General Relativity and Gravitation, Vol. 12, No. 2, 1980

Gravitational Energy-Momentum: The Einstein Pseudotensor Reexamined I T. N. PALMER 2 Department o f Astrophysics, Oxford University, South Parks Road, Oxford, U.K. Received May 25, 1979

Abstract By using a suitable two-point scalar field, a covariant formulation of the Einstein pseudotensor is given. A unique choice of scalar field is made possible by examining the role of linear and angular momentum in their correct geometric context. It is shown that, contrary to many text-book statements, linear momentum is not generated by infinitesimal coordinate transformations on space-time. Use is made of the nonintersecting lifted geodesics on the tangent bundle, T~, to space-time, to define a globally regular three-dimensional Lagrangian submanifold of TN, relative to an observer at some point z in space-time. By integrating over this submanifold rather than a necessarily singular spaceilke hypersurface, gravitational linear and angular momentum, relative to z, are defined, and shown to have sensible physical properties.

On this the Einstein centenary year, it is appropriate to reexamine a topic in general relativity that was certainly of great interest to Einstein himself. The concept of energy-momentum in relativity theory has been the subject of considerable investigation, dating back almost to the inception of the theory itself [1 ]. In view of the apparent failure of the various pseudotensors to describe "local" gravitational energy-momentum, the consensus is now that the energy-momentum concept is only meaningful for asymptotically flat spacetimes. However, as a result of studies, particularly b y Dixon [ 2 - 4 ] , on equations of motion of extended bodies in general relativity, we can now show that the 1This essay received an honorable mention from the Gravity Research Foundation for the Year 1979-Ed. Present address: Meteorological Office, Bracknell, Berkshire, U.K. 149 0001-7701/80/0200-0149503.00/0 9 1980 Plenum Publishing Corporation

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gravitational energy-momentum content of a region of space-timeis well defined. Certain preliminary results of this, and some detailed calculations referred to in this article, are contained in Reference 5. In this paper, we follow a hypothetical discussion between Sagredus, who presents the orthodox view, and Salvatius, who attempts to show that orthodoxy should reexamine its case. Sagredus. I can find many gravitational pseudotensors, yet they are all inadequate. Consider, for example, the Einstein pseudotensor aL 167rta b - _ _

aagcd - L6a b

~Ob&a)

where Oa denotes partial differentiation, and L

a b - Fa b c P ab a ) g c d = (PcdPab

is the first-order Lagrangian. This is in Canonical form, and therefore superficially compatible with other field-theoretic energy-momentum tensors, yet it is completely noncovariant. Indeed, at any point in space-time one can find a frame of reference where ta b is identically zero. Salvatius. I agree that a definition of gravitational energy-momentum at a point of space-time is meaningless, though in a sense this is true for any radiative field. What I propose is that we consider the energy-momentum of a region of space-time. Sagredus. This does not help. If I integrate ta b over a spacelike region, for example, I am still left with a noncovariant object. Salvatius. No, the object in fact can be made covariant. If we consider a suitable scalar field, f, with arguments at two points of space-time, z and x, say, then we can define a coordinate system with origin at z such that the coordinates of x are

X m *=V m (2)f(z, x) ~ f m (z, x)

(1)

where Vm (z) denotes covariant differentiation at z, and * denotes equality in the coordinate system. In the following, indices a, b, c, d act at the point x, and are raised and lowered by the metric at x. Indices m and n act at z, and are raised and lowered by the metric at z. Now, let/c a be given by the ten-parameter family of "coordinate" vector fields a rnn k a * S a m am +X[m6n]W

for arbitrary am , Wmn , and where, by the above conventions Xn *=g n m ( z ) f m ( z , x ) = fn(Z,X)

Using these vector fields the Einstein pseudotensor can be written in the form aL 16nta(k) - O(Oagea) s

- Lka

(2)

GRAVITATIONAL ENERGY-MOMENTUM

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Differentiating (1), with Va(X)denoting covariant differentiation at x,

