A Nonstationary Model of the Universe

(l) ,aJE312d)$Sor~[3j2((DX ); 1

V[8'~-l~:/fiilr(o,)f,)1/2 (1,) oi x3/2axf~162176 -~ '~1/2(~

fo0=

/ 0 , = - - V F' t

9

~i2 (or) 0 x312dxY3i2(o)x) lo,+Y312(o~r) , x 312d xH(l) 3/2 (o~x) I0,

)

where we have introduced the notation A1

=

J r-.r (1) ts ~

- - - -

t"

q_ f m (o~r) ix512 [ilorH,I2 (,) (o)x) q- r"rr H ~I2 ~') (COX)]dx}, tt

.

I

s

4n'io~ it(sj~(cor) xSi2dx[ilorS~12(o~x)__i,ry-3/2(cox)l+gC~/2(o~r)

zal 2 ---- - - ~ ' -

0

i

x

ttxo,rssi2

r

1

Using away from matter the gauge transformation

/ ~ - , . / ~ + D ~a~+ D ia~ -- wiD ma% we impose two c o n d i t i o n s

on t h e c o m p o n e n t s o f t h e g r a v i t a t i o n a l

field:

f=O, /oo=0. In order that the gauge transformation not violate the condition Dn fnm = O, the gauge fourvector must satisfy away from matter the equation DnDna m = 0. Choosing gauge vectors of the form ~O ~ Ct~ ~ - O,

~ . j oi x 3 / 2 d x {1o~ [f~12 ( ~ x ) + o ~ x f v2 ( ~ x ) l - - i ~ x l.Y31~ (~x)}, r o = ~'-2"~f '7i '~,c,),l~ t....x 0 a

2hi raIi)%~rx ~ X5[2

(~x)+ifrr~al2(~x)]dx,

it is easy to see that all components of the nonstatic gravitational field away from matter vanish:

L~=0 Thus, in the case of a nonstatic source with spherically symmetric distribution and motion of matter the gravitational field away from matter is a static field with components given by formulas (20.7) and (20.8). 23.

A Nonstationary Model of the Universe

The field theory of gravitation makes it possible to construct nonstationary models of the universe capable of describing the effect of the cosmological red shift and free of divergences of Newtonian type. These models correspond to a flat universe. It should be noted that in the field theory of gravitation a model of the unverse characterizes only the part of it with linear dimension r ~ cT, where T is the age of the universe. From this point of view "creation" of the universe means that in the past the density of matter in a given, sufficiently large portion of the universe was sufficiently high. Subsequent evolution of this region of the universe can be described by the model considered. All other regions of the universe can hereby evolve independently of the evolution of the given region and even by completely different laws. However, observation of them in the field theory of gravitation is possible. Astronomical observations show [8, 23] that matter in the universe is distributed in a very inhomogeneous manner: the main mass of matter is contained in planets and stars; only a

1801

small portion of the total mass goes to interstellar gas and radiation. However, on averaging over regions of space with linear dimensions which are considerably greater than the distance between clusters of galaxies, the density of matter of that part of the universe accessible to observation is found to be a constant quantity not dependent on the position of the center of the region of averaging. From a physical point of view it is thus natural as a first step to consider the model of a homogeneous, isotropic universe. In this approach the inhomogeneity of the distribution of matter occurring in averaging over smaller regions of space (clusters of galaxies, galaxies, etc.) can be considered by introducing small inhomogeneous perturbations on the background cosmological field of a homogeneous universe. A homogeneous, isotropic universe is described by the interval ( 2 3 . I)

d S ~= U (t) dt ~ - V (t) [dx 2 + dy 2 + dz2].

We consider matter in the universe as an ideal fluid with density of the energy--momentum tensor (16.7). Because of the homogeneity and isotropicity of the universe we have =

us=O,

(t), p p (t), uo:pO, uOuOgoo=l. ~

=

The components of the density of the energy--momentum tensor of matter then take the form

]/-~ --U--';

Too = e Using the expression time:

(23.2)

TC~= --P ~ / u v ~c~.

(23.1) for the interval, we determine the connection of Riemannian

0

space--

r~=0; r~=0; (23.3)

F0~ = ~ 9,

r $ = ~9- ~ ,

where the dot d e n o t e s simple d i f f e r e n t i a t i o n

r~=0,

w i t h r e s p e c t to t .

S u b s t i t u t i n g e x p r e s s i o n s (23.2) and (23.3) i n t o the e o v a r i a n t c o n s e r v a t i o n e q u a t i o n of the d e n s i t y of t h e energy-momentum t e n s o r of m a t t e r ( 1 6 . 9 ) , we o b t a i n @ (eV~)~-p~t dt The solution of Eq.

