General Relativity and Gravitation, Vol. 2, No. 3 (1971), pp. 223-234.

A Non-Covariant Theory of Gravitation, II

N A T H A N R O S E N

Department of Physics, Israel Institute of Technology, Haifa, Israel

Received 27 October 1970

Abstract

This is a continuation of a previous paper, in which a theory of gravitation was developed based on the existence of a preferred frame of reference and a preferred time coordinate in the universe. The gravitational field equations are derived with the help of a variational principle containing three constants. Two relations among the constants are introduced, leaving one of them arbitrary. This constant does not affect the precession of the peri- helion of Mercury but does affect the behaviour of gravitational waves. By changing one of the relations among the constants, one can account for the discrepancy in the preces- sion of the perihelion associated with the oblateness of the sun, as found by Dicke and Goldenberg.

1. Field Equations

In a previous paper [4], to be referred to hereafter as I, an at tempt was made to set up a theory o f gravitation based on the existence o f a privileged frame of reference and a privileged time coordinate in the universe. On the basis o f the assumptions made, the line-element in this preferred coordinate system was taken to have the fo rm

ds 2 = qb2 dt 2_ ~2(dx z + dy2 + dz~) (1)

with �9 and ~g functions o f the coordinates to be determined by the field equations. The gravitational field was thus described by two scalars which were simply related to the metric tensor in the privileged coordinate system. The problem was to choose the appropriate field equations. For this purpose, the Einstein field equations were taken as the starting point, and modifications were then introduced in order to satisfy certain requirements. The present paper involves the same basic point o f view, but the procedure for arriving at the field equations is somewhat more general. The notat ion used here is the same as in I.

Let us obtain the field equations in the preferred coordinate system with the help o f a variational principle. I f one studies the form o f the expression that was used in I, Section 6, one is led to take as a generalization,

J = f j dr (2)

with J = ~ J l +/3j2 + r J a (3)

Copyright �9 1971 Plenum Publishing Company Limited. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo- copying, microfilming, recording or otherwise, without written permission of Plenum Publishing Company Limited.

224 NATHAN ROSEN

where f , .lie g t''7-t t, W,. 7 -t 0

1=t--g) 7J2 =-~ T,o T , o - ~ T,~ T,~ (4a)

g~"O, # 0, v_ T3 W J2----(-g) 1/2 02 - ~ 0 , 0 0 , 0 - - ~ , / c 0 , 0 , / c (4b)

. . . . O7 t - ~ O,0 ~ 0 - O , ~ T,~ (4c)

and a,/3, 9' are constants. For arbitrary infinitesimal variations 30 and 8T vanishing on the bound-

ary of the region of integration, let us write

so that

3J=z f ao+ a- N(o) N(s) 3W} (_g)l/2 d.,. (5)

02 3.1 1 3J N(0) = 27 ja 30' N(s) = 2 0 3T (6)

Following I, let us now take as the field equations

N~oj = - 8#Too (7a)

N(s) = - 8rrT~ (7b) and

T~, . ; . = 0 (8)

where T~, is the energy-momentum density tensor of the matter or other non-gravitational fields.

Equation (8) is taken over from the general theory of relativity and repre- sents the energy-momentum relations for the matter and other fields, with the effect of the gravitational field taken into account. From it one can show that the motion of a test particle in a gravitational field is given by the equations of the geodesic. A plausibility argument for the form of equations (7a, b) is that, with their help, equation (8) leads to conservation laws for energy, momentum and angular momentum. This will be discussed in Section 3.

Now the field equations contain the coefficients o~,/3, ~,, and the question that must be considered is: what values should one give them? The procedure to b e followed here will be to assign them values so as to obtain agreement with Newtonian mechanics and with general relativity in certain special cases.