~ma = Va(X)f m ( z , x ) - I m a ( z , x ) and ,-1

-1

~am =f%,(z,x)

where farn is the matrix inverse o f f m a (which is assumed to exist, hence the word "suitable" in an earlier paragraph). Furthermore $

-['mab = V b ( x ) f m a(Z, x) = fmab(Z , X)

(3)

and using this ,

-1

~bgca-- 2 ga(cf m a ) V o f am

(4)

It is shown in Reference 5 that associated withfmab(Z, Jc) is a nonlocal, , nonmetric covariant derivative Va, ,and in terms of this derivative, the righthand side of (4) can be written as Vbgca 9 Va is the unique covariant derivative that reduces to ~a in the coordinate system defined by f. Finally, the tenparameter family of coordinate vector fields can be written as -1

-1

k a ( x ) = f a m a m (z) + f[ m f a n ] wrnn(z)

(5)

for arbitrary a m (z), wmn (z). Hence using (3), (4), and (5), and integrating (2) over x, the set of points x representing a spacelike region of space-time, we form a ten-parameter covafiant object at z. Sagredus. Yes, I agree that covariance is not a problem, but your choice of f ( z , x) is not unique; indeed, modulo the Poincar6 group, there are as many suitable functions f a s there are coordinate systems. Your construction has achieved nothing. Salvatius. You are right, though what we have done sets the scene for a unique construction. Let us approach the problem more geometrically. Spacetime comprises the manifold ~ll, the metric g, and the metric connection V. This connection defines a decomposition of the tangent space at any point p in the tangent bundle T~ll of ~ into a vertical and horizontal subspace. The connection also defines a horizontal vector field W on T~IIwhose integral curves, the socalled lifted geodesics, are everywhere nonintersecting (see for example Reference 6). Hence for z E ~1I and Uz C Tz ~t, W defines a regular (Lagrangian) submanifold Az c T~I by the smooth mapping

Xz: Uz

' Az

which takes an initial point in Uz a unit parameter distance along an integral curve of W. In general, Az may cut a fiber of T ~ more than once, and above a point of ~ conjugate to z, A z will not be transverse to these fibers. Hence a compound mapping of Xz with the bundle projection ~rwill only be regular when mapping onto a normal neighborhood of z. Let us temporarily restrict ourselves

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to such a neighborhood, though later in our discussion we will extend the formalism globally. Now consider the ten-parameter family of vector fields given by (5). What is the underlying concept we are trying to express by using these vector fields? The first term on the right-hand side of (5) attempts to generate linear momentum, the second term angular momentum. A generator of angular momentum, relative to z, lies naturally in the vertical subspaces V(p), where p is in the fiber above z. In terms of a basis Vm = ~/OX m

of V(p), generators of the Lorentz group, Lrn n (and hence generators of angular momentum), are given by Lmn = X[ m 8/OXn ]

The vector fields ka(x) generating angular momentum are then given by mapping Lmn into Tx ~. The previous discussion gives us this mapping-it is Qr, o Xz ,). The horizontal subspace H ( p ) is complementary to V(p) in the sense that Tp T $ = V(p) 9 H ( p ) and therefore a basis H m of H(p) is a natural candidate for the four generators of linear momentum. Again, the corresponding vectors fields ka(x) are obtained by mapping Hm by Or, o Xz,). Let us now relate this with the previous two-point scalar field construction. For X ~ Tz~ , then (~ o x~) ( x ) = x

if X m = -om(z,X)

where o(z, x ) is the world function [7]. It may be shown that [5], in terms of o, Vm is mapped under ~ - & , o Xz,) to -1 t~ ( Vm ) = - 0 am a/ax a (6) so that Lmn is mapped to ~1

r (L,.,,) = o Im o % j ~/~x ~

(7)

and finally H m is mapped to -1

I~ (H m ) = - 0 anonrn a/Ox a

(8)

[ohm (z, x) ~ vn(z)Vm (z) o(z, x)]. We can now relate this to our previous discussion by putting f(z, x) = - o(z, x)

The coordinates referred to by = are, of course, normal coordinates. It is important to examine the vector field ka(x) in this geometric light.

