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Acta Physica Academiae Scientiarum Hungaricae, Tomus 42 (3), pp. 245--250 (1977)
COSMOLOGICAL UNIVERSES WITH SPHERICAL SYMMETRY
By
SHRI RAM* and J . N. S. KASHYAP
DEPARTMENT OF MATHEMATICS, BANARAS HINDU UNIVERSITY, VARANASI- 221005, INDIA
(Received 8. III . 1977)
In this paper Einstein's field equations for perfect fluid distribution with a speeifir equation of state ate imposed to the most general spherieally symmetric space-time defined by SYNCE in Kruskal eoordinates. A functional relationship is obtained which culminates in a pre- scription for building the most general eosmological solutions. The de Sitter cosmological uni- verse is obtained in a systematie way.
1 . I n t r o d u e t i o n
The mos t general spherical ly s y m m e t r i c met r ic def ined b y SYNCE in
Kruska l coordinates , is [1]:
ds 2 = - - 2 f d u d v -~- r2(dO 2 -}- sin2Od~ 2) , (1.1)
where f and r are func t ions of (u, v). The coordinates x 1 = u, x 4 ---- v are t aken on a 2-space U 2. The coordinates x 2 = O and x 3 = �9 belong to a uni t sphere S 2. Fo r this met r ie SY~cE has explored the solut ions of Eins te in ' s f ie ld equat ions in vacuo in t e r m s of sui table coordinates ; and ob ta ined Schwarzschild solution in a sys t ema t i c way . The present au tho r s have imposed E ins t e in - -Maxwe l l equa t ions to this l ine-e lement and o b t a i n e d a funet ional re la t ionship be tween the general funet ions f and r which enables us to get several solut ions under specific coordinate sys tem. The Re issner - -Nords t r i~m solution is shown to be a special case [2]. The present authors also h a v e ob ta ined an explici t solution
of Eins te in-Scalar zero-res t -mass field equat ions [3]. I n th is p a p e r we have ob ta ined a func t iona l re la t ionship be tween the
f u n c t i o n s f a n d r in (1.1) sat isfying the Eins te in ' s f ield equat ions for the perfect f lu id wi th the equa t ion of s t a te Q + p ---- 0. B y using sui table a r b i t r a r y func- t ions occurr ing in the re la t ionship we can e rea te spherical ly s y m m e t r i c cosmo- logical universes w o r t h y of s tudy . For jus t i f i ca t ion the de Si t te r universe is
ob t a ined in a sy s t ema t i c way.
* Applied Mathematic8 Seetion, Institute of Teehnology, Banaras Hindu University, Varanasi - - 22 1005, India
Acta Physica Aeaaemiae Seienliarum Hungarieae 42, 1977
24• SHRI RAM and J. N. S. KASHYAP
2. Field equations
The energy-momentum tensor f o r a perfect fluid distribution is given by
Tij = (e + p)v,vj - - Pgij (2.1) together with
g i j v i l ) j : 1 ,
p being the pressure, ~ the density and vi = (v 1, O, 0, v4) the flow rector which represents the motion of the fluid in the u-direction. The s equations
R i / - - 1 R - ~ gij -4- Ag~j = - - 8~Tij (2.2)
for the metric (1.1) are as follows:
2r14 + 2rlr4/fr, q- X/r ~ q- A = K [ - - (~ + p ) v l v 4 / ! - f ] , (2.3)
2r~4/# q- (f14 - - f l f 4 / f ) / f 2 q- A = - - K p , (2.4)
2(r n -- r~fz / f ) / f r = K(p -4-p)v~l/f, (2.5)
2(r~~ - r4A/ f )q = K(e + p)v~/ f , ( 2 . 6 )
where A is the cosmological constant and K = --8~. The necessary condition for perfect fluid distribution ensures us the equality of the Einstein tensor components G~, G~, G~ and G3/GJ4t i---- R~ - - 1/2~JR). Therefore e -4- p : 0 is the only admissible physical situation for which the Eqs. (2.3)--(2.6) may gire solutions. Imposing the condition ~ + p : 0 the above equations reduce to
2ra4/fr -4- 2rxr4/fr 2 q- 1/r 2 = K p - - A , (2.7)
2rl4/fr q- (f~4 _ f l f ~ / f ) / f 2 = K p - - A , (2.8)
rll -- r~f~/f = 0, (2.9)
r a , - r , A / f = O. (2.10)
The suffixes 1 and 4 after the symbols f and r denote ordinary differentiation with respeet to u and v respeetively.
