An Introduction to General Relativity - Pablo...

Gravitation: Tensor CalculusAn Introduction to General Relativity

Pablo Laguna

Center for Relativistic AstrophysicsSchool of Physics

Georgia Institute of Technology

Notes based on textbook: Spacetime and Geometry by S.M. CarrollSpring 2013

Pablo Laguna Gravitation: Tensor Calculus

Manifolds

Gravity as geometry

Manifolds

Vectors and Tensors

The Metric

Causality

Tensor Densities

Differential Forms

Integration


Gravity as Geometry

According to Einstein:

The metric tensor describing the curvature of spacetime is thedynamical field responsible for gravitation.

Gravity is not a field propagating through spacetime.

Gravitational interactions are universal (Principle of equivalence)


Weak Principle of Equivalence (WEP)

The inertial mass and the gravitational mass of any object are equal

~F = mi ~a~Fg = −mg∇Φ

with mi and mg the inertial and gravitational masses, respectively.

According to the WEP: mi = mg for any object. Thus, the dynamics ofa free-falling, test-particle is universal, independent of its mass; thatis, ~a = −∇Φ

Weak Principle of Equivalence (WEP)

The motion of freely-falling particles are the same in a gravitationalfield and a uniformly accelerated frame, in small regions of spacetime


Einstein Equivalence Principle

In small regions of spacetime, the laws of physics reduce to those ofspecial relativity; it is impossible to detect the existence of agravitational field by means of local experiments.

Due to the presence of the gravitational field, it is not possible tobuilt, as in SR, a global inertial frame that stretches throughspacetime.

Instead, only locally inertial frames are possible; that is, inertialframes that follow the motion of individual free-falling particles ina small enough region of spacetime.

Spacetime is a mathematical structure that locally looks likeMinkowski or flat spacetime, but may posses nontrivial curvatureover extended regions.


Manifolds

Generally speaking, a manifold is a space that with curvature andcomplicated topology that locally looks like Rn.

Examples:

Rn itself. R is a line and R2 a plane.

The n-sphere, Sn; that is, the locus of all points some fixeddistance from the origin in Rn+1. S1 is a circle and S2 sphere.

The n-torus T n. T 2 is the surface of a doughnut.

A Riemann surface of genus g. That is, a n-torus with g holes.

A set of continuous transformations such as rotations in Rn.

The direct product of two manifolds is a manifold.


Manifolds

identify opposite sides

genus 0 genus 1 genus 2


Not manifolds


Map: Given two sets M and N, a map φ : M → N is arelationship which assigns, to each element of M, exactly oneelement of N.

Composition: Given two maps φ : A→ B and ψ : B → C, wedefine the composition ψ ◦ φ : A→ C by the operation(ψ ◦ φ)(a) = ψ(φ(a)). So a ∈ A, φ(a) ∈ B, and thus(ψ ◦ φ)(a) ∈ C.

One-to-one map: A map φ : M → N such that each element of Nhas at most one element of M mapped into it.

Onto map: A map φ : M → N such that each element of N hasat least one element of M mapped into it.


Examples: Consider a function φ : R→ R. Then φ(x) = ex isone-to-one, but not onto; φ(x) = x3 − x is onto, but not one-to-one;φ(x) = x3 is both; and φ(x) = x2 is neither.

Given a map φ : M → N, the set M is known as the domain ofthe map φ, and the set of points in N which M gets mapped intois called the image of φ.

For some subset U ⊂ N, the set of elements of M which getmapped to U is called the preimage of U under φ, or φ−1(U).

A map which is both one-to-one and onto is known as invertibleand there exists a corresponding inverse map φ−1 : N → M by(φ−1 ◦ φ)(a) = a.


