Cosmology and Gravitation || Bimetric General Relativity Theory

BIMETRIC GENERAL RELATIVITY THEORY

Nathan Rosen

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

I. INTRODUCTION

The Einstein general theory of relativity is the most beautiful structure in all of theoretical physics. It has been remarkably successful in describing gravitational phenomena. It has provided a basis for constructing models of the universe. It has also provided a conceptual framework for discussing large-scale phenomena in general.

If there is room for criticism of the general relativity theory, it is in connection with the question of singularities. A satisfactory theory should be free from singularities, for a singularity implies a breakdown of the physical laws. Now in general relativity one encounters singularities in two situations:

1. Cosmological models based on plausible assumptions expand from an initial singularity, the "big bang". It has been suggested that this singularity would be eliminated if quantum effects were taken into account. However, no one has succeeded up to now in showing how this could be accomplished.

2. The Schwarzschild solution for the field of a particle has a singularity at the location of the particle (in addition to a strange geometry inside the Schwarzschild sphere, where space and time coordinates appear to exchange roles). A sufficiently massive star undergoing gravitational collapse will (according to the generally prevailing opinion) contract until it reaches this singularity at the center.

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P. G. Bergmann et al. (eds.), Cosmology and Gravitation© Plenum Press, New York 1980

384 N. ROSEN

The purpose of the present work is to consider the possibility of modifying the general relativity theory so as to avoid the above singularities. For this one needs a guiding idea. The present approach is based on the fact that the universe appears to have a fundamental rest frame. This is the reference frame in which the Hubble effect and the black-body background radiation are isotropic. Now it is true that general relativity provides cosmological models characterized by such a rest frame. However, one can try to modify the theory by taking into account the existence of this rest frame - a preferred frame of reference - in the foundations of the theory.

It is proposed to modify the general theory of relativity by introducing a second metric tensor into the theory, so that one gets a bimetric form of general relativity. The second metric is to be associated with the fundamental reference frame of the universe.

II. BIMETRIC GENERAL RELATIVITY

Let us begin by considering the general bimetric formalism. It is assumed that, for a given coordinate system with coordinates x~, at each point of space-time there exist two metric tensors, g~v and Y~v ' corresponding respectively to the line elements

ds 2 = g~vdX~dXv , (2.1)

and 2 ~ v dcr = Y~vdx dx • (2.2)

The tensor g~v' the physical or gravitational metric, describes gravitation, and it interacts w~th matter. With its help one can define the Christoffel symbol f~v\ and hence covariant differentiation (g-differentiation) denoted by a semicolon (;), and one can form the curvature tensor RA~vcr. The tensor Y~v ' the background metric, can be regarded as, in a certain sense, determining inertial forceX' With its help one can define the Christoffel symbol r~v and corresponding covariant differentiation (Y-differentiation) denoted by a bar (I) , and one can form the curvature tensor PA~vcr

It is found that there exist two interesting addition theorems. The first one has the form

{~v} = r~v + ~~v ' where is a tensor having the same form as {~vr , but

(2.3)

BIMETRIC GENERAL RELATIVITY THEORY 385

with ordinary partial derivatives replaced by y-derivatives,

!J. A. 1 ),0 ( ~v = 2 g g~olv + gvol~ - g~vlo) . (2.4)

The second one can be written

RA. ~vo

Here again derivatives

= pA. + KA. ~vo Wo·

A. K ~vo has the same form as replaced by y-derivatives,

(2.5)

A. R ~ver' but with ordinary

KA. = _!J.A. + !J.A. _!J.A.!J.a +!J.A.!J.a (2.6) ~ver ~vler ~erlv era ~v va ~er

As a special case, one can have pA~vo = 0 , so that y~v describes a flat space-time. This was proposed many years ago [1,2,3] as a modification of the formalism of general relativity, but one that does not change the physical contents of this theory. In this case one has

(2.7)

so that the curvature tensor can be written in two forms, either with ordinary derivatives or with y-derivatives. Contracting, one gets

R W

K W

(2.8)

so that the Einstein field equations can also be written in two forms. Working with y-derivatives has certain advantages. For example, one can obtain the Einstein field equations for empty space from the variational principle

where the scalar K is given by

In. K = (gjy) ,

(2.9)

(2.10)

and g~v is varied while y~v is kept fixed. Since the integrand is a scalar density, one can derive an energy-momentum density tensor for the gravitational field in place of the pseudo-tensor that one has in the conventional form of the general relativity theory.

