COMPARISON ENERGY I.

OF

VARIOUS

GRAVITATIONAL

TENSORS M.

Dozmorov

UDC 530.12 : 531.51

ha this p a p e r we m a k e a c o m p a r i s o n of v a r i o u s e n e r g y t e n s o r s of the gravitational field, o b tained by introducing v a r i o u s a u x i l i a r y s t r u c t u r e s into the discussion. Notwithstanding d i f f e r e n c e s in initial positions, the best-known investigations have led to the s a m e result. In conventional t r e a t m e n t s the lack of c o v a r i a n c e in the definition of quantities that a r e c o n s e r v e d in the g e n e r a l t h e o r y of r e l a t i v i t y has been thought to be a n e c e s s a r y c h a r a c t e r i s t i c of a g r a v i t a t i o n a l field -- a m a n i f e s t a t i o n of the principle of equivalence. This position w a s r e j e c t e d by B e r g m a n n [1], who showed how one can obtain c o v a r i a n t c o n s e r v e d quantities by introducing a u x i l i a r y s t r u c t u r e s into the d i s c u s s i o n (a second m e t r i c t e t r a d s , a field of p r e f e r r e d v e c t o r s , etc. ). I m m e d i a t e l y a f t e r this, B e r g m a n n ' s p r o g r a m w a s r e a l i z e d by a n u m b e r of i n v e s t i g a t o r s . The p r e s e n t p a p e r is devoted to a c o m p a r i s o n of r e s u l t s of these investigations. Following the lines m a r k e d out by B e r g m a n n , K o m a r [2] obtained the g e n e r a l i z e d e n e r g y v e c t o r E~ [~1 = 1 (~.,; .,_ ~.~;:,);~,

(1)

that s a t i s f i e s the c o n s e r v a t i o n law E:~::,. = O,

(2)

At the s a m e t i m e the global s c a l a r quantity

is c o n s e r v e d . K o m a r a s s u m e s that (1) r e p r e s e n t s the e n e r g y density and that (3) r e p r e s e n t s the total e n e r g y of the s y s t e m if one c h o o s e s as the p r e f e r r e d v e c t o r field K i l l i n g ' s t i m e - l i k e v e c t o r field, d e t e r m i n e d by the conditions ;~;., -4- "

= 0

(4)

o r equivalently : ; ~ : = 0 , ~.,;~: --R,~,,r',~.

(5)

Such a choice of the p r e f e r r e d v e c t o r field is dictated by the analogy with L o r e n t z - i n v a r i a n t t h e o r i e s in f i a t s p a c e w h e r e the s y m m e t r y of the field equations with r e s p e c t to infinitesimal coordinate t r a n s f o r m a tions x~ = x# + E~/~ (motions) g e n e r a t e s c o n s e r v a t i o n laws having a c l e a r p h y s i c a l meaning. Of c o u r s e , making such an analogy lacks sufficient foundation, but it is worthy of notice. It should be pointed out that a t i m e - l i k e Killing field does by no m e a n s e x i s t in any R i e m a n n space, but only in static o r s t a t i o n a r y s p a c e s . In o t h e r c a s e s it has been p r o p o s e d [3] that quasi-Killing v e c t o r fields be e m p l o y e d , s a t i s f y i n g weakened v e r s i o n s of conditions (4), (5), f o r e x a m p l e , the c o n f o r m a l - K i U i n g conditions ~,~: ~, ~

+ ~",;',,

=

"t.g~,,

(6)

All-Union Scientific R e s e a r c h Institute of Opticophysical M e a s u r e m e n t s . T r a n s l a t e d f r o m I z v e s t i y a VUZ. F i z i k a , No. 6, pp. 12-15, June, 1973. Original a r t i c l e submitted F e b r u a r y 2, 1972.

9 19 75 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

746

o r v e c t o r fields which, like Killing fields, a r e n o r m a l to the e x t r e m e h y p e r s u r f a c e s .

;~ =

0.

