptions

Non-baryonic cold dark matter, subscripted c:

δc − k vc = 3 Φ , (23.26a)

vc +a

avc = −kΨ . (23.26b)

Radiation, which includes both photons and neutrinos:

Θ0 − kΘ1 = Φ , (23.27a)

Θ1 +k

3Θ0 = − k

3(Ψ− 2Θ2) , (23.27b)

Θ2 = − 4k

9neσTaΘ1 . (23.27c)

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23.4 Equations for the simplest set of assumptions 369

Einstein energy and quadrupole pressure equations:

− k2Φ− 3a

aF = 4πGa2(ρcδc + 4ρrΘ0) , (23.28a)

k2(Ψ− Φ) = − 32πGa2ρrΘ2 , (23.28b)

where

F ≡ a

aΨ + Φ . (23.29)

In place of the Einstein energy equation (23.28a) it is sometimes convenient to use the Einstein momentum

equation

−kF = 4πGa2(ρcvc + 4ρrΘ1) , (23.30)

which, because the matter and radiation equations (23.26) and (23.27) already satisfy covariant energy-

momentum conservation, is not an independent equation.

The radiation quadrupole Θ2, equation (23.27c), derived in Chapter 24, equation (24.94), is proportional

to the comoving mean free path lT to electron-photon (Thomson) scattering,

lT ≡1

neσTa, (23.31)

where σT is the Thomson cross-section. The quadrupole becomes important only near recombination, where

the increasing mean free path to electron-photon scattering leads to dissipation of photon-baryon sound

waves. In the simple treatment of this Chapter, neutrinos are being lumped with photons, and of course

neutrinos do not scatter, but rather stream freely. However, radiation is gravitationally sub-dominant near

recombination, so not much error arises from treating neutrinos as gravitationally the same as photons near

recombination. To make comparison with observed CMB fluctuations, the important thing is to follow

the evolution of the photon multipoles, and for this purpose the radiation quadrupole defined by equa-

tion (23.27c), without any correction for neutrino-to-photon ratio, is the appropriate choice.

In much of the remainder of this Chapter, that is, excepting in §23.7 and §23.14, the radiation quadrupole

will be set to zero,

Θ2 = 0 , (23.32)

which is equivalent to neglecting the effect of damping. If the radiation quadrupole vanishes, then the

Einstein quadrupole pressure equation (23.28b) implies that the scalar potentials Ψ and Φ are equal,

Ψ = Φ . (23.33)

In any case, the radiation quadrupole is always small, so that Ψ ≈ Φ to a good approximation.

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370 Cosmological perturbations: a simplest set of assumptions

23.5 Unperturbed background

In the unperturbed background, the unperturbed dark matter density ρc and radiation density ρr evolve

with cosmic scale factor as

ρc ∝ a−3 , ρr ∝ a−4 . (23.34)

The Hubble parameter H is defined in the usual way to be

H ≡ 1

a

da

dt=

a

a2, (23.35)

in which overdot represents differentiation with respect to conformal time, a ≡ da/dη. The Friedmann

equations for the background imply that the Hubble parameter for a universe dominated by dark matter

and radiation is

H2 =8πG

3(ρc + ρr) =

H2eq

2

(

a3eq

a3+a4eq

a4

)

(23.36)

where aeq and Heq are the cosmic scale factor and the Hubble parameter at the time of matter-radiation

equality, ρc = ρr.

The comoving horizon distance η is defined to be the comoving distance that light travels starting from

zero expansion. This is

η =

∫ a

0

da

a2H=

2√

2

aeqHeq

(√

1 +a

aeq− 1

)

=2√

2

aeqHeq

(

a/aeq

1 +√

1 + a/aeq

)

. (23.37)

In the radiation- and matter-dominated epochs respectively, the comoving horizon distance η is

η =

√2

aeqHeq

(

a

aeq

)

∝ a radiation-dominated ,

2√

2

aeqHeq

(

a

aeq

)1/2

∝ a1/2 matter-dominated .

(23.38)

The ratio of the comoving horizon distance η to the comoving cosmological horizon distance 1/(aH) is

ηaH =2√

1 + a/aeq

1 +√

1 + a/aeq

, (23.39)

which is evidently a number of order unity, varying between 1 in the radiation-dominated epoch a ≪ aeq,

and 2 in the matter-dominated epoch a≫ aeq.

Exercise 23.2 Matter-radiation equality.

1. Argue that the redshift zeq of matter-radiation equality is given by

1 + zeq =a0

aeq= ? Ωmh

2 , (23.40)

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23.6 Generic behaviour of non-baryonic cold dark matter 371

where Ωm is the matter density today relative to critical. What is the factor, and what is its numerical

value? The factor depends on the energy-weighted effective number of relativistic species gρ, Exercise 10.18.

Should this gρ be that now, or that at matter-radiation equality?

2. Show that the ratio Heq/H0 of the Hubble parameter at matter-radiation equality to that today is

Heq

H0=√

2Ωm(1 + zeq)3/2 . (23.41)

Solution. The redshift zeq of matter-radiation equality is given by

1 + zeq =Ωm

Ωr=

45c5~3ΩmH20

4π3Ggρ(kT0)4= 8.093× 104 Ωmh

2

gρ= 3200

( gρ

3.36

)−1(

Ωmh2

0.133

)

, (23.42)

where T0 = 2.725 K is the present-day CMB temperature, and gρ = 2 + 6 78

(

411

)4/3= 3.36 is the energy-

weighted effective number of relativistic species at matter-radiation equality.

23.6 Generic behaviour of non-baryonic cold dark matter

Combining equations (23.26) for the dark matter overdensity and velocity gives

(

d2

dη2+a

a

d

)

(δc − 3Φ) = −k2Ψ = −k2Φ , (23.43)

where the last expression follows because Ψ = Φ to a good approximation. In the absence of a driving

potential, Φ = 0, the dark matter velocity would redshift as vc ∝ 1/a, and the dark matter density equa-

tion (23.26a) would then imply that δc = kvc ∝ a−1. In the radiation-dominated epoch, where η ∝ a,

this leads to a logarithmic growth in the overdensity δc, even though there is no driving potential, and the

velocity is redshifting to a halt. In the matter-dominated epoch, where η ∝ a1/2, the dark matter overdensity

δc would freeze out at a constant value, in the absence of a driving potential.

Exercise 23.3 Generic behaviour of dark matter. Find the homogeneous solutions of equation (23.43).

Hence find the retarded Green’s function of the equation. Write down the general solution of equation (23.43)

as an integral over the Green’s function.

Solution. The general solution of equation (23.43) is

δc(a)−3Φ(a) = A0+A1 ln

(√1 + a+ 1√1 + a− 1

)

+2k2

∫ a

0

ln

[

(√

1 + a+ 1)

(√

1 + a− 1)

(√

1 + a′ − 1)

(√

1 + a′ − 1)

]

Φ(a′)a′ da′√1 + a′

, (23.44)

where A0 and A1 are constants.

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372 Cosmological perturbations: a simplest set of assumptions

23.7 Generic behaviour of radiation

Combining equations (23.27) for the radiation monopole, dipole, and quadrupole gives(

3d2

dη2+ 2√

3κd

dη+ k2

)

(Θ0 − Φ) = − k2 (Ψ + Φ) = − 2 k2Φ , (23.45)

where in the last expression the approximation Ψ = Ψ has again been invoked. The coefficient κ of the linear

derivative term in equation (23.45) is a damping coefficient

κ ≡ 4k2lT

9√

3, (23.46)

where lT is the comoving electron-photon scattering (Thomson) mean free path, equation (23.31), The mean

free path is small, Exercise 23.6, except near recombination. In the absence of a driving potential, Φ = 0,

and in the absence of damping, κ = 0, the radiation oscillates as Θ0 ∝ e±iωη with frequency ω =√

13 k. In

other words, the solutions are sound waves, moving at the sound speed

cs =ω

k=

1

3. (23.47)

Define the conformal sound time by

ηs ≡ csη =η√3. (23.48)

In terms of the conformal sound time ηs, the differential equation (23.45) becomes(

d2

dη2s

+ 2κd

dηs+ k2

)

(Θ0 − Φ) = − 2k2Φ . (23.49)

As you will discover in Exercises 23.7 and 23.5, equation (23.49) describes damped oscillations forced by the

potential on the right hand side.

Exercise 23.4 Generic behaviour of radiation. Find the homogeneous solutions of equation (23.49)

in the case of zero damping, κ = 0. Hence find the retarded Green’s function of the equation. Write down

the general solution of equation (23.49) as an integral over the Green’s function. Convince yourself that

Θ0 − Φ oscillates about −2Φ.

Solution. The general solution of equation (23.49) is, with α ≡ kηs,

Θ0(α)− Φ(α) = B0 cosα+B1 sinα− 2

∫ α

0

sin(α− α′)Φ(α′) dα′ , (23.50)

where B0 and B1 are constants.

Exercise 23.5 Behaviour of radiation in the presence of damping. Now suppose that there is a

small damping coefficient, κ≪ k. Try a solution of the form Θ0 −Φ ∝ eR

ω dηs in equation (23.49). Suppose

that the frequency ω changes slowly over a period, ω′ ≪ ω2, so that ω′ can be set to zero. Show that

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23.8 Regimes 373

the homogeneous solutions of equation (23.49) are approximately Θ0 − Φ ∝ e−R

κ dηs ± ikηs . Hence find the

retarded Green’s function, and write down the general solution to equation (23.49).

Solution. See §24.17. The general solution of equation (23.49) is, with α ≡ kηs and β ≡∫

κ dηs,

Θ0(α) − Φ(α) = e−β (B0 cosα+B1 sinα) − 2

∫ α

0

e−(β−β′) sin(α− α′)Φ(α′) dα′ , (23.51)

where B0 and B1 are constants.

23.8 Regimes

In the remainder of this Chapter, approximate analytic solutions are developed that describe the evolution

of perturbations in the matter and radiation in various regimes. The regimes are:

1. Superhorizon scales, §23.9.

2. Radiation-dominated:

a. adiabatic initial conditions, §23.10;

b. isocurvature initial conditions, §23.11.

3. Subhorizon scales, §23.12.

4. Matter-dominated, §23.13.

5. Recombination §23.14.

6. Post-recombination §23.15.

7. Matter with dark energy §23.16.

8. Matter with dark energy and curvature §23.17.

23.9 Superhorizon scales

At sufficiently early times, any mode is outside the horizon, kη < 1. In the superhorizon limit kη ≪ 1, the

evolution equations (23.27)–(23.28) reduce to

δc = 3Φ , (23.52a)

Θ0 = Φ , (23.52b)

− 3a

aF = 4πGa2(ρcδc + 4ρrΘ0) . (23.52c)

The first two of these equations evidently imply that the dark matter overdensity δc and radiation monopole

Θ0 are related to the potential Φ by

δc = 3Φ + constant , (23.53a)

Θ0 = Φ + constant . (23.53b)

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374 Cosmological perturbations: a simplest set of assumptions

aeq arec

Cosmic scale factor →

Com

ovin

gdi

stan

ce→

superhorizon

radiation

radi

atio

nba

ckgr

ound

mat

ter

fluc

tuat

ion

matter

horizon

Figure 23.1 Various regimes in the evolution of fluctuations. The line increasing diagonally from bottomleft to top right is the comoving horizon distance η. Above this line are superhorizon fluctuations, whosecomoving wavelengths exceed the horizon distance, while below the line are subhorizon fluctuations, whosecomoving wavelengths are less than the horizon distance. The vertical line at cosmic scale factor aeq ≈a0/3200 marks the moment of matter-radiation equality. Before matter-radiation equality (to the left), thebackground mass-energy is dominated by radiation, while after matter-radiation equality (to the right), thebackground mass-energy is dominated by matter. Once a fluctuation enters the horizon, the non-baryonicmatter fluctuation tends to grow, whereas the radiation fluctuation tend to decay, so there is an epochprior to matter-radiation equality where gravitational perturbations are dominated by matter rather thanradiation fluctuations, even though radiation dominates the background energy density. The vertical lineat a∗ ≈ a0/1100 marks recombination, where the temperature has cooled to the point that baryons changefrom being mostly ionized to mostly neutral, and the Universe changes from being opaque to transparent.The observed CMB comes from the time of recombination.

In effect, the dark matter velocity vc and radiation dipole Θ1 are negligibly small at superhorizon scales,

vc = Θ1 = 0 . (23.54)

Plugging the solutions (23.53) into the Einstein energy equation (23.52c), and replacing derivatives with

respect to conformal time η with derivatives with respect to cosmic scale factor a,

∂η= a

∂a= a2H

∂a, (23.55)

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23.9 Superhorizon scales 375

10−3 10−2 10−1 1 10 102 103

.0

.5

1.0

Cosmic scale factor a / aeq

Φ/Φ

(lat

e)

isocurvature

adiabatic

Figure 23.2 Evolution of the scalar potential Φ at superhorizon scales, from radiation-dominated to matter-dominated. The scale for the potential is normalized to its value Φ(late) at late times a≫ aeq.

with the Hubble parameter H from equation (23.36) gives the first order differential equation, in units

aeq = Heq = 1,

2a(1 + a)Φ′ + (6 + 5a)Φ + 4C0 + C1a = 0 , (23.56)

where prime ′ denotes differentiation with respect to cosmic scale factor, d/da, and the constants C0 and C1

are

C0 = Θ0(0)− Φ(0) , C1 = δc(0)− 3Φ(0) . (23.57)

The constants C0 and C1 are set by initial conditions. There are adiabatic and isocurvature initial conditions.

Inflation generically produces adiabatic fluctuations, in which matter and radiation fluctuate together

δc(0) = 3 Θ0(0) = − 32 Φ(0) adiabiatic . (23.58)

Notice that a positive energy fluctuation corresponds to a negative potential, consistent with Newtonian

intuition. Isocurvature initial conditions are defined by the vanishing of the initial potential, Φ(0) = 0. This,

together with equations (23.56) and (23.57), implies the isocurvature initial conditions

Φ(0) = Θ0(0) = 0 , δc(0) = − 8 Φ′(0) = − 8 Θ′0(0) isocurvature . (23.59)

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376 Cosmological perturbations: a simplest set of assumptions

Subject to the condition that Φ remains finite at a → 0, the adiabatic and isocurvature solutions to equa-

tion (23.56) are, in units aeq = 1,

Φad =Φ(0)

10

(

9 +2

a− 8

a2− 16

a3+

16√

1 + a

a3

)

=Φ(0)

10

[

3(14 + 9a) + (38 + 9a)√

1 + a]

(

1 +√

1 + a)3 , (23.60a)

Φiso =8 Φ′(0)

5

(

1− 2

a+

8

a2− 16

a3+

16√

1 + a

a3

)

=8 Φ′(0)

5

a(

6 + a+ 4√

1 + a)

(

1 +√

1 + a)4 , (23.60b)

in which the last expressions in each case are written in a form that is numerically well-behaved for all a.