Up to this point we have not considered how the two metric tensors are related. It is natural to assume that, far from matter, where the gravitational field vanishes, g~v = y~v . However, because of the identities existing among the Einstein field equations, for a given y ,the solution for g

W llV

386 N.ROSEN

will contain four arbitrary functions. It is therefore desirable to add four equations in order to tie the two tensors together. One possibility is to take

(Kg~N)IV = 0 . (2.11)

This is a generalization of the De Donder condition often used in general relativity. It should be noted that, while the latter is noncovariant and essentially fixes the coordinate system, (2.11) is covariant and permits general coordinate transformations, under which both tensors transform. However, although Eq. (2.11) seems to be reasonable as a condition on g~v' the fact is that it is arbitrary. Other conditions are poss1ble [3].

The present approach differs from the earlier one in that the background metric is taken as describing, not flat space-time, but rather a space-time related to that of the universe.

Let us begin with some general considerations. In trying to form a picture of the universe, let us take the standpoint that it is something unique in the sense that, although it is based on general physical laws, it involves special conditions. In choosing a background metric y~v we therefore assume that it describes a space-time which, while not flat, nevertheless has maximum symmetry, i.e., a space-time of constant curvature, so that

p A.~va

and hence

p W

3 2Y~V a

where a is a (positive) constant.

In this case, by a suitable choice of coordinates, one write da2 in three forms, corresponding to various values the spatial curvature k ,

k = 1 da2 dt2 a2cosh2(t/a)(dx2 + sin2xcID2)

k 0 do2 = dt2 e2t/ a (dr2 + r 2cID2)

k = -1 do2 = dt2 a2sinh2(t/a) (dx2 + sinh2XcID2)

where

These correspond to three different ways of splitting up the space-time into space and time.

(2.12)

(2.13)

can of

(2.14)

(2.15)

(2.16 )

(2.17)


In choosing among these possibilities, let us be guided by "the principle of finiteness", according to which one should avoid singularities or infinities in the universe. One would expect the space-time described by g~v to have the same topology as that described by Y~v' in order to avoid singularities. If we consider an isotropic model of the universe, as described by g~~, and look for one containing a finite amount of matter (say, a finite number of baryons), then the model must be spatially closed (k = 1). This then must also be the case for the space described by y~v. Hence we take (2.14), with k = 1. The corresponding metric can be regarded as providing a framework,or skeleton, for the universe, so that in the absence of matter g~v = Y~v' With matter present g~v' as determined by the field equations, will be different from v •

,~v

In view of the fact that now (2.7) and (2.8) do not hold, let us modify the Einstein gravitational field equations by replacing ~v by K~v, This means that, if we derive the field equations from a variational principle, the latter should contain a term like the integrand in (2.9) to give the left side of the field equations, rather than the corresponding term involving Christoffel symbols that one has in general relativity. We therefore have, as the field equations,

KV _ l C;vK 8 TV ~ 2 ~ = - ~ ~ . (2 . 18)

Here T~ is the energy-momentum density tensor of the matter or other non-gravitational fields. Indices are raised and lowered with the help of g unless otherwise indicated.

W

Writing

(2.19)

and taking (2.13) into account, we can put the field equations into the form

GV = SV _ 8~TV ~ ~ ~

where GV is the Einstein tensor and ~

SV _ ~ av _ l C;V as ~ - 2 Y~ag 2 ~YaSg

a In view of the Bianchi identity,

G~;V- 0 ,

we have

(2.20)

(2.2l)

(2.22)

388

In empty space

SV = 0 • ].l;V

(TV = 0) , ].l

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(2.23)

(2.24)

It is natural to take this to hold everywhere, so that, from (2.23) ,

TV = 0 , (2.25) ].l;V

as in the Einstein theory. Now from (2.21) one finds

(2.26)

Hence (2.24) gives the condition (2.11) that was considered earlier. However, now it is not an arbitrary choice; it follows from the form of the field equations.