Consider the maps φ : Rm → Rn that takes an m-tuple(x1, x2, . . . , xm) to an n-tuple (y1, y2, . . . , yn) such that:

y1 = φ1(x1, x2, . . . , xm)y2 = φ2(x1, x2, . . . , xm)

···

yn = φn(x1, x2, . . . , xm)

The functions φi are Cp if they are continuous and p-timesdifferentiable.

The entire map φ : Rm → Rn is Cp if each of its componentfunctions are at least Cp.


A C0 map is continuous but not necessarily differentiable.

A C∞ or smooth map is continuous and can be differentiated asmany times as one likes.

Diffeomorphisms

Two sets M and N are diffeomorphic if there exists a C∞ mapφ : M → N with a C∞ inverse φ−1 : N → M; the map φ is then calleda diffeomorphism.


Open ball: The set of all points x ∈ Rn such that |x − y | < r forsome fixed y ∈ Rn and r ∈ R.

Open set: in Rn is a set constructed from an arbitrary union ofopen balls. In other words, V ⊂ Rn is open if, for any y ∈ V ,there is an open ball centered at y which is completely inside V .

A chart or coordinate system consists of a subset U of a set M,along with a one-to-one map φ : U → Rn, such that the imageφ(U) is open in R.


A C∞ atlas is an indexed collection of charts {(Uα, φα)} such that

1 The union of the Uα is equal to M

2 The charts are smoothly sewn together. That is, if two chartsoverlap, Uα ∩ Uβ 6= ∅, then the map (φα ◦ φ−1

β ) takes points inφβ(Uα ∩Uβ) ⊂ Rn onto φα(Uα ∩Uβ) ⊂ Rn, and all of these mapsmust be C∞ where they are defined.

U!

" ( )

" ( )

"

"

" "

" "# !

! #

!

#U#

U!

!

#

U#

-1

-1these maps are onlydefined on the shadedregions, and must besmooth there.

M

R

R

n

n


ManifoldA C∞ n-dimensional manifold is a set M along with a “maximal atlas”,one that contains every possible compatible chart.

The requirement of a maximal atlas is needed so two equivalentspaces with different atlases do not count as different manifolds.

Notice that the existence of a manifold does not depend on anembedding.


Most manifolds cannot be covered by a single chart.

U

S1

2

U1

x

x

x = -1

x1

2

3

(y , y )1 23

(x , x , x )1 32


Chain Rule

Consider two maps f : Rm → Rn and g : Rn → Rl , and theircomposition (g ◦ f ) : Rm → Rl

and each space in terms of coordinates: xa ∈ Rm, yb ∈ Rn, andzc ∈ Rl .

Chain rule:∂

∂xa (g ◦ f )c =∑

b

∂f b

∂xa∂gc

∂yb .

or∂

∂xa =∑

b

∂yb

∂xa∂

∂yb .


Vectors

Consider:

The set of all parameterized curves γ(λ) through p, i.e. all mapsγ : R→ M such that p is in the image of γ.

The space F of all the C∞ smooth maps f : M → R.

Tangent Space Tp

Each curve γ(λ) through p defines a directional derivative operator,which maps f → df/dλ. Tp is the space of directional derivativeoperators along curves through p


Vectors

Notice:

ddξ

= ad

dλ+ b

ddη

also they obey the Leibniz rule

(a

ddλ

+ bddη

)(fg) = af

dgdλ

+ agdfdλ

+ bfdgdη

+ bgdfdη

=

(a

dfdλ

+ bdfdη

)g +

(a

dgdλ

+ bdgdη

)f


Tp basis

Given a coordinate chart {xµ} in an n-dimensional manifold M, thereis a set of n directional derivatives at p given by the partial derivatives∂µ at p.

p

1

!

2

!

x x2

1


Tp basis{∂µ} at p form a basis for the tangent space Tp.

Proof: Consider a coordinate chart φ : M → Rn, a curve γ : R→ M,and a function f : M → R such that

f -1 !

!!-1

fM

RR

"

! "

f "

xµ

Rn


Tp basis

f -1 !

