Cosmological solutions of the Einstein-Yang equation and of the Einstein equation

Pergamon Press. Printed in Great Britain 0146-6364/81/030357-08$07.50/O

Chin.Astron.Astrophys.1 (1981) 357-364 Acta Astrophys.SinicaI (1987) 131-140

COSMOLOGICAL

SOLUTIONS

THE EINSTEIN

EQUATION

HAN Ji-chang

OF THE EINSTEIN-YANG

EQUATION

AND OF

Department of Mathematics, Nanjiq University

Received 1980 August 24

ABSTRACT For the Robertson-Walkermetric for a homogeneous and isotropic universe, I have derived explicit solutions of the Einstein-Yang equation (interiortorsion-freesolutionsfor a spin-less ideal fluid) and hence found certain solutions of the Einstein field equation for the radiationdominated phase of the early universe.

1.

INTRODUCTION

In modern cosmological studies, the distribution of matter in the universe is generally assumed to be homogeneous and isotropic. According to a well-known theorem in Riemannian geometry, a homogeneous and isotropic 3-dimensional Riemannian space must be one of constant positive, zero or negative curvature, for which the metric can be written in the isotropic form (dx’)’

dd-

+ (dx’)’

+ (d~~)~

‘1

[(xl)'+ (X2)'-t-(X3)2]}'

1+f

where k is the curvature of the space. The metric for the 4-dimensional Riemannian space is then ds’ -

a2b)

d? -

(1 + $

2 [dr$ + r:(d@ + sin’Od+‘)]. kr:)

For this metric, HUANG Peng and GUO Han-ying [l] and HUANG Peng [2] have discussed the properties of the solutions of the Einstein-Yang equation, and HUANG Peng [3] made detailed comparisons between these solutions and the correspondingsolutionsin General Relativity. If, in (l), we make the transformation

we then have d.+ -

dt’ -

a’(r)

-

dr2 1 + tr’

i-

~~(d@ + sin’Bd+‘)

1 .

This 4-space is one that has the maximally symmetrical 3-subspace [4]. The line-element

Einstein-Yang Equation

358

is the well-known Robertson-Walkermetric, k> 0 corresponds to a Riemannian space, k = 0, to the euclidean space, and k
PI *

a Lobachevsky space, and a(t) is an unknown fuDEtion of time

For this metric, I shall give in this paper explicit expressions for a torsion-free,

interior solution of the Einstien-Yang equation in an ideal rotating fluid, and hence find certain cosmological solutions of the Einstien equation for the radiation phase of the early universe, Similar discussions can obviously be made in terms of the metric (1).

2. COSMOLOGICAL SOLUTIONS OF THE EINSTEIN-YANG EQUATION In General Relativity, spcaetime with the presence of gravitation is a torsion-free Riemannian space, the rotation of the matter field not being regarded as source of gravitational field. From the point of view of the gauge theory of gravitational fields, [6,7], this amounts to prescibing the conditions of a non-rotating gravitational field and no torsion. Under these conditions, the gravitational gauge field equation is reduced to the Einstein-Yang equation, R P* &I1

$

g,.R

-

g,,A

(3)

-

= %I;,, (4)

where

R:,, - R,,, R I*r -

R:, $

g+,,R.+urR-

-I- RsporRPo*.

coupling All the symbols have their usual meanings (x-’ =16nk is the gravitational constant, and n is a new coupling constant). Eqn. (3) differs from the Einstein equation by the addition of the tuv-term, while (4) is the source-free Yang equation [3]. The two conservation equations are reduced to (5)

Z:. -0, t:;.- 0.

(6)

They are compatible with Eqns. (3), (4) and give no new constraints [9]. In the metric (Z), the unknown function a(t) is to be determined by the field equations. In these coordinates (z" =t, z'=r,

x2 = 8, x3 = e), the curvature tenser has only two non-zero

independent components,

dot representing time differentiation.

Einstein-Yang Equation

359

Both the Riccitensor and tpv are diagonal: Ri-

3A,

(81

R;=R:=R:=A+ZB, r: = 3(A’-

(91

B’),

Here the scalar curvature is a constant [lo], which can be written as R=6(A+

(101

B).

In co-moving coordinates, Tc can be expressed as T: - diag (P,

-p,

-p,

-p>,

(111

where o and p are the density and pressure of the fluid. Eqn (3) now reduces to -

A=-&

B+$

5

q( B’ -

&p+

ZA+B+A--

+-

AZ)

q( B’ -

(121 A’)

(13)

and Eqn. (4) reduces to the single equation 2(/4 - B)ci-((k f 2h)a

-0.

(14)

Using (7). the last can be written as &+ad~-22d3-2~dS00, Suppose a#O;

(15)

make the transformation e-Y

(161

I'* (y>O),

and (15) becomes ... YY - 35 - 2K9 = 0.