Let us consider the static case, so that �9 0-- T, 0 = 0. Then

T Su J = - ~ ~p 7",k . ~ - / 3 ~ , ~ ~ - v O k T k (9)

A NON-COVARIANT THEORY OF GRAVITATION, II 225

and one readily finds that

8J / 2T T 2 ) a g ~ = f l ~--~- ~ 0 , k ~ - ~ ~ , ~ r r (10a)

' r T , ~ T , ~ ) - f l ~ ~ , k ~ , ~ (10b)

Now let us suppose that we have a weak field which, by a suitable choice of length and time units, can be characterized by the relations

~ = 1 + r T = I + ~ (11)

where r and ~b (as well as their derivatives) are small quantities. Then the field equations (7a, b), with the help of(6) and (10a, b) can be written in the linear approximation

- / 3r e~ - ~7'r ~ = - 8 ~'T00 (12a)

�89162 1c~ + ~b, ~ = - 87rT~ (12b)

Let us assume that T~e is everywhere negligibly small, and let us write T00=p. Now, in the general relativity theory, with a certain choice of coordinates, one obtains for the case of a weak static field [3]

~b= - r (13)

Let us require this to hold in the present theory. Then (12b) gives

7,=2~ (14) and (121) can be written

(/3- ~)r ~k= 8=p (15)

From the relation between general relativity and classical mechanics we know that, in the weak field case, r is the Newtonian gravitational poten- tial which satisfies the relation

r ~ = 4~rp (16) Comparing this with (15) gives

~ = ~ + 9 (17)

The value of ~ remains to be determined. For this purpose let us investi- gate the static, spherically symmetric solution of the field equations for empty space. The equations for this case are obtained by setting (10a, b) equal to zero. If one takes q~ = ~(r), T = T(r), the two equations, with 13 and 7 chosen as above, can be combined to give the following relations

r _, ~ ' T ' ^ r ~o + ~ = 0 (18a)

u 1 T '2 ~ + 2 T~b'2 r 2 7 t 2~ ~2 = 0 (18b)

For the present purpose it is sufficient to look for a solution in the form

226 NATHAN ROSEN

of a power series in 1/r. If one takes into account equation (13) for the first order terms, one obtains

rn m s (19a) �9 = 1 - 7 + ~ 7 ~ + . . .

ml( ~ = 1 + 7 + ~ l + 7 ~ - + . . . (19b)

I f one considers a test particle moving in the field according to the equations of the geodesic, then the first order term (of order m/r) in �9 leads to motion corresponding to Newtonian mechanics. The precession of the perihelion of a planet is obtained by taking into account the second order term in �9 and the first order term in 7:. However, one readily verifies that these terms are the same here as in the case of the isotropic form of the Schwarzschild solution. Hence one obtains, to the usual accuracy, the same results as in general relativity. It turns out therefore that one gets here the same results in the three crucial tests as in the case of the general theory of relativity for arbitrary values of ~ ( # 0).

Thus the value of ~ still remains undetermined. It might be pointed out that the formalism discussed in I, Section 6, corresponds to taking ~ = - 2 (so that /3=0, y = -4) . We now see that other values are also possible.

2. Gravitational Waves

In the general case the field equations (7a, b) are found to have the form

(~ o 3qi, o ~ _1_3r 7t, o ~b,e~ {_r 2~ ~ ~ T T 9' 2T2

{~,o= . ~ U , ~ ! e . ~ [7",oo 4 kU, o~

( 7t2~~176 W=qi, o = . ~ , e g ' _ o ( 3 7 ~ , o ~

~ ' ~ waT'~)

q ~ e = - 8rrToo (20a)

~oo ~ , o ~ o ~,o ~ ~,r ~,~'~ +~ ~2 ~ q~a 2r ~ 2~2 ]

= - 8rrTk~ (20b)

If we set/3 = ~ + 2, y = 2~ and divide (20a) by r and (20b) by T~, we get equations which can then be combined to give the following relations:

1 / ~ o o 3 ~ o T , o\ 1 { r

2 7;o= =4, (Te _ToO)(21a) + ~ l ~ -+ ~ - ~-

A NON-COVARIANT THEORY OF GRAVITATION, II 227

1 (~__~o 7~,02 4. qS, 01rJ, O.11/'t, O2~ 42 - - 2 q)2 - - ~ - + 2 T ~ ]

1 / T,k~ lq),k~,~ 1 T ~ T , k \ 1 [ 3~,o2,~,k~,~ '~

Let us now consider the weak field case, with (11) holding. Then (21a, b) become, in the linear approximation,

r o o - r ~ = - 4 ~ ( p + 3p), (22a)

r oo- ~b, ~ =47rp + 127r (1 + ! ) p

where we have written 1 T . k - - 1 T k ~ P = - x ~ - x ~ , p=To ~176176

(22b)

(23)

corresponding to this approximation. Let us solve the equations by the method discussed in I, Section 7. Here

too we choose a coordinate system with the emitting system located at the origin and assume the distance r from the origin to the observer to be very large. If we let the unit vector pointing from the origin to the observer have components nk, we can write for the approximate solution of (22a, b)

m 1 dr,}t_r r (ld~/~2 f p(nkx'k)2dz'+ f (24a)

• = r + r [2 dt 2 J P(n~x'~)z d r ' + d:"} (24b)

where the integrals are to be taken at the retarded time t - r , and primes refer to the coordinates of the element of the emitting system.

