Weak-Field General Relativity Compared with Electrodynamics

Wendeline B. Everett

Oberlin College

Department of Physics and Astronomy

Advisor: Dan Styer

May 15, 2007

Contents

1 Introduction and Overview 3

2 Introduction to Tensors, Spacetime, and Relativity 6

2.1 Vectors, Dual Vectors, and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Tensor Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Maxwell’s Equations in Tensor Form 8

4 Introduction to General Relativity and the Einstein Equation 11

4.1 Vectors, Dual Vectors, and Tensors in Curved Spacetime . . . . . . . . . . . . . . . . . . . . . 12

4.2 The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Covariant Derivatives and the Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Riemann, Ricci, and Einstein Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.5 Minimal Coupling Principle and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Linearized Gravity 23

5.1 Writing the Metric using a Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 Solving Einstein’s Equation for Weak Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Weak-Field Metric Analysis 27

6.1 Verification of Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Calculating Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Riemann Tensor Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.4 Calculating Riemann, Ricci, and Einstein Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Geodesic Equation for a Weak-Field Metric 37

8 Conclusions 38

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9 Appendix A: Stress-Energy Tensor 41

10 Appendix B: Electromagnetism with Differential Forms 43

11 Acknowledgments 47

2

1 Introduction and Overview

We can draw many parallels between electromagnetism and general relativity, such as, for example, the fact

that both electrostatic and gravitostatic forces fall off as 1/r2. These parallels become especially clear when

we compare the two in tensor notation. However, while Maxwell’s equations in electromagnetism have a rela-

tively straight-forward, intuitive understanding in terms of electric and magnetic field components, a similar

intuitive understanding of general relativity is not so easily forthcoming. In contrast to electromagnetism,

the equations of general relativity are non-linear and involve many more independent degrees of freedom.

These factors complicate analysis. Furthermore, general relativity is generally studied in the language of

geometry. While other forces of nature, such as electromagnetism and the short-range nuclear forces, are

typically thought of as fields on spacetime, gravitational fields are usually analyzed as the curvature of space-

time itself. However, can we analyze general relativity in the familiar field-theoretic manner and from this

gain a more intuitive understanding through parallels with these other theories?

For electromagnetism the sources are charges and currents (the four-current), and these are related to

the fields using Maxwell’s equations. For gravity, the sources are energies and momenta (the stress-energy

tensor), which are related to the curvature of spacetime using the Einstein equation. To see the parallels

between the two theories emerge, we need to write the corresponding equations in the same form, namely,

tensor formulation. Maxwell’s equations can be written in tensor form (in flat space) as

∂µFµν = Jν (1)

∂µGµν = 0. (2)

where Fµν is the field tensor given by

Fµν =

0 −E1/c −E2/c −E3/c

E1/c 0 B3 −B2

E2/c −B3 0 B1

E3/c B2 −B1 0

(3)

and Gµν is the dual field tensor given by

Gµν =

0 B1/c B2/c B3/c

−B1/c 0 −E3 E2

−B2/c E3 0 −E1

−B3/c −E2 E1 0

. (4)

[If we assume we already know the structure of the field and dual field tensors (the positions of the

electric and magnetic components in the tensors), then writing Maxwell’s equations in tensorial form is

straightforward. If we want to delve deeper into the structure of these equations, we turn to the mechanics

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of differential forms which deals with covariant, totally antisymmetric tensors which the field and dual field

tensors are, and this is treated in Appendix B of the paper.]

The analogous equation in general relativity is the Einstein equation:

Gµν = Rµν −12Rgµν = 8πGTµν (5)

where Rµν is the Ricci tensor, R the Ricci scalar, which are both formed from contractions of the Riemann

tensor, which, given in terms of the the Christoffel symbols Γ, is

Rρσµν = ∂µΓρ

νσ − ∂νΓρµσ + Γρ

µλΓλνσ − Γρ

νλΓλµσ (6)

and the Christoffel symbols are formed from the metric gµν (which characterizes the underlying geometry of

a space) by

Γσµν =

12gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν). (7)

Essentially, the Christoffel symbols allow a conversion from structures in flat, Minkowski spacetime of special

relativity to curved spacetime of general relativity. They are necessary because the ordinary partial derivative

in an arbitrary spacetime is no longer linear (tensorial), and therefore the Christoffel symbols serve as a

correction factor to make tensorial derivatives in curved spacetime possible (the covariant derivative).

In Einstein’s equation, Tµν is the stress-energy tensor, also referred to as the energy-momentum tensor,

which is the generalization of the mass density from Newtonian gravity, and G is Newton’s gravitational

constant. (An analysis that parallels the structure of electromagnetism for the stress-energy tensor is given

in Appendix A.)

What we’d like to do is to look at the structure of Einstein equation in terms of the gravitational fields

rather than the curvature of the underlying spacetime. However, it turns out that Einstein’s equation in its

full form is a bit unwieldy (ten nonlinear partial differential equations). Therefore, we turn to the limit in

which Einstein’s equation becomes linear which does have a general solution. In the linearized limit, we can

write the metric as a sum of the Minkowski (flat) metric and a small perturbation term which we use only

to the first order because we restrict ourselves to only weak gravitational fields.

gµν = ηµν + hµν (8)

The weak-field limit can be used to find the weak-field metric given by

ds2 = −(1 + 2Φ)dt2 + (1− 2Φ)(dx2 + dy2 + dz2) = diag(−2Φ,−2Φ,−2Φ,−2Φ) (9)

where the gravitational potential, Φ, obeys the Newtonian Poisson equation.

For our analysis, we use the weak-field metric but we consider that the potential can vary with time,

instead of the usual Newtonian gravitational potential which is a function of only the space coordinates.

Therefore by comparison of eqns. 8 and 9 we see that we can write our perturbation hµν as

hµν = −2Φ(x0, x1, x2, x3)δµν . (10)

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Since we’re interested in the gravitational fields more than the potentials we define

fµ(x0, x1, x2, x3) = − ∂Φ∂xµ

(11)

which for the three space coordinates is just the usual gravitational field.

Allowing the potentials to vary with time yet still satisfying the Einstein equation, we arrive at the

restriction that the zero component of the field f must be constant for all space and time. We use lin-

earized versions of the equations for the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, and

correspondingly, Einstein tensor, to see how these quantities are formulated in terms the fields, f , rather

than in terms of the metric. Doing this for the Christoffel symbols is quite straight-forward and the result

is given in section 6.2. The Riemann tensor is subject to a number of symmetries so that we (fortunately)

don’t have to calculate all 256 components. After showing that the number of independent components is

only 20, we calculate them in the array in section 6.3. The Ricci tensor and scalar and the Einstein tensor

components are calculated on the following pages using contractions of the Riemann tensor. In essence what

these calculations perform is a rewriting of the curvature structures of general relativity that are usually

formulated in terms of geometry (the metric) in terms of fields.

What we find from these calculations is that in contrast to several texts which state that the Riemann

tensor in gravity is analogous to the field tensor in electromagnetism, our calculations show that this is

not the case. Rather, we find that it is the Christoffel symbols (which contain the fields fµ themselves as

components) that are the analogous structure.

To get a sense of how the fields result in the motions of test particles and what paths they follow,

we look at the geodesic equation, which describes the motion of particles under the influence of no forces

(beyond gravity). The results we find are given by eqns. 203 and 204 for the space and time components,

respectively. Since forming a clear qualitative understanding of these results is not easily forthcoming, we

look at the Raychaudhuri equation which relates geometrically the Ricci tensor to the paths followed by a

group of nearby geodesics. The Newtonian limit of this equation makes sense in terms of fields, but the

analysis in a less strict limit is still under current work.

Furthermore, we’d like to use the structures of general relativity now written in terms of fields to assess

to what extent gravity mirrors any of the hierarchy of electric and magnetic fields (the fact that in the static

situation electric fields dominate magnetic fields and the interrelated relationship of electric and magnetic

fields). Specifically, we know a changing electric field generates a magnetic field. We’d like to see whether

perhaps a changing gravitational field generates any higher structures in gravity, and this analysis is still

under current study.

Linearized general relativity has been an established area of study for almost 100 years and the parallels

with electromagnetism are clearly apparent. Therefore, we suspect that all the results obtained here have

been independently discovered and published previously. The object of this project has not been to uncover

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virgin territory, but to look at familiar results in new combinations from new angles. Therefore the goal of

the project is not a unique result, and as of yet, we don’t report any distinctly new findings.

2 Introduction to Tensors, Spacetime, and Relativity

General relativity is a powerful tool to deal with any arbitrary spacetime, but we can introduce useful

concepts in a comparatively simple case of general relativity, that of special relativity, where there is no

gravity and spacetime is flat. The spacetime of special relativity is known as Minkowski space. Particles in

such a space travel along curves through spacetime referred to as worldlines. In general, we can represent the

position of a particle by four components, three for space and one for time, and we can write the spacetime

distance between two events, the spacetime interval, as

(∆s)2 = −(c∆t)2 + (∆x)2 + (∆y)2 + (∆z)2 (12)

where x, y, z are the usual space components and t is time. In contrast to Newtonian mechanics, which

provides provides a clear distinction of space from time and correspondingly the notion of simultaneity of

events in time, no such distinction is possible in Special Relativity and the universal notion of simultaneity

is lost.

Using index notation with the Einstein summation convention we can write the position of a particle as

xµ where µ runs from 0 to 3 with the zeroth component standing for the time component, ct. We can then

write the infinitesimal spacetime interval as

ds2 = ηµνdxµdxν (13)

where ηµν is the 4×4 matrix, the metric for Minkowski space, with values

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (14)

Since we’re interested in specific situations in some specific coordinate system, we’d also like to know

how to go about relating measurements in one reference frame to those made in another, i.e. how to relate

the position of a particle in one frame expressed by xµ to its position in some other frame, xµ′ . Specifically,

we’re interested in frames of reference in which one moves with some fixed velocity relative to the other, since

those separated by some fixed distance are relatively trivial and those separated by some fixed (or changing)

acceleration are the subject of general, not special, relativity, or at least they must be treated in the limit in

which special relativity can approximately hold true, namely, very short periods of time. What we’re looking

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for are transformations that leave the interval between events constant. We can consider transformations

between frames of reference separated by a constant velocity (also called boosts) by multiplying the position

in one frame by a matrix

xµ′ = Λµ′

νxν . (15)

(Recall that in index notation, an upper index that is repeated as a lower index in the same equation indicates

a sum over that index.) Possibilities for Λµ′

ν are limited by the restriction that the spacetime interval should

be the same in both reference frames. Thus we write

ds2 = ηµνdxµdxν = ηµ′ν′dxµ′dxν′ = ηµ′ν′Λµ′

µdxµΛν′

νdxν (16)

so

ηµν = Λµ′

µΛν′

νηµ′ν′ . (17)

For the interval to be invariant, we must find transformation matrices such that the components of ηµν are

the same as those of ηµ′ν′ . Such transformations are known as Lorentz transformations.