In a given coordinate system, with a given background metric Y~v ' the gravitational metric guv is determined by the field equations (2.20) together with (2.11). We have here a case of over-determination, since there are 14 equations (satisfying the four Bianchi identities) for the ten components g].lV' Hence there will be severe restrictions on the initial conditions that can be imposed. In the case of the Einstein field equations one often obtains solutions which do not appear to have physical significance. One can expect that many solutions of this kind will be ruled out by the present overdetermination.

In the rest-frame of the universe, with a suitable choice of coordinates, Y].lV is given by (2.14). In any other coordinate system, Yvv is determined by the transformation equations from this coord1nate system. While the field equations are covariant, they take on their simplest form in the fundamental rest-frame.

One can expect the scale constant a appearing in the field equations to be of the order of l/H, where H is the Hubble constant, i.e., of the order of 1028 cm. If we are dealing with physical systems having dimensions that are small compared to a such as the solar system, then in general s~ is negligible in(2.20), and the field equations are equivalent to the Einstein equations. However, in cosmological problems this term should be important.


III. COSMOLOGY

Isotropic Universe

Let us consider a closed isotropic model of the universe (k = 1). Taking a coordinate system in which (2.14) holds, we can write

(3.1)

where ¢ and ~ are functions of t. As usual, let us assume that the universe is filled with matter characterized by a density pet) and a pressure pet) , so that the non-vanishing components of TV are

)l

T~ = P , (3.2)

It is convenient to introduce the dimensionless variable

x = t/ a •

The field equations (2.20) then give two relations

e-2¢(~" + tanh X)2 + e-2~sech2x

e- 2¢[2x" + 3(~' + tanh X)2 - 2¢'(ljJ' + tanh x)] +

-2~ -2,1, 2 3 -2~ -2,1, 2 + (2e ~+e ~)sech x = 2 (e ~+e ~) - 8na p ,

where a prime denotes a derivative with respect to x.

In addition we have (2.25), which gives the relation

R' p' + 3 R (p + p) o ,

where R is the radius of the universe,

R a cosh x e~

and also the condition (2.11) , which takes the form

3~' - ¢' 3tanh x (e2¢-2~ - 1) .

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

In looking for solutions of the field equations, we make some assumptions about the behavior of the universe, regarded as a unique system with special conditions. We assume that there is

390 N. ROSEN

a certain moment, which we take as t = 0 , such that the behavior of the universe is symmetric with respect to it, i.e., that the field variables (~,$,p) are even functions of t. In that case one can regard the universe as developing from t = - 00 to t = + 00 (or in the reverse direction), or one can think of it as starting from t = 0 and developing (in either direction) towards large values of t. From the second point of view, there exists an initial moment (t = 0) , and one can specify initial conditions, but if one asks what happened before that, the answer is: the same as what happened after it.

As we shall see, at the initial moment (t = 0) the density has its maximum value, and the universe expands as the time increases. As t goes to infinity, p tends to zero. One would expect that g~v should then approach Y~v' as the presence of the matter becomes unimportant, and this will be assumed to be the case. One can say therefore that we have both initial conditions (t = 0) and final conditions (t = 00) , but this refers to the way we go about solving the field equations. One can imagine a "four-dimensional" point of view in which the history of the universe (- 00 < t < 00) is regarded in its entirety and which is completely determined over its entire length.

Hence we look for a solution of the field equations for x >, 0 taking for x = 0 ,

~o ' $0

p' = ~' = $' = 0

with the additional conditions that, as x + 00, ~,$ + O. For a given value of p these conditions should determine ~o and $ . It is convenignt in the calculation to write

o

z=$-$o (3.9)

so that, for x = 0 , z o , z, o for x + 00 , Z + - $0 •

What can one say about the initial density? The present standpoint is that this is a fundamental quantity and that its order of magnitude is determined by the fundamental constants c , ~ and G From dimensional considerations one can write for the density in conventional units

and, in the general-relativity units used here,


3 -1 -1 65-2 P :: ac fi. G 'V 10 cm ,

o

391

(3.10)

where a is a coefficient of the order of unity. We have here an extremely large value, and this leads to large values for I~ol and I~ol , as we shall see.