!!-1

fM

RR

"

! "

f "

xµ

Rn

Let λ be parameter along γ, then

ddλ

f =d

dλ(f ◦ γ) =

ddλ

[(f ◦ φ−1) ◦ (φ ◦ γ)]

=d(φ ◦ γ)µ

dλ∂(f ◦ φ−1)

∂xµ=

dxµ

dλ∂µf


Tp basis

Thus,d

dλf =

dxµ

dλ∂µf

implies that ∂µ is a good coordinate basis for the tangent space Tp

Coordinate Basis

ddλ

=dxµ

dλ∂µ ⇒ e(µ) = ∂µ


Vector transformation law

Given∂µ′ =

∂xµ

∂xµ′∂µ .

then

Vµ∂µ = Vµ′∂µ′

= Vµ′ ∂xµ

∂xµ′∂µ ,

so

Vµ′ =∂xµ

′

∂xµVµ

Lorentz transformations Vµ′ = Λµ′µVµ are a special case.


Commutators

[X ,Y ](f ) = X (Y (f ))− Y (X (f ))

[X ,Y ](a f + b g) = a[X ,Y ](f ) + b[X ,Y ](g)

[X ,Y ](f g) = f [X ,Y ](g) + g[X ,Y ](f )

[X ,Y ]µ = Xα∂αYµ − Yα∂αXµ


One-forms

A Cotangent space T ∗p is the set of linear maps ω : Tp → R.

A one-form df is the gradient of a function f .

Action of df on a vector d/dλ:

df(

ddλ

)=

dfdλ

.

Recall θ(µ)(e(ν)) = δµν , thus

dxµ(∂ν) =∂xµ

∂xν= δµν

{dxµ} are a set of basis one-forms; that is, ω = ωµ dxµ.

Transformation properties:

dxµ′

=∂xµ

′

∂xµdxµ ,

andωµ′ =

∂xµ

∂xµ′ωµ .


Tensors

A (k , l) tensor T can be expanded

T = Tµ1···µkν1···νl∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl ,

Under a coordinate transformation the components of T changeaccording to

Tµ′1···µ′kν′1···ν

′l

=∂xµ

′1

∂xµ1· · · ∂xµ

′k

∂xµk

∂xν1

∂xν′1· · · ∂xνl

∂xν′lTµ1···µk

ν1···νl .

The partial derivative of a tensor is not a new tensor

∂

∂xµ′Wν′ =

∂xµ

∂xµ′∂

∂xµ

(∂xν

∂xν′Wν

)=

∂xµ

∂xµ′∂xν

∂xν′

(∂

∂xµWν

)+ Wν

∂xµ

∂xµ′∂

∂xµ∂xν

∂xν′


The Metric: gµν

The metric gµν :

(0,2) tensor,

gµν = gνµ (symmetric)

g = |gµν | 6= 0 (non-degenerate)

gµν (inverse metric)

gµν is symmetric and gµνgνσ = δµσ .

gµν and gµν are used to raise and lower indices on tensors.


gµν properties

The metric:

provides a notion of “past” and “future”

allows the computation of path length and proper time:ds2 = gµν dxµ dxν

determines the “shortest distance” between two points

replaces the Newtonian gravitational field

provides a notion of locally inertial frames and therefore a senseof “no rotation”

determines causality, by defining the speed of light faster thanwhich no signal can travel

replaces the traditional Euclidean three-dimensional dot productof Newtonian mechanics


gµν canonical formCanonical form:

gµν = diag (−1,−1, . . . ,−1,+1,+1, . . . ,+1,0,0, . . . ,0)

If gµν is the metric in a n-dimension of the manifold M, and s isthe number of +1’s in the canonical form, and t is the number of−1’s, then s − t is the signature of gµν and s + t rank.

If gµν is nondegenerate, the rank is equal to the dimension n.

If gµν is continuous, the rank and signature of the metric tensorfield are the same at every point.

