15. P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Nauka, Moscow (1967).

16. V. A. Fock, The Theory of Space, Time, and Gravitation, Pergamon (|964). 17. L. P. Eisenhart, Continuous Groups of Transformations [Russian translation], Gostekhiz-

dat, Moscow (1947). 18. A. Einstein, Collection of Scientific Works [Russian translation], Vol. I, Nauka, Mos-

cow (1965). 19. J. Barnes, "Lagrangian theory for second-rank tensor fields," J. Math. Phys., 6, No. 5,

788-794 (1965). 20. C. Fronsdal, "On the theory of higher spin fields," Nuovo Cimento, 2, No. 2, 4~6-443

(1958). 21. H. Minkowski, "Raum und Zeit," Phys. Z., 10, 104-118 (1909). 22. V. I. Ogievetsky and I. V. Polubarinov, "Interacting fields of spin 2 and the Einstein

equations," Ann. Phys., 35, No. 2, 167-208 (1965). 23. N. Rosen, "A bimetric theory of gravitation," Gen. Rel. Gravit., 4, No. 6, 435-447

(1973). 24. C. M. Will, "Experimental disproof of a class of linear theories of gravitation," Astro-

phys. J., 185, 31-42 (1973).

CHAPTER 3

DESCRIPTION OF GRAVITATIONAL EFFECTS IN THE FIELD THEORY OF GRAVITAITON

16. Post-Newtonian Approximation of the Field Theory of Gravitation

The first question which any theory of gravitation should answer is the question of the correspondence between its predictions and the results of available gravitational experiment.

Until recently the requirements on possible theories of gravitation reduced to the neces- sity of obtaining Newton's law of gravitation in the weak-field limit and also the descrip- tion of the three effects accessible to observation: the gravitational red shift in the field of the sun, the curving of a light ray passing near the sun, and the displacement of the peri- helion of Mercury.

Thus, the available requirements on possible theories of gravitation were clearly insuf- ficient, since a large number of theories satisfied them. Formulation of qualitatively new experiments was required for further choice of gravitational theories.

At the present time, in connection with the development of experimental technology, pri- marily cosmonautics, and the increase in the accuracy of measurements, new possibilities have appeared regarding more precise measurement of the orbital parameters of planets (primarily the moon), measurement of the retardation of radio signals in the gravitational field of the sun, and performance of new experiments within the solar system. These experiments make it possible to further restrict the circle of viable theories of gravitation~

Nordtvedt and Will [36] developed a formalism, called the parametrized post-Newtonian formalism to facilitate comparison of results of experiments performed within the solar sys- tem with predictions of various metric theories of gravitation (i.e., theories of gravita- tion according to which the action of a weak gravitational field on all physical process ex- cept gravitational processes is realized by a metric tensor of Riemannian space--time).

In this formalism the metric of Riemannian space--time created by some body consisting of an ideal fluid is written as the sum of all possible generalized gravitational potentials with arbitrary coefficients called post-Newtonian parameters. Using these parameters of Will-- Nordtvedt, the metric of Riemannian space--time can be written in the form

goo = 1 -- 2U + 2~U 2- (2~ + 2 + ~ + ~t) O1 + ~,A + + 2~- 2 [(3v + 1 - 2~ + ~2) o2 + (1 + ~3) o~ + 3 (v + ~4) ~d +

+ (~ - az - ~3) w ~U + ~2w~wFu~ - (2~3- ~) w~v~; ( 16.1 )

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where w ~ are the spatial components of the velocity of the reference system relative to some universal rest system. For some theories of gravitation this is the velocity of the center of mass of the solar system relative to the rest system of the universe.

The generalized gravitational potentials have the form

i __> Po dV; R=l r - r ' l , U= --~ ~ po~dv'O v

~-- - dV; ~ __,

pov.vRVRo~ Wo~= f R~ dV; U~f~z I PoR~Rf~ R~ dV;

[ ~ i~ ( x~- x'~) . . . . > i d3r'd3r ", J I r - -r" I r ' - - r" I ] r - - r" I

(16.2)

where 00 is the invariant density of the mass of the body; v ~ are the components of the three-dimensional velocity of elements of the ideal fluid; p is the isotropic pressure, P0H is the density of internal energy of the ideal fluid; R ~ = x ~ -- x '~.