After having found the solution of the field equations, one would like to compare its consequences with observation. Let us consider the Hubble effect. It should first be noticed from (3.1) that the time T as given by a clock at rest in our coordinate system is related to the coordinate time t by

T :: Lt e~dt aLx e¢dx • (3.11)

The Hubble constant H is given by

H 1 dR R dT ' (3.12)

with R as in (3.7) It follows that

h :: Ha :: e-~(~I + tanh x) (3.13)

The deceleration parameter ~ is defined as

q R d2V(dRY dT2 dT

(3.14)

and this can be written

1 + ~' ~" + sech 2 x (3.15) q

~' + tanh x (~' + tanh x) 2

In order to be able to apply the solution of the field equations to the description of the present state of the universe, we must determine two quantities, the scale parameter a and the value of t corresponding to the present time, or the corresponding value of x. For this purpose we can make use of two pieces of observatIonal da-ta, the mean density of matter in the universe [4] , p:: 4xlO- 31 gm/cms :: 3XlO- 59 cm-2 , and the Hubble constant H, for which we write [5] H- 1 :: 2xl0 10 £t-yr :: 2xl028 cm.

Dust-Filled Universe

Let us consider the case of a universe filled with dust, so that p:: 0 , this being a fairly good approximation to the present situation in the real universe. In that case (3.6) can be integrated to give

392

p = ~ = ~ e- 3W sech 3x R3 a 3

(A = const.) .

Let us take

Y = 2(¢-W)

Then (3.8) gives

1 3 Y W' = 4 y' + 2 tanh x (e - 1) .

Equation (3.5) can now be put into the form

y" - i y,2 + ~ y'tanh x (3eY + 1) +

i tanh2x (ge2Y - 10eY + 1) + 5(eY - 1)

while (3.4) can be written

i y,2 + y'tanh x (3eY - 1) +

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(3.16)

(3.17)

(3.18)

o , (3.19)

tanh2x (ge 2Y - 10eY + 1) + 2(1 - eY) = N , (3.20)

where

32'Tf 2 2¢ N = --3-- a pe ,

so that, by (3.16) and (3.17) ,

NeW-Ycosh 3x = 32'TfA/3a

From (3.20) and (3.21) one sees that

8'Tfa2po ~ e- 2¢o (1 _ eYo) ,

so that Yo < 0

Let us first investigate the solution of Eq. x = 0 by assuming that Y is large and negative exponential functions can be neglected, eY « 1 . then has the form

y" - l y' 2 + l y' tanh x + -21 tanh 2x - 5 = 0 8 2

(3.21)

(3.22)

(3.23)

(3.19) near so that the

The equation

(3.24)

The solution satisfying the initial conditions is given by


8 Y = Yo - 3 ~n cos w + 2~n cosh x ,

with w = (with cp

o

3x/8 li2 • From (3.17) and (3.18) - 1/10 = ~Yo)

cp = cp - 2~n cos w , o

and

2 1/1 = 1/10 - 3 ~n cos w - ~n cosh x .

Eq. (3.20) gives

so that, from (3.21) ,

2. 3 -2.~ 2. 8~a p = - e ~o cos w

2

Comparing this with (3.16), one finds I/I-y

8~A/a = f e 0 o.

one then gets

It should be noted that, at x = 0, N = 2 and

8TIa2. p = l e-2.CPo o 2 '

in agreement with (3.23) in the present approximation.

393

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

The clock time T, as given by Eq. (3.11) , is found from (3.26) to be

8 ~ cP T = -3- ae 0 tan w , (3.32)

and the radius of the universe, as given by (3.7), now has the form

1/1 2./3 R = ae 0 sec w (3.33)

We see that the above solution becomes singular for x = x* , wi th 3x* / 8 ~ = ~ /2 , x* = 1. 48096 ... However, in the exact equation (3.19), as x approaches x*, the exponential functions become large and the character of the solution changes, so that no singularity occurs, although the functions and their derivatives may take on very large values near x* .

The approximate solution can be valid very close to x* if - y is sufficiently large. If we write

o

394

x = x* - ~

then it is found that,

8 y Yo + "3 !l,nA +

<P <P + 0

29-nA ,

~J 2

Iji + - !l,nA -0 3

(~ > 0) ,

for ~ « 1

29-n cosh x*

!l,n cosh x* ,

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(3.34)

(3.35)

(3.36)

(3.37)

with A for ~ = imation

1: 8 2 /3~. If, for example,

-15 3.89xlO , (3.35) gives one has y = -100 , then,

o y = -10 , so that the approx-based on eY « 1 is still a good one.