Figure 23.2 shows the evolution of the potential Φ from equations (23.60), normalized to the value Φ(late)

at late times a ≫ aeq. For adiabatic fluctuations, the potential changes by a factor of 9/10 from initial to

final value, while for isocurvature fluctuations the potential evolves from zero to a final value of 85Φ′(0):

Φad(late) = 910Φ(0) = − 3

5δc(0) adiabatic , (23.61a)

Φiso(late) = 85Φ′(0) = − 1

5δc(0) isocurvature . (23.61b)

The superhorizon solutions (23.53) for the dark matter overdensity δc and radiation monopole Θ0 are,

δc = 3Θ0 = 3(

Φad − 32Φ(0)

)

adiabatic , (23.62a)

δc = 3Φiso − 8Φ′(0) , Θ0 = Φiso isocurvature . (23.62b)

23.10 Radiation-dominated, adiabatic initial conditions

For adiabatic initial conditions, fluctuations that enter the horizon before matter-radiation equality, kηeq ≫1, are dominated by radiation. In the regime where radiation dominates both the unperturbed energy and

its fluctuations, the relevant equations are, from equations (23.27), (23.28), and (23.30),

Θ0 − kΘ1 = Φ , (23.63a)

− k2Φ− 3a

aF = 16πGa2ρrΘ0 , (23.63b)

−kF = 16πGa2ρrΘ1 , (23.63c)

in which, because it simplifies the mathematics, the Einstein momentum equation is used as a substitute

for the radiation dipole equation. In the radiation-dominated epoch, the horizon is proportional to the

cosmic scale factor, η ∝ a, equation (23.38). Inserting Θ0 and Θ1 from the Einstein energy and momentum

equations (23.63b) and (23.63c) into the radiation monopole equation (23.63a) gives a second order differential

equation for the potential Φ

Φ +4

ηΦ +

k2

3Φ = 0 . (23.64)

Page 389: General Relativity, Black Holes, And Cosmology

23.10 Radiation-dominated, adiabatic initial conditions 377

0 1 2 3 4 5 6 7 8−2.0−1.5−1.0−.5

.0

.51.01.52.02.53.0

kηs /π

Θ0

−Φ

and

−2

Φ

adiabatic initial conditions

large scales (kηs,eq << 1)

small scales (kηs,eq >> 1)

0 1 2 3 4 5 6 7 8−.5

.0

.5

1.0

kηs /π

Θ0

−Φ

and

−2

Φ

isocurvature initial conditions

large scales (kηs,eq << 1)

small scales (kηs,eq >> 1)

Figure 23.3 The difference Θ0 − Φ between the radiation monopole and the Newtonian scalar potentialoscillates about − 2Φ, in accordance with equation (23.45). The difference (Θ0 − Φ) − (−2Φ) = Θ0 + Φ,which is the temperature Θ0 redshifted by the potential Φ, is the monopole contribution to temperaturefluctuation of the CMB. The top panel is for adiabatic initial conditions, equations (23.58), while the bottompanel is for isocurvature initial conditions, equation (23.59). The units of Φ and Θ0 are such that Φ(0) = −1for adiabatic fluctuations, and δc(0) = 1 for isocurvature fluctuations. In each case, the thin lines showthe evolution of small scale fluctuations, which enter the horizon during the radiation-dominated epoch wellbefore matter-radiation equality, while the thick lines show the evolution of large-scale fluctuations, whichenter the horizon during the matter-dominated epoch well after matter-radiation equality.

Equation (23.64) describes damped sound waves moving at sound speed√

13 times the speed of light. The

sound horizon, the comoving distance that sound can travel, is η/√

3, the horizon distance η multiplied by

Page 390: General Relativity, Black Holes, And Cosmology

378 Cosmological perturbations: a simplest set of assumptions

the sound speed. The growing and decaying solutions to equation (23.64) are

Φgrow =3j1(α)

α=

3(sinα− α cosα)

α3, Φdecay = − j−2(α)

α=

cosα+ α sinα

α3, (23.65)

where the dimensionless parameter α is the wavevector k multiplied by the sound horizon η/√

3,

α ≡ kη√3

=

2

3

k

aeqHeq

a

aeq, (23.66)

and jl(α) ≡√

π/(2α)Jl+1/2(α) are spherical Bessel functions. The physically relevant solution that satisfies

adiabatic initial conditions, remaining finite as α→ 0, is the growing solution

Φ = Φ(0)Φgrow . (23.67)

The solution (23.65) shows that, after a mode enters the sound horizon the scalar potential Φ oscillates with

an envelope that decays as α−2. Physically, relativistically propagating sound waves tend to suppress the

gravitational potential Φ.

For the growing solution (23.67), the radiation monopole Θ0 is

Θ0 = Φ(0)3

α3

[

(1− α2) sinα− α(

1− 12α

2)

cosα]

. (23.68)

The thin lines in the top panel of Figure 23.45 show the growing mode potential Φ and the radiation monopole

Θ0, equation (23.68). The Figure plots these two quantities in the form −2Φ and Θ0 − Φ, to bring out the

fact that Θ0 −Φ oscillates about −2Φ, in accordance with equation (23.45). After a mode is well inside the

sound horizon, α≫ 1, the radiation monopole oscillates with constant amplitude,

Θ0 =3Φ(0)

2cosα for α≫ 1 . (23.69)

The dark matter fluctuations are driven by the gravitational potential of the radiation. The solution

of the dark matter equation (23.43) driven by the potential (23.67) and satisfying adiabatic initial condi-

tions (23.62a) is

δc = 3Φ− 9Φ(0)

(

γ − 1

2+ lnα− Ciα+

sinα

α

)

, (23.70)

where the potential Φ is the growing mode solution (23.67), γ ≡ 0.5772... is Euler’s constant, and Ci(α) ≡∫ α

∞ cosxdx/x is the cosine integral. Once the mode is well inside the sound horizon, α≫ 1, the dark matter

density δc, equation (23.70), evolves as

δc = − 9Φ(0)

(

γ − 1

2+ lnα

)

for α≫ 1 , (23.71)

which grows logarithmically. This logarithmic growth translates into a logarithmic increase in the amplitude

of matter fluctuations at small scales, and is a characteristic signature of non-baryonic cold dark matter.

Page 391: General Relativity, Black Holes, And Cosmology

23.11 Radiation-dominated, isocurvature initial conditions 379

23.11 Radiation-dominated, isocurvature initial conditions

For isocurvature initial conditions, the matter fluctuation contributes from the outset, |ρcδc| > |ρrΘ0| even

while radiation dominates the background density, ρc ≪ ρr.

To develop an approximation adequate for isocurvature fluctuations entering the horizon well before

matter-radiation equality, kηeq ≫ 1, regard the Einstein energy equation (23.28a) as giving the radiation

monopole Θ0, and the Einstein momentum equation (23.30) as giving the radiation dipole Θ1. Insert these

into the radiation monopole equation (23.27a), and eliminate the δc terms using the dark matter density

equation (23.26a). The result is, in units aeq = Heq = 1,

2a(1 + a)Φ′′ + (8 + 9a)Φ′ + 2(

1 +2k2a

3

)

Φ + δc = 0 , (23.72)

where prime ′ denotes differentiation with respect to cosmic scale factor a. Equation (23.72) is valid in

all regimes, for any combination of matter and radiation. For isocurvature initial conditions, the radiation

monopole and potential vanish initially, Θ0(0) = Φ(0) = 0, whereas the dark matter overdensity is finite,

δc(0) 6= 0. For small scales that enter the horizon well before matter-radiation equality, kηeq ≫ 1, the

potential Φ is small, while δc has some approximately constant non-zero value, up to and through the time

when the mode enters the horizon, kη ≈ ka ≈ 1. In the radiation-dominated epoch, a≪ 1, and with Φ ≈ 0

(but k large and ka ∼ 1, so k2aΦ is not small) equation (23.72) simplifies to

2aΦ′′ + 8Φ′ +4k2a

3Φ + δc = 0 . (23.73)

The solution of equation (23.73) for constant δc = δc(0) is, with α given by equation (23.66),

Φ = − δc(0)√

2/3 k

1 + 12α

2 − cosα− α sinα

α3. (23.74)

The solution (23.51) for the radiation monopole Θ0 driven by the potential (23.74) is

Θ0 = − δc(0)√

2/3 k

(− 1 + α2) cosα+ (− 1 + 12α

2)(− 1 + α sinα)

α3. (23.75)

Equations (23.74) and (23.75) are the solution for small scale modes with isocurvature initial conditions

that enter the horizon well before matter-radiation equality. After a mode is well inside the sound horizon,

α≫ 1, the radiation monopole (23.75) oscillates with constant amplitude,

Θ0 = − δc(0)

2√

2/3 ksinα α≫ 1 . (23.76)

Whereas for adiabatic initial conditions the radiation monopole oscillated as cosα well inside the hori-

zon, equation (23.69), for isocurvature initial conditions it oscillates as sinα well inside the horizon, equa-

tion (23.76).

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380 Cosmological perturbations: a simplest set of assumptions

23.12 Subhorizon scales

After a mode enters the horizon, the radiation fluctuation Θ0 oscillates, but the non-baryonic cold dark

matter fluctuation δc grows monotonically. In due course, the dark matter density fluctuation ρcδc dominates

the radiation density fluctuation ρrΘ0, and this necessarily occurs before matter-radiation equality; that is,

|ρcδc| > |ρrΘ0| even though ρc < ρr. This is true for both adiabatic and isocurvature initial conditions; of

course, for isocurvature initial conditions, the dark matter density fluctuation dominates from the outset.

Even before the dark matter density fluctuation dominates, the cumulative contribution of the dark matter

to the potential Φ begins to be more important than that of the radiation, because the potential sourced by

the radiation oscillates, with an effect that tends to cancel when averaged over an oscillation.

Regard the Einstein energy equation (23.28a) as giving the dark matter overdensity δc, and the Einstein

momentum equation (23.30) as giving the dark matter velocity vc. Insert these into the dark matter density

equation (23.26a) and eliminate the Θ0 terms using the radiation monopole equation (23.27a). The result

is, in units aeq = 1,

2a2(1 + a)Φ′′ + a(6 + 7a)Φ′ − 2Φ− 4Θ0 = 0 , (23.77)

where prime ′ denotes differentiation with respect to cosmic scale factor a. Equation (23.77) is valid in all

regimes, for any combination of matter and radiation. Once the mode is well inside the horizon, kη ≫ 1,

the radiation monopole Θ0 oscillates about an average value of −Φ (since Θ0 − Φ oscillates about −2Φ, as

noted in §23.7):

〈Θ0〉 = −Φ . (23.78)

Inserting this cycle-averaged value of Θ0 into equation (23.77) gives the Meszaros differential equation

2(1 + a)a2Φ′′ + (6 + 7a)aΦ′ + 2Φ = 0 . (23.79)

The solutions of Meszaros’ differential equation (23.79) are

Φ = − 3

4

(

aeqHeq

k

)2δc

a/aeq, (23.80)

where the dark matter overdensity δc is a linear combination

δc = Cgrow δc,grow + Cdecay δc,decay (23.81)

of growing and decaying solutions, in units aeq = 1,

δc,grow = 1 +3

2a , δc,decay =

(

1 +3

2a)

ln(

√1 + a+ 1√1 + a− 1

)

− 3√

1 + a . (23.82)

For adiabatic initial conditions, the desired solution is the one that matches smoothly on to the the logarith-

mically growing solution for the dark matter overdensity δc given by equation (23.70). For modes that enter

the horizon well before matter-radiation equality, the matching may be done in the radiation-dominated

epoch a≪ 1, where the growing and decaying modes (23.82) simplify to

δc,grow = 1 , δdecay = − ln(a/4)− 3 , for a≪ 1 . (23.83)

Page 393: General Relativity, Black Holes, And Cosmology

23.13 Matter-dominated 381

Matching to the solution for δc well inside the horizon, equation (23.71), determines the constants

Cgrow = − 9 Φ(0)

[

γ − 7

2+ ln

(

4

2

3

k

aeqHeq

)

]

, Cdecay = 9 Φ(0) adiabatic . (23.84)

For isocurvature initial conditions, for modes that enter the horizon well before matter-radiation equality,

only the growing mode is present,

Cgrow = δc(0) , Cdecay = 0 isocurvature . (23.85)

The dark matter overdensity δc then evolves as the linear combination (23.81) of growing and decaying

modes (23.82). For modes that enter the horizon well before matter-radiation equality, the constants are

set by equation (23.84) for adiabatic intial conditions, or equation (23.85) for isocurvature intial conditions.

The solution remains valid from the radiation-dominated through into the matter-dominated epoch. At late

times well into the matter-dominated epoch, a≫ 1, the growing mode of the Meszaros solution dominates,

δc,grow = 32a , δc,decay = 4

15a−3/2 for a≫ 1 , (23.86)

so that the dark matter overdensity δc at late times is

δc = 32 Cgrow a for a≫ 1 . (23.87)

The potential Φ, equation (23.80), at late times is constant,

Φ = − 9

8k2Cgrow for a≫ 1 . (23.88)

For modes that enter the horizon well before matter-radiation equality, the radiation monopole Θ0 at late

times a≫ 1 is, with α ≡ kη/√

3,

Θ0 = −Φ +3

2Φ(0) cosα adiabatic , (23.89a)

Θ0 = −Φ− δc(0)

2√

2/3 ksinα isocurvature . (23.89b)

23.13 Matter-dominated

After matter-radiation equality, but before curvature or dark energy become important, non-relativistic

matter dominates the mass-energy density of the Universe.

In the matter-dominated epoch, the relevant equations are, from equations (23.26), (23.28a), and (23.30),

δc − k vc = 3 Φ , (23.90a)

− k2Φ− 3a

aF = 4πGa2ρcδc , (23.90b)

−kF = 4πGa2ρcvc , (23.90c)

Page 394: General Relativity, Black Holes, And Cosmology

382 Cosmological perturbations: a simplest set of assumptions

in which, because it simplifies the mathematics, the Einstein momentum equation is used as a substitute

for the matter velocity equation. In the matter-dominated epoch, the horizon is proportional to the square

root of the cosmic scale factor, η ∝ a1/2, equation (23.38). Inserting δc and vc from the Einstein energy and

momentum equations (23.90b) and (23.90c) into the matter density equation (23.90a) yields a second order

differential equation for the potential Φ

Φ +6

ηΦ = 0 . (23.91)

The general solution of equation (23.91) is a linear combination

Φ = Cgrow Φgrow + Cdecay Φdecay (23.92)

of growing and decaying solutions

Φgrow = 1 , Φdecay = α−5 , (23.93)

where the dimensionless paramater α is the wavevector k multiplied by the sound horizon η/√

3,

α ≡ kη√3

= 2

2

3

ka1/2

a3/2eq Heq

. (23.94)

The constants Cgrow and Cdecay in the solution (23.92) depend on conditions established before the matter-

dominated epoch. The corresponding growing and decaying modes for the dark matter overdensity δc are

δc,grow = −(

2 +α2

2

)

Φgrow , δc,decay =

(

3− α2

2

)

Φdecay . (23.95)

For modes well inside the horizon, α ≫ 1, the behaviour of the growing and decaying modes agrees with

that (23.86) of the Meszaros solution, as it should. Any admixture of the decaying solution tends quickly to

decay away, leaving the growing solution. The solution (23.51) for the radiation monopole Θ0 driven by the

potential (23.92) is a sum of a homogeneous solution and a particular solution,

Θ0 = B0 cosα+B1 sinα+ Cgrow Θ0,grow + Cdecay Θ0,decay , (23.96)

with growing and decaying modes

Θ0,grow = −Φgrow , Θ0,decay =Φdecay

12

12− 2α2 + α4 + α5 [cosα (Siα− π/2)− sinαCiα]

. (23.97)

23.14 Recombination

Exercise 23.6 Electron-scattering mean free path.