If all of the signs are positive (t = 0), gµν is called Euclidean orRiemannian or positive definite

If there is a single minus (t = 1), gµν is called Lorentzian orpseudo-Riemannian,

If gµν with some +1’s and some −1’s is called indefinite

The spacetimes of interest in general relativity have Lorentzianmetrics.


Riemann normal coordinates: At any point p there exists a coordinatesystem such that:

gµν takes its canonical form

the first derivatives ∂σgµν all vanish

the second derivatives ∂ρ∂σgµν cannot be made in general to allvanish

the associated basis vectors constitute a local Lorentz frame.

Notice: In Riemann normal coordinates, gµν at p looks, to first order,like the flat or Minkowski metric ηµν . That is, in a small enoughregions, the spacetime looks like flat or Minkowski space (localflatness theorem).


Proof: local flatness theorem

gµ′ν′ =∂xµ

∂xµ′∂xν

∂xν′gµν

Taylor expand both sides with xµ(p) = xµ′ (p) = 0

xµ =

(∂xµ

∂xµ′

)p

xµ′ +1

2

(∂2xµ

∂xµ′1∂xµ′2

)p

xµ′1 xµ′2

+1

6

(∂3xµ

∂xµ′1∂xµ′2∂xµ′3

)p

xµ′1 xµ′2 xµ′3 + · · · ,

Thus to second order

(g′)

p+(∂′g′)

px′ +

(∂′∂′g′)

px′x′

=

(∂x

∂x′∂x

∂x′g)

p+

(∂x

∂x′∂2x

∂x′∂x′g +

∂x

∂x′∂x

∂x′∂′g

)p

x′

+

(∂x

∂x′∂3x

∂x′∂x′∂x′g +

∂2x

∂x′∂x′∂2x

∂x′∂x′g +

∂x

∂x′∂2x

∂x′∂x′∂′g +

∂x

∂x′∂x

∂x′∂′∂′g

)p

x′x′ .


Proof: local flatness theorem

(g′)

p+(∂′g′)

px′ +

(∂′∂′g′)

px′x′

=

(∂x

∂x′∂x

∂x′g)

p+

(∂x

∂x′∂2x

∂x′∂x′g +

∂x

∂x′∂x

∂x′∂′g

)p

x′

+

(∂x

∂x′∂3x

∂x′∂x′∂x′g +

∂2x

∂x′∂x′∂2x

∂x′∂x′g +

∂x

∂x′∂2x

∂x′∂x′∂′g +

∂x

∂x′∂x

∂x′∂′∂′g

)p

x′x′ .

16 numbers in (∂xµ/∂xµ′ )p to bring gµ′ν′ (p) into a canonical form

40 numbers in (∂2xµ/∂xµ′1∂xµ′2 )p to zero out the 40 components in ∂σ′gµ′ν′ (p)

80 number in (∂3xµ/∂xµ′1∂xµ′2∂xµ′3 )p to zero out 80 of the 100 components in ∂ρ′∂σ′gµ′ν′ (p)

Thus, the deviation from flatness is measured by 20 coordinate-independent degrees of freedom representing thesecond derivatives of the metric tensor field (Riemann curvature tensor).


Causality

Initial-value problem or formulation: Given the appropriate initialdata or state state of a system, the subsequent dynamicalevolution of the system is uniquely determined.

Causality: Future events are a consequence of past events.

Fundamental principle: no signals can travel faster than thespeed of light.

Let (M,gµν) be a spacetime, if a continuous choice of future andpast is possible as one varies p in M, one says that M is timeorientable.

Lema: Let (M,gµν) be time orientable, then there exists asmooth non-vanishing tim-elike vector field tµ on M.


Causality

Causal curve: a curve γ that at every point p ∈ γ has a tangenttµ that is time-like or null.