3. Solution of the field equations
The Eqs. (2.9) and (2.10), after integration, provide
f = 2B(v)rl, f = 2A(u)r,,
Acta Phy~ª Academ/ae Scientiarum Hun$aricae 42, 1977
(3.1)
COSMOLOGICAL UNIVERSES WITH SPHERICAL SYMMETRY 2 4 ~
where A(u) and B ( v ) are a rb i t ra ry funet ions of u and v, respect ively. Using~ (3.1) the Eq. (2.7) can ]~e wri t ten in the a l te rnat ive forms
(,r~)~ = A(u)(~~r ~ - - 1)r~, (3.2~
(~~~)~ = B(0(~~~ ~ - - 1)r~,
where 22 = K p - - A . In tegra t ing Eqs. (3.2) we obta in
r 1 = D / r - - A(1 - - ~2r2), (3.3)
r 4 = E / r - - B(1 -- ~2r2). (3.4)
D and E being a rb i t r a ry functions of u and v, respectively, and ~2 = 22/3. Different iat ing (3.3) and (3.4) with respect to v and u, respect ively, and using (3.1) we get
D = k A , E = k B k = constant) . (3.5)
Subs t i tu t ion of (3.5) in (3.3) and (3.4) provides
r~ : - - A(1 -- k / r - - ~2r2) , (3.6)
r4 = -- B(1 -- k / r - - ~2r2) . (3.7)
On account of the Eqs . (3.6) and (3.7) the Eq . (3.1) takes the form
f = -- 2AB(1 - - k / r - - :r (3.8)~
Making use of the Eqs. (3.6), (3.7) and (3.8) the Eq. (2.8) has been found to be ident ical ly satisfied.
In Minkowskian coordinate sys tem the metr ic (1.1) takes the form
d s 2 = 4 d u d v + r2 (dO 2 + sin 2 0 d ~ 2) (3.9)~
u ---- ( r -4- O / 2 , v = ( r - - t ) / 2 .
There ate no singularities in Minkowskian spaee-t ime, and r : 0 is mere ly a t ime-l ike geodesic. When r = u + v, the Eqs. (3.1) show tha t f is constant . So in view of the Eqs. (3.2)--(3.9) we mus t t ake k : 0. Therefore , (3.6) and (3.7) reduce to
r~ = -- A(1 - - ~2r2), (3.10)
r , = - - B(1 - - 42r2). (3.11),
F r o m dr = r l d u -4- radv , we obtain
R M r / ( I ~ - - r ~) = - - A d u - - B d v , (3.12)
Acta Physica ~4r Sci~ntiarum Htmstwicat 42, 1977
2 4 8 S H R I RAM and J . N. S. K A S H Y A P
where ~2 _-- 1/R 2. Thus the problem of finding the functions f and r now re- duces to integratc the funct ional equat ions (3.10), (3.11) and (3.12).
