Localization of gravitational energy

Foundations of Physics, Vol. 15, No. 10, 1965

Localization of Gravitational Energy

Nathan Rosen 1

Received August 16, 1984

In the general relativity theory gravitational energy-momentum density is usually described by a pseudo-tensor wffh strange transformation properties so that one does not have localization o f gravitational energy. It is proposed to set up a gravitational energy-momentum density tensor having a unique form in a given coordinate system by making use o f a bimetric formalism. Two versions are considered: (1) a bimetric theory with a flat-space background metric which retains the physics of the general relativity theory and (2) one with a background corresponding to a space o f constant curvature which introduces modifications into general relativity under certain conditions. The gravitational energy density in the case of the Schwarzschild solution is obtained,

1. INTRODUCTION

In the general theory of relativity the energy-momentum density of the gravitational field is treated differently from that of other physical fields. For fields other than gravitation this is described by a tensor as, for example, the Maxwell tensor in the case of the electromagnetic field. With such a tensor there is physical significance to the localization of energy. For example, the energy density at every point remains unchanged if one carries out arbitrary transformations of the space coordinates. In the case of gravitation, on the other hand, one usually describes the energy-momentum density by means of a pseudo-tensor or "complex" with strange transformation properties under general coordinate transformations. Although, under suitable conditions, one can assign a physical significance to the total energy and momentum of the gravitational field, their localization is in

Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel.

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general meaningless. Considerable effort has been made to improve this situation, (1) but it still appears to be unsatisfactory.

The purpose of the present work is to set up an energy-momentum density tensor for the gravitational field by the use of a bimetric formalism. This idea was proposed in the past, (z3) but not enough attention was paid to the necessity of imposing conditions that would give this tensor a unique form in a given coordinate system.

The bimetric formalism is associated with the idea proposed a long time ago, (4,s) of modifying the general theory of relativity by introducing a background metric in addition to the usual metric, so that one has "bimetric general relativity." Since it was proposed to take the background metric as describing flat space-time, it was found that its introduction did not change the physical contents of the theory, but it did improve the formalism and provided a gravitational energy-momentum density tensor.

Subsequently it was proposed to take the background metric as corresponding to a space-time of constant curvature associated with the structure of the universe. (6'7) This introduced modifications into the theory, the effect of which was to remove some of the singularities appearing in solutions of the Einstein equations. However, for gravitational phenomena in the solar system the modified equations gave the same predictions as the usual equations.

Most of the present work is devoted to the bimetric theory with a flat- space background metric, since this retains the physics of the Einstein general relativity theory. However, the case of the constant-curvature background metric will also be considered.

2. T W O BIMETRIC THEORIES

The general idea of a bimetric theory is that at each point of space- time, in a given coordinate system, there are two metric tensors, a physical metric g~ describing gravitation and interacting with matter, and a background metric Y,v which may interact with the physical metric but not with matter. With the help of the metric g,v one can define the Christoffel symbols {~v} and hence covariant differentiation (g-differentiation), denoted by a semicolon, and one can form the Riemann curvature tensor R~w. In the same way from Y,v one gets the Christoffel symbols F a covariant differentiation (y-differentiation), denoted by a bar (I), and the curvature tensor P~

From the two Christoffel symbols one gets the tensor

~ = ~ r ~. (1)

Localization of Gravitational Energy 999

which has the same form as { ~ } but with ordinary derivatives replaced by 7-derivatives. This gives two useful vectors

A~; = ½g~g~lu = K~/t¢ (2)

where a comma denotes an ordinary derivative and

= (3)

and also ~v g~V A~,, = -(1/~)(~cg )l~ (4)

There is also the tensor

~- - ~- - P ~ ( 5 )

which has the same form as R ~ but again with ordinary derivatives replaced by ?-derivatives. Contracting (5) gives

K~ = R~,~- P~,~ (6)

Two different kinds of bimetric theories have been proposed. The simplest (2"4'5) is the one in which the background metric ~ describes a flat spece-time so that

Hence

and

The Einstein field equations,

P~.~ = 0 (7)

2 m 2

K.~ = Ruv (9)

1 G~,, =- Ru~ - 7guvR = - 8 g T u v (10)

with T.v the energy-momentum density tensor of the matter, can therefore be written also in the form

K~v - ½guvK = -8~T~v (11 )

where the left side is obtained from (10) by the replacement of ordinary derivatives with ^/-derivatives. It should be stressed that (t0) and (1 t) are equivalent because of (7).