V~=O.

(23.4)

(23.4) has the form

dg' lnV--=--~-2 f 8,+p(~,).

(23.5

80

In the present case the equation of minimal coupling in the form

(14.3) can conveniently be written

(23.6

grim= Ynmf l~- /nm/2 ~- /m/m; An, where we have introduced the notation

fl=

b~ f, ifsil_~_f2, -~A u == ~ yu.

I -- ~I f +

b2 f2=I+T/; For a homogeneous

universe the equations of the gravitational

field

(13.27) take the

form 700 = 70= ----0, Using definition

] a ~ = -- 16n [h=~q- y=~h001;

(13.7), we obtain f00 ~ O,

Because of isotropicity of the universe, field must have the form

1802

(23.7

f0=--O.

(23.8)

the spatial components of the gravitational

f=0 == y=~F (t).

( 23.9 )

goo=U (F), g=~=y.~V (F).

(23.10)

Then

Using the equation of minimal coupling in the form (23.6), it can be shown that Ogoo __ I

r Therefore,

field equations

3

yc*~ dU ~ n Og'~n ~- y~P dV d"--F-; " c~f~f~ dF

"

(23.7) take the form

As initial conditions for Eq.

(23.11) we choose conditions at the present time t = 0: s=So,

U----V=1,

dv dt

----2H,

(23.12)

where H is the Hubble constant. It should be especially noted that the initial conditions have been chosen proceeding from the assumption that the energy density of matter g0 z 0. Therefore, the subsequent calculation will pertain only to this case. It follows from experiments [2] that

20.109 years> @ > 7,5.109 years. With this choice of initial conditions the cosmological field at the present time is the pseudo-Euclidean background on which we consider all other physical processes. It follows from conditions

(23.12),

(23.6), and (23.9) that

F(0)=0,

dF

-gF-i~=o=

4H.

We b r i n g Eq. (23.11) to the form

Considering

the conservation equation

(23.4), we obtain #2 + G'0= - -128r~ -~ e ~VaU.

(23.13)

It is interesting to note that Eq. (23.13) is another way of writing the law of conservation of the energy density of matter and gravitational field in flat space--time. Indeed, if we use definitions (15.9) and (15.7), the coupling equation (23.6), and the components of the density of the energy-momentum tensor of matter in Riemannian space--time (23.2) and also consider Eqs. (23.8) and (23.9), then we obtain too= _

l - -3f ~ - F "2.,

t o= M--_tg

--O.

(23.14)

Therefore, the conservation law of the density of the energy--momentum tensor in flat space-time (15.1) will have the form 0t

ItS+if]=0

From this it follows that

t~+t~=const. Using the initial conditions

(23.12) and expressions 3

- - -128~ -F

"2

(23.14), we obtain 3

=~Co,

where

8~Eo

C o = 16H2 (cz-- 1), ~ -- --3--~ 9 1803

Thus, the total energy density of matter and gravitational field of the universe in flat space--time is constant at all stages of its evolution. This means that the energy of the universe during the course of evolution does not change and is only redistributed between matter and the gravitational field. Using the initial conditions, we can write the solution of Eq.

(23.13) in the form

F

t-~ --'4-H-

-I/:V"" u 0

(23.15)

13o

,

Expressions (23.5), (23.10), and (23.15) determine parametrically the entire evolution of the homogeneous, isotropic universe, including the singular state (or hot universe) with an arbitrary equation of state of matter p = p(e) and coupling equation (23.6). In expressions (23.10) and (23.1) we go over to proper time. In that time interval where U(t) is nonzero, it is possible to pass to proper time ~(t) as follows:

V-U(t)dt=dr. The interval in this case has the form

dS 2----d'~z -- V (x) [dx ~-5 dy 2-5 dz2].

(2 3.16 )

Considering the current time r(0) = 0, we obtain expressions determining the evolution of the universe in parametric form: F

1~V 4---H--

~=-

],fU'dF'__

=8 .[/-~---V, 1--~+~

,

(23.17)

0 8

InV(F)=

- - ~2

I 8 " +de' p (8')

"

(23.18)

8o

For the functions U and V in the equation of minimal coupling

(14.3) we have

U = l - - - ~ F -51(9b4-53b3)F!,

(23.19)