(9)

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GRAVITATIONAL ENERGY-MOMENTUM

From (6), (7), and (9), then (5) becomes ka(x) = t~(Vm)aa m + t ~ ( r m n ) a w mn

However, as we have discussed, the first term is not suitable as a generator of linear momentum; it should be replaced by ~ (Hm) a m , giving

k~(x) = g,(Hm)akm(z) + g'(Lm,,)avmkn(z)

(10)

where we have used the coincidence limits of a to replace a m and wmn by the values of k m and its derivative at z. An important point to realize is that qJ(Hm ) is not expressable in normal coordinates with origin at z; indeed, there is no coordinate expression for ~ (Hm). We were wrong to look for a ten-parameter family of vector fields that are constant in a coordinate system: in other words, generators of infinitesimal coordinate transformations are not suitable generators of linear momentum. Our representation in terms of the world function requires we remain in a normal neighborhood of z. In general, however, this is not necessary. Consider an observer at z with 4-velocity Um . Let • ~ correspond to that subset of Tz orthogonat to Um , and let Az be the Lagrangian submanifold Xz(• ~). Now redefine ~b as the compound mapping (p o Xz.), where p is the projection map of TqT~ into H(q), and q E Az. As before, qJ maps to zero above a point conjugate to z. Now ~ . restricted to H(q) is one-to-one onto TTr(q)~, and together with its pull-back rr* may be extended to maps of tensor fields. Hence the contravariant metric at lr(q)may be mapped to a tensor g E H(q) | H(q). Similarly we may lift V to operate on horizontal tensors at q. The point of this is that we can now integrate over the globally regular three-dimensional submanifold Az, rather than the corresponding singular spacelike hypersurface lr (Az) on space-time. Dropping the bars over g, and bearing in mind the redefinition of r the total gravitational linear momentum 1-form, and angular momentum 2-form, relative to the observer at z, are deemed by .

167rpm(Z)= f~

aL

(_-"*'~s

L~(Hm)a) d~a

A z kO~agbe

bL 16~r~mn(z)=f A z [=,--'-.c o(Lmn)gbc - Lt~(Lmn)a] [.OVagbc

dZ,a

Notice how the letters m and n, which labeled bases in TTz~, can now be thought of as abstract indices [8]. The gravitational energy-momentum of a finite (nonzero) volume of space-time is defined by starting with a suitable subset of J-Tz ~. Sagredus. You have convinced me of the uniqueness of your geometric construction, but does it have any important physical significance? For example, what happens to your gravitational energy-momentum if space-time admits a Killing vector?

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PALMER Salvatius. This is an interesting question. Dixon [2] has shown that if space-

time admits a Killing vector then it will be of the form given by (10). Hence, in view of the Lie derivative in (2), the presence of a global Killing vector will signify the vanishing of a certain component of gravitational energy-momentum. This is physically a very satisfactory result. For example, we do not expect static space-times to radiate gravitational energy, or axisymmetric space-times to radiate gravitational angular momentum. Some details of this are given in [5 ]. Sagredus. Well, you certainly have breathed new life on the Einstein tensor. Surely this will have consequences on the ability to discuss observer dependent quantum field effects in general space-times, where an isolated observer does not determine a preferred vector field. Salvatius. This is true, but it is work for the future. References

1. 2. 3. 4. 5. 6.

Einstein, A. (1916).Ann. Phys. (Leipzig), 49, 769. Dixon, W. G. (1970). Proc. R. Soc. London, A314, 499. Dixon, W. G. (1970). Proc. R. Soc. London, A319, 509. Dixon, W. G. (1974).Phii. Trans. R. Soc. London, A277, 59. Palmer, T. N. (1978). Phys. Rev. D, 18, 4399. Hawking, S. W., and Ellis, G. F. R. (1973). The Large Scale Structure o f Space-Time. (Cambridge University Press). 7. Synge, J. L. (1971). Relativity: The General Theory (North-Holland, Amsterdam). 8. Penrose, R. (1968). Battelle Rencontres. ed. DeWitt, C. M., and Wheeler, J. A. (Benjamin, New York).