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Up to this point g~ and ~',v are unrelated. Let us suppose we are dealing with matter occupying a finite region so that far from it the space-time is fiat. Let us take g~ = 7,v as the boundary condition at infinity. However, the solution of the field equations for g~. contains four arbitrary functions because of the Bianchi identities satisfied by the left members of the Einstein equations. One should therefore impose four conditions on the solution, conditions which are consistent with the boundary condition. For example, one can take

(~g~)~ = 0 (12)

which is satisfied by guy = 7~v and corresponds to the De Donder condition sometimes used in general relativity,

[(-g) ' /2g~V],v = 0 (13)

However (12) and (13) are different, in general, since (12) is a tensor equation, whereas (13) is noncovariant and restricts the coordinate system. If 7~ = t/~v, the metric of special relativity, then (12) goes over into (13). It should be remarked that, although (12) is useful and convenient, it is arbitrary, and other auxiliary conditions are possible. (2)

Since Eqs. (10) and (11) are equivalent, the introduction of the bimetric formalism does not change the physical contents of the general relativity theory, but it provides some improvements in the formalism, as we shall see.

Recently a second version of a bimetric theory was considered. (6'7) On the basis of certain arguments it was proposed to take the background metric 7,~ as describing a space-time of constant curvature (related to the large-scale structure of the universe) instead of a fiat space-time. Instead of (7) one has

Pj.j~,o = (1/a2)(~,o/~ - },~.~-~,~) (14)

where a is a constant chosen to be of the order of the size of the universe (or, more exactly, of the order of 1/H, where H is the Hubble constant). Hence

P~,v = (3/a 2) 7~,v (15)

so that, in view of (6), Eqs. (10) and (11) are now no longer equivalent. In the proposed bimetric theory one takes (11) as the field equations, rather than (10), thus obtaining a modification of the general relativity theory. Making use of (6) and (15), one can rewrite (11) in the form

G~,~= S~-- 8rcT,.. (16)


where

S~,v = (3 /a2 ) (y~- ½g~ 7~ g ~) (17)

Hereafter indices will be raised and lowered with g~. The Bianchi identities give

G~V;~-=0 (18)

Since matter is assumed to interact with gu~, and not with 7;~, one can s]how (for example, with the help of a variational principle) that

Hence one gets from (t6)

One finds that

T~'~';v = 0 (19)

S~V;v = 0 (20)

S~;,~ = (3/a2~c) 7~(~cg;~)t . (21)

Hence (20) is equivalent to (12). The situation in the present bimetric theory is different from what we had in the earlier version. There Eq. (12) was chosen arbitrarily; here it is a consequence of the field equations.

It was found, (6'7) on the basis of Eqs. (12), (14), and (t6) and a suitable choice of 71,v, that there are cosmological models without singularities. It was also found (8) that there is a solution for the field of a particle essentially agreeing with the Schwarzschild solution outside of the Schwarzschild sphere but without any singularity inside.

At the present time it does not appear to be possible to distinguish between the two bimetric theories on the basis of observation. For some the theory with the flat-space background may be more attractive because it leaves the general relativity theory essentially unchanged. For others--those who feel uneasy about having singularities in a theory--the constant-curvature background version may seem to be preferable. Since both theories provide the possibility of localizing gravitational energy, they will both be considered for this purpose. In the following section the flat- space approach (which is the simpler one) will be investigated.