4. Frmetional re la t ionsh ip
We now choose a real variable H related to r by the differential equat ion
H d H / ( H 2 _ R 2) = R2dr/(r2 _ / { 2 ) . (4.1)
The solution of this differential equat ion is
(H 2 - - R 2) = b { r - - R IR (b = c o n s t a n t ) . (4.2) ( r + R ]
From Eqs. (3.12) ana (4.1) we have
H d H / ( H 2 - R 2) = A d u + B d v . (4.3)
This equat ion provides
A = H H 1 / ( H 2 - - R2), B = H H 4 / ( H 2 - - R2). (4.4)
Assuming the solut ion of (4.3) as
H 2 - - R 2 = U 2 V 2 , (4.5)
where U and V are funct ions of u and v, respeet ively, one can obtain
A = U 1 / U , B = V ~ / V (4.6)
using (4.4). Then from (4.1) we have
rl = HHI(r2/R2 - - 1)/(H e - - R2), (4.7)
r~ = HH4(r2/R 2 - - 1)/(H 2 - /{2). (4.8)
A simple and s t ra ightforward calculation provides the expression for the func- t ion f a s
f = 2H2H1H,(r2/R2 _ 1)/(H 2 _ / { 2 ) 3 . (4.9)
Hence for the metric (1.1) satisfying the Einstein 's field equat ions (2.1), (2.2) and the equat ion of s ta te 0 + P = 0 the funct ion f is given in te rms of r b y the functional equat ion (4.9), where r is determined in t e rms of H b y (4.5) so t ha t r 2 comes out a s a funet ion of (u, v). Consequent ly the function f can be
Acta Physiea dcademiae Sdentiarmn Hungtsrieae 42, 1977
COSMOLOGICAL UNIVERSE$ WITH $PHERICAL SYMMETRY 249
writ ten in a number of equivalent forros as functions of (u, v). Choosing the suitable forms of the funetions U and V we can obtain spherically symmetr ic cosmological models wor thy of s tudy. In the next Section we shall obtain the well-known de Sit ter cosmological model for the specific choice of U and V. The constant of integrat ion b, occurring in (4.2), may be t aken uni ty .
5. Part icu lar case
Choosing U = u , V = v (5.1)
Eq. (4.5) becomes H ~ - R ~ = u % 2 . (5.2)
By (5.2) H ~ H 4 = u3v3/H 2 . (5.3)
Therefore, from (4.9) f = 2(r2/R 2 -- 1 ) / u v . (5.4)
Again, since d H = H l d u + H4dv ,
we obtain H d H / u 2 v z = du/u + dv/~. (5.5)
F rom (4.1) and (5.5) du/u + dv/v = R2dr/(r 2 - - R 2 ) . (5.6)
Define ~ by
--du/u + d v / ~ = d~ or 3 = log[ v--] + cons t . (5.7)
The Eqs. (5.6) and (5.7) together gire
4 d u d v / u v = dr2/(1 - - r2/R2) -a~~ . (5.8)
Combining the Eqs. (5.4) and (5.8) we obtain
- - 2 f d u d v = [(1 -- r2 /RZ)- ldr z - - (1 -- r~q . (5.9)
Therefore the metr ic (1.1) reduces to
ds z = (1 - - r2/R 2)-ldr2 + r2(dO z -4- sin 2 Odq ~2) -- (1 - - r2/R2)dr 2 . (5.10)
The line-element (5,10) is the de Sit ter cosmological universe which is spheric- ally s y m m e t ¡ solution of the Einstein's f ield equations (2.1) and (2.2) with
Acta Physica Academiae Sr Hungaricae 42, 1977
250 SHRI RAM and J. N. S. KASHYAP
the equat ion of state p ~ p ~-~ 0. The expression for the cons tant R is given by
I q 2 : - - ( S H p + A)/3 (5.11)
which can be considered as the radius of the universe.
The a u t h o r s ate thankfu l to Dr. K. P. SINGH for va luable suggest ions.
REFERENCES
1. J. L. SYN6E, Annali di Matematica pura ed applicata, (VI), 98, 239, 1974. 2. SHRI ~ and J. N. S. KASHYAP, Acta Phys. Hung., 41, 87, 1976. 3. SH~I RAM and J. N. S. KASHYAP, communicated.
Acta Physica .4cademiae Scientiarum Htmgarieae 42, 1977