V = I - - 8 9 F -51(bI-5302-53b3+9b4)F2. We shall investigate the solutions obtained in a neighborhood of the present moment (ITI I/4H) of proper time. We assume that in a neighborhood of the present moment of proper time the pressure is negligibly small as compared with the energy density: p ~ g. From Eq. (23.18) we therefore find that

e0 Substituting the expressions

for U, V, ~ into the integral

(23.20) (23.17) and integrating, we have

l

Solving this relation for F and substituting

into the expression for V(F), we obtain

V (~)= 1 + 2H~ + H2~2 [ ~ a - 3 -5 4(b,-5 3b2-5 363 -5 964)] -50(H3~S)

9

M e t r i c ( 2 3 . 1 6 ) w i t h t h e c o s m o l o g i c a l s c a l e f a c t o r V(T) l e a d s t o e f f e c t s o b s e r v a b l e by experiment. One o f t h e m i s t h e c o s m o l o g i c a l r e d s h i f t d i s c o v e r e d i n 1929 b y H u b b l e [ 1 7 ] . T h i s e f f e c t c o n s i s t s i n t h e r e d s h i f t o f s p e c t r a l l i n e s r a d i a t e d by d i s t a n t g a l a x i e s , and the magnitude of the shift is directly proportional to the distance from the galaxy to the earth. In the general theory of relativity t h i s e f f e c t was p r e d i c t e d b y t h e S o v i e t A. A. F r i d m a n i n 1922 [ 1 6 ] .

1804

In the field theory of gravitation the model of the homogeneous universe in a neighborhood of the present moment of time (for HT I/2 (e > I) the universe is ume but having no boundaries. In the field theory of gravitation relation -- the universe has infinite volume for any values of q0 From estimates of the mass of matter in galaxies

one of the most important a parameter q0 < I/2 (~ < I) closed, having finite volthere is no such interand ~.

[23] it follows that

~ o = 3 . 1 0 -31 1 c m 3"

In this case the magnitude of ~ is ~ = 0.06. In EinsteinVs theory the retardation parameter must then be equal to q0 = 0.03, and the universe is open and expanding without bound. However, measurements of the magnitude of the retardation parameter have given another result. Thus, for example, in [29] the conclusion was drawn that the value of qo is located in a range from 2 to 32 with the most probable value being q0 = 5. Thus, in Einstein's theory the magnitude of the retardation parameter obtained from observations is in contradiction to the observed density of matter in the galaxies which is considerably less than required for correspondence. To eliminate this lack of correspondence between the characteristics of the cosmological solution of Einstein's theory and the values of them obtained from observations, attempts are now being made both to increase the magnitude E0 (the search for the missing matter in the galaxies, the "secret of hidden matter") and to reduce the value of q0 obtained from experiment (the assumption of the presence of strong evolution of the emittance function from the magnitude of the red shift). These attempts have so far had no definite success in solving this question. In the field theory of gravitation, in contrast to Einstein's GTR, the retardation parameter is determined not only by the m e a n d e n s i t y of matter e0 [the parameter ~ = 8"~g0/(3H2)] but also by the parameters of minimal coupling; therefore, without turning to post-Newtonian experiments in the solar system, measurement of the retardation parameter q0 makes it possible to measure the quantity

3cz--4]. bl @ 3b2 + 3b3 q- 9b4 = - - T1 [q o + -2

(23.22)

The nature of the behavior of the model of the homogeneous universe in the distant past depends in an essential way on the form of the coupling equations for strong gravitational fields. If the equation V(Fz) tensor and also its spatial F = FI a singular state of ultrarelativistic particles

= 0 has real roots, then for F = FI the determinant of the metric components vanish. Therefore, it is natural to suppose that for the universe is realized. Near the singular state of the universe dominate; they have an equation of state of the form 8 p=~-.

Substituting this equation into expression

(23.18), we obtain 8~

~ = V/,

(23.23)

1805

From this it follows that if the function V(F) vanishes the density of total energy of the universe becomes infinite, and for F = Fi there is actually a singular state of the universe. Some moment of time in the past 9 = T m corresponds to the last positive root F* of the equation V(F) = 0. The time T = --Tm is naturally called the age of the universe. It is given by F:~

T"

1 "

=-TWt

V uctF "

We introduce

the time To = T + T reckoned

(23.24)

--=+~

0

from the singular

state:

F*

1 ~

V8 ap ...... I--~ + ~ ~ g u ~

~0=-$NIn a neighborhood we obtain

of the singular

state

(for F ~ F*) the relation

F*

1 !