To each metric theory of gravitation there corresponds a collection of values of the 10 parameters

There fore , f rom the po in t of view of exper iments car r ied out in the so la r system one theory of g rav i ta t ion w i l l d i f fe r from another on ly by the va lues of these parameters . To determine theor ies of g rav i ta t ion which in the post -Newton ian l im i t make i t poss ib le to de- scribe experiments carried out in the solar system it suffices to determine from these ex- periments the values of the 10 post-Newtonian parameters and to select only those theories of gravitation whose post-Newtonian approximation leads to values of the parameters that coincide with those obtained from experiments. All such theories of gravitation will then be indistinguishable from the point of view of any experiments carried out with post-New- tonian accuracy.

Further selection of a theory of gravitation adequate to reality is connected with an increase of the accuracy of measurements above the post-Newtonian level or with the search for possibi l it ies of studying the properties of gravitational waves and also phenomena in strong gravitational fields.

We shall determine the collection of values of the post-Newtonian parameters corre- sponding to the field theory of gravitation.

The equations of the gravitational field in this theory have the form

E]2 f nm: - - 16~Jnm; [] : a~O i. (16 .3 )

If we use notation (13.11), then for the tensor current jnm we obtain the following expres- sion:

Jnm : ~]~nm ~ OnOihmi__ OmOihnt.Jf_ ynmOiOlhli" (16.4)

Following Fock [11], to construct post-Newtonian approximations valid in the solar sys- tem, we consider a problem of astronomic type. We shall assume that the components of the energy--momentum tensor of matter are equa ! to zero throughout space except for certain re- gions. Within each such region the energy--momentum tensor of matter should correspond to the model of an ideal fluid we adopt and should satisfy a covariant conservation equation in Riemannian space--time. Aside from the physical properties of the model of celestial bodies, the energy--momentum tensor of matter will also depend on the metric of Riemannian space--time. Therefore, construction of the energy--momentum tensor of matter and the deter- mination of the metric tensor of Riemannian space--time must be accomplished together.

We use the circumstance that within the limits of the solar system the maximal values of the gravitational potential, the square of the characteristic velocity v 2 (the velocity

1770

of celestial bodies relative to the center of mass of the solar system), the specific pres- sure p/P0, and the specific internal energy ~ have approximately the same order of smallness ~2 where ~ ~ 10 -3 is a dimensionless parameter. Therefore, in the solar system we have the following estimates:

U=O(~D; v~O(~), (16.5)

P = 0 &-). n =0 &); p~

Moreover, we shall consider the field in the near zone, i.e., at distances from the sun con- siderably less than the length of a gravitational wave radiated by objects in the solar sys- tem which are moving with characteristic speed v ~ ~:R/% ~ R(3/3t) ~ g. In this case the change of all quantities with time is due primarily to the motion of matter. Therefore, the partial derivatives with respect to time are small as compared to the partial derivatives with respect to coordinates

~ =0(~), o Ox ~ 9 (16 .6 )

The problem of the joint determination of the energy--momentum tensor of matter and of the metric tensor of Riemannian space--time we shall solve in successive stages each of which corresponds to expansion of the exact equations of the problem in powers of the dimensionless parameter ~.

We have the following exact relations: the density of the energy-moment tensor of the ideal fluid

T'~'~---- ] / - - g [(p--k $) t t '~u~- -Pg~m], (16 .7 )

the covariant equation of continuity

i a [ f - - - fp0w]=0 (16 8)

and the conservation equation of the density of the energy--momentum tensor of matter in Rie- mannian space--time

FmT =0, (16.9)

where $ is the total energy density of the ideal fluid.

For our purposes it is more convenient to write the equations of the gravitational field (16.3) and the equation of minimal coupling (14.3) in the form

[~2%nm=--16~Anm, (16.1'0)

\ (16 11)

where we have introduced the notation

Zn==fnm--89 %=%n n, (16.12) A~=D [h~-89 y~m&~]-- a"Gh~z--o~aA~. (16 .13)

We shall expand all quantities contained in Eqs. (16.9)-(16.12) in series in the small para- meter ~. If we neglect energy losses to the radiation of gravitational waves, then these ex- pansions must also be valid when the sign of the time is reversed. In reversing the sign of the time, i.e., under the coordinate transformation x '~ = --x ~ the components v ~, A0~, g0~, 3/3x ~ change their sign to the opposite sign. Since v ~ g and 3/3x ~ ~ e, when the sign of the time is reversed the dimensionless parameter ~ also changes sign. From this it follows that if energy losses to the radiation of gravitational waves are neglected the expansions of the components v ~, X ~ T ~ A ~ contain only odd powers of the parameter e.