3., GRAVITATIONAL ENERGY WITH A FLAT-SPACE B A C K G R O U N D

Since the Einstein field equations involve second derivatives of g,v, it is natural to look for expressions for the gravitational energy-momentum

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density that contain only first derivatives. One has here an analogy with electromagnetic theory if one thinks of the metric as analogous to the electromagnetic potential. Two such expressions have been obtained, one by Einstein, (9) the other by Landau and Lifshitz. (~°) They are both pseudo-tensors, and each one can be made to vanish at an arbitrary point by a suitable choice of coordinates. However, the expression given by Landau and Lifshitz is preferable since it is symmetric and therefore enables one to describe the conservation of angular momentum.

With the help of the bimetric formalism one can convert the Landau and Lifshitz expression into a gravitational energy-momentum density tensor having a physical significance in every coordinate system. To do this, let us follow the procedure used by these authors, but with the Einstein equations (10) replaced by the bimetric field equations (11). Let us assume that ~ corresponds to a fiat space-time, so that these two sets of equations are equivalent, and the change from (10) to (11) does not change the physical contents of the theory. If we write (11) in the contravariant form, it is found that this can be written

(8rc/~c2)( - Q~I;o~ + t~) = -8nT"~ (22)

where

Q ~ o = (1 /16n) (g~ ~" - ~,~gx~) (23)

with

g ~ = xgUV, gu~ = (l /x) g,~ (24)

and t u~ is quadratic in the first y-derivatives of g"L One can therefore take as the total energy-momentum density tensor

O "~ = te2T ~ + t s*~ (25)

so that O ~ then satisfies the relation

o . v = Q X , W l ~ ~ (26)

from which it follows that

Ouvlv = 0 (27)

if one makes use of the symmetry properties of Qx"~. One can also introduce a "superpotential" y,v~ given by

yu~ = QX~Wl) " = _ y~v (28)


so that

8 ~ = Y~V°l, (29)

which is the same as (26), so that (27) follows. From the expression for K ~ and (2.2) one can calculate t uv. The result

is

where

16rct~ = g~la g~-~la + ½ g ~ l . g-%~l~ - g~al~ g~-~t~

g ~ l y ~'~#- I~ -- g~z~lc~ g~ l~ - - K,#l£,v/1£2

+ g~'Vl~ g~alt~ + ~,UVL (30)

L ~- ~ V ( A ~ A~ ~

- --~ i-;. - ~ "½~¢.~¢;/t¢ 2 (31) = ½ g ~ l ° g - I ; , - z g I~g I~_" , ,_

A line under an index indicates that it is to be raised or lowered with ~,v or g:,v after differentiation.

Let us suppose that we have a physical system in a finite region of space, and we have for it a solution of the Einstein field equations with space-time fiat at large distances. Hence g,v approaches some fiat-space raetric 7°v at infinity. Let us now choose the fiat background metric 7~ so that it also goes to 7°~ at infinity. However, this requirement does not fix 7,v, since one can still carry out transformations of 7u~ corresponding to coordinate transformations (but without transforming g~,~) which leave 7°~ at infinity unchanged. Such transformations can introduce four arbitrary functions into 7uv. Changing 7,~ in this way will change tuL In order to fix the latter one must impose four more conditions, and we have seen in the previous section that (12) can be used for this purpose.

To be sure, it was remarked that (12) is arbitrary if one has a fiat- space background, as we have here. However, it was shown that, if one has 7u~ .corresponding to a space of constant curvature characterized by a scale parameter a, then (12) is a consequence of the field equations and is not arbitrary. If we now consider fiat space-time as the limit of the curved space-time as a ~ ¢ ~ , then (12) remains valid in the limit. We take tlherefore

g~l~ = 0 (32)

as an auxiliary condition that ties the two metrics together. This fixes the tensor t u~ for a given coordinate system.

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With (32) holding, (30) has the form

167zt~ = g~'~l¢ gV-~fa + ½g~Iug-%~l~ - g~l . g~gl~

- g~"~l~_ g~'~l~ - ~c,~,~_i ~c2 + ~"~L (33)

where L is given by (31). In this form, and with (32) holding, t ~ is the energy-momentum density tensor of the gravitational field.

