zotero/storage/I2S6M2D4/.zotero-ft-cache

Physics 8.962

Massachusetts Institute of Technology Department of Physics

Spring 2000

Stress-Energy Pseudotensors and Gravitational Radiation Power
c 2000 Edmund Bertschinger. All rights reserved.

1 Introduction
In a curved spacetime there is, in general, no globally conserved energy-momentum. Aside from the case of scalars like electric charge, tensors defined in different tangent spaces cannot be added in nonflat spacetime.
However, if we are willing to drop the requirement that all our equations be tensor equations, then it is possible to define a globally conserved energy-momentum. Unlike a tensor equation, the form of the conservation laws we derive will change depending on the coordinate system. However, that makes them no less valid. Consider, for example, the three-dimensional equation ∇ · B = 0 and its integral form, dS · B = 0. When written in Cartesian coordinates these equations have an entirely different form than when they are written in spherical coordinates. However, they are equally correct in either case.
The approach we will follow is to derive a conserved pseudotensor, a two-index object that transforms differently than the components of a tensor. Unlike a tensor, a pseudotensor can vanish at a point in one coordinate system but not in others. The connection coefficients are a good example of this, and the stress-energy pseudotensors we construct will depend explicitly on the connection coefficients in a coordinate basis.
Despite this apparent defect, pseudotensors can be quite useful. In fact, they are the only way to define an integral energy-momentum obeying an exact conservation law. Moreover, in an asymptotically flat spacetime, when tensors can be added from different tangent spaces, the integral energy-momentum behaves like a four-vector. Thus, we can use a pseudotensor to derive the power radiated by a localized source of gravitational radiation.
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2 Canonical Stress-Energy Pseudotensor

The stress-energy tensor is not unique. Given any T µν such that ∇µT µν = 0, one may always define other conserved stress-energy tensors by adding the divergence of another

object:

T µν → T µν + ∇λSµνλ where Sµνλ = −Sµλν .

(1)

Clearly the stress-energy tensor need not even be symmetric. Recall the equation of local stress-energy conservation:

∇µT

µ ν

=

(−g)−1/2∂µ

√ −gT

µ ν

−

ΓλνµT

µ λ

=

0

.

(2)

Because of the Γ terms, Gauss’ theorem does not apply and the integral over a volume

does not give a conserved 4-vector.

lawHisow∂eµv(√er,−agsτwµνe)

will see, it is = 0 instead

possible

to

define

a

pseudotensor

τ

µ ν

of

∇µτ

µ ν

= 0.

The two equations

whose conservation are identical in flat

spacetime but the first one can be integrated by Gauss’ law while the second one cannot.

Moreover, there are many different conserved stress-energy pseudotensors, just as there

are many different conserved stress-energy tensors.

This section will show how to construct conserved stress-energy pseudotensors and

tensors, illustrating the procedure for scalar fields and for the metric. The key results

are given in problem 2 of Problem Set 7.

2.1 Stress-energy pseudotensor for a scalar field
We begin with a simple example: a classical scalar field φ(x) with action

S[φ(x)] = L(φ, ∂µφ) d4x .

(3)

The Lagrangian density depends on φ and its derivatives but is othewise independent

of the position. In this example we suppose that L includes no derivatives higher than

first-order, but this can equal a scalar times the

bfaecetaosril√y −gegnewrhailcizhedis.

Note that if S is a scalar, then L must needed to convert coordinate volume to

proper volume. We are not assuming flat spacetime — the treatment here is valid in

curved spacetime.

Variation of the action using δ(∂µφ) = ∂µ(δφ) yields

δS =

∂L ∂φ

δφ

+

∂

∂L (∂µφ)

∂µ(δφ)

d4x

=

∂L ∂φ

−

∂µ

∂L ∂(∂µφ)

δφ(x) d4x +

surf

∂L ∂(∂µφ)

δφ(x) d3Σµ

.

(4)

2

The surface term comes from integration by parts and is the counterpart of the pδq

endpoint contributions in the variation of the action of a particle. Considering arbitrary

field variations δφ(x) that vanish on the boundary, the action principle δS = 0 gives the

Euler-Lagrange equation

∂µ

∂L ∂(∂µφ)

−

∂L ∂φ

=

0

.

(5)

Now, by assumption our Lagrangian density does not depend explicitly on the co-

ordinates: ∂L/∂xµ = 0. This implies the existence of a conserved Hamiltonian den-

sity. To see how, recall the case of particle moving in one dimension with trajectory

q(t). In this case, time-indepence of the Lagrangian L(q, q˙) implies dH/dt = 0 where

H = q˙(∂L/∂q˙) − L.