2.1 Vectors, Dual Vectors, and Tensors

The concept of vectors is familiar but takes on a different cast in relativity. Vectors exist at a single point

in spacetime and can be thought of as tangent vectors at that point, thus the collection of all vectors at

a given point, the tangent space, all lie on some subspace which is tangent at that particular point. If we

choose to look at a given situation in some frame of reference, we can define a set of basis vectors, e(µ), and

we can then write some vector A as a linear combination of basis vectors, A = Aµe(µ). Since it is Lorentz

transformations that leave a given interval unchanged upon transformation, it is these transformations that

we use to transform the components of a vector, since the vector itself must remain invariant in different

frames of reference. A dual vector space is the space of all linear maps from a given vector space to the real

numbers. We can express a dual vector in terms of the dual basis vectors for a certain frame of reference as

ω = ωµθ(µ). Similarly, vectors themselves are maps from dual vectors to the real numbers.

One particularly useful example of a vector is the vector tangent to some curve. If we parametrize the

path through spacetime as xµ(λ) then we can write the tangent vector to that path with components

V µ =dxµ

dλ. (18)

As a generalization of the notions of vectors and dual vectors, a tensor of rank (k, l) is a multilinear map

from a set of vectors and dual vectors to the reals. Therefore a (0, 0) type tensor is a scalar, a (1, 0) type

is a vector, and a (0, 1) type is a dual vector. To define a basis for a tensor, we use the tensor product ⊗which when used with a type (k, l) tensor and a type (m,n) tensor produces a type (k + m, l + n) tensor. In

component notation we write an arbitrary tensor as

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T = Tµ1···µkν1···νl

eµ1 ⊗ · · · ⊗ e(µk) ⊗ θ(ν1) ⊗ · · · ⊗ θ(νl). (19)

We can define the transformation of a tensor by generalization from vectors and dual vectors.

Tµ′1···µ′k

ν′1···ν′l= Λµ′1

µ1 · · ·Λµ′k

µkΛν1ν′1· · ·Λνl

ν′lTµ1···µk

ν1···νl. (20)

One very useful tensor is the Kronecker delta, δµρ , a type (1, 1) tensor which relates the metric to the

inverse metric, the metric written with upper indices, ηµν :

ηµνηνρ = δµρ . (21)

Another useful tensor is the Levi-Civita symbol, usually expressed as a (0, 4) tensor:

εµνρσ =

+1 if µνρσ is an even permutation of 0123

−1 if µνρσ is an odd permutation of 0123

0 otherwise.

(22)

2.2 Tensor Manipulation

There are a few important ways of manipulating tensors to review that will become useful later. First, we

can perform a contraction, which turns a type (k, l) tensor into a (k− 1, l− 1) tensor. This is done through

summing over an upper and lower index such as

Sµρσ = Tµνρ

σν . (23)

We can change a lower index into an upper one and vice versa by using the metric and inverse metric.

Therefore, from a tensor Tµναβ we can form a number of new tensors such as

Tµνγβ = ηγαTµν

αβ (24)

T νγ αβ = ηγµTµν

αβ . (25)

A tensor is symmetric with respect to a pair of its indices if the tensor remains unchanged under exchange

of those indices. Similarly, a tensor is said to be antisymmetric with respect to a pair of its indices if the

value changes sign upon exchange of those indices. It is easy to show that symmetry and antisymmetry are

properties of the tensor itself, not of its components in a particular frame of reference.

3 Maxwell’s Equations in Tensor Form

To gain a deeper appreciation for the similarities between electromagnetism and general relativity, we first

must examine the form electromagnetism takes when written in tensor notation. The four Maxwell’s equa-

tions are quite familiar:

8

∇ ·E =1ε0

ρ (26)

∇ ·B = 0 (27)

∇×E = −∂B∂t

(28)

∇×B = µ0J + µ0ε0∂E∂t

(29)

where ρ(~r, t) is the charge density as a function of space and time and J(~r, t) is the current density. The

usefulness of Maxwell’s equations is that they relate the electric and magnetic fields on one side to their

sources, ρ(~r, t) and J(~r, t), on the other. They show how some orientation of charges in spacetime corresponds

with the associated electric and magnetic fields, and we’ll see that the Einstein equation in General Relativity

performs a very similar function.

We can define an important antisymmetric tensor, the electromagnetic field tensor, as

Fµν =

0 −E1/c −E2/c −E3/c

E1/c 0 B3 −B2

E2/c −B3 0 B1

E3/c B2 −B1 0

. (30)

Although we don’t discuss the reasons behind its structure here, a brief discussion of them can be found in

Appendix B.

To begin rewriting Maxwell’s equations using the electromagnetic field tensor, first we define the current

4-vector as Jµ = (ρ, Jx, Jy, Jz). We can then write eqns. 26-29 in component notation

εijk∂jBk − ∂0Ei = J i (31)

∂iEi = J0 (32)

εijk∂jEk + ∂0Bi = 0 (33)

∂iBi = 0. (34)

Here we have used Latin indices i, j, k to refer to only space components, so that they run 1-3, whereas

Greek letters will be used for all components, 0-3, including the time component. Also, the notation ∂µ is a

shorthand notation indicating

∂µ =∂

∂xµ, (35)

and εijk is the 3-dimensional Levi-Civita symbol, so that we write the curl of a vector in terms of the

Levi-Civita symbol as

(∇×V)i = εijk∂jVk (36)

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for some vector V , since permutations of j and k for a given i will produce the desired alternation of positive

and negative derivative terms.

Looking at the field tensor written with upper indices (found by Fµν = ηαµηβνFαβ) we can see that we

can express the components of E and B as

F 0i = Ei (37)

F ij = εijkBk. (38)

We can then use this to rewrite the first two Maxwell’s equations, eqns. 31 and 32 as

∂jFij − ∂0F

0i = J i (39)

∂iF0i = J0. (40)

Using the antisymmetry of the field tensor, we may rewrite the first equation, eqn. 39, as

∂jFij + ∂0F

i0 = J i (41)

and we may combine these first two Maxwell’s equations into a single tensor equation:

∂µFµν = Jν . (42)

It can be shown similarly that the second two Maxwell’s equations can be written in component tensorial

form as

∂[µFνλ] = 0 (43)

where the square brackets indicate an antisymmetrization with respect to the indices µ, ν, λ in which the

permutations of indices are added in an alternating sum (terms with an odd number of permutations from

the ordering µνλ are added with a minus sign and those with an even number of permutations are added

with a plus sign). Using the antisymmetry of the field tensor we can write the antisymmetrization as

∂µFνλ + ∂νFλµ + ∂λFµν = 0. (44)

However, we can also write the second two Maxwell’s equations in a slightly different and somewhat more

intuitive form using a tensor defined as the dual field tensor Gµν to the field tensor:

Gµν =

0 B1/c B2/c B3/c

−B1/c 0 −E3 E2

−B2/c E3 0 −E1

−B3/c −E2 E1 0

. (45)

It turns out that we can relate the field tensor to the dual field tensor by the substitution E/c −→ B and

B −→ −E/c, and the mechanics of this relation come through operations in the field of differential forms,

discussed briefly in Appendix B.

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Comparing the first two equations of Maxwell’s equations with the last two, we see that writing the field

components using the dual field vector has done exactly what we would like, namely that it has swapped

the electric field terms for the magnetic field terms and vice versa, with the change of some minus signs.

Therefore, we can write the second two Maxwell’s equations in a very similar form to the first two, namely

∂Gµν

∂xµ= 0. (46)

Overall, we have reduced the Maxwell’s equations from four to two equations, one for the equations with

source charges and currents, and one for equations that are source-free:

∂µFµν = Jν (47)

∂µGµν = 0. (48)

4 Introduction to General Relativity and the Einstein Equation

As we move from the relatively straightforward Minkowski spacetime to some arbitrary spacetime that could

have curvature, a bit of care is needed in defining what we mean by the structures mentioned in the last

section. To describe more complicated and curved spacetimes, we use the concept of manifolds. We are

accustomed to working in flat, Euclidean space, Rn, the set of n-tuples (x1, ..., xn), often accompanied by

a positive, definite metric that has components δij . A (differentiable) manifold corresponds to a space that

may globally be very complicated and curved but one which on the local level resembles Euclidean space in

the way that functions and coordinates operate (though the metric may not be the same). Therefore, we can

think of a differentiable manifold as a space that can be composed of small pieces of Rn patched together

to form the manifold as a whole.

The contrast of global compared to local structure in manifolds is precisely fitting for the structure of

general relativity. In essence, general relativity states that the gravitational fields caused by mass-energy is

the same as the curving of spacetime, and this concept is embodied by Einstein’s Equivalence Principle. In

its weak form, the principle states that “The motion of freely falling particles are the same in a gravitational

field and a uniformly accelerated frame, in a small enough region of space and time.” A freely falling particle

is one which is influenced by gravity but no other forces, and in the context of general relativity, we define

such reference frames as inertial reference frames. The idea behind the weak formulation of the principle is

the idea that if one were to perform experiments inside a box it would be impossible to tell whether the box

was in a gravitational field or in a frame with uniform acceleration, so long as the box was small enough that

the gravitational field was uniform in space (and therefore no inhomogeneities in the field would cause tidal

effects) and the experiment was performed for a short enough period of time that any field inhomogeneities

would not be evident. Therefore, on any manifold we can always define locally inertial frames of reference,

but unlike Minkowski space, on a global scale this is no longer possible.

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4.1 Vectors, Dual Vectors, and Tensors in Curved Spacetime

Without embedding our manifold in a higher dimensional space, we lack our previous notion of a vector

as tangent to the curves passing through a given point. It is appropriate instead to use the notion of the

directional derivative along curves through a specific point. We define a space that is made up of all the

smooth functions on a given manifold, and then use the fact that for each curve passing through a desired

point we can uniquely assign a directional derivative operator (for a function f we have the assignment

f → df/dλ at a certain point, where λ is the parameter along the curve), and the space of all of these

directional derivatives is equivalent to the notion of a tangent space, the collection of tangent vectors at that

point. We can then find a basis for tangent vectors in a given coordinate system as

d

dλ=

dxµ

dλ∂µ, (49)

which shows that the partials {∂µ} are a good set a of basis vectors for the vector space of directional

derivatives. We can recognize the components of d/dλ as the same as those defined above for the spacetime

of special relativity, eqn. 18, providing further legitimation for using directional derivatives as a replacement

for the tangent space of Minkowski space. The only difference is that here, for an arbitrary spacetime, we

use the basis vectors e(µ) = ∂µ. We can find the transformation of basis vectors from one reference frame to

another simply by applying the chain rule

∂µ′ =∂xµ

∂xµ′∂µ. (50)

This relation produces the transformation rule for the components of a tangent vector as

V µ = V µ′∂µ′ (51)

= V µ′ ∂xµ

∂xµ′∂µ (52)

so that

V µ′ =∂xµ′

∂xµV µ. (53)

We see that eqn. 53 is compatible with Lorentz transformations in special relativity; indeed, transfor-

mations in flat Minkowski space are simply a specific example of a transformation of the form of eqn. 53.

Once again we can consider dual vectors to be the linear maps from our newly redefined tangent space to

the real numbers. In the same way that the partial derivatives along coordinate axes can be considered as

a natural basis for tangent space, the gradients of coordinate functions xµ form a natural basis for dual

vectors, expressed as dxµ. We therefore find the transformation properties of dual basis vectors to be

dxµ′ =∂xµ′

∂xµdxµ (54)

and for the components,

ωµ′ =∂xµ

∂xµ′ωµ. (55)

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Generalizing to an arbitrary type (k, l) tensor, we find

T = Tµ1···µkν1···νl

∂µ1 ⊗ · · · ⊗ ∂µk⊗ dxν1 ⊗ · · · ⊗ dxνl (56)

defined similarly to eqn. 19, and in general, we refer to upper indices as contravariant and lower indices as

covariant. By extension we can find the transformation properties of tensors to be

Tµ′1···µ′k

ν′1···ν′l=

∂xµ′1

∂xµ1

· · · ∂xµ′k

∂xµk

∂xν1

∂xν′1· · · ∂xν

1

∂xν′1Tµ1···µk

ν1···νl, (57)

similar to eqn. 20.