Let us go back to the energy relation (27). It is a tensor equation. However, it is not a conservation law, since it does not involve an ordinary divergence. If one carries out a coordinate transformation so as to get 7,~ = r/~, the metric of special relativity, then (27) goes over into

O~'v~=o (34)

so that we have an ordinary divergence. If this is integrated over a volume V bounded by a surface S, one gets with the help of Gauss's theorem

(d/dt)fvO~°d3x=-jsO"~dS k (k= 1,2, 3) (35)

describing the conservation of energy (/~=0) and of momentum (p = 1, 2, 3). Moreover, if one makes use of (29), taking into account the antisymmetry shown in (28), one can write, again with the help of Gauss's theorem,

fv 0~0 d3X = JS y~ok dSic

so that the energy and momentum can be determined from the behavior of the metric far from the system.

With 7,,=t/u~, one can define the angular-momentum-density Lorentzian tensor

(36)

Taking the divergence, we get from (34)

m~"v ~ = O ~a - O ~u (37)

and since O "a= 0;%

m~"~ = 0 (38)


describing the conservation of angular momentum. In view of the explicit appearance of the coordinates in (36) one cannot convert in ~uv into a general tensor.

Let us go back to the gravitational-energy tensor t ~ of (33) and consider its physical significance. In the Einstein general relativity the metric gu~ describes both the gravitational and the inertial fields in accordance wJith the principle of equivalence. When we write the equations of motion of a test particle in the form

d2x~ dx~ dx~ (39) d,

the right side, the (four-dimensional) force per unit mass acting on the particle, may have a contribution from the true gravitational field arising from the presence of masses and a contribution from the inertial field arising from the acceleration of the frame of reference. In a small neighborhood one can choose coordinates for which {~} = 0, corresponding to a freely faJ[ling reference frame, so that these two contributions cancel each other, On the other hand, in the bimetric theory the background metric ~'.v describes the inertial field, and the two metrics together enable one to separate out the true gravitational field. One can write (39)

dZx ~ dx ~ dx ~ dx ~ dx ~ ds 2 = - r ~ r 3 ds ds A~a-~s ds (40)

The first term on the right is the inertial force, and the second is the true gravitational force. Since the latter is a tensor, it cannot be made to vanish by any choice of coordinates unless it vanishes for all coordinates. The tensor #v describes the energy and momentum distribution of the true gravitational field.

In working with t ~ there are two possible procedures. One possibility is to begin with a given background tensor y~v and look for a solution of the Einstein equations [in the form of either (10) or (11)] for g~ satisfying the auxiliary condition (12) and being asymptotically equal to Y.v. The ot]her possibility is to begin with a solution of the Einstein equations for g~v which is asymptotically flat and then look for 7u~ which satisfies (12) and is asymptotically equal to g~ . For this purpose we can make use of (1) and (4) so as to write (12) in the form

g ~ = 0 (41)

o r

g~Pl'~ = g~{ ~ } (42)

as an equation for ?:~,~.

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As a simple example, let us consider the Schwarzschild solution, and let us write

ds2 = g .v dx ~ dxV, d ~2 = 7~v dx~ d xv (43)

Corresponding to the first approach, let us take the usual polar coordinates so that

with

dot 2 : d t 2 _ dr 2 _ r 2 dg2 2

d [ 2 2 = d O 2 "t- sin 2 0 d~b 2

(44)

(45)

Then the form of the Schwarzschild solution for which (12) holds is found to be given by (4)

ds 2= [ ( r - m ) / ( r + m ) ] dt 2 - [ ( r + m ) / ( r - m ) ] dr 2 - ( r+rn)2dg22 (46)

On the other hand, we can take the Schwarzschild solution in the usual form,

ds 2 = (1 - 2 m / R ) dt 2 - (1 - 2 m / R ) -1 d R 2 _ R 2 d[22 (47)

It is obvious that now we must have

da 2 = dt 2 - d R 2 - ( R - - m) 2 dO 2 (48)

in order to satisfy (12) or (42). If one calculates the energy density t oo by using (33), one finds that for