Causal future J+(S): Given any S ⊂ M, the causal future of S isgiven by J+(S) ⊂ M that can be reached from S by following afuture-directed causal curves.

Chronological future I+(S): Given any S ⊂ M, the causal futureof S is given by I+(S) ⊂ M that can be reached from S byfollowing a future-directed time-like curves.

Notice: I+(S) ⊂ J+(S)

The causal past J−(S) and chronological past I−(S) are definedanalogously.


Causality

Achronal: A subset S ⊂ M is called achronal if no two points inS are connected by a time-like curve.

Future domain of dependence D+(S): Given a closed achronalset S, D+(S) is the set of points p ∈ M such that every pastdirected inextendible (goes on forever) causal curve through pintersects S.

Future Cauchy horizon H+(S): The boundary of D+(S) (nullsurface). Notice: S ⊂ D+(S) ⊂ J+(S)


Causality

Information at S is sufficient to predict the situation in p.

Domain of dependence: D(S) = D+(S) ∪ D−(S)

Cauchy surface Σ: A closed achronal surface Σ is said to be aCauchy surface if the domain of dependence is the entiremanifold, i.e. D(Σ) = M

A space-time (M,gµν) which possesses a Cauchy surface issaid to be globally hyperbolic.


Closed Timelike Curves

Closed Timelike Curve: A forward directed curve that is everywheretime-like and intersects itself.

Example: Consider the 2-dimensional spacetime (M,gµν) withcoordinates {t , x} with topology R× S1; that is, with coordinates (t , x)and (t , x + 1) identified, one can show that the metric

ds2 = − cos (λ) dt2 − 2 sin (λ) dt dx + cos (λ) dx2

with λ = cot−1 t has closed time-like curves for t > 0 and a Cauchyhorizon surface at t = 0.


Tensor Densities

Levi-Civita symbol:

εµ1µ2···µn =

+1 if µ1µ2 · · ·µn is an even permutation of 01 · · · (n − 1) ,−1 if µ1µ2 · · ·µn is an odd permutation of 01 · · · (n − 1) ,0 otherwise .

It has the same components in any coordinate system

It is not a tensor since it does not to change under coordinatetransformations.


Tensor Densities

Given some n × n matrix Mµµ′ , the determinant |M| obeys

εµ′1µ′2···µ′n |M| = εµ1µ2···µn Mµ1µ′1

Mµ2µ′2· · ·Mµn

µ′n .

If Mµµ′ = ∂xµ

∂xµ′ ,

εµ′1µ′2···µ′n =

∣∣∣∣∣∂xµ′

∂xµ

∣∣∣∣∣ εµ1µ2···µn

∂xµ1

∂xµ′1∂xµ2

∂xµ′2· · · ∂xµn

∂xµ′n.

Notice: It transforms almost as a tensor. Objects which transform inthis way are known as tensor densities.


Tensor Densities

gµ′ν′ =∂xµ

∂xµ′∂xν

∂xν′gµν

by taking the determinant in both sides one gets

g(xµ′) =

∣∣∣∣∣∂xµ′

∂xµ

∣∣∣∣∣−2

g(xµ) .

thus, g is not a tensor.

Tensor density weight: The weight of a density is given by the powerof the Jacobian. E.g. the Levi-Civita symbol is a tensor density ofweight 1 and the determinant of the metric g is a scalar density ofweight -2.


Levi-Civita tensor

Given the Levi-Civita symbol, we can then define

Levi-Civita tensor:

εµ1µ2···µn =√|g| εµ1µ2···µn .

which will transform like a tensor.


Differential Forms

Differential forms: A differential p-form is a (0,p) tensor which iscompletely antisymmetric.

Examples:

Scalars are 0-forms

Dual vectors are 1-forms

The electromagnetic tensor Fµν is a 2-form

εµνρσ is a 4-form.


Differential Forms

Λp(M) is the space of all p-form fields over a manifold M.