In a field theory q(t) becomes φ(x) and there are d = 4 (for four spacetime dimensions)

parameters for the field trajectories instead of just one. Therefore, instead of dH/dt = 0,

the conservation law will read ∂µHµ = 0. However, given d parameters, there are d conservation laws not one, so there must be a two-index Hamiltonian density Hµν such that ∂µHµν = 0. Here, ν labels the various conserved quantities.
To construct the Hamiltonian function one must first evaluate the canonical momen-

tum. For a single particle, p = ∂L/∂q˙. For a scalar field theory, the field momentum is

defined similarly:

πµ

=

∂L ∂(∂µφ)

.

(6)

In a simple mechanical system, the Hamiltonian is H = pq˙ − L. For a field theory the

Lagrangian is replaced by the Lagrangian density, the coordinate q is replaced by the

field, and the momentum is the canonical momentum as in equation (6).

The divided

cbayn√o−nigc:al

stress-energy

pseudotensor

is

defined

as

the

Hamiltonian

density

τ

µ ν

≡

(−g)−1/2

[(∂ν φ)πµ

−

δ

µ ν

L]

.

(7)

The reader may easily check that, as a consequence of equation (5) and the chain rule

∂µf (φ, ∂νφ) = (∂f /∂φ)∂µφ + [∂f /∂(∂νφ)]∂µ(∂νφ), the canonical stress-energy pseudoten-

sor obeys

(−g)−1/2∂µ

√ −gτ

µ ν

=0.

(8)

2.2 Stress-energy pseudotensor for the metric
The results given above are easily generalized to an action that depends on a rank (0, 2) tensor field gαβ instead of a scalar field. Let us suppose that the Lagrangian density depends only on the field and its first derivatives: L = L(gαβ, ∂µgαβ). (This excludes the Einstein-Hilbert action, which depends also on the second derivatives of the metric. We will return to this point in the next section.) The Euler-Lagrange equations are simply equation (5) with φ replaced by gαβ.

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Because our field has two indices, the canonical momenta have two more indices than

before:

πµαβ = ∂L .

(9)

∂µ (gαβ )

The stress-energy pseudotensor for the metric, hence for the gravitational field, is there-

fore

τ

µ ν

≡

(−g)−1/2

(∂ν gαβ)πµαβ − δµν L

.

(10)

It obeys equation (8).

2.3 Covariant symmetric stress-energy tensor

Given the results presented above, it is far from obvious that there should be a conserved stress-energy tensor. How does one obtain a well-defined stress-energy tensor that obeys a covariant local conservation law?
Let us start from the action for the metric, with a Lagrangian density that may depend on gµν and on any finite number of derivatives. Then, after integration by parts, one may write the variation of the metric as the functional derivative plus surface terms:

δS[gµν] =

δS δgµν

√ δgµν(x) −g

d4x

+ surface terms .

(11)

(As long as we are varying only the metric, we are free to use either gµν or its inverse gµν. Variations of the two are related by δgµν = −gµαgνβδgαβ.)
Now we use the fact that the action is a scalar, hence invariant under arbitrary coordinate transformations. We make an infinitesimal coordinate transformation xµ → xµ − ξµ(x), which transforms the metric components gµν → gµν + ∇µξν + ∇νξµ. Other tensor fields are also modified by this coordinate transformation: ψ → ψ + Lξψ where Lξ is the Lie derivative and ψ is any other field (e.g. the electromagnetic potential Aµ). However, we assume that other fields obey their equations of motion so that δS/δψ = 0
and therefore δψ = Lξψ makes no change to the action. Thus, we obtain

0 = δS =

2

δS δgµν

∇µξν

√−g

d4x

=

−

∇µ

2

δS δgµν

ξν √−g d4x .

(12)

Since this must hold for any ξν(x), we obtain ∇µT µν = 0 with T µν ≡ 2δS/δgµν = −2gµα gν β δS/δgαβ .
The same derivation, without the requirement that other fields obey their equations of motion, when applied to the Einstein-Hilbert action implies ∇µGµν = 0.
Note that the derivation of a conserved stress-energy tensor is quite different from the
derivation of a conserved pseudotensor. However, both rely on the fact of coordinate-
invariance. In the pseudotensor case this arises in the assuption that L depends on

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position only through the fields; ∂µL = 0 at fixed values of the fields and their derivatives. This led to a set of conserved Hamiltonian densities. In the tensor case, coordinate-

invariance was used to demand that δS = 0 under a diffeomorphism gµν → gµν + Lξgµν, with the implication that ∇µ(δS/δgµν) = 0.
One may ask, are the two approaches more directly related? Does one follow from

the other?

