Some expanding and shearing viscous fluid cosmological models in general relativity

S O M E E X P A N D I N G A N D S H E A R I N G V I S C O U S F L U I D

C O S M O L O G I C A L M O D E L S I N G E N E R A L R E L A T I V I T Y

RAJ BALI a n d D E E P A K RAJ J A I N

Department of Mathematics, University of Rajasthan, Jaipur, India

(Received 29 October, 1987)

Abstract. Two cylindrically-symmetric cosmological models representing viscous fluid distributions when free-gravitational field of type D where coefficient of shear viscosity is assumed to be proportional to the rate of expansion, are obtained. The behaviour of the models in the absence of viscosity and other physical properties are also discussed.

1. Introduct ion

In a previous paper, Bali and Jain (1988) obtained a gravitationally non-degenerate

cosmological model with expanding and shearing viscous fluid in general relativity. Roy

and Prakash (1976) have obtained some viscous fluid cosmological models of plane symmetry in which free-gravitational field is of type D and coefficient of shear viscosity

is assumed to be constant. In this paper, we have obtained two cylindrically-symmetric

cosmological models with free-gravitational field of type D in which coefficient of shear viscosity is assumed to be proportional to the rate of expansion (0).

We consider the space time to the Bianchi type I as it represents the simplest anisotropic-homogeneous universe. The space-time is taken in the form

ds 2 = A 2 (dx 2 - dt 2) + B 2 dy 2 + C 2 dz 2 , (1.1)

where A, B, C are functions of t alone. The energy-momentum tensor for a viscous fluid distribution is given by Landau and Lifshitz (1963) as

T{ = (~ + p)vi vi + pgJi - ~l(v[; + vJ.i + vJvtvi, z +

+ v,v'vj,)) - (~ - ]@v~,(gl + vy), (1.2)

together with

gijviv i = - 1 . (1.3)

is the density; p, the pressure; t /and ~, the two coefficients of viscosity; and v i, the

flow vector satisfying Equation (1.3). We assume the coordinates to be co-moving, so that

v 1 = 0 = v 2 = v 3 and / ) 4 = A - 1

The field equations

R ] - 1 j 5Rg~ + hg{ = - 8nT/ (1.4)

Astrophysics and Space Science 141 (1988) 207-216. �9 1988 by Kluwer Academic Publishers.

208 R. BALI AND D. R. JAIN

for the line-element (1.1) are given by

1 [ B44 C44 B4C 4 A4B 4 A4C4] AB AC J

p A4 2~ xvl l = 8 ~ - 2 ~ ~ - (~ - ~.j ; , j , (1.5)

1[ c. A44 A4q + - A = A 2 C A AeJ

Ip 2. ,V, 1 = 8# 2~B4AB (~ - ~ ) ;'J' (1.6)

1 [ B44 A44 A42 1 Ip 2~C4 2 l ] - - - + - a = 8re (~ - g r / ) v ; , ( 1 . 7 )

A 2 B A A2._] AC

and

1 [A4B 4 A4C 4 n4c41 + + + A = 8r~,. (1.8) A z L AB A C BC ._1

The suffix 4 after the symbols A, B, and C denotes ordinary differentiation with respect to t.

2 . S o l u t i o n o f F i e l d E q u a t i o n s

For complete determination of the set (1.5)-(1.8), we need an extra condition. We assume the space-time to be Petrov-type D. This requires that, either

c ~ = ClV (a)

or

(b) Cl~ = C~23.

Conditions (a) and (b) are identically satisfied if B = C and A = C, respectively. However, we shall assume.4, B, C to be unequal on account of the assumed anisotropy.

From Equations (1.5), (1.6), and (1.7), we have

a n d

( _ ~ ) 4 + A 4 ( B 4 B4C4 1 6 m / A ( ~ ~4) 7 \ ~ - + _~) _ ~44~ ~-c _ (2.1)

22,

COSMOLOGICAL MODELS IN GENERAL RELATIVITY 209

THE FIRST MODEL The condition C( 2 = C( 3 leads to

B44 C44 A4(_~ ~ ) + 2 - - = 0. (2.3) B C A

From Equations (2.2) and (2.3), we have

( C4 ~ ) (8T~]-~ -~" ~-~) = 0 . (2.4)

Since B ~ C, so from Equation (2.4) we have

A4 8rct/A + - - = 0. (2.5)

A

Two cases arise: Case (i) r/-- constant. Case (ii) r//0 = constant = l (say). As regards case (i), the models has already been studied by Roy and Prakash (1976).