"V'CrdF

,

(23.23)

holds,

and hence

(23.25)

The e x p r e s s i o n (23.25) determines the dependence of the proper time in a neighborhood of the singular state on the gravitational f i e l d F and t h u s makes i t p o s s i b l e to determine t h e b e h a v i o r o f t h e f u n c t i o n V(T) i n t h i s n e i g h b o r h o o d . I t s h o u l d b e n o t e d t h a t i f t h e e q u a t i o n V ( F ) = 0 h a s no r e a l r o o t s , then the model of I n t h i s e a s e t h e O l b e r s p a r a d o x may b e t h e d i v e r g e n c e o f the unverse has no singular state. Indeed, the total energy of star light 0 at the present the emittance integral of all stars. moment T = 0 is [2]: 0

P=

(23.26)

I Z (T) V 2 (~) dz, --oo

where

Z(T)

is the proper

density

of emittance

z

of stars:

L) aL,

=

where n(T, L) is the density of stars with absolute emittance at time ~. Convergence of integral (23.26) requires either the existence of a singular state of the universe [V(FI) = 0] for finite FI, as a result of which integral (23.26) is effectively cut off at the lower limit for some T = T(FI), or sufficiently rapid decay of the quantity V(T) with increasing

~V 2 (~) Z (~) ~ o, We introduce

I ~ I-~ ~ .

(23.27)

the notation

bl q- 3b2 q- 3b3 q- 9b4 == 7"~V, 3ba ~- 9b4 = k. Using

this notation,

we rewrite

expression

(23.19)

T F ~,

(23.28)

in the form

V=I

" FL

(23.29)

We now study the effect of the coefficients k and w on the nature of the behavior of the model of the universe. We hereby require that there be no paradox of Olbers type in the theory nor physical singularities of the metric of the universe for finite values of the energy density of matter. We shall first establish conditions under which the first of these requirements is satisfied. As follows from expressions (23.17) and (23.19), in the field theory of gravitation with minimal coupling the cosmological gravitational field F is a monotone function of T, and hence as IT] * ~ the quantity [FI grows without bound. Now as IFl + ~ the scale factor V(F) not only does not decrease but also grows without bound. Since in the present case

1806

the condition (23.27) cannot be satisfied, to eliminate a paradox of Olbers type, the model of the homogeneous universe in the field theory of gravitation with minimal coupling must possess a singular state. This condition requires that the equation V(F1) = 0 have real roots. Using the expression

(23.29),

it is easy to see that it can be satisfied only for ~ q o > 3 - - ~ ,

In this case both roots of the function V are positive. to the singular state of the universe, is

(23.32)

The least of these, which corresponds

F,_l--~t--4w w

The magnitude of the root F* in the range of values of w (23.32) its

is contained within the lim-

2~F*U~--2+k>~.

The character of evolution of the universe in a neighborhood of the singular state is essentially determined by the magnitude of the parameter w. Thus, for example, for w = !/4 from expressions (23.25), (23.23), and (23.29) we have

V~4o/~,

e~'~o8/3, U-------5+4k.

For w = 0 we have

VN-~4o/S, e ~ ~o8/5, U-----2+~. Thus, for large values of the parameter w faster growth of the scale factor V in a neighborhood of the singular state of the universe occurs with the passage of time. For comparison we point out that in the general theory of relativity [7] for any type of model of the universe in a neighborhood of the singular state there are the estimates

V~.,,-[o~~ - ~ 0 -2.

1807

The character of the behavior of the functions U and V in a neighborhood of the singular state essentially determines the flux densities and spectral characteristics of the residual electromagnetic, neutron, and gravitational radiation. The frequency of the electromagnetic and neutron radiation and hence also their temperature hereby vary both as a result of the influence of the cosmological gravitational field and as a result of the Doppler effect. The frequency and temperature of the gravitational radiation can vary only as a result of the Doppler effect. Therefore, measurement of the flux densities and spectral characteristics of these residual radiations affords the possibility of uniquely resolving the question of the nature of the cosmological red shift of the frequency of electromagnetic radiation emitted by distant galaxies. To determine the age of the universe we need an equation of state of matter. Since we do not know a precise equation of state of matter, we shall estimate the age of the universe approximately. We first note that for any equation of state of matter the inequality 0 < p ~ ~/3 is satisfied. Because of expressions (23.18) and (23.29) this means that for 0 F < F* there is the estimate ~"

Vv Therefore, for the age of the universe

(23.24) we have

TI>T>T2, where F*

I!

r1=~-~

1!