Page 395: General Relativity, Black Holes, And Cosmology

23.15 Post-recombination 383

1. Define the neutron faction Xn by

Xn ≡nn

np + nn, (23.98)

where the proton and neutron number densities np and nn are taken to include protons and neutrons in

all nuclei. For a H plus 4He composition, the proton and neutron number densities are

np = nH + 2n4He , nn = 2n4He . (23.99)

Show that the primordial 4He mass fraction defined by Y4He ≡ ρ4He/(ρH + ρ4He) satisfies

Y4He = 2Xn . (23.100)

The observed primordial 4He abundance is Y4He = 0.24, implying

Xn = 0.12 . (23.101)

2. Define the ionization fraction Xe by

Xe ≡ne

np(23.102)

where again the proton number density np is taken to include protons in all nuclei, not just in hydrogen.

Show thatne

ρb=Xe(1−Xn)

mp(23.103)

where mp is the mass of a proton or neutron.

3. Show that the (dimensionless) ratio of the comoving electron-photon scattering mean free path lT to the

comoving cosmological horizon distance c/(aeqHeq) at matter-radiation equality is

aeqHeqlTc

≡ aeqHeq

cneσTa=

16πGmp

3cσTXe(1−Xn)Heq

Ωm

Ωb

(

a

aeq

)2

=28.94 h−1

Xe(1 −Xn)

H0

Heq

Ωm

Ωb

(

a

aeq

)2

=0.002

Xe

(

a

aeq

)2

, (23.104)

the Hubble parameter Heq at matter-radiation equality being related to the present-day Hubble parameter

H0 by equation (23.41).

23.15 Post-recombination

Recombination frees baryons and photons from each other’s grasp.

Exercise 23.7 Growth of baryon fluctuations after recombination.

Page 396: General Relativity, Black Holes, And Cosmology

384 Cosmological perturbations: a simplest set of assumptions

23.16 Matter with dark energy

Some time after recombination, dark energy becomes important. Observational evidence suggests that the

dominant energy-momentum component of the Universe today is dark energy, with an equation of state

consistent with that of a cosmological constant, pΛ = −ρΛ. In what follows, dark energy is taken to have

constant density, and therefore to be synonymous with a cosmological constant. Since dark energy has a

constant energy density whereas matter density declines as a−3, dark energy becomes important only well

after recombination.

Dark energy does not cluster gravitationally, so the Einstein equations for the perturbed energy-momentum

depend only on the matter fluctuation. However, dark energy does affect the evolution of the cosmic scale

factor a. In fact, if matter is taken to be the only source of perturbation, then covariant energy-momentum

conservation, as enforced by the Einstein equations, implies that the only addition that can be made to the

unperturbed background is dark energy, with constant energy density. To see this, consider the equations

governing the matter overdensity δm and scalar velocity vm (now subscripted m, since post-recombination

matter includes baryons as well as non-baryonic cold dark matter), together with the Einstein energy and

momentum equations:

δm − k vm = 3 Φ , (23.105a)

vm +a

avm = −kΦ , (23.105b)

− k2Φ− 3a

aF = 4πGa2ρmδm , (23.105c)

−kF = 4πGa2ρmvm . (23.105d)

The factor 4πGa2ρm on the right hand side of the two Einstein equations can be written

4πGa2ρm =3a3

0H20Ωm

2a, (23.106)

where a0 and H0 are the present-day cosmic scale factor and Hubble parameter, and Ωm is the present-

day matter density (a constant). Allow the Hubble parameter H(a) ≡ a/a2 to be an arbitrary function

of cosmic scale factor a. Inserting δm and velocity vm from the Einstein energy and momentum equa-

tions (23.105c) and (23.105d) into the matter equations (23.105a) and (23.105b), and taking the overdensity

equation (23.105a) minus 3a/a times the velocity equation (23.105b), yields the condition

a4 dH2

da+ 3a3

0H20Ωm = 0 , (23.107)

whose solution is

H2

H20

=Ωm

(a/a0)3+ ΩΛ (23.108)

for some constant ΩΛ. This shows that, as claimed, if only matter perturbations are present, then the unper-

turbed background can contain, besides matter, only dark energy with constant density ρΛ = H20ΩΛ/(

83πG).

Page 397: General Relativity, Black Holes, And Cosmology

23.17 Matter with dark energy and curvature 385

The result is a consequence of the fact that the Einstein equations enforce covariant conservation of energy-

momentum.

With the Hubble parameter given by equation (23.108), the matter and Einstein equations (23.105) yield

a second order differential equation for the potential Φ, in units a0 = 1:

2a(Ωm + a3ΩΛ)Φ′′ + (7Ωm + 10a3ΩΛ)Φ′ + 6a2ΩΛΦ = 0 . (23.109)

The growing and decaying solutions to equation (23.109) are, in units a0 = 1,

Φgrow =5ΩmH

20

2

H(a)

a

∫ a

0

da′

a′3H(a′)3, Φdecay =

H

a. (23.110)

The factor 52ΩmH

20 in the growing solution is chosen so that Φgrow → 1 as a → 0. The growing solution

Φgrow can be expressed as an elliptic integral. The corresponding growing and decaying solutions for the

matter overdensity δm are, again in units a0 = 1,

δm,grow =

(

3− 2k2a

3ΩmH20

)

Φgrow − 5 , δm,decay =

(

3− 2k2a

3ΩmH20

)

Φdecay . (23.111)

For modes well inside the horizon, kη ∼ ka1/2/H0 ≫ 1, the relation (23.111) agrees with that (23.117) below.

23.17 Matter with dark energy and curvature

Curvature may also play a role after recombination. Observational evidence as of 2010 is consistent with the

Universe having zero curvature, but it is possible that there may be some small curvature.

If the curvature is non-zero, then strictly the unperturbed metric should be an FRW metric with curvature.

However, a flat background FRW metric remains a good approximation for modes whose scales are small

compared to the curvature, that is, for modes that are well inside the horizon, kη ≫ 1. For modes well inside

the horizon, the time derivative of the potential can be neglected, Φ = 0. With matter, curvature, and dark

energy present, and for modes well inside the horizon, the equations go over to the Newtonian limit:

δm − k vm = 0 , (23.112a)

vm +a

avm = −kΦ , (23.112b)

− k2Φ = 4πGa2ρmδm . (23.112c)

The factor 4πGa2ρm in the Einstein equation can be written as equation (23.106). The matter and Einstein

equations (23.112) yield a second order equation for the matter overdensity δm, in units a0 = 1:

δm +a

aδm −

3ΩmH20

2

δma

= 0 . (23.113)

Equation (23.113) can be recast as a differential equation with respect to cosmic scale factor a:

δ′′m +

(

H ′

H+

3

a

)

δ′m −3ΩmH

20

2

δma5H2

= 0 , (23.114)

Page 398: General Relativity, Black Holes, And Cosmology

386 Cosmological perturbations: a simplest set of assumptions

where H ≡ a/a2 is the Hubble parameter, and prime ′ denotes differentiation with respect to a. In the case

of matter plus curvature plus dark energy, the Hubble parameter H satisfies, again in units a0 = 1,

H2

H20

= Ωma−3 + Ωka

−2 + ΩΛ , (23.115)

where Ωm, Ωk, and ΩΛ are the (constant) present-day values of the matter, curvature, and dark energy

densities. The growing and decaying solutions to equation (23.114) are

δm,grow ≡ a g(a) =5ΩmH

20

2H(a)

∫ a

0

da′

a′3H(a′)3, δm,decay =

H

H0. (23.116)

The potential Φ is related to the matter overdensity δm by, again in units a0 = 1, equation (23.112c),

Φ = −3ΩmH20

2k2

δma

. (23.117)

The observationally relevant solution is the growing mode. The growing mode is conventionally given a

special notation, the growth factor g(a), because of its importance to relating the amplitude of clustering at

various times, from recombination up to the present. For the growing mode,

δ ∝ a g(a) , Φ ∝ g(a) . (23.118)

The normalization factor 52ΩmH

20 in equation (23.116) is chosen so that in the matter-dominated phase

shortly after recombination (small a), the growth factor g(a) is

g(a) = 1 . (23.119)

Thus as long as the Universe remains matter-dominated, the potential Φ remains constant. Curvature or

dark energy causes the potential Φ to decrease.

It should be emphasized that the growing and decaying solutions (23.116) are valid only for the case of

matter plus curvature plus constant density dark energy, where the Hubble parameter takes the form (23.115).

If another kind of mass-energy is considered, such as dark energy with non-constant density, then equations

governing perturbations of the other kind must be adjoined, and the Einstein equations modified accordingly.

The growth factor g(a) may expressed analytically as an elliptic function. A good analytic approximation

is (Carroll, Press & Turner 1992, Ann. Rev. Astron. Astrophys. 30, 449)

g ≈ 5Ωm

2[

Ω4/7m − ΩΛ +

(

1 + 12Ωm

) (

1 + 170ΩΛ

)

] , (23.120)

where Ωa are densities at the epoch being considered (such as the present, a = a0).

Page 399: General Relativity, Black Holes, And Cosmology

24

∗Cosmological perturbations: a more carefultreatment of photons and baryons

The “simple” treatment of cosmological perturbations in the previous Chapter is sufficient to reveal that the

photon-baryon fluid at the time of recombination shows a characteristic pattern of oscillations. Translating

this pattern into something that can be compared to observations of the CMB requires a more careful

treatment that follows the evolution of photons using a collisional Boltzmann equation.

Cosmologists conventionally refer to atomic matter — anything that acts by either strong or electromag-

netic forces — as baryons (from the greek baryos, meaning heavy), even though they mean by that not only

baryons (protons, neutrons, and other nuclei), but also electrons, which are leptons (from the greek leptos,

meaning light). The designation baryons does not include relativistic species, such as photons and neutrinos,

nor does it include non-baryonic dark matter or dark energy. The designation “baryons” is nonsensical, but

has stuck.

Although baryons are gravitationally sub-dominant, having a mass density about 1/5 that of the non-

baryonic dark matter, they play an important role in CMB fluctuations. Most importantly, before recombi-

nation atomic matter is ionized, and the free electrons scatter photons, preventing photons from travelling

far. Recombination occurs when the temperature drops to the point that electrons combine into neutral

atoms, releasing the photons to travel freely across the Universe, into astronomers’ telescopes.

Electron-photon scattering keeps photons and baryons tightly coupled, so that up to the time of recom-

bination they oscillate together as a photon-baryon fluid. As recombination approaches, the baryon density

becomes increasingly important relative to the photon density. The baryons, which provide mass but no

pressure, decrease the sound speed of the photon-baryon fluid, and their gravity enhances sound wave com-

pressions while weakening rarefactions. As recombination approaches, the mean free path of photons to

scattering increases, which tends to damp sound waves at short scales. All these effects — a decreased sound

speed, an enhancement of compression over rarefaction, and damping at small scales — produce observable

signatures in the power spectrum of temperature fluctuations in the CMB.

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388 ∗Cosmological perturbations: a more careful treatment of photons and baryons

24.1 Lorentz-invariant spatial and momentum volume elements

DO THIS BETTER. To define an occupation number in a Lorentz-invariant fashion, it is first necessary

to define Lorentz-invariant volume elements of space and momentum. With respect to an orthonormal

tetrad, volume elements transform as they do in special relativity. With respect to an orthonormal tetrad,

a Lorentz-invariant spatial 3-volume element can be constructed as

d4x

dτ=dt

dτd3x = E d3x . (24.1)

Lorentz invariance of Ed3x is evident because the tetrad-frame 4-volume element d4x is a scalar, and likewise

the interval dτ of proper time is a scalar. Similarly, with respect to an orthonormal tetrad, a Lorentz-invariant

momentum-space 3-volume element can be constructed as∫

E>0

2 δD(E2 − p2 −m2) d4p =

E>0

2 δD(E2 − p2 −m2) dE d3p =d3p

E. (24.2)

Lorentz-invariance of d3p/E is evident because d4p is a Lorentz-invariant momentum 4-volume, and the

argument E2 − p2 −m2 of the delta-function is a scalar. From the Lorentz invariance of Ed3x and d3p/E it

follows that the product

d3xd3p (24.3)

of spatial and momentum 3-volumes is Lorentz-invariant.

24.2 Occupation numbers

WARNING: d3x IS TETRAD-FRAME VOLUME, BUT x IS A COORDINATE. Each species of energy-

momentum is described by a dimensionless occupation number, or phase-space distribution, a function

f(η,x,p) of conformal time η, comoving position x, and tetrad-frame momentum p, which describes the

number dN of particles in a tetrad-frame element d3xd3p/(2π~)3 of phase-space

dN(η,x,p) = f(η,x,p)g d3xd3p

(2π~)3, (24.4)

with g being the number of spin states of the particle. The tetrad-frame phase-space element d3xd3p/(2π~)3

is dimensionless and Lorentz invariant, and the occupation number f is likewise dimensionless and Lorentz

invariant. The tetrad-frame energy-momentum 4-vector pm of a particle is

pm ≡ emµdxµ

dλ= E,p = E, pi , (24.5)

where λ is the affine parameter, related to proper time τ along the worldline of the particle by dλ ≡ dτ/m,

which remains well-defined in the limit of massless particles, m = 0. The tetrad-frame energy E and

momentum p ≡ |p| for a particle of rest mass m are related by

E2 − p2 = m2 . (24.6)

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24.3 Occupation numbers in thermodynamic equilibrium 389

The tetrad-frame components of the energy-momentum tensor Tmn of any species are integrals over its

occupation number f weighted by the product pmpn of 4-momenta:

Tmn =

f pmpn g d3p

E(2π~)3. (24.7)

The energy-momentum tensor Tmn defined by equation (24.7) is manifestly a tetrad-frame tensor, thanks

to the Lorentz-invariance of the momentum-space 3-volume element d3p/E.

24.3 Occupation numbers in thermodynamic equilibrium

Frequent collisions tend to drive a system towards thermodynamic equilibrium. Electron-photon scatter-

ing keeps photons in near equilibrium with electrons, while Coulomb scattering keeps electrons in near

equilibrium with ions, primarily hydrogen ions (protons) and helium nuclei. Thus photons and baryons

can be treated as having unperturbed distributions in mutual thermodynamic equilibrium, and perturbed

distributions that are small departures from thermodynamic equilibrium.