(7)

Then, in p a r t i c u l a r , the total e n e r g y (3) will be positive. T h e r e a r e also o t h e r definitions of quasi-Killing v e c t o r fields. However, aside f r o m insufficient justification of these conditions, t h e r e r e m a i n s a doubt a s to w h e t h e r they do not impose r e s t r i c t i o n s on the g e o m e t r y of the manifold itself. As the investigations [4] have shown, in a n u m b e r of c a s e s e x p r e s s i o n (3) can be interpreted as the total e n e r g y of the s y s t e m , but (1) does not have the meaning of e n e r g y density. This defect was r e m o v e d in the succeeding investigations of P i r a n i , Denen, Cattaneo, et al. Following P i r a n i [5], we a s s u m e that t h e r e is a privileged r e f e r e n c e f r a m e in the space, whose b a s i s test bodies a r e under the action of external f o r c e s that completely c o m p e n s a t e the action of the gravitational field. If the field of the velocity f o u r - v e c t o r uP of the b a s i s of the r e f e r e n c e f r a m e f o r m s a n o r m a l c o n g r u e n c e , then on the basis of the above d i s c u s s i o n one can r e g a r d f ~ - u ~ ; v u U as the f o r c e (intensity) of the gravitational field and, in analogy with the Newtonian t h e o r y of gravitation, the total gravitational m a s s within a closed s u r f a c e S with n o r m a l np is defined as E ['#l = -- 1

(8)

~fi~n,~dS.

"K

A f t e r simple t r a n s f o r m a t i o n s this e x p r e s s i o n r e d u c e s to the f o r m

E[u~ l = -1- ( ( w~;' = w;~%~ u~ d V , X

(9)

t

w h e r e the integration is c a r r i e d out o v e r the space bounded by the s u r f a c e S. The e x p r e s s i o n obtained is s i m i l a r to K o m a r ' s r e s u l t (3). The p r i n c i p l e s for selecting the p r e f e r r e d fields a r e also s i m i l a r . In p a r t i c u l a r , for static and stationary s y s t e m s the p r e f e r r e d v e c t o r field in (9) m u s t be tangential to the Killing v e c t o r field [6]

% = ~u~, ~ = V%~, ~ .

(10)

This p r e f e r r e d r e f e r e n c e f r a m e can be said to be at r e s t relative to the s o u r c e of the gravitational field. A s the calculation of the global e n e r g y in (3) and (9) can be reduced to the integral of a superpotential o v e r an infinitely r e m o t e surface

e

.,f

-

ds,

eI.,

l

-

these r e s u l t s must coincide, it being the case that ~ ~ 1 in the asymptotic region. This is easily verified, if only by the example of the Schwartzschild solution [6, 7]. T h e r e is also a significant difference. In c o n t r a s t to K o m a r ' s r e s u l t , in the approach of P i r a n i and Denen there is meaning not only to the total e n e r g y , but to its localization as well. In fact, in Denen's t r e a t m e n t the integraad in (9) is the e n e r g y d e n sity -- the t i m e - l i k e component of the e n e r g y - m o m e n t u m v e c t o r =--(W,--u

):~, h = E ~ u , ~

(ii)

x

T h e r e f o r e Eq. (9) can also be employed for the calculation of the energy in a region of space which c o n tains no s o u r c e s 0Rpv = 0). This is also d e m o n s t r a t e d by the example of the Schwartzsehild m e t r i c [6]. E x p r e s s i o n (3) is not suitable for this purpose and f o r such a region it gives a nuI1 r e s u l t [8]. (It is true that here the calculation was c a r r i e d out using the MSller--Mitskevich p s e u d o t e n s o r , but in the ease of Schwartzschild' s solution it coincides with K o m a r ' s expression. ) As in K o m a r ' s investigations, here t h e r e is an important u n r e s o l v e d question: in the g e n e r a l case what v e c t o r field u# should be c o n s i d e r e d p r e f e r a b l e for (9) ? Cattaneo [9] a s s u m e d the condition hl'" L L hl~, = O, h ~ = gi,.~ - - t~,~ tt~ , ~zu