Now we consider the case (ii) which leads to

/ (a4 B4 ~-~) q = A \A- + --B + " (2.6)

Equations (2.5) and (2.6) lead to

A4- a fl ( ~ '~ ~ ) ' (2.7)

where

8nl /~ - - - ( 2 . 8 )

8=/+ 1

Putting B C = # a n d B / C = v in Equations (2.2) and (2.7) we have

~v4/v)~4 - 2 A4 (2.9)

(Uv4/v) a

and

A4 _ fl ~4 (2.10) A #

Equation (2.10) on integration leads to

A = ~/~-~, (2.11)


where 7 is a constant. Equation (2.9) on integration leads to

114 = k2.-(2B+ 1), (2.12) v

where m 2 is constant and k 2 = m272. Now putting B C = , and B / C = v and using

Equations (2.11) and (2.12) in Equation (2.1), we have

__/~ (~44 , 2 ~ . 1~ ,2 1 ~ 4 4 ,2 k2(2/~ q_ 1) _(2fl+ 1. ,4 }

1 ~q- k4.-2(213+1)+ 2k2.-2,213+1) "_~} 1 ( ~ - - k4.-2(2.0+1) ) 4 / . . 2 - 4 \ . 2

= 1 6 n l ( 1 - N ( f i + � 8 9 , (2.13) . 2

leading to

~.44 "}- 2B"42 = 0 ;

where fl = 8ra/(8r~/+ 1). Inserting .4 = f ( , ) in (2.14), we have

f = K . - 2t~,

K being constant of integration. F rom Equation (2.15), we have

, = N ( K t + M) 1/(2/~+ 1),

(2.14)

(2.15)

(2.16)

M being also a constant and N = (2fi + 1) 1/(2~+ 1). From (2.12) and (2.16), we have

v = P ( K t + M) k~/tK(2/~+ i)1, (2.17)

in which P is also a constant of integration. Hence,

and

B 2 = #v = N P ( K t + M ) (K+~2)/tx(2~+ 1)] (2.18)

C2 = _# = _N ( K t + M) ~ k2)/ttc(2/~ + 1)] , (2 .19) v P

A 2 = ~ 2 . - 2 f l = 7 2 N - 2 # ( K t + M)-2#/(2/~+ 1) (2.20)

After suitable transformation of coordinates, the metric transformation reduces to the

form ds2 = a 2 T - 2fl/(2fl+ 1)(dx 2 _ d T 2) + TOV+k2)/tK(2t~+ 1)] dy2 +

+ T ( K - k2)/[K(21~+ 1)1 dz2. (2.21)

where

a -

COSMOLOGICAL MODELS IN GENERAL RELATIVITY 211

3. Geometrical Features

The pressure and density for the model (2.21) are given by

gr ip= T - [2(1 +fl) / (2jS+ 1)J

4 K 2 a 2 ( 2 / ? + I) 2 {K 2 - 4 K k 2 f l - k 4 _ 8K2fi 2} +

16zr~/ (K + k 2) T - [ ( f l+ 1)/(2fl+ 1)] + 8 ~ ( ~ 2/,]) ?] A ..1. _ _ _ a 2K(2/?+ 1) l

(3.1)

and

T - [2(1 +/01(2/~+ D] 8TC~ = ( g 2 - k 4 - 4 i lK 2) + A .

4a2K2(2fl + 1) 2

The model (2.21) has to satisfy the reality conditions (Ellis, 1971)

(i) (~ + p) > 0 ,

(ii) (g + 3p) > 0 .

The condition (i) leads to

(3.2)

T - [2(1 + fl)/(2B+ 1)]

2 a Z K 2 ( 2 f l + 1) 2 ( K e - k 4 - 2 K k z - 2 i l K 2 - 4K2/? 2) +

161tr/(K + k 2) + T -[(1+B)/(21~+l)] + 8rc(~ - -~t/)_> 0 . (3.3)

2aK(2/? + 1)

The condition (ii) leads to

(K 2 - k 4 - / ? K 2 - 3 K k 2 f l - 8KZfl 2) + 24aK~rr l (K + k 2) x

x (2/?+ 1)T(1 +~)/(2~+ 1) > 2A + a2K2(2fl + 1) 2 Tr2(1 +a)/(2a+ l~J (3.4)

From Equation (3.3), we find the condition on T during which the model exists. However, from (3.4) we find the condition on A, and

~l = 10,

where

0 - 1 - / ? T - [(1 +fl)/(2fl+ 1)] (3.5) a(1 + 2/?)