In thermodynamic equilibrium at temperature T , the occupation numbers of fermions, which obey an

exclusion principle, and of bosons, which obey an anti-exclusion principle, are

f =

1

e(E−µ)/T + 1fermion ,

1

e(E−µ)/T − 1boson ,

(24.8)

where µ is the chemical potential of the species. In the limit of small occupation numbers, f ≪ 1, equivalent

to large negative chemical potential, µ → −large, both fermion and boson distributions go over to the

Boltzmann distribution

f = e(−E+µ)/T Boltzmann . (24.9)

Chemical potential is the thermodynamic potential assocatiated with conservation of number. There is a

distinct potential for each conserved species. For example, photoionization and radiative recombination of

hydrogen,

H + γ ↔ p+ e , (24.10)

separately preserves proton and electron number, hydrogen being composed of one proton and one electron.

In thermodynamic equilibrium, the chemical potential µH of hydrogen is the sum of the chemical potentials

µp and µe of protons and electrons

µH = µp + µe . (24.11)

Photon number is not conserved, so photons have zero chemical potential,

µγ = 0 , (24.12)

which is closely associated with the fact that photons are their own antiparticles.

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390 ∗Cosmological perturbations: a more careful treatment of photons and baryons

24.4 Boltzmann equation

The evolution of each species is described by the general relativistic Boltzmann equation

df

dλ= C[f ] , (24.13)

where C[f ] is a collision term. The derivative with respect to affine parameter λ on the left hand side of

the Boltzmann equation (24.13) is a Lagrangian derivative along the (timelike or lightlike) worldline of a

particle in the fluid. Since both the occupation number f and the affine parameter λ are Lorentz scalars,

the collision term C[f ] is a Lorentz scalar. In the absence of collisions, the collisionless Boltzmann equation

df/dλ = 0 expresses conservation of particle number: a particle is neither created nor destroyed as it moves

along its wordline.

The left hand side of the Boltzmann equation (24.13) is

df

dλ= pm∂mf +

dpi

∂f

∂pi= E∂0f + pi∂if +

dp

dλ· ∂f∂p

+dp

∂f

∂p. (24.14)

Both dp/dλ and ∂f/∂p vanish in the unperturbed background, so dp/dλ ·∂f/∂p is of second order, and can

be neglected to linear order, so that

df

dλ= E∂0f + pi∂if +

dp

∂f

∂p. (24.15)

The expression (24.15) for the left hand side df/dλ of the Boltzmann equation involves dp/dλ, which in

free-fall is determined by the usual geodesic equation

dpk

dλ+ Γk

mn pmpn = 0 . (24.16)

Since E2−p2 = m2, it follows that the equation of motion for the magnitude p of the tetrad-frame momentum

is related to the equation of motion for the tetrad-frame energy E by

pdp

dλ= E

dE

dλ. (24.17)

The equation of motion for the tetrad-frame energy E ≡ p0 is

dE

dλ= −Γ0

mn pmpn = Γ0i0 p

iE + Γ0ij pipj . (24.18)

From this it follows that

d ln p

dλ=E

p2

dE

dλ= E

(

Epi

pΓ0i0 + pipjΓ0ij

)

= E

(

− a

a2+Epi

pΓ0i0 + pipj

1

Γ0ij

)

, (24.19)

where in the last expression the tetrad connection Γ0ij , equation (22.14b), has been separated into its

unperturbed and perturbed parts −(a/a2)δij and1

Γ0ij .

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24.4 Boltzmann equation 391

In practice, the integration variable used to evolve equations is the conformal time η, not the affine

parameter λ. The relation between conformal time η and affine parameter λ is

dλ= pη = em

ηpm = (δnm + ϕm

n)0

enηpm =

1

a

[

E(1− ϕ00)− piϕi0

]

, (24.20)

whose reciprocal is to linear order

dη=

a

E

(

1 + ϕ00 +pi

Eϕi0

)

. (24.21)

With conformal time η as the integration variable, the equation of motion (24.19) for the magnitude p of

the tetrad-frame momentum becomes, to linear order,

d ln p

dη= − a

a

(

1 + ϕ00 +pi

Eϕi0

)

+Epi

paΓ0i0 + pipja

1

Γ0ij . (24.22)

With respect to conformal time η, the Boltzmann equation (24.13) is

df

dη=∂f

∂η+ vi∇if +

d ln p

∂f

∂ ln p=dλ

dηC[f ] , (24.23)

with dλ/dη from equation (24.21), and d ln p/dη from equation (24.22). Expressions for dλ/dη and d ln p/dη

in terms of the vierbein perturbations in a general gauge are left as Exercise 24.1. In conformal Newtonian

gauge, the factor dλ/dη, equation (24.21), is

dη=

a

E(1 + Ψ) . (24.24)

In conformal Newtonian gauge, and including only scalar fluctuations, the factor d ln p/dη, equation (24.22),

is

d ln p

dη= − a

a+ Φ− Epi

p∇iΨ . (24.25)

To unperturbed order, the Boltzmann equation (24.23) is

d0

f

dη=∂

0

f

∂η− a

a

∂0

f

∂ ln p=

a

EC[

0

f ] , (24.26)

where C[0

f ] is the unperturbed collision term, the factor a/E coming from dλ/dη = a/E to unperturbed

order, equation (24.21). The second term in the middle expression of equation (24.26) simply reflects the

fact that the tetrad-frame momentum p redshifts as p ∝ 1/a as the Universe expands, a statement that is

true for both massive and massless particles.

Subtracting off the unperturbed part (24.26) of the Boltzmann equation (24.23) gives the perturbation of

the Boltzmann equation

d1

f

dη=∂

1

f

∂η+ vi∇i

1

f − a

a

∂1

f

∂ ln p+d ln(ap)

∂0

f

∂ ln p=

a

EC[

1

f ] +

1

dηC[

0

f ] . (24.27)

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392 ∗Cosmological perturbations: a more careful treatment of photons and baryons

In conformal Newtonian gauge, the perturbed part of dλ/dη is

1

dη=

a

EΨ . (24.28)

In conformal Newtonian gauge, and including only scalar fluctuations, d ln(ap)/dη is

d ln(ap)

dη= Φ− Epi

p∇iΨ . (24.29)

Exercise 24.1 Boltzmann equation factors in a general gauge. Show that in a general gauge, and

including not just scalar but also vector and tensor fluctuations, equation (24.21) is

dη=

a

E

[

1 + ψ +pi

E(∇iw + wi)

]

, (24.30)

while equation (24.22) is, with only scalar fluctuations included,

d ln p

dη= − a

a+ φ+

Epi

p

[

−∇iψ +

(

∂η+a

a

m2

E2

)

(∇iw + wi)

]

+ pipj[

−∇i∇j(w − h)− 12 (∇iWj +∇jWi) +∇jwi + hij

]

. (24.31)

24.5 Non-baryonic cold dark matter

Non-baryonic cold dark matter, subscripted c, is by assumption non-relativistic and collisionless. The un-

perturbed mean density is ρc, which evolves with cosmic scale factor a as

ρc ∝ a−3 . (24.32)

Since dark matter particles are non-relativistic, the energy of a dark matter particle is its rest-mass energy,

Ec = mc, and its momentum is the non-relativistic momentum pic = mcv

ic.

The energy-momentum tensor Tmnc of the dark matter is obtained from integrals over the dark matter

phase-space distribution fc, equation (24.7). The energy and momentum moments of the distribution define

the dark matter overdensity δc and bulk velocity vvvc, while the pressure is of order v2c , and can be neglected

to linear order,

T 00c ≡

fcmcgc d

3pc

(2π~)3≡ ρc(1 + δc) , (24.33a)

T 0ic ≡

fcmc vic

gc d3pc

(2π~)3≡ ρcv

ic , (24.33b)

T ijc ≡

fcmc vic v

jc

gc d3pc

(2π~)3= 0 . (24.33c)

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24.5 Non-baryonic cold dark matter 393

Non-baryonic cold dark matter is collisionless, so the collision term in the Boltzmann equation is zero,

C[fc] = 0, and the dark matter satisfies the collisionless Boltzmann equation

dfc

dη= 0 . (24.34)

The energy and momentum moments of the Boltzmann equation (24.23) yield equations for the overdensity

δc and bulk velocity vc, which in the conformal Newtonian gauge are

0 =

dfc

dηmc

gc d3pc

(2π~)3=

∂η

fcmcgc d

3pc

(2π~)3+∇i

fcmcvic

gc d3pc

(2π~)3−∫ (

a

a− Φ

)

∂f

∂ ln pmc

gc d3pc

(2π~)3

=∂ρc(1 + δc)

∂η+∇i(ρcv

ic) + 3

(

a

a− Φ

)

ρc , (24.35a)

0 =

dfc

dηmcv

ic

gc d3pc

(2π~)3=

∂η

fcmcvic

gc d3pc

(2π~)3+∇j

fcmcvicv

jc

gc d3pc

(2π~)3

−∫ (

a

a− Φ +

Epj

p∇jΨ

)

∂f

∂ ln pmcv

i gc d3pc

(2π~)3

=∂ρcv

ic

∂η+ 4

(

a

a− Φ

)

ρcvic + ρc∇iΨ . (24.35b)

The Φρcvic term on the last line of equation (24.35b) can be dropped, since the potential Φ and the velocity

v ic are both of first order, so their product is of second order. Subtracting the unperturbed part from

equations (24.35a) and (24.35b) gives equations for the dark matter overdensity δc and velocity vvvc,

δc + ∇ · vvvc − 3Φ = 0 , (24.36a)

vvvc +a

avvvc + ∇Ψ = 0 . (24.36b)

Transform into Fourier space, and decompose the velocity 3-vector vvvc into scalar vc and vector vvvc,⊥ parts

vvvc = −ikvc + vvvc,⊥ . (24.37)

For the scalar modes under consideration, only the scalar part of the dark matter equations (24.36) is

relevant:

δc − kvc − 3Φ = 0 , (24.38a)

vc +a

avc + kΨ = 0 . (24.38b)

Equations (24.38) reproduce the equations (23.26) derived previously from conservation of energy and mo-

mentum.

Exercise 24.2 Moments of the non-baryonic cold dark matter Boltzmann equation. Confirm

equations (24.35).

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394 ∗Cosmological perturbations: a more careful treatment of photons and baryons

24.6 The left hand side of the Boltzmann equation for photons

In the unperturbed background, the photons have a blackbody distribution with temperature T (η). Define

Θ to be the photon temperature fluctuation

Θ(η,x,pγ) ≡ δT (η,x,pγ)

T (η). (24.39)

In the unperturbed background, the photon occupation number is

0

fγ =1

epγ/T − 1. (24.40)

Since pγ ∝ T ∝ 1/a, the unperturbed occupation number is constant as a function of pγ/T . The definition

Θ ≡ δT/T = δ lnT of the photon perturbation is to be interpreted as meaning that the perturbation to the

occupation number of photons is (the partial derivative with respect to temperature ∂/∂ lnT is at constant

photon momentum pγ)

1

fγ =∂

0

∂ lnTδ lnT =

∂0

∂ lnTΘ , (24.41)

in which it follows from equation (24.40) that

∂0

∂ lnT=

0

fγ(1 +0

fγ)pγ

T. (24.42)

The photon Boltzmann equation in terms of the occupation number fγ can be recast as a Boltzmann

equation for the temperature fluctuation Θ through

d1

dη=

d

∂0

∂ lnTΘ

=d

∂0

∂ lnT

Θ +∂

0

∂ lnT

dη=

∂0

∂ lnT

dη, (24.43)

in which the first term of the penultimate expression vanishes because ∂0

fγ/∂ lnT is a function of pγ/T only,

and pγ/T is, to unperturbed order (which is all that is needed since the term is multiplied by Θ, which is

already of first order), independent of time, d(pγ/T )/dη = 0, since pγ ∝ T ∝ a−1:

d

∂0

∂ lnT= − d ln(pγ/T )

∂20

∂ lnT 2= 0 . (24.44)

In terms of the temperature fluctuation Θ, the perturbed photon Boltzmann equation (24.27) is

dη=∂Θ

∂η+ pi

γ ∇iΘ−a

a

∂Θ

∂ ln pγ− d ln(apγ)

dη=

(

a

pγC[

1

fγ ] +

1

dηC[

0

fγ ]

)/

∂0

∂ lnT, (24.45)

where the d ln(apγ)/dη term gets a minus sign from ∂0

f/∂ ln pγ = −∂0

f/∂ lnT .

The unperturbed photon distribution is in thermodynamic equilibrium, so the unperturbed collision term

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24.7 Spherical harmonics of the photon distribution 395

in the photon Boltzmann equation (24.45) vanishes, C[0

fγ ] = 0, as found in Exercise 24.3 below. The

photon distribution is modified by photon-electron (Thomson) scattering, §24.10. Since the electrons are

non-relativistic, to linear order collisions change the photon momentum but not the photon energy. As a

consequence, the temperature fluctuation is a function Θ(η,x, pγ) only of the direction pγ of the photon

momentum pγ , not of its magnitude, the energy pγ . This is shown more carefully below, equation (24.78).

Hence the derivative ∂Θ/∂ ln pγ in the photon Boltzmann equation (24.45) vanishes to linear order. Thus

the photon Boltzmann equation (24.45) reduces to

dη=∂Θ

∂η+ pi

γ ∇iΘ−d ln(apγ)

dη=

a

pγC[

1

fγ ]/ ∂

0

fγ∂ lnT

. (24.46)

In conformal Newtonian gauge, and including only scalar fluctuations, d ln(apγ)/dη is given by equa-

tion (24.29), so the photon Boltzmann equation (24.45) becomes

dη=∂Θ

∂η+ pi

γ ∇iΘ− Φ + piγ∇iΨ =

a

pγC[

1

fγ ]/ ∂

0

∂ lnT. (24.47)

24.7 Spherical harmonics of the photon distribution

It is natural to expand the temperature fluctuation Θ in spherical harmonics. As seen below, equa-

tions (24.55), the various components of the photon energy-momentum tensor Tmnγ are determined by the

monopole, dipole, and quadrupole harmonics of the photon distribution. Scalar fluctuations are those that

are rotationally symmetric about the wavevector direction k, which correspond to spherical harmonics with

zero azimuthal quantum number, m = 0. Expanded in spherical harmonics, and with only scalar terms

retained, the temperature fluctuation Θ can be written

Θ(η,k, pγ) =

∞∑

ℓ=0

(−i)ℓ(2ℓ+ 1)Θℓ(η,k)Pℓ(k · pγ) , (24.48)

where Pℓ are Legendre polynomials, §24.21. The scalar harmonics Θℓ are angular integrals of the temperature

fluctuation Θ over photon directions pγ :

Θℓ(η,k) = iℓ∫

Θ(η,k, pγ)Pℓ(k · pγ)dopγ

4π. (24.49)

Expanded into the scalar harmonics Θℓ(η,k), the left hand side of the photon Boltzmann equation (24.46)

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396 ∗Cosmological perturbations: a more careful treatment of photons and baryons

is, in conformal Newtonian gauge,

dΘ0

dη= Θ0 − kΘ1 − Φ , (24.50)

dΘ1

dη= Θ1 +

k

3

(

Θ0 − 2k2Θ2

)

+k

3Ψ , (24.51)

dΘℓ

dη= Θℓ +

k

2ℓ+ 1[ℓΘℓ−1 − (ℓ + 1)Θℓ+1] (ℓ ≥ 2) . (24.52)