The expansions of the tensor current A nm and the field X nm we write in the form

O) (2)

x"~=x~ +x~ + 9 9 9 (16.14) (o) (i)

A=~ = A~'} - A~ i f - . . - ( 16.15 )

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(o) 0) (2) where the components of the zeroth A n~ , first A n~ , and second A n~ the following order of smallness:

(o) O) ~o~=0(0; AOO=O(1); ~"~=-0(1) ; A~

(1) (1) (2) A*~ =O (e~); AOO=O (e2); A~ =O (e~);

(2) (2) Aoo=O (~4); A=~--~ O (~4).

approximations have

(16.16)

With consideration of expansions (16.14), (16.15) and estimate (16.6) we can rewrite the equations of the gravitational field (16.10) in the form of a series of successive ap- proximat ions :

(t) (o) A2%n,n = __ 16nA~m;

(2) (1) ~2 0) A2% nm= - - 16hA nm+ 2 ~t ~ AX rim.

From expressions (13.11) and (16.11) we obtain

h"m=Tnm+~lT~iz i~+T~iz t~ ]

(16 .17)

(16.18)

(16.19)

b, + b= ^ .Tn,n + 4

4 ynmTslYsi"

For the tensor current A nm we then have

(0) ] (o) F(O) 21 yn,nTstysi- ;j A nm -~- - - A [T.m--

A "n'O) .a* [(o),nz -1 ra(~ 1 n FmTsiO) (0) _m O)n (0)s,.. [(~)nm 1 0). b, (0) (1) (0) O) (16 .20) =~[r - -~ .T v,,]+o ( ~, )+o (r j )--A~ ---fy'mTs'v,,+-y-(T'nx,~+T"%/9-

b,+b2 ~) (I) (o) (1) (1) (o). /b~ (0) ] Y ntnTs~ J

where A =--3a~ a.

To find the post-Newtonian parameters it suffices for us to determine the coefficients gas to accuracy g2, the components g0~ to accuracy ~3, and the component g00 to accuracy ~4. It follows from the coupling equations (16.11) that for this it is necessary to determine the components of the field XaB up to E 2, X ~ up to ~3, and X ~176 up to e 4.

In the initial approximation we assume that the metric tensor of Riemannian space--time coincides with the metric tensor of pseudo-Eucl idean space--time, i.e., we neglect gravita- tional forces. Equations (16.8) and (16.9) then take the form

o-~ (og ') =O (82),

OnT"O = 0 (e3), O,~T "~ = 0 (82).

Cons ider the es t imates (16 .5 ) , f rom these equat ions we have

T~=p0[1 +O(82)1, T(~=poO(e2) , T ~ ~[1 +O(e~)].

There fore , the components of the tensor cur rent A nm in zeroth approx imat ion can be wr i t ten in the form

(o) 1 (,~oo~ (~ 1 __o. AOO = - - ~ A9o; - - - - A (poV~); A " =-~- y""a9o. ( 1 6.21 )

From Eq. (16.17) we then obtain

(16.22)

As a result, the components of the metric tensor of Riemannian space--time (16.11) in first approximation can be written in the form

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goo = 1 - - iU + O (84); g0~ =4V~ [1 +O (sz)]; (16.23)

Knowledge of the metr i c in th i s approx imat ion makes i t poss ib le to determine the com- ponents of the energy-momentum tensor of mat ter in the next approx imat ion . Using the expres - s ions (16 .23) , we f ind that

- - g = 1 + 2U -t- 0 @4), /~0 = 1 + U - - v i= ,~v Fo OU . 00=--~-+o

To determine the values of the post-Newtonian parameters of our theory, it is necessary to go over to the coordinate system in which the post-Newtonian expansion of the metric (16.1) is written. If we make the coordinate transformation

Ytn=Xn-~$n(X) (16 .31)

and assume that

$~ (x) = 0 (~), ~o (x) ~ 0 (~3), then metric (16.30) in the new coordinate system has the form

goo = goo -- 20o~o + 0 (e6);

g'o~ = go~ - 0o~ - 0~o + 0 (89;

g~ = g~.~ ~ 0~.~-- Of~o~ + 0 (e4).

As a "canonical" coordinate system a system of coordinates is usually chosen in which the nondiagonal components of the spatial part of the metric tensor gni are equal to zero,

g12 ---- g13 - - g23 = 0

and, moreover, the component g00 does not contain terms of the form

or--2 9oRdV.