If we now use the relation (based on I, Section 7)

then we get

f T kgdr ' - I d 2 f 2 dt ~ pr '2 dr' (25)

- r - 2 r di ~ p[(n~x'k)2+r '2] dr' (26a) t--r

( 2 6 5 )

If the coordinate system is chosen so that the observer is on the x-axis,

228 NATHAN ROSEN

with x = r, then these expressions can be written

m l { d ~ fp(x,2+r,2)dr,},_, (27a) 4= - 7 - ~

~=m+l {~-~z f p [x'2+(l+2) r'~]dr'}t_. (27b)

For very large values of r, one can consider 1/r as nearly constant, and hence one can write

r162 ~=~( t -x ) (28) so that

4,0 = - r 1, r r n (29a)

~b, 0 = - ~k, 1, ~b, 00 = ~, 11 (29b)

Let us now consider the question of the detection of gravitational waves. Some discussion of this was given in I, Section 7, based on the work of J. Weber [5]. It was mentioned that, if the idealized detector consists of two masses connected by a spring and oscillating along the line joining them, then if the detector oscillates in the x-direction, its motion depends on the curvature tensor component Rt010, while if it oscillates in the y- direction, the motion depends on R2020. In the case in which the waves are propagating in the x-direction, these directions of oscillation correspond to longitudinal and transverse directions, respectively.

The expression for the component of the Riemann tensor R~ono (I, Section 7) becomes in the weak field case

Rono~ =,L e,,- 8~"r oo

For the plane wave moving in the x-direction, one gets by (29a)

R(t) ~ R1010 = 4 , 00 - - ~b, 00

(30)

Roe) - R2o2o = Rao3o = - ~b, to

(31a)

(31b)

where the symbols Ro) and R(e) have been introduced to denote the longitudinal and transverse components respectively. Let us now consider some special cases:

(a) Longitudinal Source Suppose that the source is such that

e.f dt 2 p(y'2 + z '2) dr'=O

I f we let

KO) = 7

(32)

(33)

A NON-COVARIANT THEORY OF GRAVITATION~ II 229

then

R ( z ) = - (2+ ! )K ,~ ) (34a)

R ( t ) = - ( 1 + 1) K(,) (34b)

In general, both components will be present. However, for a = - �89 the wave will be all transverse, and for c~ = - 1 it will be all longitudinal.

(b) Transverse Source Suppose that the source is such that

dt-- ~ px '2 d r ' = 0 (35)

If we now let

K(t)=l {~--~-~4 f p(Y'2+z'2) dr'}~_~ (36)

then

R ( ~ , = - ( 1 + 1) K(t) (37a)

[l+q R ( t ) = - - ~2 ~] K(t) (37b)

Here, again, both components will generally be present, but for a = - 1 the wave will be transverse, and for c~ = - 2 it will be longitudinal.

(c) Spherical Source In the case of a source which has spherical symmetry, one can define

K(s)=l {~--~4 f pr'2 d,'}t_x (38)

Then

{4+1A R(O = - ~3 a] K(s) (39a)

(2+1] R(t) = - k3 o~] K(s) (39b)

In this case, for a = - �88 one gets a transverse wave, for ~ = - .~ a longitudinal one. Otherwise, the wave will display both longitudinal and transverse properties.

We see that the behaviour of gravitational waves depends strongly on the value of a. In principle, the value of a could be determined from observa- tions of gravitational waves emitted by physical systems.

