Some Five-Dimensional Bianchi Type-III String Cosmological ...

Bulg. J. Phys. 38 (2011) 380?389

Some Five-Dimensional Bianchi Type-III String Cosmological Models in General Relativity

G.C. Samanta1, S.K. Biswal2, G. Mohanty3,4, 1Mathematics Group, Birla Institute of Technology and Science (BITS) PilaniK. K. Birla, Goa Campus-403726, India 2Department of Physics, Gandhi Institute for Technological Advancement (GITA), Madanpur-752054, India 3P.G. Department of Mathematics, Sambalpur University, Jyoti Vihay-768019, India 4703, Rameswarpatna, Bhubaneswar-751002, India

Received 22 July 2011 Revised 29 December 2011

Abstract. In this paper we have constructed some five-dimensional Bianchi type-III cosmological models in general relativity when source of gravitational field is a massive string. We obtained different classes of solutions by considering different functional forms of metric potentials. It is also observed that one of the models is not physically acceptable and the other models possess big-bang singularity. The physical and kinematical behaviors of the models are discussed.

PACS codes: 04.30.Nk, 98.80.Cq, 98.70Vc

1 Introduction

The Kaluza-Klein theory is attractive because it has an elegant presentation in terms of geometry. In certain sense, it looks just like ordinary gravity in free space, except that it is phrased in five dimensions instead of four. Kaluza [1] and Klein [2,3] attempted to unify gravitation and electromagnetism. An interesting possibility known as the "cosmological reduction process" is based on the idea that at very early stage all dimensions in the universe are comparable. Later, the scale of the extra dimension becomes so small as to be unobservable by experiencing contraction. Such cosmological models were investigated by Forgacs and Horvath [4]. Chodos and Detweller [5] showed that, the extra dimensions are unobservable due to dynamical contraction to very small scale, while the three other spatial dimensions expand isotropicaly as a consequence of cosmological evolution in the frame work of a pure gravitational theory of Kaluza-Klein. Freund [6] explained the smallness of the extra dimensions of the universe through

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Five-Dimensional Bianchi Type-III String Cosmological Models

the dynamical evolution in 10 and 16-dimensional super gravity models. Guth [7] and Alvarez and Gavela [8] observed that during the contraction process extra dimensions produce large amount of entropy, which provides an alternative resolution to the flatness and horizon problem, as compared to usual inflationary scenario. Gross and Perry [9] showed that the five-dimensional Kaluza-Klein theory of unified gravity and electromagnetism admits soliton solutions. Further, they explained the inequality of the gravitational and inertial masses due to the violation of Birkoff's theorem in Kaaluza-Klein theories, which is consistent with the principle of equivalence. Appelquist and Chodos [10] and RandjbarDaemi et al. [11] claimed through solution of the field equations that there is an expansion of four-dimensional space-time while fifth dimension contracts to the unobservable plankian length scale or remains constant as needed for the real universe.

At present the relativistic considered that the cosmic string theory is important in the early stages of the evolution of the universe before the particle creation. Moreover the study of cosmic strings in elementary particle physics raised from the gauge theories with spontaneous broken symmetry. After the big-bang, it is believed that the universe might have experienced a number of phase transitions by producing vacuum domain structures such as domain walls, strings and monopoles. Cosmic strings may act as gravitational lenses and give rise to density perturbations leading to formation of galaxies. Chatterjee [12] constructed massive string cosmological model in higher-dimensional homogeneous space time. Krori et al. [13] constructed Bianchi type-I string cosmological model in higher-dimensional field and obtained that matter and strings coexist throughout the evolution of the universe. Rahaman et al. [14] obtained exact solutions of the field equations for a five-dimensional space-time in Lyra manifold when the source of gravitation are massive strings. Mohanty et al. [15] constructed five-dimensional string cosmological models in Barber's second self creation theory of gravitation and they showed that, one of the models degenerates into two different string cosmological models in Einstein's theory corresponding to variable G and constant G. Recently Mohanty and Samanta [16] constructed five-dimensional string cosmological models with massive scalar field in Lyra manifold and showed that the models avoid initial singularity in presence of massive scalar field. In this paper we have taken an attempt to study the role of string in five-dimensional Bianchi type-III space-time in general theory of relativity.

2 Metric and Field Equations

Here we consider the five-dimensional Bianchi type-III metric in the form

ds2 = dt2 - A2dx2 - B2e-2axdy2 - C2dz2 - D2d2,

(1)

where A(t), B(t), C(t) and D(t) are the scale factors.