These conditions make it possible to uniquely determine the four-vector to the required accuracy. In our case to pass to a "canonical" coordinate system it is necessary to choose the following four-vector ~n:

~ (x)=O, to (x) . . . . .

Using the equation of continuity (16.8), we obtain

1~ f '2 at 9oRdV.

(16 .32)

I w

As a result, we have the following expression for the metric tensor of the effective Riemannian space--time

goo = 1 -- 2U + 2~U 2 - - 4@1 + 4 (~ - - 2) 02 - - 2r 604 + 0 (e~); 7 1

go~ = ~ V~ + 7 W~ + 0 (es); (16 .33)

g~---- y~ [ 1 + 2U--I- 0 (e4)l.

Thus, the post-Newtonian approximation of the field theory of gravitation leads to the metric of Riemannian space--time (16.33) which contains only one unknown constant B.

For the case where the source o~ the gravitational field is a static spherical symmetric body of radius a this metric assumes the form

M s M ~ g~176 1- - 2 f + 2~ -~--~- 0 ('7~-); ( t6 .34)

MS

where M is the gravitational mass of the source of the field:

M=4~ g Oo [1 +g+_~_j+2(2--~)U ]r~dr. (16.35) 0

Using the Newtonian virial theorem for static bodies

1

1774

and also relation (16.27) between the conserved and invariant mass densities, we bring ex- pression (16.35) to the form

G

+ ] 0

As we shall see below, for coincidence of the post-Newtonian expressions for the gravi- tational mass (16.36) and the inertial mass (17.13) of a static, spherically symmetric body it is necessary to set B = I. Then the post-Newtonian parameters of the field theory of gravitation have the values

u a i=~2=~3=0, ~i-----~=~3-----$4=~w=0. (16.37)

For comparison, we point out that in the general theory of relativity the post-Newtonian parameters have the same values [36].

It should be noted that the vanishing of the parameters ~ and ~ was long considered a property of Einstein's theory alone and was considered one of its achievements. However, as we see, in the field theory of gravitation these parameters are also equal to zero. The re- maining parameters in the general theory of relativity and the field theory of gravitation are equal to one.

Since the post-Newtonian parameters of Einstein's general theory of relativity and the field theory of gravitation coincide, these two theories will be indistinguishable from the point of view of any experiments carried out with post-Newtonian accuracy of measurements in the gravitational field of the solar system.

As shown in [30], vanishing of the three parameters a has a definite physical meaning: any theory of gravitation in which el = a2 = a3 = 0 does not possess a preferred universal rest system in the post-Newtonian limit. In this case, on passing from a universal rest sys- tem to a moving system, the metric of Riemannian space--time in the post-Newtonian limit is form-invariant, and the velocity w a of the new coordinate system relative to the rest system does not explicitly enter the metric.

It follows from expressions (16.37) that in the field theory of gravitation a universal preferred rest system is lacking.

Linear dependence of the parameters ~ and ~ also has a definite physical meaning. As shown in [18], if the relations

~1----~3=0; ~z--~,--2~w=O; $2----~w; %,-}-~,-~2~--0; (16.38) 3~4 %- 2~- -0 ; ~t - ] -2~ = 0

are satisfied from the post-Newtonian equations of motion it is possible to determine quan- tities which in the post-Newtonian approximation do not depend on time. However, interpreta- tion of these quantities as the energy--momentum and angular momentum of the system (i.e., as integrals of the motion) is possible only in those theories of gravitation which possess con- servation laws of the energy--momentum tensor of matter and the gravitational field.

Thus, for example, in Einstein's theory relations (16.38) are satisfied, but detailed analysis shows that quantities not depending on time in the post-Newtonian approximation are not integrals of the motion of a system consisting of matter and gravitational field.

In the field theory of gravitation an isolated system has all I0 conservation laws in their usual sense which lead to 10 integrals of the motion of the system; therefore, in the post-Newtonian approximation the field theory of gravitation has I0 quantities not depending on time. The fact that the relations (16.38) are satisfied in the field theory of gravita- tion confirms this conclusion.

17. Post-Newtonian Integrals of the Motion in the Field Theory

of Gravitation

In the field theory of gravitation the gravitational field considered in pseudo-Euclidean space--time behaves like other physical fields. It possesses energy--momentum and contributes to the density of the total energy--momentum tensor of the system. The covariant conservation law of the density of the total energy--momentum tensor in pseudo-Euclidean space-dime written in a Cartesian coordinate system has the usual meaning

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