However, to get the solution for x larger than x* one must solve the equations numerically. This was done for Eq. (3.19), for a number of values of Yo ,with z obtained by integrating from x = 0 the relation

1 3 Y z' = 4 y' + I tanh x (e - 1) . (3.38)

Some interesting results were obtained:

The functions are finite near x* but vary very rapidly with x. In particular, for large IYol y has an extremely sharp maximum (> 0) near x* and then, remaining positive, decreases to zero as x increases. On the other hand, <p and 1jJ, lnltially negative, increase rapidly near x* but remain negative in this region. In the range 20 ~ - Yo ~ 100 and for x ~ 1.5 the functions y, <p and Iji are nearly independent of Yo' It is found that Iji remains negative everywhere, tending to zero as x increases, while <p reaches zero in the neighborhood of x = 2.1 , attains a small positive maximum (~0.02) near x = 2.5 and then decreases to zero as x increases.

one

and

The value of Yo determines <p and Iji o 0 In the above range

finds

3 + 4.1 ljio + 4.1 <Po=IYo Yo ,

(3.30) then gives

8rrA/a ;" 91 . If we write 0 = 8rra2p, then we have approximately

o = 91e- 3ljisech 3x .

(3.39)

(3.40)

(3.41)


Going back to (3.13), we see that

(3.42)

depends only on x. For the values of p and H mentioned earlier D/h2 0.3 From the numerical calculations one finds that for this value x = 2.6 approximately. From (3.13) one gets h = 1.16 , so that a = 2.3xl0 28 cm. This also gives Ht = 3.0 . From (3.11) one gets HT = 1.2 , so that the age of the universe is only slightly larger than l/H. For x = 2.6 , one finds q = - 0.67. However, it should be recalled that these results are based on the dust-filled model, which cannot provide a good description of the early stages of our universe. Using a more appropriate model for the early history of the universe and then going over to the dust-filled model would probably lead to different values of the above parameters. It should also be remarked that there appears to be considerable uncertainty in the value of q as estimated from recent observations [5] .

o

ex = 1 , with the

As for so that p above valug l/Jo = - 93.5

the initial conditions, if in (3.10) we take = 4X10 65 cm- 2 , then from (3.31) and (3.39), of ~, one gets Yo = - 97.6 , so that ¢o

Incidentally, one finds for the initial - 142.3 ,

value of R ,

(3.43)

which is comparable to the size of an elementary particle.

In the region 0 ~ x ~ x* the values of T/a and Ria are extremely small. Only for x > x* do they become appreciable. In Fig. 1 Ria as a function of T/a is shown by the curve labelled "DUST".

For large values of x , compared to 1 and tanh x = 1

y" + 6y' + 9y o ,

for which y, ¢ and Il/JI are small Eq. (3.19) can be written

(3.44)

the solution of which is given by

y (3.45)

Equation (3.20) becomes

N = 2y' + 6y , (3.46)

and, if (3.45) is substituted into it, one gets

N 2C2e -3X (3.47)

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0.5 1.0 T/o

Fig. 1. The radius of the universe R as a function of the clock time T for the' dust-filled model (DUST) and the radiation-filled model CRAD) , with a the scale parameter.

On the other hand, from (3.21) and (3.16) we have, for large values of x,

N 256'TTA -3X ~e

Using (3.40) we then get .

C = 485 2

(3.48)

(3.49)

If one fits y and y' , as obtained from (3.45) , to the values of y and y' obtained numerically for, say, x = 5 , one finds

- 900 , (3.50)

while C2 is close to the value in (3.49) . A simple calculation then gives, approximately,

- 3X (- 305 + 121 x)e , (3.51)


(145 - 121 x)e- 3X (3.52)

Radiation-Filled Universe

The dust-filled model considered above is appropriate to the present state of the universe, since the mean pressure of the matter is small. In the distant past, when the density was much greater, the pressure was also large and, as remarked earlier, the assumed model cannot give a good approximation to the situation that existed at that time. During the very early history of the universe, when the density, pressure and temperature were extremely high, there was a period when the universe was dominated by radiation. One can describe this situation by means of a radiation-filled model of the universe. This will be briefly considered here.