F*

VUaF V1-~+~Vg

'

VUd~

tl

Analysis shows that numerical values of the quantities Tx a n d T2 d e p e n d i n a n e s s e n t i a l way o n t h e v a l u e s o f t h e p a r a m e t e r s w and k and vary within broad limits as these parameters change. The m i n i m u m v a l u e o f t h e a g e o f t h e u n i v e r s e for the range of variation of the param e t e r s ( 2 3 . 3 2 ) a n d ( 2 3 . 3 3 ) i s a c h i e v e d f o r w = O, k ~% 9 / 4 , 3 H / 4 /> T 7> 2 H / 9 . As t h e m a g n i t u d e s of the parameters w and k increase in the range of variation (23.32) and (23.33), the age of the universe increases monotonically. T h u s , f o r e x a m p l e , f o r w = 1 / 4 , k = 12 we h a v e

It should be noted that in this case by inequalities (20.9), (23.32), and (23.33) the parameters o f m i n i m a l c o u p l i n g b l a n d b2 a n d a l s o ba a n d b4 a r e n o t p a i r w i s e equal to zero. Moreover, if one of the parameters bl, b2, b3, b4 is equal to zero, then all the remaining parameters are necessarily nonzero. II.. 4 - - ~ c e .

In this case the roots of the function V have different signs, and hence "expansion" of the universe in the future is replaced by "contraction," and it returns to the singular state. This change occurs when the scale factor V attains the value !

V~I--T6>

1.

Therefore, in the present model of the universe the cosmological red shift of the frequency in the future for nearby galaxies will become a blue shift, and the region of space containing galaxies with this shift will increase constantly during the course of time. The magnitude of the root F* corresponding to the initial state of the universe is contained within the limits

0--c~. It should be noted that the metric of the universe will have no singularities between these singular states provided that the function U has no roots in the region F2 < F < F*. The character of the evolution of the universe in a neighborhood of the singular state and also the age of the universe in this case are essentially determined by the values of the parameters w and k, whereby analysis shows that the age of the universe may be either larger or smaller than the quantity I/H. Thus, in the field theory of gravitation nonstationary, homogeneous models of the universe describe the cosmological red shift and admit both monotonic and nonmonotonic behavior. The character of the behavior of the model and the lifetime of the universe depend in an essential way on the magnitude of the retardation parameter q0; for values of q0 contained within the limits 4 -- 3~/2 ~ q0 ~ 3 -- 3~/2 the universe will expand for an indefinitely long time, while for q0 > 4 -- 3~/2 during the course of time "expansion" yields to "contraction" and the universe returns to the singular state. Since the latest data on measurement of the quantity q0 contains large indeterminacy, 2 < q0 < 32, at present it is not possible to unequivocally settle the question of which of these two types of models of the universe is actually realized. LITERATURE CITED I 9

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C. M. Will and K. Nordtvedt, Jr., "Preferred-frame theories and an extended PPN formalism,'! Astrophys. J., 177, No. 3, 757-774 (1972). H. G. Williams, R. H. Dicke, P. L. Bender, C. O. Alley, W. E. Carter, D. G. Currie, D. H. Eckhard, J. E. Faller, W. M. Kaula, J. D. Mulholland, H. H. Plotkin, S. K. Poultney, P. J. Shelus, E. C. Silverberg, W. S. Sinclair, M. A. Slade, and D. T. Wilkinson, "New test of the equivalence principle for lunar ranging," Phys. Rev. Lett., 36, No. 11, 551-554 (1976). CHAPTER 4 WEAK GRAVITATIONAL WAVES IN THE FIELD THEORY OF GRAVITATION

One of the most important problems of the theory of gravitation and of all contemporary physics is the problem of radiation and reception of gravitational waves. The recent heightened interest of investigators in this problem has led to the appearance of a large number of theoretical and experimental works directed both at improvement of the procedures and techniques of experiment and at calculation of possible radiators and detectors of gravitational waves. The major interest in these questions is due to the fact that the problem of gravitational waves is important both at the theoretical and applied levels. At present several versions of gravitational theories have been proposed which provide a rather satisfactory description of the available post-Newtonian experiments but which differ from one another in the description of gravitational waves. Therefore, experimental proof of the existence of gravitational waves and the study of their properties may afford the possibility not only of choosing a tbreory adequate to reality but also of further improving it. Moreover, the existence of gravitational waves and the possibility of receiving them will open up gravitational-wave astronomy and new channels of communication. The problem of the radiation and reception of gravitational waves in the field theory of gravitation contains a number of aspects. In the present chapter we shall concentrate our principal attention only on several of these: we investigate wave solutions of the field theory of gravitation in the weak-field approximation, study gravitational radiation of binary systems, and also indicate a number of gravitational-wave experiments making it possible to verify the predictions of the field theory of gravitation and of Einstein's general theory of relativity regarding the properties of weak gravitational waves in the presence of external gravitational fields.

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