The rotation co is identically zero and the shear is given by

(~2 T-[2(l+fl)/(2fl+l)][ (K + 3k2 + 2flK)Z + ( K - 3k2 + 2/?K) 2] = 1 + (3.6)

2a z 36K2(2/? + 1) 2 �9


The non-vanishing components of conformal curvature tensor are given by

C112~__ C13 = l p 2 3

I K2 _ k 4 + 2ilK21 T2#/(2B+ 1) fl + . (3.7)

The effect of viscosity is to retard the expansion in the model. The expansion in the model stops when fl = -1 . The flow vector represents shearing, geodetic, and non- rotating universe in general. For large values of T, the space-time is conformaUy fiat. Since l imr~ ~ a/0 ~ 0; the model does not approach isotropy for large values of T.

The metric (2.21) is the absence of viscosity is given by

ds: = a z (dx 2 - d T 2) + T (K + k2)/r dye + T(Ir k2)/K dz z . (3.8)

The pressure, density, scalars of expansion, shear and the component of conformal curvature tensors are given by

K 2 _ k 4 87rp = A, (3.9)

4K2aET z

K 2 _ k 4 8he = - - + A, (3.10)

4KEa2T 2

1 0 = - - (3.11)

aT '

(K + 3k2) = + ( r - 3k=) 2] a z - 2 a l T 2 1+ -36K-2 7 '

(3.12)

c ; ; - = i t 2 . r , - + I,, - k 4 ) -i'~23 = 6a 2 k 2K 2 j -

(3.13)

The model in the absence of viscosity represents expanding, shearing, geodetic, and non-rotating universe in general. For large values of T, the space-time is conformally fiat.

4. T h e S e c o n d M o d e l

The condition C 12 = C~ leads to

- - + . ( 4 . 1 ) 4 C B C a IkB

COSMOLOGICAL MODELS IN GENERAL RELATIVITY


B~4 = - 8 7rr/A

and B

n44B C44-284(~C n ~)"


c = B(/~ - / r ;

K and L being constants of integration. Now if we use the assumed condition

~=lO'~'l(@ g Jr ~)

in Equation (4.2), we have

n 4 87"~l ( ~ ~ ) -- Jr , B 8 M + 1

Equation (4.5) on integration then leads to

8 = ~ ( A C ) - ~ ,

where 7 being a constant of integration and

8nl 8 - - - -

8 M + 1


B = 7i/(a +8) [ A ( L - Kt )] -t~/~l +~)~


B 4 K (a 4 B )4 = ( A4 B4) t ~gt '

213

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

which on integration gives

A 4 B 4 _ M ; (4.9)

A B L - K t

M being a constant of integration. From Equatins (4.7) and (4.9), we have

A 4 = M f l + M + K ~ 1

A (2/~ + 1) L - K t (4.10)

2 1 4 R. BALI AND D, R. IAIN


A = m(L - K t ) - [(Mp+M+K,8)/K(2fl+ l)l (4.11)


B = ]71/(1 +,8) m - [,8/(1 +,8)1 [(L - K t ) - [ , 8 ( K - M ) / K ( 2 , 8 + 1 ) 1 ] , (4.12)

C = 71 / (1 +,8)m- [,8/(1 +,8)] [(L - K O [(,SK+,SM+K)/K(2,8+ 1) ] ] . (4.13)

After suitable transformation of coordinates, the metric transformation reduces to

d s 2 = T - t2(Mfl+M+K,8)/K(2,8+ l ) l ( d x 2 _ d T 2) + T - [2,8(K--M)/K(2fl+ l ) l dy2 +

+ T [ 2 ( K , 8 + K + flM)/K(2fl + 1)1 d z 2 " (4.14)

5. Some Physical and Geometrical Features

The pressure and density for model (4.14) are given by

T [ 2 ( M - K ) (/3 + 1) /K (2,8 + 1 )]

8~p = - (K +/~K + /~M) 2 - K2(2fl + 1) 2

_ 16Z~v/Tt(M-K)(e+ 1)/K(2,8+ 1)] ~ f l ( K - m ) ~ + 8 g ( ~ - - 2 t / ) ~ - - A (K(2fl + 1)) 1

and

8/'Cg = - -

(5.1)

T I Z ( M - K ) (13+ 1 ) / K ( 2 f l + 1)1

K2(2fl + 1) 2 [(Kfl + tiM) 2 + 2fl(K 2 + M 2) + KM] + A .