For isotropic radiation one has the equation of state

If this is introduced into Eq. (3.6), one gets

p B _,+,1, "t - e 'I' sech" -'+ a a

(B = const.)

(3.53)

(3.54)

Let us make use of the variable y as before. If we add (3.4) and (3.5) taking into account (3.53), (3.17) and (3.18), we get

y" - i y,2 + y'tanh x (6eY+l) +

tanh2x (ge2Y-lOeY+l) + 4(eY-l) = 0 . (3.55)

From (3.4) we have (3.20) with N given by (3.21) . However, in light of (3.54) we now have

Ne 21jJ - y cosh'+x = 32nB/3a2

If we assume eY « 1 , Eq. (3.55) takes the form

y" _ 1:.. y' 2 + y' tanh x + tanh 2 x - 4 4

o

(3.56)

(3.57)

The solution satisfying the initial conditions is then given by

y

¢ =

y - 49-n cos u + 29-n cosh x , o

¢ - 39-n cos u , o

(3.58)

(3.59)

398

= ~ - ~n cos u - ~n cosh x , o

with u = -k

2 2 X •

Eq. (3.20) gives

N

so that

3 Y 2'/' 3 -Yo-2~O 4 8na2p = 4 Ne- - ~ = I e cos u

On the other hand, from (3.54) and (3.60)

8nB -4~O 8na2p = --- e cos 4 u .

Hence

and

a2

3 -y +2~ o 0

Ie

3 -YO-2~0 8na2Po = I e

One also gets

T = - !z <1>0

2 ae [ sin u 1 -- + ~n(sec u + tan u) cos2u

and

R = ~o

ae sec u .

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(3.60)

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

(3.66)

(3.67)

Obviously the above solution becomes singular for x = x* wi th 2-!z x* = n/2 , x* = 2.22144... Here again such a singularity does not occur in the solution of the exact equations, although the functions and their derivatives may take on large values for x near to x*.

Solutions of the equations were obtained numerically. It was found that qualitatively the behavior was somewhat similar to that for the dust-filled model. One must have Yo negative and large. One then finds that y increases very rapidly near x* reaching a sharp positive maximum and then decreasing to zero as x increases. The functions <I> and ~ also increase rapidly near x* ,but ~ remains negative while <I> attains a small positive maximum, both functions then tending to zero as x goes to infinity. For large values of Iy I , say, in the range 60 ~ Iy I ~ 160 one finds that, with Ox ~ 2.222 , the functions o


are nearly independent of the value of Yo in this range of y

One also finds that

o 1

1jJ 0 = "2 Yo + 3.6 . (3.68)

The relation between Rand T is shown in Fig. 1 by the curve labelled "RAD".

It is found that in the range 10-50 ~ T/a::: 10-9 one can write in

R a

good approximation

7.15(i)~· If one takes a = 2x 102 8 cm and p

one gets from (3.65) and (3.68) 0

- 145.7 , ,J.. - 142.1 , 'f'o

(3.69)

4x 106 5 as before,

1jJ = - 69.25 o (3.70)

This gives R = 0.0168 cm. o

From these initial values one gets

p 3. 9x 10107

R'+ (3.71)

with R in cm and p in erg cm- 3 On the other hand, if we consider a time T such that muon-pair annihilation has taken place, but the universe is still lepton-dominated, one can write [6]

where 8 is the radiation Stefan-Boltzmann constant Comparing this with (3.71)

8 1. 8x 10 3 0

R

Now (3.69) can be written

R = 1. 8x10 2 0 T ~ ,

temperature (in oK) and 0 (7.59x10- 15 erg cm- 3 deg-'+ ).

one gets

(3.72)

is the

(3.73)

(3.74)

with R in cm and T in sec. Substituting into (3.73) gives

8 (3.75)

The time dependence of the radiation temperature at this stage in the development of the universe is practically the same as in the general relativity theory, and it appears to account for the present cosmic abundance of helium and other light elements [7] .

400 N. ROSEN

We see from this cosmological model that, on the basis of the present field equations, one can obtain a satisfactory description of the early stages of the universe without a singular initial state.