24.8 Energy-momentum tensor for photons

Perturbations to the photon energy-momentum tensor involve integrals (24.7) over the perturbed occupation

number of the form, where F (p) is some arbitrary function of the momentum direction p,

1

fγ p2γ F (pγ)

2 d3pγ

pγ(2π~)3=

∂0

∂ lnTp2

γ

2 4πp2γdpγ

pγ(2π~)3

ΘF (pγ)dopγ

4π= 4ργ

ΘF (pγ)dopγ

4π, (24.53)

in which the last expression is true because

∂0

∂ lnTp2

γ

2 4πp2γdpγ

pγ(2π~)3= 4

0

fγ pγ

2 4πp2γdpγ

(2π~)3= 4ργ , (24.54)

which follows from ∂0

fγ/∂ lnT = − ∂0

fγ/∂ ln pγ and an integration by parts. The perturbation of the photon

energy density, energy flux, monopole pressure, and quadrupole pressure are

1

T 00γ = 4 ργ Θ0 , (24.55a)

ki

1

T 0iγ = i 4 ργ Θ1 , (24.55b)

13 δij

1

T ijγ = 4

3 ργ Θ0 , (24.55c)(

32 kikj − 1

2 δij

)

1

T ijγ = − 4 ργ Θ2 . (24.55d)

24.9 Collisions

For a 2-body collision of the form

1 + 2 ↔ 3 + 4 , (24.56)

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24.9 Collisions 397

the rate per unit time and volume at which particles of type 1 leave and enter an interval d3p1 of momentum

space is, in units c = 1,

C[f1]g1 d

3p1

E1(2π~)3=

|M|2[

− f1f2(1∓ f3)(1 ∓ f4) + f3f4(1 ∓ f1)(1∓ f2)]

(2π~)4δ4D(p1 + p2 + p3 + p4)g1 d

3p1

2E1(2π~)3g2 d

3p2

2E2(2π~)3g3 d

3p3

2E3(2π~)3g4 d

3p4

2E4(2π~)3. (24.57)

All factors in equation (24.57) are Lorentz scalars. On the left hand side, the collision term C[f1] and

the momentum 3-volume element d3p1/E1 are both Lorentz scalars. On the right hand side, the squared

amplitude |M|2, the various occupation numbers fa, the energy-momentum conserving 4-dimensional Dirac

delta-function δ4D(p1 + p2 + p3 + p4), and each of the four momentum 3-volume elements d3pa/Ea, are all

Lorentz scalars.

The first ingredient in the integrand on the right hand side of the expression (24.57) is the Lorentz-invariant

scattering amplitude squared |M|2, calculated using quantum field theory (F. Halzen & A. D. Martin 1984

Quarks and Leptons, Wiley, New York, p. 91).

The second ingredient in the integrand on the right hand side of expression (24.57) is the combination of

rate factors

rate(1 + 2 → 3 + 4) ∝ f1f2(1∓ f3)(1 ∓ f4) , (24.58a)

rate(1 + 2 ← 3 + 4) ∝ f3f4(1∓ f1)(1 ∓ f1) , (24.58b)

where the 1 ∓ f factors are blocking or stimulation factors, the choice of ∓ sign depending on whether the

species in question is fermionic or bosonic:

1− f = Fermi-Dirac blocking factor , (24.59a)

1 + f = Bose-Einstein stimulation factor . (24.59b)

The first rate factor (24.58a) expresses the fact that the rate to lose particles from 1 + 2→ 3 + 4 collisions

is proportional to the occupancy f1f2 of the initial states, modulated by the blocking/stimulation factors

(1 ∓ f3)(1 ∓ f4) of the final states. Likewise the second rate factor (24.58b) expresses the fact that the

rate to gain particles from 1 + 2 ← 3 + 4 collisions is proportional to the occupancy f3f4 of the initial

states, modulated by the blocking/stimulation factors (1∓ f1)(1∓ f2) of the final states. In thermodynamic

equilibrium, the rates (24.58) balance, Exercise 24.3, a property that is called detailed balance, or microscopic

reversibility. Microscopic reversibility is a consequence of time reversal symmetry.

The final ingredient in the integrand on the right hand side of expression (24.57) is the 4-dimensional

Dirac delta-function, which imposes energy-momentum conservation on the process 1 + 2 ↔ 3 + 4. The

4-dimensional delta-function is a product of a 1-dimensional delta-function expressing energy conservation,

and a 3-dimensional delta-function expressing momentum conservation:

(2π~)4δ4D(p1 + p2 + p3 + p4) = 2π~ δD(E1 + E2 + E3 + E4) (2π~)3δ3D(p1 + p2 + p3 + p4) . (24.60)

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398 ∗Cosmological perturbations: a more careful treatment of photons and baryons

Exercise 24.3 Detailed balance. Show that the rates balance in thermodynamic equilibrium,

f1f2(1 ∓ f3)(1∓ f4) = f3f4(1∓ f1)(1∓ f2) . (24.61)

Solution. Equation (24.61) is true if and only if

f11∓ f1

f21∓ f2

=f3

1∓ f3f4

1∓ f4. (24.62)

But

f

1∓ f = e(−E+µ)/T , (24.63)

so (24.62) is true if and only if

−E1 + µ1

T+−E2 + µ2

T=−E3 + µ3

T+−E4 + µ4

T, (24.64)

which is true in thermodynamic equilibrium because

E1 + E2 = E3 + E4 , µ1 + µ2 = µ3 + µ4 . (24.65)

24.10 Electron-photon scattering

The dominant process that couples photons and baryons is electron-photon scattering

e+ γ ↔ e′ + γ′ . (24.66)

The Lorentz-invariant transition probability for unpolarized non-relativistic electron-photon (Thomson) scat-

tering is

|M|2 = 12πm2eσT~

2[

1 + (pγ · pγ′)2]

= 16πm2eσT~

2[

1 + 12P2(pγ · pγ′)

]

, (24.67)

where P2(µ) is the quadrupole Legendre polynomial, §24.21, and

σT =8π

3

(

e2

mec2

)2

(24.68)

is the Thomson cross-section.

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24.11 The photon collision term for electron-photon scattering 399

24.11 The photon collision term for electron-photon scattering

Electron-photon scattering keeps electrons and photons close to mutual thermodynamic equilibrium, and

their unperturbed distributions can be taken to be in thermodynamic equilibrium. The unperturbed photon

collision term for electron-photon scattering therefore vanishes, because of detailed balance, Exercise 24.3,

C[0

fγ ] = 0 . (24.69)

Thanks to detailed balance, the combination of rates in the collision integral (24.57) almost cancels, so can

be treated as being of linear order in perturbation theory. This allows other factors in the collision integral

to be approximated by their unperturbed values.

The photon collision term for electron-photon scattering follows from the general expression (24.57). To

unperturbed order, the energies of the electrons, which are non-relativistic, may be set equal to their rest

masses, Ee = me. Since photons are massless, their energies are just equal to their momenta, Eγ = pγ . The

electron occupation number is small, fe ≪ 1, so the Fermi blocking factors for electrons may be neglected,

1 − fe = 0. The number of spins of the incoming electron is two, ge = 2, because photons scatter off both

spins of electrons. On the other hand, the number of spins of the scattered electron and photon are one,

ge′ = gγ′ = 1, because non-relativistic electron-photon scattering leaves the spins of the electron and photon

unchanged. These considerations bring the photon collision term for electron-photon scattering to, from the

general expression (24.57),

C[1

fγ ] =1

16

|M|2[

− fefγ(1+fγ′)+fe′fγ′(1+fγ)]

(2π~)4δ4D(pe+pγ−pe′−pγ′)2 d3pe

me(2π~)3d3pe′

me(2π~)3d3pγ′

pγ′(2π~)3.

(24.70)

The various integrations over momenta are most conveniently carried out as follows. The energy-conserving

integral is best done over the energy of the scattered photon γ′, which is scattered into an interval doγ′ of

solid angle:∫

2π~ δD(Ee + Eγ − Ee′ − Eγ′)d3pγ′

Eγ′(2π~)3= pγ′

doγ′

(2π~)2≈ pγ

doγ′

(2π~)2. (24.71)

The approximation in the last step of equation (24.71), replacing the energy pγ′ of the scattered photon by

the energy pγ of the incoming photon, is valid because, thanks to the smallness of the combination of rates

in the collision integral (24.70), it suffices to treat the photon energy to unperturbed order. As seen below,

equation (24.76) the energy difference pγ−pγ′ between the incoming and scattered photons is of linear order

in electron velocities.

The momentum-conserving integral is best done over the momentum of the scattered electron, which is e′

for outgoing scatterings e+ γ → e′ + γ′, and e for incoming scatterings e+ γ ← e′ + γ′. In the former case

(e+ γ → e′ + γ′),∫

(2π~)3δ3D(pe + pγ − pe′ − pγ′)d3pe′

Ee′ (2π~)3=

1

me, (24.72)

and the result is the same, 1/me, in the latter case (e + γ ← e′ + γ′). The energy- and momentum-

conserving integrals having been done, the electron e′ in the latter case may be relabelled e. So relabelled,

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400 ∗Cosmological perturbations: a more careful treatment of photons and baryons

the combination of rate factors in the collision integral (24.70) becomes

− fefγ(1 + fγ′) + fefγ′(1 + fγ) = fe(− fγ + fγ′) . (24.73)

Notice that the stimulated terms cancel. The energy- and momentum-conserving integrations (24.71)

and (24.72) bring the photon collision term (24.70) to

C[1

fγ ] =pγ

16πm2e~2

|M|2fe(− fγ + fγ′)2 d3pe

(2π~)3doγ′

4π. (24.74)

The collision integral (24.74) involves the difference − fγ + fγ′ between the occupancy of the initial and

final photon states. To linear order, the difference is

− fγ + fγ′ = −0

f(pγ) +0

f(pγ′)−1

f(pγ) +1

f(pγ′) =∂

0

∂ lnT

[

pγ − pγ′

pγ−Θ(pγ) + Θ(pγ′)

]

. (24.75)

The first term (pγ − pγ′)/pγ arises because the incoming and scattered photon energies differ slightly. The

difference in photon energies is given by energy conservation:

pγ − pγ′ = Ee′ − Ee

=

(

me +p′2e2me

)

−(

me +p2

e

2me

)

=(pe + pγ − pγ′)2 − p2

e

2me

=(pγ − pγ′) · (2pe + pγ − pγ′)

2me

≈ (pγ − pγ′) · ve , (24.76)

the last line of which follows because the photon momentum is small compared to the electron momentum,

pγ ∼ T ∼ mev2e = peve ≪ pe . (24.77)

Because the photon energy difference is of first order, and the temperature fluctuation is already of first

order, it suffices to regard the temperature fluctuation Θ as being a function only of the direction pγ of the

photon momentum, not of its energy:

Θ(pγ) ≈ Θ(pγ) . (24.78)

The linear approximations (24.76) and (24.78) bring the difference (24.75) between the initial and final

photon occupancies to

− fγ + fγ′ =∂

0

∂ lnT[(pγ − pγ′) · ve −Θ(pγ) + Θ(pγ′)] . (24.79)

Inserting this difference in occupancies into the collision integral (24.74) yields

C[1

fγ ] =pγ

16πm2e~2

∂0

∂ lnT

|M|2fe [(pγ − pγ′) · ve −Θ(pγ) + Θ(pγ′)]2 d3pe

(2π~)3doγ′

4π. (24.80)

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24.11 The photon collision term for electron-photon scattering 401

The transition probability |M|2, equation (24.67), is independent of electron momenta, so the integration

over electron momentum in the collision integral (24.80) is straightforward. The unperturbed electron density

ne and the electron bulk velocity vvve are defined by

ne ≡∫

0

fe

2 d3pe

(2π~)3, nevvve ≡

0

fe ve2 d3pe

(2π~)3. (24.81)

Coulomb scattering keeps electrons and ions tightly coupled, so the electron bulk velocity vvve equals the

baryon bulk velocity vvvb,

vvve = vvvb . (24.82)

Integration over the electron momemtum brings the collision integral (24.80) to

C[1

fγ ] =nepγ

16πm2e~2

∂0

∂ lnT

|M|2 [(pγ − pγ′) · vvvb −Θ(pγ) + Θ(pγ′)]doγ′

4π. (24.83)

Finally, the collision integral (24.83) must be integrated over the direction pγ′ of the scattered photon. In-

serting the electron-photon scattering transition probability |M|2 given by equation (24.67) into the collision

integral (24.83) brings it to

C[1

fγ ] = neσTpγ

∂0

∂ lnT

[

1 + 12P2(pγ · pγ′)

]

[(pγ − pγ′) · vvvb −Θ(pγ) + Θ(pγ′)]doγ′

4π. (24.84)

The pγ′ · vvvb term in the integrand of (24.84) is odd, and vanishes on angular integration:∫

[

1 + 12P2(pγ · pγ′)

]

pγ′doγ′

4π= 0 . (24.85)

Similarly, the angular integral over the quadrupole of quantities independent of pγ′ vanishes:∫

P2(pγ · pγ′) [pγ · vvvb −Θ(pγ)]doγ′

4π= 0 . (24.86)

The collision integral (24.84) thus reduces to

C[1

fγ(x, pγ)] = neσTpγ

∂0

∂ lnT

pγ · vvvb(x)−Θ(x, pγ) +

[

1 + 12P2(pγ · pγ′)

]

Θ(x, pγ′)doγ′

, (24.87)

where the dependence of various quantities on comoving position x has been made explicit. Now transform

to Fourier space (in effect, replace comoving position x by comoving wavevector k). Replace the baryon bulk

velocity by its scalar part, vvvb = ikvb. To perform the remaining angular integral over the photon direction

pγ′ , expand the temperature fluctuation Θ(k, pγ′) in scalar multipole moments according to equation (24.48),

and invoke the orthogonality relation (24.134). With µγ defined by

µγ ≡ k · pγ , (24.88)

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402 ∗Cosmological perturbations: a more careful treatment of photons and baryons

these manipulations bring the photon collision integral (24.87) at last to

C[1

fγ(k, pγ)] = neσTpγ

∂0

∂ lnT

[

− iµγvb(k)−Θ(k, µγ) + Θ0(k)− 12Θ2(k)P2(µγ)

]

. (24.89)

24.12 Boltzmann equation for photons

Inserting the collision term (24.89) into equation (24.47) yields the photon Boltzmann equation for scalar

fluctuations in conformal Newtonian gauge,

dη=∂Θ

∂η− ikµγΘ− Φ− ikµγΨ = neσTa

[

− iµγvb −Θ + Θ0 − 12Θ2P2(µγ)

]

. (24.90)

Expanded into the scalar harmonics Θℓ(η,k), the photon Boltzmann equation (24.46) yields the hierarchy

of photon multipole equations

Θ0 − kΘ1 − Φ = 0 , (24.91a)

Θ1 +k

3(Θ0 − 2Θ2) +

k

3Ψ =

1

3neσTa (vb − 3Θ1) , (24.91b)

Θ2 +k

5(2Θ1 − 3Θ3) = − 9

10neσTaΘ2 , (24.91c)

Θℓ +k

2ℓ+ 1[ℓΘℓ−1 − (ℓ + 1)Θℓ+1] = −neσTaΘℓ (ℓ ≥ 3) . (24.91d)

The Boltzmann hierarchy (24.91) shows that all the photon multipoles are affected by electron-photon

scattering, but only the photon dipole Θ1 depends directly on one of the baryon variables, the baryon bulk

velocity vb. The dependence on the baryon velocity vb reflects the fact that, to linear order, there is a

transfer of momentum between photons and baryons, but no transfer of number or of energy.