4.2 The Metric

The primary characterizing feature of a manifold is the metric and it is here that we really begin to see the

effects of curvature. The metric in general relativity is referred to with the symbol gµν while ηµν is reserved

only for the metric of Minkowski space. Components of the metric are restricted to be symmetric and if

they are also non-degenerate, meaning that the determinant of the tensor does not vanish, then the inverse

metric gµν may be defined as

gµνgνγ = δµγ . (58)

Similar to its use in special relativity, the metric in general relativity can be used to raise and lower the

indices on tensors. The metric for general relativity embodies many of the concepts we’re familiar with in

both flat spacetime and in Newtonian gravity. It encompasses many of the innate properties of a given space

it represents, such as it allows for the measurement of path length and proper time and therefore a sense of

past and future, it determines the shortest spacetime distance between two events and therefore determines

the paths that free test particles will travel. Furthermore, it generalizes the Newtonian gravitational potential

Φ.

The notion of the metric as encompassing the notion of path length becomes clear when we reexamine

the definition of the spacetime interval we used for Minkowski space, eqn. 13: ds2 = ηµν dxµdxν . We can

now recognize that the dxµ are just the dual basis vectors in general relativity, and we can write

ds2 = gµνdxµdxν (59)

as the interval for any arbitrary spacetime. From this we see that the notion of metric and interval, or

distance, become inherently related and interchangeable. The metric also embodies the sense of geometric

curvature of a given spacetime. In Minkowski space, we can always make a change of coordinates so that the

components of the metric are all constant, because Minkowski space is flat. However, having the components

of the metric vary with the coordinates chosen is not sufficient for the metric to describe a space with

13

curvature. For example, if we choose to describe flat space using spherical coordinates, we can transform

from Minkowski space in rectangular coordinates:

ds2 = −(cdt)2 + dx2 + dy2 + dz2 (60)

using the transformations

x = r sin θ cos φ (61)

y = r sin θ sinφ (62)

z = r cos θ (63)

so that the interval becomes

ds2 = −(cdt)2 + dr2 + r2dθ2 + r2 sin2 θdφ2. (64)

Comparing with eqn. 59 we can see that the components of the metric will not all be constant and will depend

on the coordinates even though we are still describing flat spacetime. Therefore, although if we’re dealing

with flat spacetime we can in principle always find the coordinate system in which the metric components

are all constant, in practice doing this may be difficult and so we will want to find an additional way of

representing the curvature of spacetime besides the metric.

The metric also includes within it a sense of the Equivalence Principle. We can write the metric in a

particularly useful formulation, known as canonical form, in which the components become

gµν = diag(−1,−1, . . . ,−1,+1,+1, . . . ,+1, 0, 0, . . . , 0) (65)

where “diag” indicates that the only nonzero components are the diagonal elements, with the values listed.

This form looks very familiar since the Minkowski metric is most often cited in canonical form, as above in

eqn. 14. It turns out it is always possible to write any arbitrary metric in this form. However, usually this can

only be done at a single point on the manifold or in some small neighborhood in which the first derivatives

of the metric vanish, though in general the second derivatives do not vanish. The coordinates used to do

this are referred to as locally inertial coordinates, incorporating the fact that any spacetime regardless of

curvature (so long as it is a differentiable manifold) looks like flat Minkowski space permitted we look in a

small enough volume for a short enough duration of time.

We know how to relate points that are close enough to each other that we can use special relativistic

principles; however, how do we go about relating distinct points that are not so close together? Since we

define tensors as maps from vectors and dual vectors to the real numbers, how do we go about relating these

maps at different points on a manifold? This issue is of extreme importance, since we’d like to be able to

take derivatives along curves in the manifold, but the question becomes: with respect to what should we take

the derivative? How do we measure the rate of change of something when we don’t have a way of relating

the same structure at different points along the curve, points which do not lie in the same tangent space?

14

As a basis for comparison, we define the concept of parallel transport, which is the idea of transporting a

vector along a curve while keeping both its direction and magnitude unchanged. In order to do this, we need

a well-defined metric. It turns out that in Minkowski space we can perform such a transformation between

two points along any curve through spacetime with the same result, but in curved spacetime the outcome is

path-dependent. There is no easy way out of this, and we’ll have to cope with the fact that there is no natural

choice for how to parallel transport a vector in curved space, and therefore we will always be limited in our

attempts to relate quantities from distinctly different points in a manifold. However, if we already have a

path in mind, or we look infinitesimally and constrain our interest to the transportation of a vector in a given

direction, we can develop the concept of transporting a vector along a path while keeping it as constant as

possible. In turn, we use this notion of parallel transport as a reference point for defining the derivative as

a deviation away from parallel propagation. The structure underlying the ability to compare nearby points

is the concept of the “connection,” a kind of correction factor for the fact that curved spacetime is not flat,

forcing us to find a way to incorporate the effects of curvature into our equations.

4.3 Covariant Derivatives and the Metric Connection

We can take an ordinary partial derivative, but it turns out that the partial derivative is not a tensorial

operator, in other words, the result of such an operator will not always be a tensor, some of the terms may

be nonlinear. However, If our structures are not tensorial then ideas such as that “we can look at the same

situation from multiple frames of reference and observe the same fundamental structures invariantly” are no

longer true. On a qualitative level, we can use the concept of parallel propagation to define a rate of change

of a vector by comparing to what it would have been if it had just been parallel transported, as mentioned

above. On a more quantitative level, clearly, we need to find a replacement for the partial derivative, and

this comes in the form of the covariant derivative. We require that the covariant derivative provide the same

function in flat space with inertial coordinates as the partial derivative, but that it transforms like a tensor

in any arbitrary manifold. We see that in flat spacetime the partial derivative is a map from type (k, l)

tensors to type (k, l + 1) tensors using the example:

∂µSνγ = T ν

µ γ . (66)

Therefore we require that the covariant derivative be a map from (k, l) tensors to type (k, l + 1) tensors

with the restriction that the map be linear and obey the Leibniz (product) rule. If it obeys the Leibniz rule,

then it is always possible to always write the covariant derivative as the partial derivative plus some linear

transformation that acts as a correction term to make the result covariant. So for a vector with components

µ, we define a set of n matrices (Γµ)ρσ where n is the dimensionality of the underlying spacetime. We refer

to Γρµσ as the connection coefficients, and we can now write the covariant derivative of (the components of)

a tangent vector as

∇µV ν = ∂µV ν + ΓνµσV σ. (67)

15

Similarly, we can write the covariant derivative of a dual vector as

∇µων = ∂µων − Γσµνωσ. (68)

Because the the connection serves to cancel out the nonlinear part of the ordinary partial derivative, therefore

leaving the covariant derivative tensorial, Γ by definition is not itself a tensor.

For tensors in general, we can generalize from tangent vectors and dual vectors so that for a type (k, l)

tensor, there are k terms of +Γ and l terms of −Γ:

∇σTµ1µ2···µkν1ν2···νl

= ∂σTµ1µ2···µkν1ν2···νl

+Γµ1σλTλµ2···µk

ν1ν2···νl+ Γµ2

σλTµ1λ···µkν1ν2···νl

+ · · ·

−Γλσν1

Tµ1µ2···µk

λν2···νl− Γλ

σν2Tµ1µ2···µk

ν1λ···νl− · · · . (69)

For a given manifold with metric gµν we can define a single unique connection by restricting the con-

nection to be torsion-free, meaning that it is symmetric in its lower indices, and has the property of metric

compatibility, meaning that the covariant derivative of the metric with respect to that connection is always

zero everywhere, ∇ρgµν = 0. We can find that these restrictions produce the existence of a single, unique

connection by writing out metric compatibility with all possible permutations of indices:

∇ρgµν = ∂ρgµν − Γλρµgλν − Γλ

ρνgµλ = 0 (70)

∇µgνρ = ∂µgνρ − Γλµνgλρ − Γλ

µρgνλ = 0 (71)

∇νgρµ = ∂νgρµ − Γλνρgλµ − Γλ

νµgρλ = 0. (72)

If we subtract the second two equations from the first and use the symmetry of the connection we find

∂ρgµν − ∂µgνρ − ∂νgρµ + 2Γλµνgλρ = 0 (73)

We solve for the connection by multiplying by the inverse metric gσρ to find

Γσµν =

12gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν), (74)

which are referred to as the Christoffel connection, and whose components are called the Christoffel symbols.

As a correction to the partial derivative to make it account for the differences between flat and curved

spacetime, the connection in many ways embodies the amount and nature of the curvature of a particular

spacetime, and as we’ll see in section 6, it also forms the most important parallel structure to the fields of

electromagnetism.

4.4 Riemann, Ricci, and Einstein Tensors

Recall that when a vector is parallel transported around a closed loop in curved spacetime the vector will

undergo a transformation dependent on the path chosen, a rotation from its initial orientation. The amount

16

of this transformation is directly the result of the amount of curvature of the space. Therefore, to define a

structure that further embodies the curvature of a space, we can think of propagating a vector V µ around

a tiny loop defined by two infinitesimal vectors, Aµ and Bν , so that we go first in the direction of Aµ, then

in the direction of Bν , and then backward along Aµ and Bν so that we end up where we started. Since the

action of parallel transport is coordinate independent, we should be able to express the change in the vector

V µ around the loop using a tensor, which is a linear transformation. Therefore, we can write the expected

expression for the change δV ρ experienced by the vector V µ around the loop in the form

δV ρ = RρσµνV σAµBν (75)

where the tensor Rρσµν is called the Riemann tensor or curvature tensor. If we move around the loop in the

opposite direction, corresponding to swapping Aµ and Bν in eqn. 75, we should just get the opposite of what

we had originally. Thus we know that the Riemann tensor must be antisymmetric in its last two indices.

We’d like to find an expression for the Riemann tensor in terms of the Christoffel symbols (which are

themselves directly related to the metric). To do this, we look at the commutator of two covariant derivatives.

Each covariant derivative along a given direction measures the amount that a vector varies compared with

if it had been parallel transported, since the covariant derivative of a vector in the direction in which it is

parallel transported is zero. Therefore, the commutator of covariant derivatives in two different directions

will essentially measure the difference in the resulting transformation of the vector if the vector had been

transported first in one direction and then the other or vice versa. If we expand the expression for the

covariant derivative using its definition we find

[∇µ,∇ν ]V ρ = ∇µ∇νV ρ −∇ν∇µV ρ (76)

= ∂µ(∇νV ρ)− Γλµν∇λV ρ + Γρ

µσ∇νV σ − (same terms with µ and ν swapped) (77)

= ∂µ∂νV ρ + (∂µΓρνσ)V σ + Γρ

νσ∂µV σ

−Γλµν∂λV ρ − Γλ

µνΓρλσV ρ

+Γρµσ∂νV σ + Γρ

µσΓσνλV λ − (same terms with µ and ν swapped). (78)

Expanding, relabeling some dummy indices, and canceling some terms through antisymmetrization, we can

rewrite the expression as

[∇µ,∇ν ]V ρ = (∂µΓρνσ − ∂νΓρ

µσ + ΓρµλΓλ

νσ − ΓρνλΓλ

µσ)V σ + (Γλνµ − Γλ

µν)∇λV ρ (79)

The last term is the torsion tensor, which will be zero for the Christoffel symbols, since they are always

torsion-free. Since the left hand side of the expression is a tensor, then the expression in the parentheses

on the right hand side must also be a tensor. Therefore, we can write the value of the Riemann tensor in

component form as

Rρσµν = ∂µΓρ

νσ − ∂νΓρµσ + Γρ

µλΓλνσ − Γρ

νλΓλµσ (80)

17

and this form can be shown to agree with the expression given in eqn. 75.