(12)

and the determining criterion for the preferred fields. At the same time, the expression for the density of gravitational energy that he introduced (similar to (8)-(I0)) is of fixed sign (negative) and vanishes only in flat space-time. As in the case (6), (7), there remains a doubt as to whether such a vector field can be found in every Riemann space. Not long ago investigations were made concerning the construction of the energy tensor of a gravitational field in tetrad formalism. MSller [i0] was the first to apply tetrad formalism in the construction

747

of an e x p r e s s i o n for the e n e r g y of a g r a v i t a t i o n a l field. However, his e x p r e s s i o n was of a nontensorial c h a r a c t e r . An e n e r g y t e n s o r in the t e t r a d f o r m a l i s m w a s obtained by Rodichev [11]. As a t e t r a d is not uniquely d e t e r m i n e d by the m e t r i c of a s p a c e , it is n e c e s s a r y to introduce additional c a l i b r a t i o n conditions f o r its concretization. In R o d i c h e v ' s theory these conditions a r e s i m i l a r to the c a l i b r a t i o n conditions f o r the potential in L o r e n t z e l e c t r o d y n a m i c s h~;~ = 0.

(13)

At the s a m e t i m e the equations of the g r a v i t a t i o n a l field also a s s u m e the Maxwellian f o r m , while the total e n e r g y t e n s o r is w r i t t e n as 2

~,~ = . . . .

uz

o:,

(14)

w h e r e y # a a a r e R i e c i ' s coefficients of rotation, connected with the nonholonomity object by the r e l a t i o n ship y # a a = C a n # + C~aa + C#aa. The s a m e g e n e r a l e x p r e s s i o n of type (14) (without the imposition of c a l i bration conditions) has been obtained by F r o l o v [12]. E x p r e s s i o n (14) is a t e n s o r with r e s p e c t to a r b i t r a r y coordinate T r a n s f o r m a t i o n s . However, under t e t r a d t r a n s f o r m a t i o n s it v a r i e s a c c o r d i n g to a n o n t e n s o r i a l law. This would be e n t i r e l y natural if one could always a s s i g n a p h y s i c a l meaning to t r a n s f o r m a t i o n s of the r e f e r e n c e f r a m e by t e t r a d t r a n s f o r m a t i o n s . It is c l e a r that in d i f f e r e n t noninertial f r a m e s o b s e r v e d values of p h y s i c a l c h a r a c t e r i s t i c s m u s t be different. However, a t e t r a d t r a n s f o r m a t i o n can also always be r e g a r d e d as a s i m p l e change of the origin f r o m which the E u l e r angles f o r the velocity v e c t o r of the o b s e r v e r a r e calculated. It is natural to conclude [13] that such a r e n o r m a l i z a t i o n m u s t have no effect on r e s u l t s obtained by m e a n s of (14). The dependence of v a l u e s , obtained by m e a n s of (14), on r o t a t i o n s of the s p a t i a l p a r t of orthogonal r e f e r e n c e f r a m e s a p p e a r s e s p e cially absurd. All this m a k e s it doubtful w h e t h e r any p r o g r e s s is achieved by this a p p r o a c h as c o m p a r e d with the m e t r i c theory, which e m p l o y s p s e u d o t e n s o r s [14]. The defect in e x p r e s s i o n (14), cited above, has been r e m o v e d in a subsequent e x p r e s s i o n f o r the e n e r g y t e n s o r , p r o p o s e d in [15]. The new e x p r e s s i o n , which can be called the truncated e n e r g y t e n s o r , h a s the f o r m Oa ~ = - - o" C :'~,;,,

C,,o,~

=~

(h..;~.--ha~;.). E1~[ /r"d - - - - - l (/:o;~

')t

h4,,;~);..