For the dipole, ℓ = 1, electron-photon scattering drives the electron and photon bulk velocities into near

equality,

vb ≈ 3Θ1 , (24.92)

so that the right hand side side of the dipole equation (24.91b) is modest despite the large scattering factor

neσTa. The approximation in which the bulk velocities of electrons and photons are exactly equal, vb = 3Θ1,

and all the higher multipoles vanish, Θℓ = 0 for ℓ ≥ 2, is called the tight-coupling approximation. The tight-

coupling approximation was already invoked in the “simple” model of Chapter 23.

For multipoles ℓ ≥ 2, the electron-photon scattering term on the right hand side of the Boltzmann

hierarchy (24.91) acts as a damping term that tends to drive the multipole exponentially into equilibrium

(the solution to the homogeneous equation Θℓ + neσTaΘℓ = 0 is a decaying exponential). As seen in

Chapter 23, in the tight-coupling approximation the monopole and dipole oscillate with a natural frequency

of ω = csk, where cs is the sound speed. These oscillations provide a source that propagates upward

to higher harmonic number ℓ. For scales much larger than a mean free path, k/(neσTa) ≪ 1, the time

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24.13 Diffusive (Silk) damping 403

derivative is small compared to the scattering term, |Θ| ∼ csk|Θ| ≪ neσTa|Θ|, reflecting the near-equilibrium

response of the higher harmonics. For multipoles ℓ ≥ 2, the dominant term on the left hand side of the

Boltzmann hierarchy (24.91) is the lowest order multipole, which acts as a driver. Solution of the Boltzmann

equations (24.91) then requires that

Θℓ+1 ∼k

neσTaΘℓ for ℓ ≥ 2 . (24.93)

The relation (24.93) implies that higher order photon multipoles are successively smaller than lower orders,

|Θℓ+1| ≪ |Θℓ|, for scales much larger than a mean free path, k/(neσTa)≪ 1. This accords with the physical

expectation that electron-photon scattering drives the photon distribution to near isotropy.

24.13 Diffusive (Silk) damping

The tight coupling between photons and baryons is not perfect, because the mean free path for electron-

photon scattering is finite, not zero. The imperfect coupling causes sound waves to dissipate at scales

comparable to the mean free path. To lowest order, the dissipation can be taken into account by including

the photon quadrupole Θ2 in the system (24.91) of photon multipole equations, but still neglecting the higher

multipoles, Θℓ = 0 for ℓ ≥ 3. According to the estimate (24.93), this approximation is valid for scales much

larger than a mean free path, k/(neσTa)≪ 1. The approximation is equivalent to a diffusion approximation.

In the diffusion approximation, the photon quadrupole equation (24.91c) reduces to

Θ2 = − 4k

9neσTaΘ1 . (24.94)

Substituted into the photon momentum equation (24.91b), the photon quadrupole Θ2 (24.94) acts as a source

of friction on the photon dipole Θ1.

24.14 Baryons

The equations governing baryonic matter are similar to those governing non-baryonic cold dark matter,

§24.5, except that baryons are collisional. Coulomb scattering between electrons and ions keep baryons

tightly coupled to each other. Electron-photon scattering couples baryons to the photons.

Since the unperturbed distribution of baryons is in thermodynamic equilibrium, the unperturbed collision

term vanishes for each species of baryonic matter, as it did for photons, equation (24.95),

C[0

fb] = 0 . (24.95)

For the perturbed baryon distribution, only the first and second moments of the phase-space distribution

are important, since these govern the baryon overdensity δb and bulk velocity vvvb. The relevant collision term

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404 ∗Cosmological perturbations: a more careful treatment of photons and baryons

is the electron collision term associated with electron-photon scattering. Since electron-photon scattering

neither creates nor destroys electrons, the zeroth moment of the electron collision term vanishes,

C[1

fe]2d3pe

me(2π~)3= 0 . (24.96)

The first moment of the electron collision term is most easily determined from the fact that electron-photon

collisions must conserve the total momentum of electron and photons:

C[1

fe]meve2 d3pe

me(2π~)3+

C[1

fγ ] pγ2 d3pγ

pγ(2π~)3= 0 . (24.97)

Substituting the expression (24.87) for the photon collision integral into equation (24.97), separating out

factors depending on the absolute magnitude pγ and direction pγ of the photon momentum, and taking

into consideration that the integral terms in equation (24.87), when multiplied by pγ , are odd in pγ , and

therefore vanish on integration over directions pγ , gives

C[1

fe]meve2 d3pe

me(2π~)3= neσT

∂0

∂ lnTp2

γ

2 4πp2γdpγ

pγ(2π~)3

[− pγ · vvvb + Θ(pγ)] pγdoγ

4π. (24.98)

The integral over the magnitude pγ of the photon momentum in equation (24.98) yields 4ρ, in accordance

with equation (24.54). Transformed into Fourier space, and with only scalar terms retained, the collision

integral (24.98) becomes, with µγ ≡ k · pγ ,

k ·∫

C[1

fe]meve2 d3pe

me(2π~)3= 4ργneσT

[iµγvb + Θ] µγdoγ

=4

3iργneσT (vb − 3Θ1) . (24.99)

The result is that the equations governing the baryon overdensity δb and scalar bulk velocity vb look like

those (24.38) governing non-baryonic cold dark matter, except that the velocity equation has an additional

source (24.99) arising from momentum transfer with photons through electron-photon scattering:

δb − kvb − 3Φ = 0 , (24.100a)

vb +a

avb + kΨ = − neσTa

R(vb − 3Θ1) , (24.100b)

where R is 34 the baryon-to-photon density ratio,

R ≡ 3ρb

4ργ= Ra

a

aeq, Ra =

3gρΩb

8Ωm≈ 0.21 , (24.101)

with gρ = 2 + 6 78

(

411

)4/3= 3.36 being the energy-weighted effective number of relativistic particle species

at around the time of recombination (Exercise 10.18). The parameter R plays an important role in that it

modulates the sound speed in the photon-baryon fluid, equation (24.110) below.

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24.15 Viscous baryon drag damping 405

24.15 Viscous baryon drag damping

A second source of damping of sound waves, distinct from the diffusive damping of §24.13, arises from the

viscous drag on photons that results from a small difference vb − 3Θ1 between the baryon and photon bulk

velocities. This damping is associated with the finite mass density of baryons, and vanishes in the limit of

small baryon density. However, for realistic values of the baryon density, viscous baryon drag damping is

comparable to diffusive damping.

Equation (24.100b) for the baryon bulk velocity may be written

vb − 3Θ1 = − R

neσTa

(

vb +a

avb + kΨ

)

. (24.102)

The right hand side of this equation is small because the scattering rate neσT is large, so to lowest order

vb = 3Θ1, the tight-coupling approximation. To ascertain the effect of a small difference in the baryon

and photon bulk velocities, take equation (24.102) to next order. In the circumstances where damping is

important, which is small scales well inside the sound horizon, kηs ≫ 1, the dominant term on the right hand

side of equation (24.100b) is the time derivative vb. With only this term kept, equation (24.102) reduces to

vb − 3Θ1 ≈ −R

neσTavb ≈ −

3R

neσTaΘ1 . (24.103)

Substituting the approximation (24.103) into the left hand side of the baryon velocity equation (24.100b)

gives

vb +a

avb + kΨ = 3

(

Θ1 +a

aΘ1 +

k

)

− 3R

neσTaΘ1 , (24.104)

in which the last term on the right hand side is the small correction from the non-vanishing baryon-photon

velocity difference, equation (24.103). As in the approximation (24.103), only the dominant term, the one

arising from the time derivative vb, has been retained in the correction, and in differentiating the right

hand side of equation (24.103), the time derivative of 3R/(neσTa) has been neglected compared to the time

derivative of Θ1. The final simplification is to replace the second time derivative of the photon dipole in

the correction term by its unperturbed value, Θ1 ≈ −c2sk2Θ1, an approximation that is valid because the

correction term is already small. Here cs is the sound speed, found in the next section to be given by

equation (24.110). The result is

vb +a

avb + kΨ = 3

(

Θ1 +a

aΘ1 +

k

)

+3Rc2sk

2

neσTaΘ1 . (24.105)

This equation (24.105) is used to develop the photon-baryon wave equation in the next section, §24.16.

24.16 Photon-baryon wave equation

Combining the photon monopole and dipole equations (24.91a) and (24.91b) with the baryon momentum

equation (24.100b), and making the diffusion approximation (24.94) for the photon quadrupole, and the

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406 ∗Cosmological perturbations: a more careful treatment of photons and baryons

approximation (24.105) for the baryon bulk velocity, yields a second order differential equation (24.113) that

captures accurately the behaviour of the photon distribution. The equation is that for a damped harmonic

oscillator, forced by the gravitational potential terms on its right hand side.

Adding the momentum equations (24.91b) and (24.100b) for photons and baryons yields an equation that

expresses momentum conservation for the combined photon-baryon fluid,

Θ1 +k

3(Θ0 − 2Θ2) +

k

3Ψ +

R

3

(

vb +a

avb + kΨ

)

= 0 . (24.106)

Setting the photon quadrupole Θ2 equal to its diffusive value (24.94), and substituting the baryon velocity

equation (24.105), brings the photon-baryon momentum conservation equation (24.106) to

Θ1 +k

3Θ0 +

8k2

27neσTaΘ1 +

k

3Ψ +R

(

Θ1 +a

aΘ1 +

k

)

+R2c2sk

2

neσTaΘ1 = 0 . (24.107)

Equation (24.107) rearranges to[

d

dη+

R

1 +R

a

a+

k2

3neσTa(1 +R)

(

8

9+

R2

1 +R

)]

Θ1 +k

3(1 +R)Θ0 +

k

3Ψ = 0 , (24.108)

where equation (24.110) has been used to replace the sound speed cs. Finally, eliminating the dipole Θ1 in

favour of the monopole Θ0 using the photon monopole equation (24.91a) yields a second order differential

equation for Θ0 − Φ:

d2

dη2+

[

R

1 +R

a

a+

k2

3neσTa(1 +R)

(

8

9+

R2

1 +R

)]

d

dη+

k2

3(1 +R)

(Θ0−Φ) = − k2

3(1 +R)[(1 +R)Ψ + Φ] .

(24.109)

Equation (24.109) is a wave equation for a damped, driven oscillator with sound speed

cs =

1

3(1 +R), (24.110)

which is the adiabatic sound speed cs for a fluid in which photons provide all the pressure, but both

photons and baryons contribute to the mass density. The term proportional to a/a on the left hand side of

equation (24.109) can be expressed as, since R ∝ a,R

1 +R

a

a= − 2

cscs

. (24.111)

Define the conformal sound time ηs by

dηs ≡ csdη , (24.112)

with respect to which sound waves move at unit velocity, unit comoving distance per unit conformal time,

dx/dηs = 1. Recast in terms of the conformal sound time ηs, the wave equation (24.109) becomes

d2

dη2s

+

[

− c′s

cs+

k2csneσTa

(

8

9+

R2

1 +R

)]

d

dηs+ k2

(Θ0 − Φ) = − k2 [(1 +R)Ψ + Φ] , (24.113)

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24.17 Damping of photon-baryon sound waves 407

where prime ′ denotes derivative with respect to conformal sound time, c′s = dcs/dηs.

The “simple” photon wave equation (23.45) derived in Chapter 23 is obtained from the wave equa-

tion (24.113) in the limit of negligible baryon-to-photon density, R ≈ 0.

24.17 Damping of photon-baryon sound waves

The terms proportional to the linear derivative d/dηs in the wave equation (24.113) are damping terms,

the first being an adiabatic damping term associated with variation of the sound speed, and the others

being dissipative damping terms associated respectively with the finite mean free path of electron-photon

scattering, and with viscous baryon drag. Lump these terms into a damping parameter κ defined by

κ ≡ 1

2

[

− c′s

cs+

k2csneσTa

(

8

9+

R2

1 +R

)]

. (24.114)

The damping parameter κ varies slowly compared to the frequency of the sound wave, so κ can be treated

as approximately constant. The homogeneous wave equation (equation (24.113) with zero on the right hand

side) can then be solved by introducing a frequency ω defined by

Θ0 − Φ ∝ eR

ω dηs . (24.115)

The homogeneous wave equation (24.113) is then equivalent to

ω′ + ω2 + 2κω + k2 = 0 . (24.116)

Since the damping parameter κ is approximately constant (and the comoving wavevector k is by definition

constant), ω′ is small compared to the other terms in equation (24.116). With ω′ set to zero, the solution of

equation (24.116) is

ω = − κ ± i√

k2 − κ2 ≈ −κ ± ik , (24.117)

where the last approximation is valid since the damping rate is small compared to the frequency, κ ≪ k.

Thus the homogeneous solutions of the wave equation (24.113) are

Θ0 − Φ ∝ e−R

κ dηs ± ikηs . (24.118)

In the present case, the first of the sources (24.114) of damping is the adiabatic damping term

κa ≡ −1

2

c′scs

. (24.119)

The integral of the adiabatic damping term is∫

κa dηs = − 12 ln cs, whose exponential is

e−R

κa dηs =√cs . (24.120)

This shows that, as the sound speed decreases thanks to the increasing baryon-to-photon density in the

expanding Universe, the amplitude of a sound wave decreases as the square root of the sound speed.

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408 ∗Cosmological perturbations: a more careful treatment of photons and baryons

The remaining damping terms are the dissipative terms

κd ≡k2cs

2neσTa

(

8

9+

R2

1 +R

)

. (24.121)

The integral of the dissipative damping terms is∫

κd dηs =k2

k2d

, (24.122)

where kd is the damping scale defined by

1

k2d

≡∫

cs2neσTa

(

8

9+

R2

1 +R

)

dηs =

1

6neσTa(1 +R)

(

8

9+

R2

1 +R

)

dη . (24.123)

The resulting damping factor is

e−R

κd dηs = e−k2/k2d . (24.124)

Thus the effect of dissipation is to damp temperature fluctuations exponentially at scales smaller than the

diffusion scale kd.