From the Riemann curvature tensor we can get a better sense of the relationship between the metric and

the curvature of the space it describes. If there exists a coordinate system in which the components of the

metric are constant, then the Riemann tensor vanishes, and conversely, if the Riemann tensor vanishes, then

we can always construct a coordinate system in which the components of the metric are all constant. In

either case, we know for sure we are dealing with flat spacetime.

There are some important properties of the Riemann tensor which can aid in calculating its components.

We show them using the Riemann tensor with all lowered indices. We have already shown that the tensor

is antisymmetric in its last two indices,

Rρσµν = −Rρσνµ, (81)

and it turns out that the tensor is also antisymmetric in its first two indices,

Rρσµν = −Rσρµν . (82)

The tensor is symmetric with exchange of the first pair of indices for the second pair,

Rρσµν = Rνµρσ, (83)

and the cyclic sum of permutations in the last three indices vanishes,

Rρσµν + Rρµνσ + Rρνσµ = 0. (84)

There are several other structures that are defined using the Riemann tensor which will become useful

later in introducing the Einstein equation. The contraction

Rµν = Rλµλν (85)

defines the Ricci tensor Rµν , which is symmetric as a result of the symmetries of the Riemann tensor. From

the definition of the Riemann tensor in terms of the Christoffel symbols, eqn. 80, and the definition of the

Ricci tensor as a contraction of the Riemann tensor, eqn. 85, we can write the Ricci tensor in terms of the

Christoffel symbols as

Rµν = ∂λΓλµν − ∂νΓλ

µλ + ΓλµνΓσ

λσ − ΓλµσΓσ

νλ. (86)

The trace of the Ricci tensor, the Ricci scalar, is

R = Rµµ = gµνRµν . (87)

4.5 Minimal Coupling Principle and Geodesics

Now that we have all the structures needed to describe curvature, we start on the real task of general relativity,

which is, “How is curvature related to mass distribution?” We need to know: “How does a gravitational

18

field affect the behavior of matter?” and conversely, “How does the distribution of matter determine the

gravitational field?” For Newtonian gravity, these two questions are answered by

~a = −∇Φ (88)

which relates the acceleration of an object in a gravitational potential, Φ, and

∇2Φ = 4πGρ, (89)

Poisson’s equation, which relates the mass density ρ to the produced gravitational potential, where G is

Newton’s gravitational constant. For general relativity, the two questions become: “How does the curvature

of spacetime affect matter in such a way as to manifest gravity?” and “How does energy-momentum cause

the curving of spacetime?” The minimal coupling principle provides a straight-forward approach for the

generalization of principles from flat to curved spacetime. It instructs that we rewrite a law of physics

known to be true in inertial frames in flat spacetime in a coordinate-invariant form (so that the result is a

tensorial equation) and then show that the result remains valid in curved spacetime, which we do through

comparison at the Newtonian limit. For example, we can look at the movement of a freely-falling particle,

which in flat spacetime means that it travels on straight-line paths. Therefore, the second derivative of the

position of the particle with respect to the parameter along the path is zero:

d2xµ

dλ2= 0. (90)

Using the chain rule we can writed2xµ

dλ2=

dxν

dλ∂ν

dxµ

dλ. (91)

To move from this equation to one that is tensorial, we simply replace the partial derivative with the covariant

derivative to getdxν

dλ∇ν

dxµ

dλ=

d2xµ

dλ2+ Γµ

ρσ

dxρ

dλ

dxσ

dλ(92)

which we recognize as the geodesic equation,

d2xµ

dλ2+ Γµ

ρσ

dxρ

dλ

dxσ

dλ= 0. (93)

We can confirm that the minimal coupling principle works in this case by finding the geodesic equation

from a different method, using what we know about parallel transport. To define parallel transport more

explicitly, for a curve xσ(λ) in order for an arbitrary tensor Tµ1µ2···µkν1ν2···νl to be parallel transported, the

components of the tensor must remain constant along the curve so that

d

dλTµ1µ2···µk

ν1ν2···νl=

dxσ

dλ

∂

∂xσTµ1µ2···µk

ν1ν2···νl= 0. (94)

To ensure the result is tensorial, we replace the partial derivative with a covariant one, and define the

directional covariant derivative asD

dλ=

dxσ

dλ∇σ. (95)

19

Therefore, for the tensor T to be parallel transported we say that(D

dλT

)µ1µ2···µk

ν1ν2···νl

=dxσ

dλ∇σTµ1µ2···µk

ν1ν2···νl= 0. (96)

With a more concrete structure of parallel transport, we can define a geodesic as a path xµ(λ) along which

a tangent vector dxµ/dλ is parallel transported to itself. So long as the connection we use is the Christoffel

connection, we can ensure that this definition and the definition of a geodesic as the shortest spacetime

distance between two points, the generalization of straight lines in flat spacetime, (found using the principle

of least action) are equivalent. Therefore, we can write for a geodesic

D

dλ

dxµ

dλ= 0 (97)

or equivalently, using the definition of the directional covariant derivative,

d2xµ

dλ2+ Γµ

ρσ

dxρ

dλ

dxσ

dλ= 0, (98)

the same as eqn. 93.

4.6 Einstein’s Equation

Finally we are at the heart of general relativity: the Einstein equation, which describes how gravitational

fields and curvature of spacetime respond to energy and momentum, just as Maxwell’s equations describe

how electric and magnetic fields respond to charge and current. Just like Newton’s second law, which can be

demonstrated but not derived from more basic concepts, the Einstein equation is ultimately too fundamental

to be derived from first principles, so we’ll just state it and then show how it works:

Rµν −12Rgµν = 8πGTµν (99)

where Rµν is the Ricci tensor, R the Ricci scalar, Tµν the so-called stress-energy tensor, also referred to

as the energy-momentum tensor, which is the generalization of the mass density from Newtonian gravity,

and G Newton’s gravitational constant. Sometimes the left hand side of the equation is written as a single

tensor, called the Einstein tensor, for convenience:

Gµν = Rµν −12Rgµν . (100)

Since under relativity we think of energy as the same as mass, the stress-energy tensor, Tµν , embodies both

energy and momentum of a given source distribution. The 00 component of the tensor is the energy density,

and the 0i and i0 components are the momentum density in the ith direction. The ij components are the

“stress” part of the stress-energy tensor and can be thought of as forces or pressures, for example, for a fluid,

the stress tensor becomes the surface pressure of the fluid, the force exerted per unit area. We treat the

structure of the stress-energy tensor as a parallel to electromagnetic source vectors in Appendix A.

20

One concern with the Einstein equation is ensuring that it satisfies conservation of energy, which for

general relativity takes the form

∇µTµν = 0. (101)

In addition to the four symmetries obeyed by the Riemann tensor, it also obeys a principle known as the

Bianchi identity:

∇[λRρσ]µν = ∇λRρσµν +∇ρRσλ +∇σRλρ = 0. (102)

If we contract twice on the Bianchi identity, we have

∇µRρµ =12∇ρR. (103)

Comparing this with the definition of the Einstein tensor, we can see that the twice-contracted Bianchi

identity is equivalent to

∇µGµν = 0. (104)

Therefore, if we look at the Einstein equation, we can see that energy conservation is always satisfied if we

use the Einstein tensor, which is why we use the particular combination of Ricci tensor and scalar found in

eqn. 99. Delving deeper into the Einstein equation, we see that the curvature on the left hand side is related

to the energy-momentum on the right by a proportionality constant, 8πG, which will be fixed by comparison

with Newtonian gravity.

Our main concern vis-a-vis the minimal coupling principle is ensuring that Einstein’s equation reduces

to the Poisson equation of Newtonian gravity, ∇2Φ = 4πGρ, which it replaces in moving from special to

general relativity. To look at the Einstein equation in the Newtonian limit, we restrict ourselves to only

weak gravitational fields, small velocities, and negligible pressures in the stress-energy tensor. We can write

the stress-energy tensor for a perfect fluid source of energy-momentum as

Tµν = (ρ + p)UµUν + pgµν (105)

where Uµ is the four-velocity of the fluid and ρ and p are the rest-frame energy and momentum densities,

respectively. Since we’re looking in the Newtonian limit, we can neglect the pressure, and therefore, the

stress-energy tensor reduces to

Tµν = ρUµUν . (106)

We can work in the rest frame of the fluid we’re working with so that

Uµ = (U0, 0, 0, 0), (107)

and we normalize the four-velocity using

gµνUµUν = −1. (108)

We consider our weak gravitational field to be represented by a small perturbation away from a flat Minkowski

metric, so that we can write gµν = ηµν + hµν where hµν is a small perturbation term, and it turns out

21

for Newtonian gravity, the 00 component of the perturbation is h00 = −2Φ where Φ is the Newtonian

gravitational potential. We’ll deal with where this comes from later when discussing the weak-filed metric

and general relativity in the linearized limit. We approximate that U0 = 1 since the energy density ρ is close

to zero for situations close to flat spacetime. Therefore, in the Newtonian limit

T00 = ρ. (109)

and all the other components are negligible. If we contract over both sides of the Einstein equation, we find

that

R = 8πGT, (110)

where T is the stress-energy scalar just as R is the Ricci scalar, and we can use this to write an equivalent

form of the Einstein equation as

Rµν = 8πG

(Tµν −

12Tgµν

). (111)

Therefore using eqn. 109 we find the 00 component of the Ricci tensor to be

R00 = 4πGρ (112)

Next we find the relationship between the Ricci tensor components and the perturbation factors, hµν , using

the definition of the Ricci tensor in terms of the Christoffel symbols, eqn. 86, and the definition of the

Christoffel symbols in terms of the metric. Since in the Newtonian limit we’re only concerned to within the

first order in the perturbation of the metric, we can neglect the factors of (Γ)2 in eqn. 86. Thus we calculate

R00 as

R00 =∂Γλ

00

∂xλ− ∂Γλ

0λ

∂x0(113)

= ∂λ

(12ηλσ (∂0hσ0 + ∂0hσ0 − ∂σh00)

)(114)

= −12δij∂i∂jh00 (115)

= −12∇2h00 (116)

where we have neglected the time derivatives of the Christoffel symbols in the first line and those of the

metric in the second line, since in the Newtonian limit we consider only static fields. (Note that this will no

longer be the case in the linearized limit). Therefore, we can write using eqn. 112

∇2h00 = −8πGρ (117)

from which we can see, with h00 = −2Φ, that Einstein’s equation yields precisely the Poisson equation of

Newtonian gravity:

∇2Φ = 4πGρ. (118)

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5 Linearized Gravity

Einstein’s equation comprises ten partial differential equations for the ten unknown coefficients of the metric

gµν , which, unlike Maxwell’s equations, are nonlinear. Although there is some progress to be made with

Einstein’s equation in its complete form, namely solutions such as flat space and for the spherically symmetric

and static Schwarzchild metric (often situations where symmetries can make solving the Einstein equation

easier), there is no general solution to the equation. However, for situations that are very close to flat

spacetime, we can find a general solution. Working in this limit is quite common, though as computational

techniques become more extensive and developed, the need for this limit to simplify calculation is sometimes

no longer necessary. The linearized limit is not so strict as the Newtonian limit, in which we are constrained

only to weak fields, small velocities, and small pressures in the stress-energy tensor. Here, we are restricted

similarly to weak fields, but we allow for the possibility of large velocities and pressures.