230 NATHAN ROSEN

3. Energy and Momentum Relations

Let us now consider the energy and momentum densities of the gravita- tional field. Because of the connection between the field equations (7a, b) and the variational integral relation (5), one can show by the procedure given in I, Sections 2 and 6, that equation (8) can be put into the form

[(-g)~/2rl,~ + tt, q,~=O, (40) where

a,r _ a j 16~rt,,"= . - ~ - ~ ~ , + - ~ - T , t , - J 8 7 (41)

J" d(P, v ' UT, v

I f one defines as the grav i ta t iona l energy-momentum density

Ot,~=(-g)- l /2g, at[,a (42)

then, with the help of (3) and (4a, b, c), one obtains

2~ a~ .+Z.. (a~ ~, .+~, . ~,~) 1 6 r r O ~ . = - ~ 7 - ' ~ + ~ , t , , ~v~e '~ '

- ( - g ) - l / ~ , l g e ~ (43) so that

Oj,~ = 0v# (44)

Corresponding to I, Section 6, let us write for the total energy-momentum density

0 ~ = T#v + 0/*~ Then (40) can be written

(45)

(46a)

(46b)

(q~zT~80~), ~ = 0

( r ~ = 0

Let us now define the angular momentum density

M~* ,~ = x ~ 8 ~ - x,"O '~ (47)

Then, if TJ ~ = T ~j, one finds from (46b) that

(q~ WSM~k~),, = 0 (48)

Since equations (46a, b) and (48) involve ordinary partial derivatives, they represent conservation relations for energy, momentum and angular momentum.

Let Us now consider the gravitational energy density. We have from (43)

16~'0oo= ~ - ~ + q)'~ '

[ r T,o. q~r ~U,r~\ + y L - ~ W - . ~-~ ] (49)

A NON-COVARIANT THEORY OF GRAVITATION, II 231

If we now set/3 = a + 2, y = 2a, this can be written

{q~,o 2 , ~ , ~ , ~ /~ ,o 7X, o\ 2 16. ,000 = 2 t t v + v -)

r We see that, for this to be positive definite, one must have ~ 1> 0.

The energy current density is given, according to (43), by

16rr0~ ~-~2 7t'~ e+2~2 ~ ~ ~ , e + ~ ( ~ , ~ 7 t , ~ + q ~ ~ o ' ,~Wo), (51)

With the above values of/3 and ~,, one can write this

/ ~ o 7 x , o \ / ~ 7 ssA (52)

Let us consider the case of a weak field in the form of a plane gravita- tional wave propagating in the x-direction, so that (28) and (29a, b) hold. Then from (50) and (52) one gets

000= 0 0 1 = L 8rr [2r + ~(r + ~b')2] (53)

where a prime denotes a derivative with respect to the argument. We have here the equality of 0 ~176 and 001, since the wave is propagating with unit velocity, i.e., the velocity of light. I f the energy density is positive, then energy will flow in the direction of the wave propagation, i.e., energy will be emitted by the radiating system. This will certainly be the case for a>~0.

For a negative value of ~, the right-hand member of (53) need not be positive. However, one might ask whether it is possible to have 00o and 001 positive definite in such a case, provided ~ and ~b are related in a way corresponding to radiation from a material system, as discussed in the previous section. It is readily verified that this is not possible in the general case.

Here the question arises whether the concepts of gravitational energy density and energy current density have a physical meaning, that is, whether gravitational energy is localizable. I f one takes the standpoint that the energy current density associated with a radiating system does not have any physical significance, but only the total energy flux, then one can conclude that negative values of ~ are admissible. This can be seen from the following considerations.

Let us write

Q'~= [ D fPx"x'~dT']t_ r (54)

232 NATHAN ROSEN

Then, from (26a ,b), we have

1 q~, o = - 2r (n~n~O k + Q~g) (55a)

If we denote the energy current density in the direction of the observer by 0% then we can make use of (53) to write

oor=~12(n,nkQ'~+Q~g)2+4 (Qk~) ~] (56) ~zlrr I ~ )

If we integrate this over the surface of a sphere of radius r with its centre at the origin, so that the components n~ take on all possible values, we obtain for the rate of outward energy flow through the surface

1 {Q'~Q ~m ntnmdQ} S= ]-C~ f n,nkntnm dt2 + 2Qk~Q~m f

1 (1 +2)(Qkk)2 (57) +~

where dO is an element of the solid angle subtended at the origin. Carrying out the integrations, one gets

1 o~to~t 4- [13 4 - 1 ] S = (Q~)~ (58) 30

We see from this that, if one demands only that S be positive definite, that is, that a physical system should lose, and not gain, energy by emitting gravitational waves, one can have negative values for ~, provided

1 13 /> 15

Possible values that suggest themselves are ~=- (15 /13 ) , for which the second term in (58) vanishes, and ~ = -(5/4) , for which the two terms have equal coefficients.