With adiabatic and diffusion damping included, the homogeneous solutions to the wave equation (24.113)

are approximately

Θ0 − Φ ∝ √cs e−k2/k2d e±ikηs . (24.125)

The driving potential on the right hand side of the wave equation (24.113) causes Θ0 − Φ to oscillate not

around zero, but rather around the offset − [(1 +R)Ψ + Φ]. At the high frequencies where damping is

important, this driving potential also varies slowly compared to the wave frequency. To the extent that the

driving potential is slowly varying, the complete solution of the inhomogeneous wave equation (24.113) is

Θ0 + (1 +R)Ψ ∝ √cs e−k2/k2d e±ikηs . (24.126)

As will be seen in Chapter 25, the monopole contribution to CMB fluctuations is not the photon monopole

Θ0 by itself, but rather Θ0+Ψ, which is the monople redshifted by the potential Ψ. This redshifted monopole

is

Θ0 + Ψ = −RΨ +A√cs e

−k2/k2d e±ikηs . (24.127)

Exercise 24.4 Diffusion scale. Show that the damping scale kd defined by (24.123) is given by, with a

normalized to aeq = 1,

H2eq

k2d

=8√

2πGmp

9cσT(1−Xn)Heq

Ωm

Ωb

∫ a

0

a2

Xe

√1 + a(1 +R)

(

8

9+

R2

1 +R

)

da , (24.128)

the Hubble parameter Heq at matter-radiation equality being related to the present-day Hubble parameter

H0 by equation (23.41). If baryons are taken to be fully ionized, Xe = 1, which ceases to be a good

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24.18 Ionization and recombination 409

approximation near recombination, then the integral can be done analytically. At times well after matter-

radiation equality,

f(a) ≡∫ a

0

a2

√1 + a(1 +R)

(

8

9+

R2

1 +R

)

da → 2

5a5/2 for a≫ 1 , (24.129)

independent of the value of the constant Ra ≡ R/a. Conclude that, neglecting the effect of recombination

on the electron fraction Xe,

H2eq

k2d

=6.821 h−1

1−Xn

H0

Heq

Ωm

Ωbf(a/aeq) = 4.9× 10−4f(a/aeq) ≈ 0.0016

(

a

a∗

)5/2

, (24.130)

where a∗ is the cosmic scale factor at recombination.

24.18 Ionization and recombination

24.19 Neutrinos

Before electron-positron annihilation at temperature T ≈ 1 MeV, weak interactions were fast enough that

scattering between neutrinos, antineutrinos, electrons, and positrons kept neutrinos and antineutrinos in

thermodynamic equilibrium with baryons. After ee annihilation, neutrinos and antineutrinos decoupled,

rather like photons decoupled at recombination. After decoupling, neutrinos streamed freely.

24.20 Summary of equations

Non-baryonic cold dark matter, baryons, photons, neutrinos:

δc − kvc = 3 Φ , (24.131a)

vc +a

avc = −kΨ , (24.131b)

δb − kvb = 3Φ , (24.131c)

vb +a

avb = − kΨ− neσTa

R(vb − 3Θ1) , (24.131d)

Θ0 − kΘ1 = Φ , (24.131e)

Θ1 +k

3(Θ0 − 2Θ2) = − k

3Ψ +

1

3neσTa (vb − 3Θ1) , (24.131f)

2k

5Θ1 = − 9

10neσTaΘ2 , (24.131g)

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410 ∗Cosmological perturbations: a more careful treatment of photons and baryons

Einstein energy and quadrupole pressure equations:

− k2Φ− 3a

aF = 4πGa2(ρcδc + ρbδb + 4ργΘ0 + 4ρνN0) , (24.132a)

k2(Ψ− Φ) = − 32πGa2(ρrΘ2 + ρνN2) , (24.132b)

24.21 Legendre polynomials

The Legendre polynomials Pℓ(µ) satisfy the orthogonality relations∫ 1

−1

Pℓ(µ)Pℓ′(µ) dµ =2

2ℓ+ 1δℓℓ′ (24.133)

and∫

Pℓ(z · a)Pℓ′(z · b) doz =4π

2ℓ+ 1Pℓ(a · b) δℓℓ′ , (24.134)

the recurrence relation

µPℓ(µ) =1

2ℓ+ 1[ℓPℓ−1(µ) + (ℓ+ 1)Pℓ+1(µ)] , (24.135)

and the derivative relationdPℓ(µ)

dµ=

ℓ+ 1

1− µ2[µPℓ−1(µ)− Pℓ+1(µ)] . (24.136)

The first few Legendre polynomials are

P0(µ) = 1 , P1(µ) = µ , P2(µ) = − 1

2+

3

2µ2 . (24.137)

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25

Fluctuations in the Cosmic MicrowaveBackground

25.1 Primordial power spectrum

Inflation generically predicts gaussian initial fluctuations with a scale-free power spectrum, in which the

variance of the potential is the same on all scales,

〈Φ(x′)Φ(x)〉 ≡ ξΦ(|x′ − x|) = constant , (25.1)

independent of separation |x′ − x|. A scale-free primordial power spectrum was originally proposed as

a natural initial condition by Harrison and Zel’dovich (Harrison, 1970, PRD 1, 2726; Zel’dovich, 1972,

MNRAS, 160, 1P), before the idea of inflation was conceived. During inflation, vacuum fluctuations generate

fluctuations in the potential which become frozen in as they fly over the horizon. The amplitude of these

fluctuations remains constant as inflation continues, producing a scale-free power spectrum.

The power spectrum PΦ(k) of potential fluctuations is defined by

〈Φ(k′)Φ(k)〉 ≡ (2π)3δD(k′ + k)PΦ(k) , (25.2)

the power spectrum PΦ(k) being related to the correlation function ξΦ(x) by (with the standard convention

in cosmology for the choice of signs and factors of 2π)

PΦ(k) =

eik·xξΦ(x) d3x , ξΦ(x) =

e−ik·xPΦ(k)d3k

(2π)3. (25.3)

The scale-free character means that the dimensionless power spectrum ∆2Φ(k) defined by

∆2Φ(k) ≡ PΦ(k)

4πk3

(2π)3(25.4)

is constant.

Actually, the power spectrum generated by inflation is not precisely scale-free, because inflation comes to

an end, which breaks scale-invariance. The departure from scale-invariance is conventionally characterized

by a scalar spectral index, the tilt n, such that

∆2Φ(k) ∝ kn−1 . (25.5)

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412 Fluctuations in the Cosmic Microwave Background

Thus a scale-invariant power spectrum has

n = 1 (scale-invariant) . (25.6)

Different inflationary models predict different tilts, mostly close to but slightly less than 1.

25.2 Normalization of the power spectrum

It is convenient to normalize the power spectrum of potential fluctuations to its amplitude at the recombi-

nation distance today, k(η0 − η∗) = 1, and not to the initial potential Φ(0) (which vanishes for isocurvature

initial conditions), but rather to the late-time matter-dominated potential Φ(late) at superhorizon scales,

∆2Φ(late)(k) = A2

late [k(η0 − η∗)]n−1 . (25.7)

The relation between the superhorizon late-time matter-dominated potential Φ(late) and the primordial

potential Φ(0) is, equations (23.61),

Φ(late) =

910Φ(0) adiabatic ,

85Φ′(0) isocurvature .

(25.8)

25.3 CMB power spectrum

The power spectrum of temperature fluctuations in the CMB is defined by

Cℓ(η0) ≡ 4π

∫ ∞

0

|Θℓ(η0,k)|2 d3k

(2π)3. (25.9)

As seen in Chapter §23, during linear evolution, scalar modes of given comoving wavevector k evolve with

amplitude proportional to the initial value Φ(0,k) of the scalar potential (or of its derivative Φ′(0,k), for

isocurvature initial conditions). The evolution of the amplitude may be encapsulated in a transfer function

Tℓ(η, k) defined by

Tℓ(η, k) ≡Θℓ(η,k)

Φ(late,k), (25.10)

where Φ(late) is the superhorizon late-time matter-dominated potential. By isotropy, the transfer function

Tℓ(η, k) is a function only of the magnitude k of the wavevector k. The power spectrum of the CMB observed

on the sky today is related to the primordial power spectrum PΦ(late)(k) or ∆2Φ(late)(k) by

Cℓ(η0) = 4π

∫ ∞

0

|Tℓ(η0, k)|2 PΦ(late)(k)4πk2dk

(2π)3= 4π

∫ ∞

0

|Tℓ(η0, k)|2 ∆2Φ(late)(k)

dk

k. (25.11)

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25.4 Matter power spectrum 413

25.4 Matter power spectrum

The matter power spectrum Pm(k) is defined by

〈δm(k′)δm(k)〉 ≡ (2π)3δD(k′ + k)Pm(k) . (25.12)

At times well after recombination, the matter power spectrum Pm(η, k) at conformal time η is related to the

potential power spectrum (25.2) by, equation (23.117), in units a0 = 1,

Pm(η, k) =

(

2a

3ΩmH20

)2

k4 PΦ(η, k) =

(

2a

3ΩmH20

)2(2π)3

4πk∆2

Φ(k, η) . (25.13)

At superhorizon scales, the potential Φ(η, k) at conformal time η is related to the late-time matter-dominated

potential by

Φ(η) = g(a)Φ(late) for kη ≪ 1 , (25.14)

where g(a) is the grown factor defined by equation (23.116). For a power-law primordial spectrum (25.5),

the matter power spectrum at the largest scales goes as

Pm(η, k) ∝ kn , (25.15)

which explains the origin of the scalar index n.

25.5 Radiative transfer of CMB photons

To determine the harmonics Θℓ(η0,k) of the CMB photon distribution at the present time, return to the

Boltzmann equation (24.90) for the photon distribution Θ(η,k, µ), where µ ≡ k · p:

Θ− ikµγΘ + neσTaΘ = Φ + ikµγΨ + neσTa[

− iµγvb + Θ0 − 12Θ2P2(µγ)

]

. (25.16)

This equation is also called the radiation transfer equation. The terms on the right are the source terms.

Define the electron-photon (Thomson) scattering optical depth τ by

dη≡ −neσTa , (25.17)

starting from zero, τ0 = 0, at zero redshift, and increasing going backwards in time η to higher redshift. The

photon Boltzmann equation (25.16) can be written

eikµγη+τ d

(

e−ikµγη−τ Θ)

= S0 − iµγS1 + (−iµγ)2S2 , (25.18)

where Si are source terms

S0 ≡ Φ− τ(

Θ0 + 14Θ2

)

, (25.19a)

S1 ≡ −kΨ− τvb = −kΨ− 3τΘ1 , (25.19b)

S2 ≡ −τ 34Θ2 , (25.19c)

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414 Fluctuations in the Cosmic Microwave Background

where in S1 the tight-coupling approximation vb = 3Θ1 has been used to replace the baryon bulk velocity

vb with the photon dipole Θ1. Thus a solution for the photon distribution Θ(η0,k, µγ) today is, at least

formally, an integral over the line of sight from the Big Bang to the present time,

Θ(η0,k, µγ) =

∫ η0

0

[

S0 − iµγS1 + (−iµγ)2S2

]

e−ikµγ (η−η0)−τ dη . (25.20)

The −iµγ dependence of the source terms inside the integral can be accomodated through

(−iµγ)ne−ikµγ (η−η0) =

(

1

k

∂η

)n

e−ikµγ (η−η0) , (25.21)

which brings the formal solution (25.20) for Θ(η0,k, µγ) to

Θ(η0,k, µγ) =

∫ η0

0

e−τ

(

S0 + S11

k

∂η+ S2

1

k2

∂2

∂η2

)

e−ikµγ (η−η0) dη . (25.22)

The dipole source term S1 contains a −Ψ term which it is helpful to integrate by parts:∫ η0

0

−e−τΨ∂

∂ηe−ikµγ (η−η0) dη =

[

−e−τΨe−ikµγ(η−η0)]η0

0+

∫ η0

0

∂η

(

e−τΨ)

e−ikµγ (η−η0) dη

= −Ψ(η0) +

∫ η0

0

e−τ(

Ψ− τΨ)

e−ikµγ (η−η0) dη . (25.23)

With the source terms written out explicitly, the present-day photon distribution Θ(η0,k, µγ), equation (25.20),

is

Θ(η0,k, µγ) + Ψ(η0,k)

=

∫ η0

0

[

e−τ(

Ψ + Φ)

− τ e−τ

(

Θ0 + Ψ +1

4Θ2 + 3Θ1

1

k

∂η+

3

4Θ2

1

k2

∂2

∂η2

)]

e−ikµγ (η−η0) dη . (25.24)

The spherical harmonics Θℓ(η,k) of the photon distribution have been defined by equation (24.48). The

e−ikµγ (η−η0) factor in the integral in equation (25.24) can be expanded in spherical harmonics through the

general formula

eik·x =∞∑

ℓ=0

iℓ(2ℓ+ 1)jℓ(kx)Pℓ(k · x) , (25.25)

where jℓ(z) ≡√

π/(2z)Jℓ+1/2(z) are spherical Bessel functions. Resolved into harmonics, equation (25.24)

becomes

[Θℓ(η0,k) + δℓ0Ψ(η0,k)]

=

∫ η0

0

[

e−τ(

Ψ + Φ)

− τ e−τ

(

Θ0 + Ψ +1

4Θ2 + 3Θ1

1

k

∂η+

3

4Θ2

1

k2

∂2

∂η2

)]

jℓ [k(η0 − η)] dη . (25.26)

Introduce a visibility function g(η) defined by

g(η) ≡ −τ e−τ , (25.27)

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25.6 Integrals over spherical Bessel functions 415

whose integral is one,∫ η0

0

g(η) dη =

∫ 0

∞−e−τ dτ =

[

e−τ]0

∞ = 1 . (25.28)

The visibility function is fairly narrowly peaked around recombination at η = η∗. In this approximation of

instantaneous recombination,

[Θℓ(η0,k) + δℓ0Ψ(η0,k)] =

∫ η0

0

e−τ(

Ψ(η,k) + Φ(η,k))

jℓ [k(η0 − η)] dη ISW

+ [Θ0(η∗,k) + Ψ(η∗,k)] jℓ [k(η0 − η∗)] monopole

− 3 Θ1(η∗,k) j′ℓ [k(η0 − η∗)] dipole

+ Θ2(η∗,k)

14jℓ [k(η0 − η∗)] + 3

4j′′ℓ [k(η0 − η∗)]

quadrupole .

(25.29)

Here prime ′ on jℓ denotes a total derivative, j′ℓ(z) = djℓ(z)/dz. The first and second derivatives of the

spherical Bessel functions are

j′ℓ(z) =ℓ

zjℓ(z)− jℓ+1(z) , j′′ℓ (z) =

ℓ(ℓ− 1)− z2

z2jℓ(z) +

2

zjℓ+1(z) . (25.30)

Notice that the monopole term (on both sides of equation (25.29)) is not Θ0 but rather Θ0 + Ψ, which

is the temperature fluctuation redshifted by the potential Ψ. On the left hand side, the δℓ0 term arises

from the redshift at our position today, but this monopole perturbation just adds to the mean unperturbed

termperature, and is not observable.

25.6 Integrals over spherical Bessel functions

Computing the photon harmonics Θl by integration of equation (25.26), or the CMB power spectrum Cℓ

in the instantaneous recombination approximation by integration of equation (25.11) with (25.29), involves

evaluating integrals of the form∫ ∞

0

f(z)g(qz)dz

z, (25.31)

with

g(z) ≡ jℓ(z)zn−1 or g(z) ≡ jℓ(z)jℓ′(z)zn−1 . (25.32)

Such integrals present a challenge because of the oscillatory character of the functions g(z). This section

presents a method to evaluate such integrals reliably. Integrals with different ℓ are related by recursion

relations (25.38) that permit rapid evaluation over many ℓ. Details are left to Exercise 25.2.