Because there are many possible stress-energy tensors and pseudotensors, I am not

sure whether a general relationship holds between them all, except to say that all stress-

energy conservation laws are implied by the field equations. However, in particular cases

the relationship is more straightforward. In the usual derivation of the electromagnetic

stress-energy tensor, for example, one starts with the canonical approach and obtains a

τ

µ ν

that

is

not

symmetric

(e.g.

Jackson

1975,

section

12.10).

A

term

is

then

added

like

that of equation (1) so as to symmetrize the stress-energy. The result is the same as the

covariant stress-energy −2δS/δgµν. We will consider the case of the gravitational action

in the next section.

3 Schr¨odinger Action and Stress-Energy Pseudotensor

The Einstein-Hilbert action is linear in the Ricci scalar, which is linear in the second derivatives of the metric:

16πG SEH[gµν] = =

d4x √−g gµνRµν

d4x

√ −g

gµν

∂αΓαµν − ∂µΓααν − γµν

,

(13)

where

γµν

≡

ΓαβµΓβαν

−

Γαµν

γ

β αβ

.

(14)

For convenience we also define γ ≡ √−g gµνγµν. Then, following Dirac (1975), we rewrite

the integrand of the Einstein-Hilbert action:

√ −g

gµν

Rµν

=

√ ∂α −g

gµν Γαµν − gαν Γµµν

+

Γααν

√ ∂µ( −g

g

µν

)

−

Γαµν

√ ∂α( −g

gµν

)

−

γ

= ∂αwα + γ , wα ≡ √−g gµν Γαµν − gαν Γµµν .

(15)

The divergence term contributes only a surface term that does not affect the functional derivative. Thus, the Einstein equations follow from a modified action known as the Schr¨odinger action (Schr¨odinger 1950):

SGS[gµν ]

=

1 16πG

d4

x

√ −g

gµν γµν

=

d4x LGS(gαβ, ∂µgαβ)

(16)

5

where γµν is defined in equation (14). Note that γµν is not a tensor, and therefore SGS is not a scalar. It differs from the Einstein-Hilbert action by a surface term which is also not a scalar but which makes no contribution to the equations of motion. Thus, the Schr¨odinger action has the same functional derivative as the Einstein-Hilbert action:

δSGS δgµν

=

Gµν

.

(17)

One may regard the Schr¨odinger action as the Einstein-Hilbert action minus the second-derivative terms. From the viewpoint of tensors, these terms are crucial because one can always choose coordinates so that the connection coefficients vanish at a point, hence γµν = 0 at a point. However, one cannot do this everywhere (unless the spacetime is globally flat) and therefore one cannot transform SGS away. At most, one can add boundary terms that have no effect on the equations of motion.
Given our strong emphasis on tensors in general relativity, one may well ask whether it is valid to use an action that is not a scalar under arbitrary coordinate transformations. The answer at the classical level is yes, of course, as long as it gives the correct equations of motion. From this perspective, the Schr¨odinger action is just as good as the EinsteinHilbert action. Moreover, it depends only on the metric and its first derivatives (through the definition of the connection coefficients in a coordinate basis). As such, it enables us to construct a stress-energy psuedotensor.
To derive the conserved stress-energy pseudotensor, we follow the approach of Section 2. After some algebra we get the canonical field momentum,

16πG

πµαβ √−g

=

Γµαβ

− gµ(αgβ)κΓλκλ − gαβgµκgλσΓ[κλ]σ

,

(18)

where the metric is used to raise and lower indices on the connection coefficients: Γαβµ = gακgβλΓκλµ, etc.
Before giving the stress-energy pseudotensor, we note another relation between the
Einstein-Hilbert and Schr¨odinger Lagrangians. From equations (15) and (18), we find 16πG gαβ πµαβ = −wµ and therefore

LEH = LGS − ∂µ gαβ πµαβ .

(19)

Thus, the two Lagrangians are related by a simple transformation equivalent to L → L − (d/dt)(pq) for the elementary mechanics of a particle in one dimension. This extra term obviously contributes nothing but boundary terms to the variation of the action.
Now we give the Schr¨odinger stress-energy pseudotensor, which follows from equation (10):

16πG

τ

µ ν

=

2gαβΓµακΓκβν −2gα(µΓλ)αν Γκλκ+2gα[µΓλ]αλΓκνκ−δµν gαβ

ΓκλαΓλκβ − ΓκαβΓλκλ

.

