Pergamon Press. Printed in Great Britain 01466364/81/03035708$07.50/O
Chin.Astron.Astrophys.1 (1981) 357364 Acta Astrophys.SinicaI (1987) 131140
COSMOLOGICAL
SOLUTIONS
THE EINSTEIN
EQUATION
HAN Jichang
OF THE EINSTEINYANG
EQUATION
AND OF
Department of Mathematics, Nanjiq University
Received 1980 August 24
ABSTRACT For the RobertsonWalkermetric for a homogeneous and isotropic universe, I have derived explicit solutions of the EinsteinYang equation (interiortorsionfreesolutionsfor a spinless 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 wellknown theorem in Riemannian geometry, a homogeneous and isotropic 3dimensional 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 4dimensional Riemannian space is then ds’ 
a2b)
d? 
(1 + $
2 [dr$ + r:(
[email protected] + sin’Od+‘)]. kr:)
For this metric, HUANG Peng and GUO Hanying [l] and HUANG Peng [2] have discussed the properties of the solutions of the EinsteinYang 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
~~(
[email protected] + sin’Bd+‘)
1 .
This 4space is one that has the maximally symmetrical 3subspace [4]. The lineelement
EinsteinYang Equation
358
is the wellknown RobertsonWalkermetric, 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 torsionfree,
interior solution of the EinstienYang 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 EINSTEINYANG EQUATION In General Relativity, spcaetime with the presence of gravitation is a torsionfree 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 nonrotating gravitational field and no torsion. Under these conditions, the gravitational gauge field equation is reduced to the EinsteinYang 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 tuvterm, while (4) is the sourcefree 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 nonzero
independent components,
dot representing time differentiation.
EinsteinYang 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 comoving 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~22d32~dS00, Suppose a#O;
(15)
make the transformation eY
(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 $Ry=
2K.
(18)
The solution of this equation has different forms according as R is greater than, equal to, or less than zero. These cases will be discussed separately. 1. R=O
The solution of (18) is
Y = KZ + C,r I C2, Cl, Cg being two constants of integration. Using (16), we have 112
3
K?
I C,t + C,.
(191
Using (7) and (19) we get, separately from (12) and (13),
’ 
12KC2 + 3C: Z(K1’  C,t  CI)l
*Starting here, K is written for k
x + ZAX, translator
(20)
EinsteinYang Equation
360
4KCI f C: x 2(Kt’  Cg  Cx)’
’ =
2AX.
(211
When t > 0, K, Cl, C2 must satisfy the conditions in TABLE 1 in order that a2ftl
be greater
than zero.
TABLE 1 (1) Cl + 4C2K<0 (2) Cl + 4cgK=o (3) Cl + 4C#>O
all t>O t#C1/2K t
X=0 _.
(1) Cl'0 (2) Cl
t > Cz/CJ t < c,/c,
K>O
(1) C12+4C2K>0
t1
KC0
t2 (tl
*t1, 2.
(")
are the two real roots of a2(t) = 0.
The solution of (18) has the form
R>O
or a’ 
C3e
3 d
+
C+e
J$ +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 +
pX(:+
x
C,e

+
C3C,)
1
E,
43 + y
(23)
,
1
2n)+$(6Y+&($$C3C,)
(
c3c
J
ii ”
+
C,e
&
+ q1.
(241
R
19
Since J$ a'(t)a e
4
J$ setting Y  e , then (22) is equivalent to C3yaIFy
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 yCscos~r+CadnJ$I+~,
hence r.z2= c5cos Cg, C6 being
J
R  t + C6sin 3
(251
Rt+E J
3
R’
integration constants. Using (7) and (ZS), (12) and (13) now give
EinsteinYang Equation
P X($
+ 2A)(6X+~R)(~CI+~C:~
I+
R tt3
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 recapitulated in the following Theorem 1
For the RobertsonWalkermetric (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 EinsteinYang equations and the corresponding solutions in General Relativity. By "corresponding solutions", I refer to the interior of a spinless ideal fluid which satisfied in the same manner the conditions of homogeneity and isotropy in the 3space of the 4dimensional spacetime, the requirement of being torsionfree 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)
EinsteinYang
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 “bigbang”
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
radiationdominated.
(26),
Then R=O,
01 S
and
respect
because
the
are
(12))
universe.
eqns.
(24),
(15),
that
more complicated,
do have
above,
find
EinsteinYang
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
EinsteinYang
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 [email protected] 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 RobertsonWalker
for the radiationdominated
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
EinsteinYang Equation
REFERENCES HUANG Peng, GUO Hanying, 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 Chongming, E'udmrXuebao (1978) No. 4, 67. GUO Hanying, WU Yongshi, ZHANG Yuanzhong, Kexue Tongbao 18 (1973) 72. ZOU Zhenlong et al, Zhongguo lfexue(1979) No. 4, 366. YANG C.N., Phys. Rev. lett. 33 (1974) 445. ZOU Zhenlong, CHEN Shi, HE %oxiu, GUO Hanying, Acta Astron.SInica17 (1976) 147. English Translation in this journal 1 (1977) 292. CHEN Shi, HE Zuoxiu, ZOU Zhenlong, GUO Hanying, 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."