General Relativity and Gravitation, Vol. 8, No. 9 (1977), pp. 787-793

On the Cosmical Constant

RAMESH CHANDRA

Department of Mathematics, Saint Andrew's College, Gorakhpur, Gorakhpur, India

Received January 18, 1977

Abstract

On the grounds of the two correspondence limits, the Newtonian limit and the special theory limit of Einstein field equations, a modification of the cosmical constant has been proposed which gives realistic results in the case of a homogeneous universe. Also, according to this modification an explanation for the negative pressure in the steady-state model of the universe has been given.

w Introduction

The dynamics of general relativity for cosmological purposes are expressed in Einstein's field equations as [1]

87rG Ri i - 1Rg i ]+Agi j - c4 Ti] ( i , ] = 1 , 2 , 3 , 4 ) (1.1)

The constant A is referred to as the "cosmical constant ," which, like R, has the dimensions of a space curvature, namely, (length) -2 . The effect of a positive A is to counteract gravity, and Einstein needed this because he wanted to con- struct a static model o f the universe. Now we need it even more, because, to fit

the observed facts [2], we want to construct an acceleratingly expanding model. Let us reexamine the field equations (1.1) on the grounds that [3]

The relativistic theory of gravitation has to agree in two different limits, with both special relativity (in the absence of gravitation) and Newtonian gravitation theory (in the weak- field and nortrelativistic limits).

First of all we take the case of the Newtonian limit. It can be seen that the Newtonian analog of the field equations (1.1) is given by [4]

V2q~ + c2A = 4~Gp (1.2)

787 This journal is copyrighted by Plenum. Each article is available for $7.50 from Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011.

788 CHANDRA

where $ is the Newtonian potential and p is the density content of the universe. According to the cosmological principle the universe is homogeneous, hence the universe would have a uniform potential. Therefore, the potential is independent of the space coordinates and equation (1.2) gives a constant density of the uni- verse as

c=A O = (1.3)

4rrG

This is the situation of Einstein's static universe. A decade later, Hubble pre- dicted the expansion of the universe and Einstein abandoned the cosmical term calling it "the biggest blunder of my life" [5].

An alternative view has been given in the steady-state theory of cosmology [6] for the constant density of the universe. According to this theory the con- tinuous creation of matter is taking place throughout the intergalactic regions. But in this case the vast negative pressure [7] of the universe remains un- explained.

Again, if we neglect the cosmical term A, then from the equation (1.3) we must have p = 0 and hence the universe is empty, which is not true since it has been observed that a ~ 2 X 10 g/cm a [8].

Thus, we see that we neither neglect nor accept the term A in its present form in the field equations (1.1). Also it can be seen from equation (1.3) that if A is a function of "cosmic time" then the density p is also a function of "cosmic time," and then we will be able to explain the expansion of the universe.

Now we take the case of the special theory limit. The metric of the space- time is given by

ds 2 = c2 d t ~ - d x 2 - d y 2 - d z 2 (1.4)

As usual we consider the case of perfect fluid for which the energy- momentum tensor is

T i j = ( p / c 2 + O ) u i u ] - g q p ( i , ] = 1 , 2 , 3 , 4 ) (1.5)

where

u i - d x d y d z d c t d r = d t ( 1 - 0 2 / c 2 ) 112 (1.6) d r ' d r ' d r ' d r '

On substituting values from (1.4)-(1.6) into (1.1), we get [9]

Ac 4 - Ac 2

P = 8rrG' P 8rrG (1.7)

Here we see that the density and pressure are of opposite sign except when A = 0. Also the density obtained from equation (1.3) and the density obtained from equation (1.7) are of opposite sign for nonzero value of A.