(44) and (46) one gets

16rot °° = -m Z[ (1 + m/r)2/r4(1 - m / r )2] (14 - 16m/r + 6m2/r 2) (49)

whereas for (47) and (48) one has

16nt °° = - EmZ/R4( 1 - 2 m / R ) 2 ( 1 - m / R ) 6 ] ( 14 - 4 4 m / R + 3 6 m 2 / R z) (50)

These two expressions are of course equal with

R = r + m (51)

4. GRAVITATIONAL ENERGY WITH A CONSTANT-CURVATURE BACKGROUND

Let us now derive the gravitational energy-momentum density tensor for the case in which the background metric describes a space of constant curvature, so that (14) holds.


Accordingly, let us go back to the field equations (1 !), but written in the contravariant form. One notes that

2(K u~' - ½g"~K) = gUOg~,+j~+ + g~g~°l+~ - g~g~lo+

-- g~g~l~+ -- g~gV~g~ g~lo+

+ g ~ g ~ g ~ g ~ l ~ + + qS~'v (52)

w]~ere q5 ~ is an expression involving only first derivatives. It will be recalled that ";-differentiation is now noncommutative. Corresponding to (22), one can write the field equations

l c , ~ + t ~ ) (8rc/~2)( - ½Q;"v~l;.o - 7z Io~.

1 erkttT( +rv'c v~: 1 t r v z [ r~l tr~ rr~tff "~ - - + Zs ~6 Io+--g I+-)+a~ ~+ I+~--s +o+)--8rcT~ (53)

with QZU~ given by (23). With the help of (14) and the relation

' ~ " ~ - e u ~ P ~ ( 5 4 )

this can be put into the form

~ '~ = K2T "~ + t ~ + (1 /16rca2) (~"v~7~ ~ - 4 ~ , v ~ 7 ~ ) (55)

where

~ v = Q;+U~+;++ = Q;+~v++~ (56)

A calculation gives

so that, if one defines

one has

~ v l v = (1 /a2 ) (?~Q~a~) l ~

OUr = ~uv _ (1/a 2) ~ QUery

O~vlv = 0

where the total energy-momentum density tensor is given by

19~ +v = t c 2 T t+v q - t uv _ (3/167+a 2) ~u~v~7~ ~

corresponding to (25).

(57)

(58)

(59)

(60)

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It will be recalled that in the present case the field equat ions lead to the condi t ion (32). Hence t #~' is again given by (33). We can define the gravi ta t ional e n e r g y - m o m e n t u m density tensor

0"v = t 'v - (3/167za2) g~g~ '7~ , (61)

so that

0 #v = 1£2T 'uv 't- 0 #v (62)

The second term on the right of (61) can be regarded as the gravi ta t ional e n e r g y - m o m e n t u m density arising f rom the interact ion between the physical metr ic g~v and the background metr ic 7m,.

I t is clear that the present discussion is significant only if one is dealing with cosmological problems. F o r systems tha t are small on the scale of the universe, such as the solar system, one can neglect the curvature associated with the background metric; hence one can neglect terms involving 1/a in the above equations. This brings us back to the discussion of the previous section.

R E F E R E N C E S

1. C. MNler, Mat. l;ivs. Medd. Dan. Vid. Selsk. 35, No. 3 (1966). 2. N. Rosen, Ann. Phys, (N.Y.) 22, 1 (1963). 3. F. H. J. Cornish, Proc. R. Soc. London A 282, 358 (1964). 4. N. Rosen, Phys. Rev, 57, 147 (1940). 5. A. Papapetrou, Proc. R. lrish Acad. A 52, 11 (1948). 6. N. Rosen, Gen. Relativ. Gravit. 12, 493 (1980). 7. N. Rosen, Found. Phys. 10, 673 (1980). 8. D. Falik and N. Rosen, Astrophys. J. 239, 1024 (1980). 9. A. Einstein, Ann. Phys. 49, 769 (1916).

10. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1971), 3rd edn., p. 304.

Localization of gravitational energy

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