5.1 Writing the Metric using a Perturbation

The weakness of the gravitational field means that, as before, we can write the metric as a sum of the

Minkowski metric plus a perturbation term, so that

gµν = ηµν + hµν (119)

where all of the components of hµν are much less than one, and where ηµν is the usual Minkowski metric

with components ηµν = diag(−1, 1, 1, 1). The inverse metric is

gµν = ηµν − hµν (120)

since because hµν is small, we can ignore all factors higher than first order in this perturbation term.

First we can calculate the Christoffel symbols for such a metric.

Γρµν =

12gρλ(∂µgνλ + ∂νgλµ − ∂λgµν) (121)

=12ηρλ(∂µhνλ + ∂νhλµ − ∂λhµν) (122)

since first order partial derivatives of the Minkowski metric vanish, and we can neglect the factor of hρλ in

the inverse metric since we are only concerned with first order perturbations.

In a vacuum, essentially the equivalent of Maxwell’s equations without source, the Einstein equation

reduces to

Rµν = 0. (123)

When we substitute our metric, eqn. 119, into this equation, the expansion has two terms, one for the

Minkowski metric and one for the perturbation. The first term vanishes, since the Ricci tensor for flat

23

spacetime is zero everywhere. The second term, δRµν , is the first-order perturbation of the Ricci tensor,

which is linear in hµν . Therefore, in the linearized limit, the Einstein equation in vacuum is

δRµν = 0. (124)

We can calculate δRµν in terms of the perturbation hµν directly using the Christoffel symbols.

δRµν = ∂λΓλµν − ∂νΓλ

µλ (125)

since the factors of (Γ)2 are negligible to first order. Substituting the equation for the Christoffel symbols

in terms of the perturbation the linearized vacuum Einstein equation becomes

δRµν =12[−2hµν + ∂µVν + ∂νVµ] = 0 (126)

where 2 is the d’Alembertian operator

2 ≡ ηµν∂µ∂ν = − ∂2

∂t2+ ~∇2 = −∂2

t + ∂2x + ∂2

y + ∂2z (127)

and the vector Vµ is a particular combination of perturbation terms

Vµ ≡ ∂λhλµ −

12∂µhλ

λ (128)

where

hλµ = ηλσhσµ. (129)

5.2 Gauge Transformation

Just as in electromagnetism where one can find an infinite number of solutions for the vector and scalar

potentials in Maxwell’s equations, so long as these solutions are related to one another by a factor with zero

curl (that is, the gradient of any scalar), thereby leaving the electromagnetic field tensor the same, in the case

of linearized gravity we similarly cannot be assured of finding a unique solution to the linearized Einstein

equation, since we can find alternate formulations for the perturbations while leaving the Riemann tensor

constant. Therefore, in order to solve the linearized Einstein equation, we must impose some restrictions

on the coordinates chosen. Fortunately we are free to choose whichever ones make a particular calculation

simpler. We know that we have chosen coordinates so that the components of the Minkowski metric are

diag(−1, 1, 1, 1) and the components of hµν are small perturbations of this metric; however, it is still possible

to choose slightly different coordinates that would leave the Minkowski metric unchanged but would vary

the components of the perturbation slightly. We can consider therefore a change of coordinates of the form

x′µ = xµ + ξµ(x) (130)

24

where ξµ(x) are four functions that are roughly the same size as the perturbations hµν . Undergoing a change

of coordinates for the metric we have the usual transformation

g′µν(x′) =

∂xλ

∂x′µ∂xσ

∂x′ν gλσ(x). (131)

We see that from the transformation properties of the metric, we can find a transformation of the metric

that has the same form as our original metric, eqn. 119, but with slightly different perturbations

h′µν = hµν − ∂µξν − ∂νξµ. (132)

This transformation is referred to as the gauge transformation due to its similarity to the traditional gauge

invariance of electromagnetism, which changes the electromagnetic vector and scalar potentials by ~A −→~A +∇Λ and Φ −→ Φ− ∂tΛ, without changing the value of the field tensor Fµν = ∂νAν − ∂νAµ. Similarly,

the gauge transformation in the general relativistic case does not change the value of the Riemann tensor

but does alter the perturbations. Since the ξµ(x) are four arbitrary but small functions, we can choose them

so that they simplify the transformation of the Einstein equation. Namely, we choose them so that

V ′µ(x) = 0 (133)

whence eqn. 126 becomes

δR′µν = −1

22h′

µν . (134)

Since we can assume that we are in such a coordinate system where eqn. 133 is true, the linearized Einstein

equation under the gauge transformation becomes

2hµν = 0 (135)

along with the gauge condition of eqn. 133

Vµ ≡ ∂µhλµ −

12∂µhλ

λ = 0. (136)

5.3 Solving Einstein’s Equation for Weak Fields

We can also linearize the Einstein equation with a source term. If we rewrite the Ricci tensor using only the

perturbation, we find

δRµν =12

(∂µ∂λhλ

ν + ∂ν∂λhλµ − ∂µ∂νhλ

λ −2hµν

)(137)

which when contracted to find the Ricci scaler gives

δR = ∂µ∂νhµν −2hλλ. (138)

Therefore, the Einstein tensor to first order as a whole is

δGµν = δRµν −12ηµνδR (139)

=12(∂µ∂λhλ

ν + ∂ν∂λhλµ − ∂µ∂νhλ

λ −2hµν − ηµν∂ρ∂λhρλ + ηµν2h). (140)

25

The linearized field equation, therefore, is the Einstein equation, Gµν = 8πGTµν where Gµν is given by

eqn. 140 and Tµν , the energy-momentum tensor, is calculated to the zeroth order in hµν . We don’t include

higher order terms since the amount of energy-momentum present must itself be small in order for the

weak-field assumption to hold true.

To make the problem of finding solutions to the linearized Einstein equation a little simpler, we can split

up the metric perturbation into pieces which when transformed will only transform into themselves and thus

can be treated somewhat independently. We see that under spatial rotations, the 00 component is a scalar,

the 0i components (and similarly i0) transform as a three-vector, and the ij components form a symmetric

spatial tensor with two indices. Therefore we can write the metric perturbation as

h00 = −2Φ (141)

h0i = ωi (142)

hij = 2sij − 2Ψδij (143)

in which Ψ encodes the trace of the perturbation, and the ij components are broken down further into a

trace and traceless part, where sij is traceless. Therefore, the metric as a whole can be written as

ds2 = −(1 + 2Φ)dt2 + ωi(dtdxi + dxidt) + [(1− 2Ψ)δij + 2sij ] dxidxj (144)

where we can recognize in the time component, the perturbation, −2Φ, away from a flat Minkowski metric.

We can now use this form of the metric to simplify the linear Einstein equation. After calculating the

Christoffel symbols, Riemann and Ricci tensor components, and the value of the Ricci scalar, it can be

shown that the components of the Einstein tensor, then, are

δG00 = 2∇2Ψ + ∂k∂lskl (145)

δG0j = −12∇2ωj +

12∂j∂kωk + 2∂0∂jΨ + ∂0∂ks k

j (146)

δGij = (δij∇2 − ∂i∂j)(Φ−Ψ) + δij∂0∂kωk − ∂0∂(iωj)

+2δij∂20Ψ−2sij + 2∂k∂(is

kj) − δij∂k∂ls

kl. (147)

where circular brackets around indices indicate a symmetrization of those indices (sum over permutation

of those indices). As above, where we used a gauge transformation by choosing suitable values for the

functions ξµ(x) that produced the simplest form of the linearized Einstein equation without source, here we

can choose a particular set of ξµ(x) to find a more convenient form of the linearized Einstein equation with

source. What we’d like to do is solve the linearized Einstein equation for the limit in which we assume that

our gravitational fields are weak, our sources are static, but we allow test particles to travel at any velocity

— the same limits we imposed earlier in finding the linearized Einstein equation, slightly less restrictive

than the Newtonian limit. Whereas when above we confirmed the validity of the Einstein by looking at the

Newtonian limit we found that only the g00 component came into effect, all the others were negligible, in

26

the slightly less restrictive limit here, the spatial components of the metric will become important. It turns

out that the most convenient gauge to choose leaves the components of the Einstein equation as

δG00 = 2∇2Ψ = 8πGδT00 (148)

δG0j = −12∇2ωj + 2∂0∂jΨ = 8πGδT0j (149)

δGij = (δij∇2 − ∂i∂j)(Φ−Ψ)− ∂0∂(iωj) + 2δij∂20Ψ−2sij = 8πGδTij . (150)

Just as we did above to confirm the Einstein equation in the Newtonian limit, here we can model our

static gravitational sources as a perfect fluid with no pressure, essentially as dust. Again we’ll work in the

rest frame of the source so that our energy-momentum tensor of the form eqn. 105 becomes

δTµν = ρUµUν = diag(ρ, 0, 0, 0), (151)

same as in eqn. 106 for the Newtonian limit. We now go about solving the gauge-transformed, linearized

Einstein equation with source, eqns. 148-150. Since our sources are static, all time derivatives can be dropped

and we use our stress-energy tensor eqn. 151 to solve for ωj , Ψ, and sij . The resulting metric is

ds2 = −(1 + 2Φ)dt2 + (1− 2Φ)(dx2 + dy2 + dz2) = diag(−2Φ,−2Φ,−2Φ,−2Φ) (152)

where the gravitational potential, Φ, obeys the Newtonian Poisson equation.

6 Weak-Field Metric Analysis

For our analysis, we’ll look at the the weak-field metric, eqn. 152, but we’ll assume that we can allow the

gravitational potential to vary with time, allowing our source particles to take on any velocities so long as

the weak-field condition is still satisfied. Comparing eqn. 152 with eqn. 119 we can write the perturbation

of the metric as

hµν = −2Φ(x0, x1, x2, x3)δµν . (153)

6.1 Verification of Einstein’s Equation

First we verify that allowing the potential to vary with time still solves the Einstein equation. Remember

that the linearized Einstein equations for a vacuum are, from eqn. 126,

2hµν − ∂µVν − ∂νVµ = 0 (154)

with

Vµ ≡ ∂λhλµ −

12∂µhλ

λ. (155)

27

For this perturbation, hλµ = −2Φηλ

µ and hλλ = −4Φ, so

Vµ = −2ηλµ

∂Φ∂xλ

+ 2∂Φ∂xµ

. (156)

Therefore,

V0 = 4∂Φ∂x0

(157)

Vi = 0 (158)

where i runs over space coordinates, i = 1, 2, 3.