(20)

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Including all terms in the action (matter plus gravitational), the equations of motion imply (problem 2 of Problem Set 7)

(−g)−1/2∂µ

√ −g

(τ

µ ν

+

T

µ ν

)

=0

(21)

where

T

µ ν

is

the

stress-energy

tensor

of

the

matter.

Equation

(21)

may

be

regarded

as

a

statement of energy conservation for gravitation and matter, since it can be integrated

over a 4-volume bounded by surfaces of constant x0 to give

dPν dx0

=−

√ dS ni −g

τiν + Tiν

,

Pν(x0) ≡

d3x

√ −g

V

τ

0 ν

+

T

0 ν

(22)

where the surface integral is taken at fixed x0 over the surface (with normal one-form ni) bounding the 3-volume V . Although equations (21) and (22) are not tensor equations, they are exact in every coordinate system.

4 Gravitational Radiation Emitted Power

We can use the stress-energy pseudotensor to determine the energy flux density of gravitational radiation in a calculation similar to the derivation of the Poynting flux for electromagnetic radiation. The full calculation is not presented here, although we set it up. We will assume that the gravity waves are weak and the spacetime is nearly Minkowski. We ignore the non-radiative scalar and vector parts of the linearized metric and consider only the transverse-traceless part due to gravitational radiation:

ds2 = −dt2 + (δij + 2sij)dxidxj , si i = 0 , ∂isi j = 0 .

(23)

Note that MTW and most other authors write hij = 2sij; I’ve inserted the factor of 2 so that sij is the strain matrix and not twice the strain. The Minkowksi metric is used to raise and lower all indices. (See the 8.962 notes Gravitation in the Weak-Field Limit.)
With this metric, the nonzero connection coefficients are

Γ0 ij = Γi 0j = Γi j0 = ∂tsij ,

Γk ij

=

1 2

(∂isj

k

+

∂j sik

−

∂k sij )

.

(24)

Substituting this into equation (20) gives the energy density and energy flux density

16πGτ

0 0

=

(∂tsij)2 + (∂ksij)2 − 2(∂isjk)(∂jsik) ,

16πGτ i 0 = 4(∂tsjk)(∂jsik) − 2(∂tsjk)(∂isjk) .

(25)

The reader may check applying quation (21),

wtheatu,sein√v−acgu≈um1

where (−∂t2 + ∂2)sij in linear theory.

=

0,

∂tτ

0 0

+

∂iτ

i 0

= 0.

In

7

The stress-energy pseudotensor we have derived is not symmetric. It may be symmetrized by adding a derivative term (cf. Jackson 1975, section 12.10):

tµν

=

τ

µ ν

+

√1−g

∂λ

√−g Sλµν

where Sλµν = −Sµλν .

(26)

Landau and Lifshitz (1975, section 96) give an alternative procedure for deriving a symmetry stress-energy pseudotensor. Their tensor is quite complicated and I don’t know the Sλµν that transforms it to the Schr¨odinger pseudotensor. A symmetric stress-energy pseudotensor is useful because it allows one to formulate a conservation law for angular momentum. I will not go into that here (see MTW chapters 19 and 20).
Another advantage of the Landau-Liftshitz pseudotensor is the simple form it takes for a plane gravitational wave:

tµν

=

1 32πG

(∂µhαβ )(∂ν hβ α)

.

(27)

Consider a plane wave propagating in the 1-direction. The nonzero strain components

are s22 = −s33 and s23. In vacuum, these components are functions of t−x1 and therefore

∂1sij = −∂tsij. It follows immediately that, in the transverse-traceless gauge of equation

(23),

t00

=

−t10

=

t11

=

1 8πG

s˙2ij

(28)

where a dot denotes ∂t. In section 36.7 of the text, MTW use this together with the solution of the wave equation for sij (including the source) to derive the gravitational radiation power crossing a sphere of radius r. For a nonrelativistically moving source

the results is the famous quadrupole formula, equation (36.23) of MTW.

References
[1] Dirac, P. A. M. 1975, General Theory of Relativity (New York: Wiley).
[2] Jackson, J. D. 1975, Classical Electrodynamics, 2nd Edition (New York: Wiley).
[3] Landau, L. D. and Lifshitz, E. M. 1975, The Classical Theory of Fields (Oxford: Pergamon Press).
[4] Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1970, Gravitation (San Francisco: Freeman).
[5] Schr¨odinger, E. 1950, Space-Time Structure (Cambridge University Press).

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