(17)*

Integrating, we have 9 -

CY = -2K,

C a constant. Using (lo), (7), (16), this becomes 9 -$Ry=

-2K.

(18)

The solution of this equation has different forms according as R is greater than, equal to, or less than zero. These cases will be discussed separately. 1. R=O

The solution of (18) is

Y = -KZ + C,r -I- C2, Cl, Cg being two constants of integration. Using (16), we have 112

3

-K?

-I- C,t + C,.

(191

Using (7) and (19) we get, separately from (12) and (13),

’ -

12KC2 + 3C: Z(K1’ - C,t - CI)l

*Starting here, K is written for k-

x + ZAX, translator

(20)

Einstein-Yang Equation

360

4KCI f C: x 2(Kt’ - Cg - Cx)’

’ =

2AX.

(211

When t > 0, K, Cl, C2 must satisfy the conditions in TABLE 1 in order that a2ftl

be greater

than zero.

TABLE 1 (1) Cl + 4C2K<0 (2) Cl + 4cgK=o (3) Cl + 4C#>O

all t>O t#C1/2K t
X=0 _.

(1) Cl'0 (2) Cl
t > -Cz/CJ t < -c,/c,

K>O

(1) C12+4C2K>0

t1
KC0

t2 (tl
*t1, 2.

(")

are the two real roots of a2(t) = 0.

The solution of (18) has the form

R>O

or a’ -

C3e

3 d-

+

C+e

-J$ +6K R’

6'3,6'4being integration constants. Using (7) and (22). (12) and (13) then give p s

x (5

x

(

+

C3e

211) + (6X + qR) (T

$

-

fl +

p--X(:+

x

C,e

-

+

C3C,)

-1

E,

43 + y

(23)

,

1

2n)+$(6Y+&($-$C3C,)

(

c3c

J

ii ”

+

C,e

-&

+ q-1.

(241

R

19

Since -J$ a'(t)a e

4

J$ setting Y - e , then (22) is equivalent to C3ya-I-Fy

f Cd>

0

and the conditions that

K, C3, C4 must satisfy can be derived in a similar manner. 3. R
The solution of (18) now assumes the form y--Cscos~r+CadnJ--$I+~,

hence r.z2= c5cos Cg, C6 being

J-

R - t + C6sin 3

(251

--Rt+E J-

3

R’

integration constants. Using (7) and (ZS), (12) and (13) now give

Einstein-Yang Equation

P -X($

+ 2A)-(6X+~R)(~CI+~C:--~

----I+

R ---t-t3

Provided Cs2 +C,*#O,

(26)

C&n

(cscosJ

x

361

Chsin

-$I+-,

6K

-2 1

J

.

(27)

let c5

&a= 2/

c;

+

cosa =

, c:

d&

then a*(t) is greater than zero only if

The explicit expressions for the cosmological solution have now been found. These can be re-capitulated in the following Theorem 1

For the Robertson-Walkermetric (2), the cosmological solution of the Einstein-

Yang equations can be classified into 3 types:

(i) When R=O,

the solution is given by (19)

-(21), in which If,CI, Cg must satisfy the conditions listed in TABLE 1; ii) when R>O,

the

solution is given by (22) - (24), in which K, C3, Cq must satisfy similar conditions to those in TABLE 1; iii) when

R< 0,

the solution is given by (25)- (27), in which K, C5, Cg must

satisfy the inequality (28).

3.

SOME COSMOLOGICAL SOLUTIONS OF THE EINSTEIN EQUATION

I shall nowconsiderthe relations between the cosmological solutions of the Einstein-Yang equations and the corresponding solutions in General Relativity. By "corresponding solutions", I refer to the interior of a spin-less ideal fluid which satisfied in the same manner the conditions of homogeneity and isotropy in the 3-space of the 4-dimensional spacetime, the requirement of being torsion-free being satisfied anyway in General Relativity. To this end, let us discuss some cosmological solutions of the Einstein field equation. Einstein's field equation and the conservation equation are Rw T::.

g,vR -

$ =

g,,A

= -

+

T,,,,

0.

(29) (39)

For (2) and (ll), those are reduced respectively to

i* +

K +

-$-Aa*

-

--&pa’,

2ad + K + ri’ + Aa* -

-

- ’ pa2, 2x

(31) (32)

Einstein-Yang

362

Equation

p+3(p+p)$=0. Comparing terms

(31),

(32)

and have

differences

While

the

the

former

expressions

under

some special

and (12))

(13),

theory,

(15)

the

cosmological

early

phase

With p = 3g

of

solution

(7)

o = 3p and pa4 = const.,

A=-R/4

in

discuss

separately

1.

A=0

Using

(20)

and

-KI’

2(Kt2

tell

(33)

then

hence

(19),

hence

(19)

us that

is 2.

bang

to,

greater

reduces

a2 =

equations

(31),

According

now seek

to

from

equation

for

curvature

give

(32),

satisfied if

(32).

we set

p and o will

than

and less If

R is showing

the

than there

any

On the other

A = 0 in

satisfy

(20))

(33).