(5.2)

The model has to satisfy the reality condition (Ellis, 1971)

(i) e + p > 0 ,

(ii) e + 3p > 0.

The condition (i) leads to

T [ 2 ( M - K ) ( f l+ 1 ) /K(2 ,8+ 1)1

K2(2fl + 1) 2 [(K + flK + tiM) 2 + (Kfl + tiM) 2 + 2fl(K 2 + M 2) + KM)] +

+ 16nr/Tt(~-K) (t~+ 1)/K(2,8+ 1)] fi(K - M) K(2fl + 1)

8 . ( ~ - 2 ) ~_< o . (5.3) gt/ l

The condition (ii) leads to

T[2(M-- K ) (B + l ) / K ( 2 f l + 1)]

K2(2fl + 1) 2 [3(K + flK + tiM) 2 + (Kfi + tiM) 2 + 2fl(K 2 + M 2 + KM)] +

+ 48rcrITI(M-K)(,8+ 1)/K(2,8+ 1)] fl(K - M )

K(Zfi + 1) ~7<2A ; 24~(~ - ~ ) 7 - (5.4)

C O S M O L O G I C A L M O D E L S IN G E N E R A L R E L A T I V I T Y 215

rl = 10, where 0 is the scalar of expansion in the model, given by

0 = (K - M) (1 - fl) Tt(M_K)(a + 1)/K(2fl+ 1)] (5.5) K ( 2 f i + 1)

Equation (5.3) gives condition on T, during which the model exists while from (5.4) we find the condition on A. The rotation co is identically zero and the shear is given by

T [ 2 ( M -- K ) (1 + fl)/K (2 f l + 1 )1

a 2 = [ K 2 + M 2 + M K ] . (5.6) 3K 2

The non-vanishing components of conformal curvature tensors are given by

11,- '13 _ = = -

M

3K _ _ _ _ T[ (2~t~+ 21t,l--K)/K(2fl+ 1)1 (5.7)

The effect of viscosity is to retard the expansion in the model. From Equation (5.5) we find that M > K. The flow vector represents shearing, geodetic, and non-rotating universe in general. Since

lim -a# 0, T---~ oo 0

the model does not also approach isotropy for large values of T in this case. The pressure, density, scalars of expansion, shear, and the non-vanishing components

of conformal curvature tensor in the absence of viscosity for the model (4.14) are given by

T2M/K 8 / ~ p - [K 2 ] - A = - T ( 2 M - 2 K ) / K - - A, (5.8)

K 2 T 2

T 2 ~ / K M 8 ~ - [ K M ] + A = - - - T ( 2 M - 2 K ) / K + A, (5.9)

K 2 T 2 K

K - M 0 - T ( M - K) /K , (5.10)

K

T 2 ( M - K )/K a 2 - (K 2 + M 2 + M K ) , (5.11)

3K 2

M 1 / ' ~ 1 3 C ~ = C ~ = - ~ 1 3 = - - - T ( z ~ t - IC)/K . (5.12) 3K

From Equation (5.10), we found that K < 0. Hence, model (4.14) in the absence of viscosity represents expanding, shearing, geodetic, and non-rotating universe in general.


Since lim~_~ co a/0 ~ 0. Therefore, the model does not approach isotropy for large values o f T.

Model (4.14) is also conformaly flat for large values o f T.

References

Bali, Raj and Jain, Deepak Raj: 1988, Astrophys. Space Sci. (in press). Ellis, G. F. R.: 1971, in R. K. Sachs (ed.), General Relativity and Cosmology, Academic Press, New York,

p. 117. Landau, L. D. and Lifshitz, E. M.: 1963, Fluid Mechanics 6, 505. Roy, S. R. and Prakash, S.: 1976, J. Phys. A:Math. Gen. 9, 261.

Some expanding and shearing viscous fluid cosmological models in general relativity

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