(15)

"~

It is a genuine t e n s o r with r e s p e c t to a r b i t r a r y coordinate t r a n s f o r m a t i o n s and spatial t e t r a d rotations. The fact that it has a n o n t e n s o r i a l c h a r a c t e r with r e s p e c t to a r b i t r a r y t e t r a d rotations might be c o n s i d e r e d a s h o r t c o m i n g in view of the p r e c e d i n g r e m a r k s ; however, we shall now change the i n t e r p r e t a t i o n of this r e s u l t somewhat. The change of r e f e r e n c e f r a m e is d e s c r i b e d m a t h e m a t i c a l l y as [16] t~~ = ~"~, u ~,

w h e r e ~u~ is the affinor that e s t a b l i s h e s the c o r r e s p o n d e n c e between t h e s e v e c t o r fields. relationship tt ~ = h~,

(16i If the implied (17)

is used in (15), we a r r i v e at the r e s u l t of P i r a n i and Denen, d i s c u s s e d e a r l i e r . The difference c o n s i s t s in a new and m o r e u n i v e r s a l c r i t e r i o n f o r the d e t e r m i n a t i o n of the p r e f e r r e d r e f e r e n c e f r a m e (13). Now it is e a s y to u n d e r s t a n d the n o n c o v a r i a n c e of e x p r e s s i o n (15) with r e s p e c t to t e t r a d t r a n s f o r m a t i o n s . The point is that u n d e r a t e t r a d rotation, involving a t i m e - l i k e v e c t o r , r e l a t i o n (17) is violated, and e x p r e s s i o n (15) should not be c o n s i d e r e d to be the e n e r g y t e n s o r , but r a t h e r the analogous P i r a n i - D e n e n definition (11). They coincide only for the c a s e (17). If a change in the p h y s i c a l r e f e r e n c e f r a m e (16) is m a d e , (15) c h a n g e s a c c o r d i n g to a n o n t e n s o r i a l law, a s m i g h t be expected: T h u s , investigations c a r r i e d out by v a r i o u s a u t h o r s f r o m different points of view have led to the s a m e r e s u l t . The d i v e r g e n c e o c c u r s only in the definition of the p r e f e r r e d r e f e r e n c e f r a m e and this question now d e s e r v e s the c l o s e s t attention. In conclusion the author e x p r e s s e s deep thanks to P r o f . V. L Hodichev f o r d i s c u s s i o n of r e s u l t s of the work.

748

LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12. 13. 14. 15. 16.

CITED

P . G . Bergmann, Phys. R e v . , 75, 680 (1949); 11__2, 287 (1958); 12__4, 274 (1961). A. K o m a r , Phys. R e v . , 113, 934 (1959). A. K o m a r , Phys. R e v . , 127, 1411 (1962); 12__99, 1873 (1963). ]: M. Dozmorov, P r o b l e m s in the T h e o r y of Gravitation [in Russain], No. 1A, All-Union Scientific R e s e a r c h Institute, Moscow (1972). F. P i r a n i , L e s T h 6 o r i e s de la Gravitation, CNRS, P a r i s (1962). G. Denen, The E i n s t e i n Collection 1969-1970 [in Russian], under the editorship of i E. T a m m and G. i Naan, Nauka, Moscow (1970). M. Moss, Nuovo Cimento, B5__.77,257 (1968). A . B . K e r e s e l i d z e and V. S. Kiriya, in collection: C u r r e n t P r o b l e m s in Gravitation [in Russian], Tbilisi State University P r e s s , Tbilisi (1967), p. 500. C. Cattaneo, Ann. Inst. H. Poincar6, A__i, 1 (1966). C. Mbller, Mat. Phys. Skr. Dansk.Vid. Selsk., i, No. i0 (1961). V. i Rodichev, Izv. Vuzov SSSR, Fizika, No. I, 142 (1965). B.N. Frolov, Bulletin of the Moscow State University, Physics Series, No. 2, 56 (1964). V. i Rodichev, The Einstein Collection, 1968 [in Russian], under the editorship of i E. Tamm and G. i Naan, Nauka, Moscow (1968). Yu. G. Sbytov, in collection: C u r r e n t P r o b l e m s in Gravitation [in Russian], Tbilisi State University P r e s s , Tbilisi (1967), p. 127. V. L Rodichev and G. L Zadonsky, Izv. Vuzov SSSR, Fizika, No. 10, 57 (1971). V. L Rodichev, The Einstein Collection 1971 [in Russian], Nauka~ Moscow (1972), p. 88.

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