C O S M I C A L C O N S T A N T 789

Thus, we see that the presence of the cosmical constant A in equations (1.1) is unphysical. Therefore, it is better to modify the cosmical constant A in such a way that the new field equations satisfy (i) the equivalence principle and (ii) the two correspondence limits.

w The Modified Cosmical Constant

As we have seen, the presence of the term A in the field equations (1.1) gives unphysical results. In spite of this the field equations give a number of nonstatic solutions that satisfy the observed facts [2, 10]. Therefore, the modification of the term A in the field equations (1.1) must be governed by the following con- ditions: (i) The new field equations must satisfy the equivalence principle; (ii) they must satisfy the two correspondence limits; and, (iii) the nonstatic so- lutions must be compatible with observed facts.

A simple change we can make is to replace the term A by the ten invariant quantities defined by

A(ij) = Aft/) = A (ij) = A (ji) = A(i) (j) ~'(i) = A(i) (i,] = 1, 2, 3, 4) (2.1)

where A(i/) are different functions of the cosmical time t for i, f = 1,2, 3 (space components) and the rest of the components (time components) are constants. Then the new field equations will be given by

87rG R i j - 1 R g i j + A ( i j ) g i j - e4 T i j (i,] = 1,2, 3, 4) (2.2)

The quantities A(ij) defined in equations (2.1) are not the components of a tensor because they do not change on coordinate transformations, though some components of A(6 ) are functions of cosmical time. The adoption of a common cosmical time (which reminds one of the absolute time of Newton) is made pos- sible by the fact that a spatially homogeneous universe would have a uniform gravitational potential.

Whether the field equations (2.2) satisfy the equivalence principle or not, it is sufficient to show that the quantity R } - �89 Rg~ + A(d.~g ~ is divergence free [11]. It can be seen as follows:

~A(i) �9 A ( i ) ~i'i _ t'Di 1 i A(i) .g! = 0 + t,~,ff) i (R~- �89 +,~(j)6j , ; i - , . . j - -~Rg;);i+e~q);, I -~xi g]=O

To determine the Newtonian limit of the field equations (2.2), we see that for slow motion of the medium the equations reduce to

1 Rll - ~Rgl l + Ao1)gl l = 0

R22 - 1Rg22 + A(22)g22 = 0 (2.3)

1 Ra3 - -~Rg33 + A(aa)ga3 = 0

790 CHANDRA

and

- 8n Gp (2.4) R 44 - IRg44 + a ( 4 4 ) g 4 4 - C2

For weak gravitational fields the only significant term of Rij is R444

(R444 ~ -V2r then from (2.3) and (2.4) we get

V2r + [a(t) +/3] c 2 = 87rGp (2.5)

where

&11) = A(22) = A(33) = a(t) > 0

and (2.6)

A(44) = -/3 = const < 0

Thus for a homogeneous universe, namely ~b = const, the density p would be

[~(t) +/31 c 2 O - (2.7)

8~rG

which varies with cosmical time. Hence for an expanding universe, c~(t) must be a decreasing function of cosmical time. Also a(t) and/3 do not vanish simulta- neously beca~,,se in this case p = 0, which is not.in agreement with observations [8].

Considering special theory limit of the field equations (2.2), it can be seen that the density and pressure for a perfect fluid will be given by [9]

/3c ~ ~ ( O c 4 P - 87rG' P - 8riG (2.8)

where a(t) and/3 are given in equations (2.6). Equations (2.8) show that both density and pressure are positive. Thus we see that the field equations (2.2) are a suitable replacement of the

field equations (1.1) for cosmological purposes.

w Equations Governing the Uniform Models

Now we are in a position to check the field equations (2.2) to see whether or not they provide nonstatic solutions compatible with the observed facts (the ex- panding universe). For this we proceed with the Robertson-Walker metric [12]

R2(t) ds2 = c2dt2 [1 + �88 2 ]2 [ dr2 + r2d02 + r2 sin2 Ode 2 ] (3.1)

where K is the curvature index and takes the value + 1, - 1, or 0, according to whether the universe is closed, open, or flat, respectively. R (t) is a scale factor for measuring distances in the nonstatic universe.