IV. THE FIELD OF A PARTICLE

Let us now consider the spherically symmetric gravitational field of a particle at rest in our coordinate system. It was pointed out earlier that, for a small physical system, the present field equations would, in general, agree with the Einstein field equations. However, this may not always be the case; if the field is very intense, the two sets of equations may give different predictions. This could happen, for example, in the case of the field of the particle near the Schwarzschild radius.

One would expect that the field in the vicinity of the particle should not be influenced appreciably by the large-scale curvature of the universe. Hence, in order to simplify the calculations, let us neglect the spatial curvature, and let us take y~v as given by (2.15), with k = 0, instead of (2.14), with k = 1. This has the additional effect of changing the time dependence: in place of cosh(t/a) we now have exp(t/a) . However, for t/a ~ 1 the qualitative behavior will be the same.

Even with this simplification we are still faced with a difficult situation: the field equations are partial differential equations (with t and r as independent variables) and are hard to solve. Let us therefore carry out the coordinate transformation

t

r

1 t' + 2 a £n (1 - r '2 /a2 )

r'e-t'/a (1 _ r'2/a2)-~

(4.1)

(4.2)

Then (2.15) goes over into the static de Sitter line element, which, if we drop the primes, has the form

(4.3)

The spherically symmetric field of a particle at the origin is then described by the line element

(4.4)

where A, ~ and V are functions of r .

Let us take the field equations (2.20) with TV = 0 . These are now ordinary differential equations. Afte¥ some


rearrangement one can write them, with a prime now denoting a derivative with respect to r,

(l_r2/a2) (fl"+ l fl,2_ 1:. A'fl'+3]1'/r-A'/r+l/r2-e',-]1/r2) 4 2

2 3 A-V A-]1 I = (l/a ) (rfl'+ 2 e -2e + 2) , (4.5)

(l_r2/a2) ( 1:. ]1,2+ 1:. ]1'v'+]1'/r+V'/r+l/r2-e A- fl /r2) 4 4

2 3 A-V A-]1 7 = (1/ a ) (r]1' - 2 e - 2 e + 2) , (4.6)

(l_r2/a2)( 1:. ]1"+ 1:. v"+ 1:. ]1,2+ 1:. V,2_ 1:. A']1'-1:. A'V' + 2 2 4 4 4 4

I I 1 + "4 ]1'v'+fl'/r- 2 A'/r+ 2 v'/r)

= (1/a2) (r]1'- ~ rA'+ ~ rv'- ~ eA-V+ ~) . (4.7)

We also have the divergence condition (2.11), which can be written

(1.r2/a2)(2]1'-A'+v'+4/r_4/-]1/r) = 2(r/a 2)(1_e\-V). (4.8)

If we take l/a = 0, the solution of the equations is well known [1] ,

so that

I - m/r dt2 _ I + m/r dr2 _ (r+m)2d~2 , I + m/r I - m/r

V -A ]1 2 e = e = (l-m/r) (l+m/r), e = (l+m/r) .

This solution can be obtained from the usual form of the Schwarzschild solution,

by the relation

r' = r + m

(4.9)

(4.10)

(4.11)

(4.12)

There is a singularity at r m, the Schwarzschild radius. If we write

y = r - m , (4.13)

402 N. ROSEN

then, for small values of y, the solution can be written

A e = 2m/y + 1 ,

e~

V e

4 - 4y/m + . .. ,

(4.14)

(4.15)

(4.16 )

If we assume that l/a is finite but small and substitute the functions of (4.10) into the right sides of the field equations, we see that, near the Schwarzschild radius, the terms involving exp(A-v) behave like (r-m)-2. Hence, even for small values of l/a, the right sides of the equations become important and cannot be neglected.

To study the behavior of the solution of our field equations near r = m , with m/a« 1, we can write the equations in the simplified form,

~"+ * ~,2_ -} A'].l'+3~'/r-A'/r+l/r2-e'I.-~/r2

+ v' /r-A' /r (4.19)

A-~ 2 A-V 2~'-A'+v'+4/r-4e /r = - 2(r/a )e , (4.20)

where terms have been neglected which are small compared to other terms having the same behavior near r = m .