Thus the ten linearized Einstein equations read:

• for µ = 0, ν = 0:

2h00 − 2∂V0

∂x0= −22Φ− 8

∂2Φ∂(x0)2

= 0 (159)

• for µ = 0, ν = i:

−∂V0

∂xi= −4

∂2Φ∂xi∂x0

= 0 (160)

• for µ = i, ν = j :

2hij = −2δij2Φ = 0 (161)

which amounts to

2Φ = 0 (162)∂2Φ

∂(x0)2=

∂2Φ∂xi∂x0

= 0. (163)

Since we’re interested in the gravitational fields themselves more than the potentials, we define

fµ(x0, x1, x2, x3) = − ∂Φ∂xµ

(164)

so that for µ = 1, 2, 3, fµ is the usual gravitational field and f0 is just the derivative of the potential with

respect to time. From the linearized Einstein equations, eqns. 162 and 163, we confirm that the weak field

Einstein equations will be satisfied so long as f0 is constant for all time and space.

28

6.2 Calculating Christoffel Symbols

Now that we’ve verified that our metric satisfies the linearized Einstein equation, we can find the Christoffel

symbols, Riemann tensor, and Ricci tensor in terms of the field components, fµ. For the Christoffel symbols

we have

δΓγµν =

12ηγσ

(∂hσµ

∂xν+

∂hσν

∂xµ− ∂hµν

∂xσ

)(165)

= −ηγσ

(δσµ

∂Φ∂xν

+ δδβ∂Φ∂xα

− δµν∂Φ∂xσ

)(166)

= +ηγσ (δσµfν + δσνfµ − δµνfσ) (167)

= ηγµfν + ηγνfµ − δµνηγσfσ (168)

To see the symmetries in the Christoffel symbols more clearly, we represent them as a one-by-four list

of four-by-four matrices , each matrix representing a value of γ = 0, 1, 2, 3. Within each individual matrix,

γ is constant and µ = 0, 1, 2, 3 advances moving downward and ν = 0, 1, 2, 3 advances moving to the right.

Recall that the Christoffel connection is always symmetric in its lower two indices, so each matrix will be

symmetric.

For γ = 0 we have

δΓ0µν = η0µfν + η0νfµ − δµνη0σfσ

=

−f0 −f1 −f2 −f3

0 0 0 0

0 0 0 0

0 0 0 0

+

−f0 0 0 0

−f1 0 0 0

−f2 0 0 0

−f3 0 0 0

−−f0 0 0 0

0 −f0 0 0

0 0 −f0 0

0 0 0 −f0

=

−f0 −f1 −f2 −f3

−f1 f0 0 0

−f2 0 f0 0

−f3 0 0 f0

.

Similarly, for γ = 1 we have

δΓ1µν = η1µfν + η1νfµ − δµνη1σfσ

=

0 0 0 0

f0 f1 f2 f3

0 0 0 0

0 0 0 0

+

0 f0 0 0

0 f1 0 0

0 f2 0 0

0 f3 0 0

−

f1 0 0 0

0 f1 0 0

0 0 f1 0

0 0 0 f1

29

=

−f1 f0 0 0

f0 f1 f2 f3

0 f2 −f1 0

0 f3 0 −f1

which is very similar to γ = 2, 3 with the appropriate substitutions, so we find the array representation for

the Christoffel symbols δΓγµν is

γ = 0 γ = 1 γ = 2 γ = 3−f0 −f1 −f2 −f3

−f1 f0 0 0

−f2 0 f0 0

−f3 0 0 f0

−f1 f0 0 0

f0 f1 f2 f3

0 f2 −f1 0

0 f3 0 −f1

−f2 0 f0 0

0 −f2 f1 0

f0 f1 f2 f3

0 0 f3 −f2

−f3 0 0 f0

0 −f3 0 f1

0 0 −f3 f2

f0 f1 f2 f3

6.3 Riemann Tensor Symmetries

To calculate the components of the Riemann tensor, first we should figure out which components are equiv-

alent to other ones through the symmetries of the tensor to avoid having to calculate all 256 elements. In a

similar way as the Christoffel symbols, we can use an array representation as an aid.

Recall that symmetries of the Riemann tensor, from eqns. 81-84, are

Rρσµν = −Rρσνµ (169)

Rρσµν = −Rσρµν (170)

Rρσµν = Rνµρσ (171)

Rρσµν + Rρµνσ + Rρνσµ = 0 (172)

We represent the tensor as a four-by-four matrix of four-by-four submatrices. µ = 0, 1, 2, 3 and ν =

0, 1, 2, 3 run vertically and horizontally, respectively, along the large matrix so that within each submatrix, µ

and ν are constant and ρ = 0, 1, 2, 3 advances vertically and σ = 0, 1, 2, 3 advances horizontally. Therefore,

as a whole, the Riemann tensor can be represented by

30

ν = 0 ν = 1 ν = 2 ν = 3

µ = 0

R0000 R0100 R0200 R0300

R1000 R1100 R1200 R1300

R2000 R2100 R2200 R2300

R3000 R3100 R3200 R3300

R0001 R0101 R0201 R0301

R1001 R1101 R1201 R1301

R2001 R2101 R2201 R2301

R3001 R3101 R3201 R3301

µ = 1

µ = 2

µ = 3

We know from the first symmetry, eqn. 169, that every submatrix is antisymmetric, so we can write for

the whole matrix0 R0100 R0200 R0300

0 R1200 R1300

0 R2300

0

0 R0101 R0201 R0301

0 R1201 R1301

0 R2301

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

31

From the second symmetry, eqn. 170, we know that the the whole matrix is antisymmetric, thus all of

the submatrices on the diagonal are zero and the subdiagonal submatrices will follow from the superdiagonal

submatrices.0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 R0101 R0201 R0301

0 R1201 R1301

0 R2301

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Therefore, using the first two symmetries we now only need to calculate the six superdiagonal elements of

the six superdiagonal submatrices, for a total of 36 components.

Using the third symmetry we can write

Rρσ01 = R01ρσ (173)

which relates the entries in the 01 submatrix (which we’ll label as a through f) to the 01 elements of the

other submatrices:

32

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 a b c

0 d e

0 f

0

0 b ? ?

0 ? ?

0 ?

0

0 c ? ?

0 ? ?

0 ?

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 d ? ?

0 ? ?

0 ?

0

0 e ? ?

0 ? ?

0 ?

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f ? ?

0 ? ?

0 ?

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

This leaves the remaining elements of the 02 submatrix unconstrained, so we’ll label them g through k.

Similarly, we can use the third symmetry in the form

Rρσ02 = R02ρσ (174)

to relate the elements of the 02 submatrix to the 02 elements of each of the submatrices so that we have0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 a b c

0 d e

0 f

0

0 b g h

0 i j

0 k

0

0 c h ?

0 ? ?

0 ?

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 d i ?

0 ? ?

0 ?

0

0 e j ?

0 ? ?

0 ?

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f k ?

0 ? ?

0 ?

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

33

We can continue this process for the 03, 12, 13, and 23 submatrices so that for the entire matrix we can

write 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 a b c

0 d e

0 f

0

0 b g h

0 i j

0 k

0

0 c h `

0 m n

0 p

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 d i m

0 q r

0 s

0

0 e j n

0 r t

0 u

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f k p

0 s u

0 v

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

This matrix has 21 independent elements, a through v. The fourth symmetry, eqn. 172, has just a single

consequence:

R0123 + R0312 + R0231 = f + m + j = 0 (175)

6.4 Calculating Riemann, Ricci, and Einstein Tensors

We can calculate the Riemann tensor components from the Christoffel symbols found above in terms of the

field components. We know the usual relation between the Riemann tensor and the Christoffel symbols, as

given by eqn. 80, and in the linearized limit we can neglect the factors of (Γ)2 so that we have

δRρσµν = ∂µδΓρ

νσ − ∂νδΓρµσ. (176)

Using this equation we can calculate the 21 independent components of the Riemann tensor (because of

the fourth symmetry there are actually only 20, but we’ll see that in this case, all of the components linked

by the fourth symmetry are zero anyway). Thus for the Riemann tensor as a whole we find:

34

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f0,0 + f1,1 f2,1 f3,1

0 f2,0 f3,1

0 0

0

0 f2,1 f0,0 + f2,2 f3,2

0 −f1,0 0

0 f3,0

0

0 f3,1 f3,2 f0,0 + f3,3

0 0 −f1,0

0 −f2,0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f2,0 −f1,0 0

0 −f1,1 − f2,2 −f3,2

0 f3,1

0

0 f3,0 0 −f1,0

0 −f3,2 −f1,1 − f3,3

0 −f2,1

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 f3,0 −f2,0

0 f3,1 −f2,1

0 −f2,2 − f3,3

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

where the notation fµ,ν means the partial derivative of fµ with respect to ν. However, since we know that

for our metric to solve the linearized Einstein equation, f0 must be constant for all time and space, the

Riemann tensor becomes0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f1,1 f2,1 f3,1

0 f2,0 f3,1

0 0

0

0 f2,1 f2,2 f3,2

0 −f1,0 0

0 f3,0

0

0 f3,1 f3,2 f3,3

0 0 −f1,0

0 −f2,0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 f2,0 −f1,0 0

0 −f1,1 − f2,2 −f3,2

0 f3,1

0

0 f3,0 0 −f1,0

0 −f3,2 −f1,1 − f3,3

0 −f2,1

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 f3,0 −f2,0

0 f3,1 −f2,1

0 −f2,2 − f3,3

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

35

Similarly, we can use eqn. 125,

δRµν = ∂λδΓλµν − ∂νδΓλ

µλ, (177)

to calculate the components of the the Ricci tensor:−f1,1 − f2,2 − f3,3 −f1,0 −f2,0 −f3,0

−f1,0 −f1,1 − f2,2 − f3,3 f1,2 − f2,1 f1,3 − f3,1

−f2,0 f1,2 − f2,1 −f1,1 − f2,2 − f3,3 f2,3 − f3,2

−f3,0 f1,3 − f3,1 f2,3 − f3,2 −f1,1 − f2,2 − f3,3

.

However, if we consider the situation in terms of the potentials rather than the fields, we see that all but the

diagonal elements vanish because, for the ij components, the partial derivative commutes so that we have

for example

f1,2 − f2,1 = ∂2∂1Φ− ∂1∂2Φ = 0 (178)

and for the 0i and i0 components the partial derivative commutes and we have restricted the 0 component

of the fields to be constant for all time and space so that for example

f1,0 = ∂0∂1Φ = ∂1∂0Φ = f0,1 = 0. (179)

So with these considerations in mind we find the components of the Ricci tensor to be−f1,1 − f2,2 − f3,3 0 0

0 −f1,1 − f2,2 − f3,3 0 0

0 0 −f1,1 − f2,2 − f3,3 0

0 0 0 −f1,1 − f2,2 − f3,3

.

Furthermore, we can calculate the components of the Gµν tensor which goes into Einstein’s equation.

Recall that

Gµν = Rµν −12gµνR = 8πGTµν . (180)

For linearized gravity we can write

δGµν = δRµν −12ηµνδR (181)

where we can calculate δR as

δR = Rµµ = ηµνδRµν (182)

= η00δR00 + η11δR11 + η22δR22 + η33δR33. (183)

Therefore, for the weak-field metric, the Ricci scalar is

δR = −2(f1,1 + f2,2 + f3,3). (184)

36

We can then write the components of δGµν as−2(f1,1 + f2,2 + f3,3) 0 0 0

0 −2(f1,1 + f2,2 + f3,3) 0 0

0 0 −2(f1,1 + f2,2 + f3,3) 0

0 0 0 −2(f1,1 + f2,2 + f3,3)

.