(21)

and

I now

is

a “big-bang”

then

at

(35) (36) the

conditions

listed

in TABLE 2 to

ensure

(3)

K>O

c, > 0

eqns.

that

a* >O.

2

t>g c, > 0

(34)

and a* is reduces

- (36)

given

satisfy

by(25).

t takes

any real

positive

value

c, ‘
eqns Again,

(31)

- (33).

applying

the

condition

for

a big-

to (37)

J

hand,

0.

I

(25)

(32) big-

a constant. that

X.

C,t)’

that

the

the

X,

K=O

Cd&

cases.

to

(2)

t=O,

the

Einstein

scalar

will

u*=-Kt2+Clt+Cz.

K
at

field,

Therefore,

resulting

(1)

A z 0, Then R
canbeexplicitlyexpressed

have

c,1)2

-

verified

field

(34)

K and Cl must satisfy

easily

great

Einstein

some particular

same form.

the

automatically

holds. the

the

(33)

Table

It

of

the

the

coupling

C,f.

3G -

2(Kt2

the

I shall

solutions

t and using (33)

A equal

C:

P=

its soluion in

contain

cause

and of

only

as a radiation of

(13)

facts

equation

available

solutions

the

and

Eqn.

we then

p-

For t>O,

(31)

cases

+

(Zl),

such

(10) to

(27),

t= 0, we must have a*=O,

and these

radiation-dominated.

(26),

Then R=O,

01 S

and

respect

because

the

are

(12))

universe.

eqns.

(24),

(15),

that

more complicated,

do have

above,

find

Einstein-Yang

latter

conditions, n = 0)

early

the

(23),

the

obtained

with

satisfying

seemingly

is

(32),

(31)

the

universe

the

(31),

Differentiating

equation

of

for

we shall

(15),

field

is

(with

solutions

radiation

(13))

solutions

explicit

However,

bang

(12),

a complementary

between

equation. whereas

(331 with

Einstein-Yang

363

Equation

(26) and (27) then give (38) (39)

When F’

-I- cl, a, 0

let 6K

then, only when K and Cg satisfy

.J?

the inequality

cos J+@ 2

2 will

a2ft)

satisfy 3.

(40)

>O

be always greater

than 0.

Again, it

is not difficult

to verify

that

(37) - (39)

(31) - (33). A<0

Then R>O,

the expression

for a2,

(ZZ),

becomes,

on requiring

a2(.tl =Cl at

t=0, (41) Eqns, (23) and (24) then give (42)

(43)

For conditions

to be satisfied

“2. R>O** of the previous

by K and C3 to ensure a2 >O, the statements

section

can be repeated.

We easily

verify

that

at the end of (41) - (43) satisfy

(31) - (33). These results Jheorem 2

can be summarized into For the Robertson-Walker

for the radiation-dominated

the following metric

(2),

the solutions

phase of the early universe

of the Einstein

can be written

equation

in the following

forms :

(1) when A=O, as eqns.

(34) - (36) in which K and Cl satisfy

(2) when A > 0, as eqns.

(37) - (393,

(3) when A < 0, as eqns.

(41) - (43) in which Ii and 6’3 satisfy

in TABLE1.

in which X and Ct: satisfy

the conditions the inequality conditions

in TABLE2. (40).

similar

to those

364

Einstein-Yang Equation

REFERENCES HUANG Peng, GUO Han-ying, Keme l'ongbao19 (1974) 512. HUANG Peng, Keme Z'ongbao20 (1975) 56. ?i (1976) 69 HUANG Peng WEINBERG, S C&Vi&ion a%?Co.smolog~ (Wileg. New York)(1972) XU Chong-ming, E'udmrXuebao (1978) No. 4, 67. GUO Han-ying, WU Yong-shi, ZHANG Yuan-zhong, Kexue Tongbao 18 (1973) 72. ZOU Zhen-long et al, Zhongguo lfexue(1979) No. 4, 366. YANG C.N., Phys. Rev. lett. 33 (1974) 445. ZOU Zhen-long, CHEN Shi, HE %o-xiu, GUO Han-ying, Acta Astron.SInica17 (1976) 147. English Translation in this journal 1 (1977) 292. CHEN Shi, HE Zuo-xiu, ZOU Zhen-long, GUO Han-ying, Zhongguo Keae (1976) 35 = Scientia Sinica -19 (1976) 199.

Translated by T. Kiang with the assistance of D. McCrea

Dr Dermott McCrea comments: "The statement that R is a constant that appeared between Eqn (33) and Eqn (34) holds much more generally than implied in the text. In fact, it follows immediately from Eqn (4) without any further conditions. Multiply (4) by guV and we have

R -Rvh;v=O .A

,

while the contracted Bianchi identities give

R" LR x;v = 2 ,x hence, R

9X

= 0, so R is a constant."