COSMICAL CONSTANT 791

By taking the large-scale viewpoint one can treat the cosmic fluid as a "perfect fluid," and then the energy-momentum tensor of the content of the universe will be given by equations (1.5). Since the cosmic fluid is at rest with respect to the comoving observer, u i = (0, 0, 0, c). If we take x i = (r, 0, ~b, ct), then from equations (1.5) and (3.1), equations (2.2) give [1]

2R /~2 K --R + ~ - + R 2 - - a(t) = -8rrGp (3.2)

and t~ 2 K /3 _ 8rrGp - - R 2 + 3 3 R2 + - (3.3)

where a(t) and/3 are given in equations (2.6) and for simplicity here we have taken c = 1.

From equations (3.2) and (3.3) we get

d (8rrGpR3~ (3.4) [-81rGp+/3+~(t)]n21~=~\ ~ /

Knowing the equation of state of the cosmic fluid and the value of a(t), equations (3.3) and (3.4) will give the geometrical evolution of the universe.

If we take

B + a(t) P = 8~rG ( 3 . 5 )

then equations (3.4) and (3.5) together give

where

/~2 =_~R 2_K+C (3.6) 3 R

87rGpR 3 c = ~ ( 3 . 7 )

3

Equation (3.6) is similar to that of Friedmann's differential equation [10] for zero pressure. But here in place of A we get -r and the pressure does not vanish, as we can see from equation (3.5). Thus, we can see that we get all the solutions of the field equations (1.1) with the help of the field equations (2.2) but with different physical conditions. This is the reason why the solutions of the field equations (1.1) are consistent with observations to some extent in spite of not being consistent with the two correspondence limits.

From the field equations (2.2) we can get other nonstatic solutions that we cannot get from the field equations (1.1) and hence we can regard a solution of the field equations (2.2) as giving a picture of the actual universe.

792 CHANDRA

w Explanation for the Steady-State Theory

In 1951 McCrea [7] suggested a dynamics for the steady-state theory based on the "perfect cosmological principle" [6]. He postulated A = 0 and of course K = 0, O = const, p = const. This implies

R = e Ht, H = (~_p)112, p = -pc 2 (4.1)

This vast negative pressure remains unexplained. It is a serious setback to the steady-state theory. But according to the field equations (2.2) it can be ex- plained by assuming ct(t) to be constant (say ao), which is obvious from equa- tion (2.7). In this case equations (3.3) and (3.4) together give

R = e H't,

ao +t~ t~ a0 +t3 8~G >/p > 8riG' 8riG > p >~ 0

(4.2)

This shows that both density and pressure are positive. This explanation gives gives a sound reason for adopting the field equations (2.2) instead of (1.1).

w Comparison with Other Theories

In cosmology other theories of space have been proposed and are being em- ployed. The most important one is the Brans-Dicke theory [13]. This theory of gravitation envisages a scalar field in addition to the metric tensor as being the cause of the gravitational force governing the cosmic expansion. The cosmologi- cal models based On this theory differ in some details from the general relativity cosmological models, and in particular predict a slightly different helium abun- dance from that predicted by general relativity, but they are inaccurate [14]. Since the field equations (2.2) are a slight modification of the field equations (1.1), it will be possible to get a model of the universe from the field equations (2.2) that has the same helium abundance as predicted by observations. Thus, we see that the modification of the cosmical constant A gives more realistic results than the results obtained from the other theories.

w Discussion

We have seen the necessity for the modification of the cosmical constant A. This modification will be meaningful only when a nonstatic solution of the field equations (2.2) governs the evolution of the actual universe. The comparison of

COSMICAL CONSTANT 793

a model to the actual universe has been made in terms of the observed parame-

ters, H (Hubble parameter) , a (density parameter) , and q (decelerat ion parame- ter). But when we use the field equations (2.2) we need to know one more quanti ty, a(t ) . Without knowing the behavior of a ( t ) , it is not possible to per- form the comparison. If a( t ) is found to be constant then it confirms the possi- bi l i ty that the universe is in steady-state, otherwise not.

The basic idea behind the modificat ion is the assumption that the universe is

homogeneous (the cosmological principle). But if the universe is not homoge- neous, then we cannot say anything about the cosmical constant A without going into greater detail.

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