The approach is to recast the integral (25.31) into Fourier space with respect to ln z, and to apply a

Fast Fourier Transform of f(z) over a logarithmic interval. This involves replacing the true f(z) with a

function that is periodic in ln z over a logarithmic interval [−L/2, L/2] of width L centred on ln z = 0. The

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416 Fluctuations in the Cosmic Microwave Background

approximation works because the functions g(z) given by equation (25.32) tend to zero at z → 0 and z →∞,

so spurious periodic duplications at small and large z contribute negligibly to the integral (25.31) provided

that the logarithmic interval L is chosen sufficiently broad. The logarithmic interval L can be broadened to

whatever extent is necessary by extrapolating the function f(z) to smaller and larger z. If f(z) is periodic

in ln z over a logarithmic interval L, then f(z) is a sum of discrete Fourier modes e2πim ln(z)/L in which m is

integral. If f(z) is a smooth function, then f(z) may be adequately approximated by a finite number N of

discrete modes m = −[(N − 1)/2], ..., [N/2], where [N/2] denotes the largest integer greater than or equal to

N/2. Under these circumstances, the function f(z) is given by a discrete Fourier expansion whose Fourier

components fm are related to the values f(zn) at N logarithmically spaced points zn = enL/N ,

f(z) =

[N/2]∑

m=−[(N−1)/2]

fme2πim ln(z)/L , fm =

1

N

[N/2]∑

n=−[(N−1)/2]

f(zn)e−2πimn/N . (25.33)

Since f(z) is real, the negative frequency modes are the complex conjugates of the positive frequency modes,

f−m = f∗m , (25.34)

and for even N (the usual choice) the outermost (Nyquist) frequency mode fN/2 is real. For a function f(z)

given by the Fourier sum (25.33), the integral (25.31) is

∫ ∞

0

f(z)g(qz)dz

z=

[N/2]∑

m=−[(N−1)/2]

fmq−2πim/L

∫ ∞

0

g(z)z2πim/L dz

z. (25.35)

For functions g(z) given by equation (25.32), the integrals over g(z) on the right hand side of equation (25.35)

can be done analytically:

∫ ∞

0

f(z)g(qz)dz

z=

[N/2]∑

m=−[(N−1)/2]

fmq−2πim/LU

(

n− 1 +2πim

L

)

, (25.36)

where for g(z) = jℓ(z)zn−1 or g(z) = jℓ(z)jℓ′(z)z

n−1 the function U(x) is respectively

Uℓ(x) ≡∫ ∞

0

jℓ(z)zx dz

z=

2x−2√πΓ[

12 (ℓ+ x)

]

Γ[

12 (ℓ− x+ 3)

] , (25.37a)

Uℓℓ′(x) ≡∫ ∞

0

jℓ(z)jℓ′(z)zx dz

z=

2x−3πΓ(2 − x)Γ[

12 (ℓ+ ℓ′ + x)

]

Γ[

12 (ℓ + ℓ′ − x+ 4)

]

Γ[

12 (ℓ − ℓ′ − x+ 3)

]

Γ[

12 (ℓ′ − ℓ− x+ 3)

] . (25.37b)

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25.7 Large-scale CMB fluctuations 417

Recurrence relations such as

Uℓ(x) = (ℓ + x− 2)Uℓ−1(x− 1) (25.38a)

=ℓ+ x− 2

ℓ− x+ 1Uℓ−2(x) , (25.38b)

Uℓℓ′(x) =ℓ+ ℓ′ + x− 2

ℓ+ ℓ′ − x+ 2Uℓ−1,ℓ′−1(x) (25.38c)

=(ℓ+ ℓ′ + x− 2)(ℓ− ℓ′ + x− 3)

2(x− 2)Uℓ−1,ℓ′(x − 1) (25.38d)

=(ℓ+ ℓ′ + x− 2)(ℓ− ℓ′ + x− 3)

(ℓ+ ℓ′ − x+ 2)(ℓ′ − ℓ+ x− 1)Uℓ−2,ℓ′(x) , (25.38e)

permit rapid evaluation of the functions Uℓ(x) or Uℓℓ′(x) as a function of ℓ and ℓ′, starting from small ℓ, ℓ′.As written, the right hand side of equation (25.36) has a small imaginary part arising from the contribution

of the outermost (Nyquist) mode, m = [N/2], but this imaginary part should be dropped since it cancels

when averaged with the contribution of its negative frequency partner m = −[N/2].

25.7 Large-scale CMB fluctuations

The behaviour of the CMB power spectrum at the largest angular scales was first predicted by R. K. Sachs

& A. M. Wolfe (1967, Astrophys. J., 147, 73), and is therefore called the “Sachs-Wolfe effect,” though why

it should be called an effect is mysterious. The Sachs-Wolfe (SW) effect is distinct from, but modulated by,

the Integrated Sachs-Wolfe (ISW) effect. The ISW effect, ignored in this section, is considered in §25.9.

At scales much larger than the sound horizon at recombination, kηs,∗ ≪ 1, the redshifted monopole

fluctuation Θ0(η∗,k) + Ψ(η∗,k) at recombination is much larger than the dipole Θ1(η∗,k) or quadrupole

Θ2(η∗,k), so only the monopole contributes materially to the temperature multipoles Θℓ(η0,k) today. The

redshifted monopole contribution to the temperature multipoles Θℓ(η0,k) today is, from equation (25.29),

Θℓ(η0,k) = [Θ0(η∗,k) + Ψ(η∗,k)] jℓ [k(η0 − η∗)] . (25.39)

At the very largest scales, kηeq ≪ 1, the solution for the redshifted radiation monopole Θ0 + Ψ at the time

η∗ of recombination is, from equation (23.62),

Θ0(η∗,k) + Ψ(η∗,k) = 2Φsuper(η∗,k)− 3

2Φ(0)

≡ ASW(η∗)

AlateΦsuper(late,k) , (25.40)

where the last expression defines the Sachs-Wolfe amplitude ASW(η∗) at recombination. In the approximation

that recombination happens well into the matter-dominated regime, so that Φsuper(η∗,k) ≈ Φsuper(late,k),

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418 Fluctuations in the Cosmic Microwave Background

from equations (23.61),

ASW(η∗)

Alate≈ ASW(late)

Alate= 2− 3Φ(0)

2Φ(late)=

13 adiabatic ,

2 isocurvature .(25.41)

In reality, recombination occurs only somewhat into the matter-dominated regime, and the solutions for

the potential Φsuper(η∗,k) from §23.9 should be used in place of the approximation (25.41). Putting equa-

tions (25.39) and (25.40) together shows that the transfer function Tℓ(η0,k), equation (25.10) that goes into

the present-day CMB angular power spectrum Cℓ(η0), equation (25.11), is

Tℓ(η0,k) =ASW(η∗)

Alatejℓ [k(η0 − η∗)] . (25.42)

If the primordial power spectrum is a power law with tilt n, equation (25.7), then the resulting CMB angular

power spectrum is, with z = k(η0 − η∗),

Cℓ(η0) = 4πASW(η∗)2

∫ ∞

0

jℓ(z)2zn−1dz

z= 4πASW(η∗)

2Uℓ,ℓ(n− 1) , (25.43)

where Uℓ,ℓ(x) is given by equation (25.37b). For the particular case of a scale-invariant primordial power

spectrum, n = 1, the CMB power spectrum Cℓ at large scales today is given by

ℓ(ℓ+ 1)Cℓ(η0) = 2πASW(η∗)2 if n = 1 . (25.44)

Thus the characteristic feature of a scale-invariant primordial power spectrum, n = 1, is that ℓ(ℓ + 1)Cℓ

should be approximately constant at the largest angular scales, ℓ ≪ η0/η∗. This is a primary reason why

CMB folk routinely plot ℓ(ℓ+ 1)Cℓ, rather than Cℓ.

25.8 Monopole, dipole, and quadrupole contributions to Cℓ

At smaller scales, kη∗ >∼ 1, not only the photon monopole Θ0(η∗, k), but also the dipole Θ1(η∗, k), and to

a small extent the quadrupole Θ2(η∗, k), contribute to the temperature multipoles Θℓ(η0, k) today, equa-

tion (25.29). The dipole is related to the monopole by the evolution equation (??) for the monopole.

kΘ1 = Θ0 − Φ . (25.45)

25.9 Integrated Sachs-Wolfe (ISW) effect

Concept question 25.1 Cosmic Neutrino Background. Just as photons decoupled at recombina-

tion, so also neutrinos decoupled at electron-positron annihilation. Compare qualitatively the expected

fluctuations in the CνB to those in the CMB.

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25.9 Integrated Sachs-Wolfe (ISW) effect 419

Exercise 25.2 Numerical integration of sequences of integrals over Bessel functions. Write code

that solves integrals (25.31) numerically for g(z) given by equation (25.32), using a Fast Fourier Transform,

equation (25.36), amd recurrence relations appropriate for the monopole, dipole, and quadrupole contribu-

tions to equations (25.26) or (25.29). To compute Uℓ(x) or Uℓℓ′(x) for the initial ℓ, ℓ′, you will need to find

code that implements the complex Gamma function. Note that most FFT codes store input and output

periodic sequences shifted by [N/2] compared to the convention (25.33). That is, an FFT code typically

takes an input sequence ordered as f(z0), f(z1), ..., f(z[N/2]), ..., f(z−2), f(z−1), with periodic identification

f(zn) = f(zn+N), and evaluates Fourier coefficients fm as

fm =1

N

N−1∑

m=0

f(zn)e−2πimn/N , (25.46)

which yields the same Fourier components fm as (25.33) but in the order f0, f1, ..., f[N/2], ..., f−2, f−1, with

periodic identification fm = fm+N .

Various recurrence relations. For g(z) = jℓ(z)zn−1, the dipole and quadrupole integrals U

(1)ℓ (x) and

U(2)ℓ (x) are related to the monopole integral Uℓ(x) (25.37a) by

U(1)ℓ (x) ≡

∫ ∞

0

j′ℓ(z)zx dz

z= (1− x)Uℓ(x− 1) , (25.47a)

U(2)ℓ (x) ≡

∫ ∞

0

[

14jℓ(z) + 3

4j′′ℓ (z)

]

zx dz

z=

ℓ(ℓ+ 1) + 2x(x− 2)

(ℓ+ x− 2)(ℓ− x+ 3)Uℓ(x) . (25.47b)

The dipole, and quadrupole integrals satisfy the recurrence relations

U(1)ℓ (x) =

ℓ+ x− 3

ℓ− x+ 2U

(1)ℓ−2(x) , (25.48a)

U(2)ℓ (x) =

(ℓ + x− 4) [l(l + 1) + 2x(x− 2)]

(ℓ − x+ 3) [l(l − 3) + 2(x− 1)2]U

(2)ℓ−2(x) . (25.48b)

For g(z) = jℓ(z)jℓ′(z)zn−1, the relations get a bit ugly — it’s a good idea to use Mathematica or a similar

program to generate the relations automatically. The various multipole integrals of interest are related to

Page 432: General Relativity, Black Holes, And Cosmology

420 Fluctuations in the Cosmic Microwave Background

the integral Uℓℓ(x) (25.37b) by

U(0,1)ℓℓ (x) ≡

∫ ∞

0

jℓ(z)j′ℓ(z)z

x dz

z=

1− x2

Uℓℓ(x− 1) , (25.49a)

U(1,1)ℓℓ (x) ≡

∫ ∞

0

[j′ℓ(z)]2zx dz

z=

4ℓ(ℓ+ 1)− x(x− 2)(x− 3)

(3 − x)(2ℓ+ x− 2)(2ℓ− x+ 4)Uℓℓ(x) , (25.49b)

U(0,2)ℓℓ (x) ≡

∫ ∞

0

jℓ(z)[

14jℓ(z) + 3

4j′′ℓ (z)

]

zx dz

z=

x [2ℓ(ℓ+ 1) + (x− 1)(x− 2)]

2(x− 3)(2ℓ+ x− 2)(2ℓ− x+ 4)Uℓℓ(x) , (25.49c)

U(1,2)ℓℓ (x) ≡

∫ ∞

0

j′ℓ(z)[

14jℓ(z) + 3

4j′′ℓ (z)

]

zx dz

z(25.49d)

=(1− x) [2ℓ(ℓ+ 1)(x− 7) + (x+ 1)(x− 3)(x− 4)]

4(x− 4)(2ℓ+ x− 3)(2ℓ− x+ 5)Uℓℓ(x− 1) ,

U(2,2)ℓℓ (x) ≡

∫ ∞

0

[

14jℓ(z) + 3

4j′′ℓ (z)

]2zx dz

z= (25.49e)

4(ℓ− 1)ℓ(ℓ+ 1)(ℓ+ 2) [12 + x(x − 2)] + x(x − 2)(x− 5) [4ℓ(ℓ+ 1)(x− 6) + (x+ 2)(x− 3)(x− 4)]

4(x− 3)(x− 5)(2ℓ+ x− 2)(2ℓ+ x− 4)(2ℓ− x+ 4)(2ℓ− x+ 6)Uℓℓ(x) .

The various recurrence relations of interest are

U(0,1)ℓℓ (x) =

(2ℓ+ x− 3)

(2ℓ− x+ 3)U

(0,1)ℓ−1,ℓ−1(x) , (25.50a)

U(1,1)ℓℓ (x) =

(2ℓ+ x− 4) [4ℓ(ℓ+ 1)− x(x − 2)(x− 3)]

(2ℓ− x+ 4) [4ℓ(ℓ− 1)− x(x − 2)(x− 3)]U

(1,1)ℓ−1,ℓ−1(x) , (25.50b)

U(0,2)ℓℓ (x) =

(2ℓ+ x− 4) [2ℓ(ℓ+ 1) + (x− 1)(x− 2)]

(2ℓ− x+ 4) [2ℓ(ℓ− 1) + (x− 1)(x− 2)]U

(0,2)ℓ−1,ℓ−1(x) , (25.50c)

U(1,2)ℓℓ (x) =

(2ℓ+ x− 5) [2ℓ(ℓ+ 1)(x− 7) + (x+ 1)(x− 3)(x− 4)]

(2ℓ− x+ 5) [2ℓ(ℓ− 1)(x− 7) + (x+ 1)(x− 3)(x− 4)]U

(1,2)ℓ−1,ℓ−1(x) , (25.50d)

U(2,2)ℓℓ (x) =

(2ℓ+ x− 6)

(2ℓ− x+ 6)U

(2,2)ℓ−1,ℓ−1(x) (25.50e)

× 4(ℓ− 1)ℓ(ℓ+ 1)(ℓ + 2) [12 + x(x− 2)] + x(x− 2)(x− 5) [4ℓ(ℓ+ 1)(x− 6) + (x+ 2)(x− 3)(x− 4)]

4(ℓ− 1)ℓ(ℓ+ 1)(ℓ − 2) [12 + x(x− 2)] + x(x− 2)(x− 5) [4ℓ(ℓ− 1)(x− 6) + (x+ 2)(x− 3)(x− 4)].


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