7 Geodesic Equation for a Weak-Field Metric

For a particle traveling along a geodesic parametrized by τ , the proper time, defined as the time measured

in the frame of reference of the particle, we can write the geodesic equation for the path that it will follow,

xσ:d2xσ

dτ2= −Γσ

αβ

dxα

dτ

dxβ

dτ. (185)

We can use the chain rule to writedxα

dτ=

dt

dτ

dxα

dt(186)

= γ

[1,

dx

dt,dy

dt,dz

dt

](187)

= γ[

1 vx vy vz

], (188)

using the definition of γ from special relativity:

γ =1√

1− v2/c2(189)

which relates a time dτ in a frame of reference moving at velocity v with respect to another frame of reference

measuring time dt so that we have (the usual special relativistic time dilation equation)

dτ =1γ

dt. (190)

Therefore, in the weak-field limit, we can write the geodesic equation for the index σ = 1 as

d2x1

dτ2= −γ2

[1 vx vy vz

]−f1 f0 0 0

f0 f1 f2 f3

0 f2 −f1 0

0 f3 0 −f1

1

vx

vy

vz

(191)

ord2x1

dτ2= γ2

[f1 − 2f0vx − vx(2~f ·~v) + f1(v2)

]. (192)

We can find similar results for the other two spatial components, resulting in

d2~x

dτ2= γ2

[~f − 2f0~v − ~v(2~f ·~v) + ~f(v2)

](193)

= γ2[~f(1 + v2)− 2~v(f0 + ~f ·~v)

](194)

= ~f − 2~vγ2(f0 + ~f ·~v). (195)

37

In the non-relativistic limit v � 1, this gives the Newtonian formula

d2~x

dt2= ~f (196)

which is exactly what we expect to get, since it relates the the acceleration of the particle to the fields, and

since we think of fields as forces acting on a test particle, in this case a mass, we find the usual Newton’s

second law: ~F = m~a.

Calculating the geodesic equation for the σ = 0 component, we find

d2x0

dτ2= f0 + 2γ2~g·~v (197)

For any function g(τ), we can write

dg

dτ=

dt

dτ

dg

dt= γ

dg

dt(198)

d2g

dτ2= γ

d

dt

[γ

dg

dt

](199)

= γ2 d2g

dt2+ γ

dg

dtγ3v

dv

dt(200)

so that the spatial components ared2~x

dτ2= γ2 d2~x

dt2+ γ4v

dv

dt(201)

and the time component isd2x0

dτ2= γ2 + γ4v

dv

dt. (202)

So, combining eqns. 195 and 201, the geodesic equations for the spatial components are

d2~x

dt2=

1γ

~f − ~v

(2f0 + f~g·~v + γ2v

dv

dt

)(203)

and combining eqns. 197 and 202 we find the geodesic equations for the time component to be

0 = f0 + γ2(−1 + 2~f ·~v)− γ4ddv

dt. (204)

8 Conclusions

We have analyzed the Einstein equations in the limit as a weak, classical field in order to gain insight into a

more intuitive understanding of the structure of the equations as a parallel to the structure of the Maxwell’s

equations of electromagnetism which function in a very similar way. We conclude that whereas some texts

refer to the the Riemann and Ricci tensors as the general relativistic embodiment of the gravitational fields,

we find that, rather, it is the Christoffel symbols which contain the field components and therefore best

represent the fields.

38

This work opens up the possibility for several ideas for future work to arrive at more conclusive interpre-

tations of the resulting treatment in the weak-field limit. Although we found the geodesic equations for test

particles in the weak-field limit, finding the actual path a particle takes with respect to the fields is difficult to

interpret. However, we can use a geometrically derived equation, the Raychaudhuri equation, for assistance.

The equation provides a direct relation between a set of geodesics, the paths that test particles will take,

and the Ricci tensor, which for us encodes the field components via the Christoffel symbols. We can imagine

a small sphere of particles moving along a group of neighboring geodesics (a congruence of geodesics) with

4-velocities Uµ = dxµ

dτ . We then monitor the expansion parameter of the ball, θ = ∇µUµ, which tells how

the sphere is shrinking or growing over spacetime. In general if we want to test the change between parallel

geodesics, we define a separation vector V µ which points from one geodesic to a neighboring one and we find

that such a vector obeysDV µ

dτ≡ Uν∇νV µ = Bµ

νV ν (205)

where

Bµν = ∇νUµ. (206)

We can interpret Bµν as measuring to what extent the separation vector fails to be parallel transported

along the group of geodesics.

We can break down the Bµν tensor into symmetric and antisymmetric parts and the symmetric part into

a trace and trace-free component and these components correspond to the the expansion θ of the sphere of

particles, the shear σµν , which is the distortion of the sphere into an ellipsoid, and the rotation ωµν , which

measures the twisting of the sphere of particles through spacetime. To see how the congruence of geodesics

varies over time we find the covariant derivative of Bµν given by

DBµν

dτ≡ Uσ∇σBµν = −Bσ

νBµσ −RλµνσUσUλ (207)

and taking the trace of this and using the decomposition of Bµν we find the Raychaudhuri equation:

dθ

dτ= −1

3θ2 − σµνσµν + ωµνωµν −RµνUµUν . (208)

This equation provides a way of directly relating the fields via the Ricci tensor to the behavior of free-falling

particles. It is derived from purely geometric properties and doesn’t need external reference to the Einstein

equation. In the simplest, Newtonian, limit we can begin with all particles at rest and close to each other so

that the rotation, expansion, and shear terms are all initially zero. If we look in locally inertial coordinates

xσ in which Uσ is in its rest frame then we have Uσ = (1, 0, 0, 0) and therefore RµνUµUν = R00. In this

limit we find eqn. 208 reduces todθ

dτ= −R00 (209)

so in terms of the field components we have

dθ

dτ= f1,1 + f2,2 + f3,3 = ~∇ · ~f. (210)

39

We find that the expansion of the sphere of particles over time is dependent on the divergence of the fields

as we would expect. However, whether it is possible to glean more useful information from this equation by

looking at a limit not so restrictive as the Newtonian one ,which cuts out all but the 00 component of the

Ricci tensor, needs more consideration.

Another issue to be pursued further is that we can find a general solution to the linearized Einstein

equation with source written as

2hµν = −16πTµν (211)

where we have used the Lorentz gauge condition in the form

∂hµν

∂xν= 0 (212)

and where we define hµν as

hµν ≡ hµν −12ηµνhλ

λ. (213)

Each component of hµν obeys a separate wave equation with its own source term and the general solution

to this problem by solving for the perturbations yields

hµν(t, ~x) = 4∫

d3x′ [Tµν(t′, ~x′)]ret|~x− ~x′|

(214)

where [Tµν(t′, ~x′)]ret indicates that the stress-energy tensor must be evaluated at the retarded time t′ =

tret ≡ t− |~x− ~x′| to take into account relativistic effects which dictate that information can travel no faster

than the speed of light.

This equation, eqn. 214, looks remarkably like the equation from electromagnetism for the vector potential

when we take into account the same kinds of relativistic effects,

~A =µ0

4π

∫d3x′ [ ~J ]ret

|~x− ~x′|. (215)

This similarity lends itself to the possibility of analyzing the structure of gravitational fields. We know that

a changing electric field generates a magnetic field, and specifically, if we look at a stationary point charge

we observe only an electric field but if we transform to a frame moving at some constant velocity relative to

the static frame of the particle, we see both electric and magnetic fields. Using similarities between eqns. 214

and 215 we can analyze whether a parallel situation for a point mass yields any similar structures, whether

we observe “gravito-electric” and “gravito-magnetic” field components. However, the details of this analysis

are still under current study.

Overall, both of these ideas for future progress aim to reach a better understanding of the structure of the

Christoffel symbols, what governs the organization of components and can they be grouped into any higher

structures? We know for electromagnetism that we group the fields into electric and magnetic components

where in the non-relativistic limit, the electric fields dominate in size over the magnetic fields. Now that we

40

know the Christoffel symbols are the generalization of the fields for general relativity, and we have a field-

theoretic structuring of them, we seek to find whether or not a similar understanding of the components of

the Christoffel symbols can be found.

9 Appendix A: Stress-Energy Tensor

We have mostly ignored the stress-energy tensor in our analysis, sticking with looking at the fields in vacuum,

away from sources that make the stress-energy tensor nonzero. However, there is interesting structure in the

stress-energy tensor as well that emerges through comparison with electromagnetism.

We analyze the stress-energy tensor for a box of free, noninteracting particles. We define the positions

of the particles using the 4-vector

xα = [ct, ~x] (216)

where ~x is the 3-vector for position, and their momenta are defined as

pα = mdxα

dτ(217)

= mdt

dτ

d

dt[ct, ~x] (218)

= mγ[c,d~x

dt] (219)

= [E/c, ~p] (220)

where ~p is the 3-momentum and γ is the usual factor from special relativity.

We know from electromagnetism that that we can write the electric charge density of a system of n

particles with positions ~xn(t) and charges qn as

ρ(~x, t) =∑

n

qnδ3(~x− ~xn(t)) (221)

and the electric current density as

~J(~x, t) =∑

n

qnd~xn

dtδ3(~x− ~xn(t)). (222)

Collectively, then, we may write the charge density and current density as a single 4-vector:

[ρ, ~J/c] =1c

∑n

qnd

dt[ct, ~xn(t)]δ3(~x− ~xn(t)) (223)

Now returning to the stress-energy tensor, we can define the tensor components for our box of n particles

in the same way as for electric charge and current densities but here instead of “electric charge density and

current” we use “four-momentum density and current.” Comparing to the structure of the current 4-vector,

41

the 0µ elements of the stress-energy tensor are for the energy components and the iµ elements are for the

momentum components. For the 00 component we have the energy density, E/c given by∑n

En

cδ3(~x− ~xn(t)). (224)

To find the 0i component we use the same method as for electric charges and find the energy current density

given by1c

∑n

En

c

d~xn

dtδ3(~x− ~xn(t)). (225)

Correspondingly, for the i0 components we want the momentum density. For the x component, px, we write∑n

pn,xδ3(~x− ~xn(t)) (226)

and similarly for the y and z components. For the ij components, then we look at the momentum current

density given by (again for the x component)

1c

∑n

pn,xd~xn

dtδ3(~x− ~xn(t)). (227)

and similarly for the y and z components.

Putting all of these together, we see that the energy-momentum tensor can be written

∑n

En

cEn

c2d~xn

dt

pn,x pn,xd~xn

dt

pn,y pn,yd~xn

dt

pn,z pn,zd~xn

dt

δ3(~x− ~xn(t)) (228)

We can rewrite the energies and momenta using relations from special relativity:

En = γnmnc2 (229)

~pn = γnmnd~xn

dt(230)

so the energy-momentum tensor can be rewritten

∑n

γnmn

c d~xn

dt1c

dxn

dtd~xn

dtd~xn

dt1c

dyn

dtd~xn

dt1c

dzn

dtd~xn

dt

δ3(~x− ~xn(t)) (231)

As it should be, the tensor is symmetric which matches the symmetry of the Ricci tensor in Einstein’s

equation. Furthermore, in the non-relativistic limit, the 00 component dominates and the 0i (and i0) are

small and the ij components are extremely small, also mirroring the symmetry of the Ricci tensor in the

weak-field limit.