Let us suppose that the solution of these equations has a singularity at r = m. (This m may be different from that characterizing the field of the particle at large distances, although of the same order of magnitude.) We look for a solution near r = m as an expansion in powers of y = r - m. One finds

A 3 2 / 11 ) (4.21) e - b (m y + - + , 8 8

e~ b2(1-y/2m + ... ) , (4.22)

V 3 e = 2 (b2m2/a2) (1-3y/m + ... ) , (4.23)

where b2 is the value of e~ for r = m


We see that this solution is differxnt from that given above for l/a = o. While the behavior of e is similar, that of eV is quite different. According to (4.16) eV vanishes at r = m and is negative for r < m, while according to (4.23) eV is very small near r = m , but remains positive as r decreases below this value. Since, for l/a = 0 , the signs of eA and eV both change as one goes into the Schwarzschild sphere, the signature of the metric rXmains unchanged. Here, with l/a f 0 , only the sign of e changes, so that there is a change in the signature - we get two time-like coordinates -, and the region r < 2m is therefore unphysical.

One can ask what happens to a test particle that falls down to the Schwarzschild sphere, the sphere of radius m. Let us compare the two cases: l/a = 0 and l/a f o. In the first case, for a particle in radial motion, very close to y = 0 , we can describe the motion by means of the variational principle

o , (4.24)

where a dot denotes a derivative with respect to s. Varying t , we have

(y/2m) 1: = C

and from the relation

(y/2m) 1:2 - (2m/y)y2

we then get

·2 = C2 Y Y - 2m

If, for s = 0, Y the motion is given by

y =

(C

= 1 ,

Yo and y

const.) ,

0 , so that C2

(4.25)

(4.26)

(4.27)

y /2m , o

(4.28 )

As s increases, S2 > 8myo ' i.e.,

y decreases and becomes negative for the particle enters the Schwarzschild sphere.

Now let us consider the case convenience, and assuming that y variational principle

o f [(m2/a2)1:2

Varying t gives

1 • "4 (m/y)y2]ds

2 l/a f 0 Taking b2 3 for is very small, we can write the

o . (4.29)

404 N. ROSEN

= (a/m)C (C = const.) , (4.30)

and putting this into the relation,

(4.31)

gives

?/y = (4/m) (C2 - 1) = (4.32)

provided we take y = - Vo given by

for y = Yo' The motion is then

y 1 2 2 -4 (v /y ) s o 0

if we take s = - 2y /v « 0) . d 0 0 '1 s 1ncreases, y ecreases unt1

again. Thus the test particle sphere and does not enter it.

(4.33)

for y = y . We see that, as it reaches ~ero and then increases

turns back at the Schwarzschi1d

If one wishes to consider the motion of a light ray instead of a particle, one must replace s by some parameter p , and one must replace the unity on the right of (4.26) and (4.31) by zero. One finds that the behavior is qualitatively the same as before: for l/a = 0 , the ray enters the Schwarzschi1d sphere, for l/a f 0 , the ray turns back from the surface of this sphere.

We see that, on the basis of the present field equations, the Schwarzschi1d sphere (r < m) is an impenetrable region, and there is no "black hole".

The coordinates We have been working with are the primed coordinates of (4.1) and (4.2). For the conditions under consideration (r'/a« 1) we have

t = t' , -t' /a r = r'e , (4.34)

and we see that our conclusions apply to the unprimed coordinates as well.

There are some topics that require investigation. One would like to know more about the field of a particle - not just its behavior near the singularity. It would be good to learn about the field of a rotating body. The question of gravitational collapse is an interesting one.


REFERENCES

1. N. Rosen, Phys. Rev. ~: 147 (1940).

2. A. Papapetrou, Proc. Roy. Irish Acad. 52A: 11 (1948).

3. N. Rosen, Ann. of Phys. ~: 1 (1963).

4. J.R. Gott III and E.L. Turner, Astrophys. J. 209: 1 (1976).

5. A. Sandage, Astrophys. J. 178: 1 (1972),

6. E.R. Harrison, Ann. Rev. of Astron. and Astrophys. 11: 155 (1973).

7. D.W. Sciama, "Modern Cosmology", Cambridge University Press, London (1971), Chaps. 12, 13.

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