The number of linearly independent p-forms on ann-dimensional vector space is n!/(p!(n − p)!).

Thus, in a 4-dimensional spacetime there is one linearlyindependent 0-form, four 1-forms, six 2-forms, four 3-forms, andone 4-form. There are no p-forms for p > n.


Wedge Product: Given a p-form A and a q-form B, the wedge productA ∧ B as

(A ∧ B)µ1···µp+q =(p + q)!

p! q!A[µ1···µp Bµp+1···µp+q ] .

the result is a (p + q)-form.

Example:(A ∧ B)µν = 2A[µBν] = AµBν − AνBµ .

Notice:A ∧ B = (−1)pqB ∧ A ,

so you can alter the order of a wedge product if you are careful withsigns.


Exterior Derivative:

(dA)µ1···µp+1 = (p + 1)∂[µ1Aµ2···µp+1] .

the result is a (p + 1)-form and thus a tensor.

Example:(dφ)µ = ∂µφ .

d(dA) = 0 ,

for any form A, a consequence that ∂α∂β = ∂β∂α.

Property:d(ω ∧ η) = (dω) ∧ η + (−1)pω ∧ (dη)

with ω a p-form and η a q-form.


Hodge duality: Hodge star operator on an n-dimensional manifold asa map from p-forms to (n − p)-forms,

(∗A)µ1···µn−p =1p!εν1···νp

µ1···µn−p Aν1···νp ,

The Hodge dual does depend on the metric of the manifold

If s is the number of minus signs in the eigenvalues of the metric,

∗ ∗ A = (−1)s+p(n−p)A ,

If A(n−p) is an (n − p)-form and B(p) is a p-form,

∗(A(n−p) ∧ B(p)) ∈ R .


Cross Product

If we restrict∗(A(n−p) ∧ B(p)) ∈ R .

to the case of 3-dimensional Euclidean space, we get

∗(U ∧ V )i = εijk UjVk .

which is the conventional cross product.

It only exists in 3-dimensions — because only in 3-dimensions do wehave a map like this from two dual vectors to a third dual vector.


E&M revised

∂µF νµ = 4πJν

∂[µFνλ] = 0

thus

dF = 0 .

There must therefore be a one-form Aµ such that

F = dA .

The one-form Aµ is the familiar vector potential of electromagnetism,

Gauge invariance: The theory is invariant under A→ A + dλ forsome scalar (zero-form) λ.

The inhomogeneous Maxwell’s equations are given by

d(∗F ) = 4π(∗J) ,


Integration

In ordinary calculus on Rn the volume element dnx transforms as

dnx ′ =

∣∣∣∣∣∂xµ′

∂xµ

∣∣∣∣∣dnx .

Identifydnx ↔ dx0 ∧ · · · ∧ dxn−1 .

thusdx0 ∧ · · · ∧ dxn−1 =

1n!εµ1···µn dxµ1 ∧ · · · ∧ dxµn ,

under a coordinate transformation, we have that

εµ1···µn dxµ1 ∧ · · · ∧ dxµn = εµ1···µn

∂xµ1

∂xµ′1· · · ∂xµn

∂xµ′ndxµ

′1 ∧ · · · ∧ dxµ

′n

=

∣∣∣∣ ∂xµ

∂xµ′

∣∣∣∣ εµ′1···µ′n dxµ′1 ∧ · · · ∧ dxµ

′n .

therefore volume element dnx transforms as a density, not a tensor.


Invariant Volume Element :√|g|dnx =

√|g|dx0 ∧ · · · ∧ dxn−1 =

√|g′|dx0′ ∧ · · · ∧ dx (n−1)′

Integral: of a scalar function φ(x) in a n-dimensional manifold M

I =

∫φ(x)

√|g|dnx

Stokes Theorem: Consider n-manifold M with boundary ∂M, and an(n − 1)-form ω on M, then ∫

Mdω =

∫∂M

ω .


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