However, it should be emphasized that, if one regards energy density and current density as physically meaningful, one would be inclined to take c~>0. Simple examples are ~= 1 (/~=3, ~=2) and ~ = 2 (/3=~=4).

4. Solar Oblateness

Dicke and Goldenberg (1967) [2] found that the sun is slightly oblate and that it therefore possesses a mass quadrupole moment. According to them, the value of the latter is such that, if its effect on planetary motion is taken into account, then the agreement between the observed precession of the

A NON-COVARIANT THEORY OF GRAVITATION, II 233

perihelion of Mercury and that predicted by the general relativity theory is destroyed, the observed being about 8 ~ less than the predicted.

Brans and Dicke (1961) [1] proposed the existence of a scalar field, in addition to the tensor field of general relativity. With a suitable choice of the constant determining the coupling of the scalar field to matter, it is possible to remove the above discrepancy. However one cannot help wondering why gravitation should require such a complicated description.

The formalism developed in the present work has led to results which, in the three crucial tests, agree with those of the Einstein general relativity theory. However, in this formalism use was made of two relations among the coefficients occurring in the variational principle, namely equations (14) and (17). Equation (14) was based on the assumption that, for a weak static field, (13) holds, as in the case of general relativity. One can try to generalize the formalism by giving up equation (13).

Let us assume that, in the case of a weak static field with T ~ = 0, one has in place of equation (13)

~= -A~ (59) where h is a constant (> 0). Going back to (12b) we see that, in place of(13),

we get y=2h~ (60)

Equation (12a) now becomes

(/3- hz~)q 5, ~k = 8rrp (61)

and for this to agree with (16) one must have

/3 = ),zc~ + 2 (62)

The exact field equations for empty space in the static case can be obtained by setting the expressions (10a, b) equal to zero, making use of (60) and (62). I f one assumes spherical symmetry and looks for a solution in the form of a power series in 1/r, one finds

g' = 1 - m / r + ~(1 + h ) m 2 / r z + . . . (63a)

7*= 1 + A m ~ r + . . . (63b)

Using this solution to calculate the precession of the perihelion of a planet, one obtains a result which depends on A. If we denote the angular velocity of precession for a given value of A by co(A), so that co(l) is the precession predicted by the general relativity theory, then it is found that

A --- ~ ~ 7 (1 - ~) (64)

We see that one can account for the results obtained by Dicke and Goldenberg if one takes h to be a little less than unity. For example, for A=6/7 one has A = 1/12=0.083.

With the parameter h present, the formalism is more complicated, but the qualitative picture is the same as before. For example, one now finds, in

234 NATHAN ROSEN

place of (50) and (52),

161r000=2 {O '~ -O'~ O , ~

A _

1~ ~ _ O , o O , ~ _ / T , o

so that, if the gravitational energy density is to be positive definite, one must have ~ > 0.

In the weak field case the field equations can be put into the form

r 00- r ~ = - 4~r(p + 3Ap), (66a)

~b, 00- ~b, k~ = - 4~rAp + 12~- (A2 + 2) p (66b)

The solution corresponding to equations (27a, b) is now given by

m 1{~_~2 fp(x,2+~r,2)d,}t_x (67a) r

~ b = ~ + ; {d~2 f p [Ax'~+ (A2+ 2) r'2] d~'}t_~ (67b)

We see that the gravitational radiation emitted by a physical system depends on both A and ~.

The rate of energy flow from a radiating system, corresponding to (58), is found to be

S = I Q~,Q~,+/1 2 1 A + I + I ) ( Q ~ ) 2 [~. A +~ (68)

Here, too, if one requires only that S be positive definite, then negative values for ~ are possible.

In conclusion, it should be remarked that, if one introduces the parameter A, one obtains a value for the deflection of light by the sun which is �89 + A) times that given by the general relativity theory.

References 1. Brans, C. and Dicke, R. H. (1961). Phys. Rev., 124, 925. 2. Dicke, R. H. and Goldenberg, H. M. (1967). Phys. Rev. Lett., 18, 313. 3. Eddington, A. S. (1924). The Mathematical Theory of Relativity, 2rid ed., p. 101,

Cambridge University Press. 4. Rosen, N. (1971). Gen. ReL and Gray., 2, 129. 5. Weber, J. (1960). Phys. Rev., 117, 306.