42

10 Appendix B: Electromagnetism with Differential Forms

Above we simply referenced the electromagnetic field and dual field tensors without explaining their structure.

It turns out this structure becomes clear once we are familiar with the mechanics of differential forms.

Differential forms are a class of tensors which we call p-forms, which are (0, p) tensors that are totally

antisymmetric. Therefore, in the language of differential forms, we call scalars 0-forms and dual vectors

1-forms. p-forms become useful because they can be manipulated without recourse to external geometry.

A few necessary concepts we’ll define are the wedge product, the exterior derivative, and the Hodge star

operator.

The wedge product of a p-form A and a q-form B forms the (p + q)-form A∧B which is found by taking

the antisymmetrized tensor product:

(A ∧B)µ1···µp+q=

(p + q)!p!q!

A[µ1···µpBµp+1···µp+q ]. (232)

Note that

A ∧B = (−1)pqB ∧A. (233)

The exterior derivative of a p-form A produces a (p+1)-form dA defined as the appropriately normalized

and antisymmetrized partial derivative:

(dA)µ1···µp+1 = (p + 1)∂[µ1Aµ2···µp+1], (234)

and the simplest example is the exterior derivative of a 0-form φ, which is just the gradient:

(dφ)µ = ∂µφ. (235)

If we want to take the exterior derivative of a wedge product, we use a modified form of the Leibniz rule

so that for a p-form A and a q-form B we have

d(A ∧B) = (dA) ∧B + (−1)pA ∧ (dB). (236)

The exterior derivative is particularly useful because although it involves only partial derivatives, the

result is tensorial. A helpful property that emerges is the fact that for any form A, the double exterior

derivative is always zero because partial derivatives commute. Therefore for any arbitrary p-form A we have

d(dA) = 0. (237)

The last operation to define is the Hodge star operator which maps, on an n-dimensional manifold, a

p-form A to a (n− p)-form ∗A:

(∗A)µ1···µn−p=

1p!

εν1···νp

µ1···µn−pAν1···νp. (238)

43

Differential forms are somewhat limited in their applicability since the results only apply to tensors that

are covariant and totally asymmetric. However, the electromagnetic field tensor is one such tensor, and the

mechanics of differential forms can help to elucidate the structure of the electromagnetic field tensor, the

arrangement of its elements between electric and magnetic field components. The structure of the field tensor

comes from the nature of the relationship between electric fields, magnetic fields, and sources. Grouping the

four Maxwell’s equations into source and source-free equations we see that

∇ ·E =1ε0

ρ (239)

∇×B− µ0ε0∂E∂t

= µ0J (240)

and

∇ ·B = 0 (241)

∇×E +∂B∂t

= 0. (242)

We can relate the field tensor to its components in some basis dxµ by

F =12Fµνdxµ ∧ dxν . (243)

. We know first off that Fµν must be antisymmetric because the wedge product of two one forms is antisym-

metric (eqn. 236). What we want is to find the appropriate values for the components of Fµν . Since we’re

looking for the divergence and curl of the electric and magnetic fields, it makes sense to look at some kind

of derivative of the field tensor components and in the case of differential forms, this will be the exterior

derivative. We first expand the field tensor in terms of its components using eqn. 243 to find

F = F01dt ∧ dx + F02dt ∧ dy + F03dt ∧ dz + F12dx ∧ dy + F13dx ∧ dz + F23dy ∧ dz (244)

using the antisymmetry of the field tensor and the wedge product. Note that because of the antisymmetry

of the wedge product, for 1-forms we can always write

dxµ ∧ dxν

{= 0 if µ = ν

6= 0 if µ 6= ν.(245)

If we take the exterior derivative of the field tensor from eqn. 244 we find, using eqn. 245

dF =∂F01

∂ydy ∧ dt ∧ dx +

∂F01

∂zdz ∧ dt ∧ dx +

∂F02

∂xdx ∧ dt ∧ dy +

∂F02

∂zdz ∧ dt ∧ dy

+∂F03

∂xdx ∧ dt ∧ dz +

∂F03

∂ydy ∧ dt ∧ dz +

∂F12

∂tdt ∧ dx ∧ dy +

∂F12

∂zdz ∧ dx ∧ dy

+∂F13

∂tdt ∧ dx ∧ dz +

∂F13

∂ydy ∧ dx ∧ dz +

∂F23

∂tdt ∧ dy ∧ dz +

∂F23

∂xdx ∧ dy ∧ dz (246)

which if we rearrange and condense using eqn. 233 we find

dF =(

∂F01

∂y− ∂F02

∂x+

∂F12

∂t

)dt ∧ dx ∧ dy +

(∂F02

∂z− ∂F03

∂y+

∂F23

∂t

)dt ∧ dy ∧ dz

+(

∂F01

∂z− ∂F03

∂x+

∂F13

∂t

)dt ∧ dx ∧ dz +

(∂F23

∂x− ∂F13

∂y+

∂F12

∂z

)dx ∧ dy ∧ dz. (247)

44

We see that in eqn. 247 the last term looks like a divergence term and the first three terms all have a

similar structure with two space derivatives and one time derivative. If we compare this with the two pairs

of Maxwell’s equations, eqns. 239-240 and eqns. 241-242, we see that we could choose either pair to fit the

components of the electric and magnetic fields to the structure of eqn. 247. For simplicity, we choose the

source-free pair and we define the ij components of the field tensor for the magnetic field components since

in the source-free Maxwell’s equations it is the magnetic field components that undergo a divergence, and

we choose the 0i components to be the electric field components since it is those that undergo a curl. The

time derivatives are then appropriately on the magnetic field components. Therefore we have, making sure

the minus signs work out, for the field tensor

Fµν =

0 −E1 −E2 −E3

E1 0 B3 −B2

E2 −B3 0 B1

E3 B2 −B1 0

(248)

which is exactly the way we defined the electromagnetic field tensor in eqn. 30, though without the factors

of 1/c which will enter to make the proportionality constants in the Maxwell’s equations come out right.

Therefore, we see that the source-free Maxwell’s equations are equivalent to writing

dF = 0. (249)

The other interpretation above for the source-free Maxwell’s equations, eqn. 43 that ∂[µFνλ] = 0, also

naturally falls out the treatment using differential forms, due to the antisymmetry of p-forms. Again we can

take the exterior derivative of the field tensor:

dF =12∂ρFµνdxρ ∧ dxµ ∧ dxν . (250)

Because the result of the exterior derivative must be totally antisymmetric, the components of dF must by

antisymmetric. Writing this out explicitly, and using the fact that Fµν is antisymmetric we see that

dF =12

13!

2 (∂ρFµν + ∂µFνρ + ∂νFρµ) dxρ ∧ dxµ ∧ dxν = 0 (251)

which we know is equal to zero from eqn. 249. Since so long as we keep µ 6= ν 6= ρ, the basis for the exterior

derivative of the field tensor will not be zero, then for Maxwell’s equation to hold, the coefficient for each

combination of µ, ν, and ρ must vanish on its own. Therefore, we have that

∂ρFµν + ∂µFνρ + ∂νFρµ = 0, (252)

the same as eqn. 43.

For the equations with source, just as above, we want to look for some mechanism that will swap the

positions of the electric and magnetic field components to produce the dual field tensor (note that above

45

we went in the reverse direction and used the field tensor for the equations with source rather than those

without, but the essential physics is the same, just the mechanism of exterior derivatives behaves slightly

differently). The Hodge star operator performs this task perfectly. The action of the Hodge star operator

on basis vectors is

∗(dxµ1 ∧ dxµ2 ∧ · · · ∧ dxµp) =

√|g|

(n− p)!εµ1···µp

νp+1···νndxνp+1 ∧ · · · ∧ dxνn (253)

where g = detgµν and

εµ1µ2···µn =1√|g|

εµ1µ2···µn (254)

and for flat space with the Minkowski metric,√|g| =

√| − 1| = 1. Therefore, for example, we have

∗(dx0 ∧ dx1) = −dx2 ∧ dx3 (255)

∗(dx0 ∧ dx3) = −dx1 ∧ dx2 (256)

∗(dx2 ∧ dx3) = dx0 ∧ dx1 (257)

and similar equations for all other combinations of 2-form basis vectors.

Therefore, taking the Hodge star of the field tensor we find

∗F = F01dy ∧ dz + F02dz ∧ dx + F03dx ∧ dy − F12dt ∧ dz + F13dt ∧ dy − F23dt ∧ dz. (258)

We see that taking the Hodge star has effected the change from the field tensor to the dual field tensor

in the treatment above. More explicitly, we can take the exterior derivative of the Hodge star dual of the

field tensor, and we see that the form of the field equations with source, eqns. 241 and 242, emerge:

d ∗ F = (~∇ · ~E)dx ∧ dy ∧ dz + (~∇× ~B − ∂0~E)xdy ∧ dz ∧ dt

+(~∇× ~B − ∂0~E)ydz ∧ dx ∧ dt + (~∇× ~B − ∂0

~E)zdx ∧ dy ∧ dt. (259)

We want to relate this to the current 4-vector which is a 1-form

J = −ρdt + J1dx + J2dy + J3dx (260)

but the exterior derivative of the field tensor (and Hodge dual field tensor) is a 3-form. Since we’re working

in 4-dimensional space, we can again use the Hodge star operator to turn the current 1-form into a 3-form

since, for example,

∗dt =12ε0 ijkdxi ∧ dxj ∧ dxk = −dx ∧ dy ∧ dz (261)

Therefore, the Hodge dual of the current 4-vector is

∗J = ρdx ∧ dy ∧ dz + J1dt ∧ dy ∧ dz + J2dt ∧ dx ∧ dz + J3dt ∧ dx ∧ dy (262)

Since the bases of each term are linearly independent, we can match the corresponding basis vectors of d ∗F

with ∗J and we find that the Maxwell’s equations with source are given by

d ∗ F = ∗J (263)

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11 Acknowledgments

As my first experience with theory and starting the project with virtually no knowledge of general relativity,

I owe immense thanks to my advisor Dan Styer for his incredible patience, knowledge, understanding, and

willingness to take on a rather unusual project both in content and time frame.

I would also like to thank my differential geometry professor, Atalay Karasu of Middle East Technical

University, Ankara, Turkey, for his insights and quirky sense of humor.

As a culmination of my five years spent at Oberlin, this project owes great thanks to the Oberlin Physics

and Astronomy Department and community for years of encouragement and support.

And of course thanks to friends and family for constant moral support.

47

References

[1] Sean M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Pearson Education,

Inc. publishing as Addison Wesley, 2004.

[2] David J. Griffiths. Introduction to Electrodynamics. Pearson Education, Inc., third edition, 1999.

[3] James B. Hartle. Gravity: An Introduction to Einstein’s General Relativity. Pearson Education, Inc.,

publishing as Addison Wesley, 2003.

[4] Ray d’Inverno. Introducing Einstein’s Relativity. Oxford University Press, 1992.

[5] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman and Co.,

1973.

[6] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of

Relativity. John Wiley & Sons, Inc., 1972.

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