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G.C. Samanta, S.K. Biswal, G. Mohanty

The universe is assumed to be filled with matter along with massive strings

Tij = uiuj - wiwj ,

(2)

uiui = -wiwi = -1,

(3)

uiwi = 0,

(4)

where is the rest energy density of the cloud of strings with particles attached to them, is the tension density of the strings and = p + , p are the rest energy of the particles. The velocity ui describes the cloud 5-velocity and wi represents the direction of anisotropy. For the weak, strong and dominant energy conditions, one finds that >0, p >0 and the sign of is unrestricted. The fifth co-ordinate is taken to be space-like and the co-ordinates are co-moving, where

u0 = u1 = u2 = u3 = 0, u4 = 1.

(5)

Without loss of generality, we choose

wi = (0, 0, 1, 0, 0).

(6)

The Einstein's field equation is given by

Rij

-

1 2

Rgij

=

-8Tij

.

(7)

With the help of equations (2) ? (6), the field equation (7) for the metric (1) can be written explicitly as

B B

+

C C

+

D D

+

BC BC

+

BD BD

+

CD CD

=

0,

(8)

A A

+

C C

+

D D

+

A C AC

+

AD AD

+

CD CD

=

0,

(9)

A A

+

B B

+

D D

+

AB AB

+

AD AD

+

BD BD

-

a2 A2

=

8,

(10)

A A

+

B B

+

C C

+

AB AB

+

A C AC

+

BC BC

-

a2 A2

=

0,

(11)

AB AB

+

A C AC

+

BC BC

+

AD AD

+

BD BD

+

CD CD

-

a2 A2

= 8,

(12)

and

A A

=

B B

.

(13)

3 Solutions of the Field Equations

From equation (13) it can be shown that A=kB, where k is an arbitrary constant. Therefore, we have five equations in six unknowns. For deterministic solutions

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Five-Dimensional Bianchi Type-III String Cosmological Models

we need one assumption. We shall explore physically meaningful solutions of the field equations (8) ? (13) by considering a simplifying assumption of the field variables A, B, C and D.

3.1 Classes of Solutions

Case 1

A = B = tn,

(14)

n is an arbitrary constant.

In this case equation (11) reduces to

C C

+

2nC tC

+

2n(n - 1) + n2 t2

-

a2 t2n

=

0.

(15)

It is easily observed that equation (15) is solvable for n = 1. The equations (14)

and (15) reduce to

A=B=t

(16)

and

t2C + 2tC + (1 - a2)C = 0.

(17)

On integration equation (17) yields

-1+ 4a2-3

C=t 2

(18a)

or

-1- 4a2-3

C=t 2

.

(18b)

Using equations (16) and (18a) in the field equation (8), we get

t2D + t(1 + 4a2 - 3)D + (-1 + 4a2 - 3)2 D = 0, 4

which on integration yields

D = tm1

or D = tm2 ,

where and

- 4a2 - 3 + m1 =

4a2 - 3 - (-1 + 4a2 - 3)2

2

- 4a2 - 3 - m2 =

4a2 - 3 - (-1 + 4a2 - 3)2

2

.

(19) (20a) (20b)

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G.C. Samanta, S.K. Biswal, G. Mohanty

Now the above solutions give two different set of models.

Set 1: Thus, the five-dimensional string cosmological model corresponding to

the solution (16), (18a) and (20a) is written as

ds2 = dt2 - t2dx2 - t2e-2axdy2 - t-1+ 4a2-3dt2 - t2m1 d2 .

(21)

The rest energy density (), string tension density (), the particle density (p), the scalar of expansion (), the shear (), the spatial volume (V ) and the deceleration parameter (q) for the model (21) are obtained as

8

=

1 4t2

3

4a2 - 3 - (-1 +

4a2 - 3)2 +

4a2 - 3

+ 4a2 - 3 4a2 - 3 - (-1 + 4a2 - 3)2 - 4a2 + 3 , (22)

8

=

m21

+ m1 + 1 - a2 t2

,

(23)

p

=

1 8 t2

(2n1

+

m1

+

n1m1

-

m21),

(24)

where

n1 = -1 +

4a2 2

-

3

,

=

n1

+

m1 t

+

2

,

(25)

2

=

1 2

2

1 3

-

1 t

2

+

1 3

-

n1 t

2

+

1 3

-

m1 t

2

,

(26)

V = tn1+m1+2e-ax, and

(27)

q

=

-(n1 + m1 + 1) n1 + m1 + 2

,

(28)

where and

1+ n1 + m1 + 1 =

4a2 - 3 - (-1 + 4a2 - 3)2

2

3+ n1 + m1 + 2 =

4a2 - 3 - (-1 + 4a2 - 3)2

2

.

The rest energy density, string tension density, expansion scalar and shear be-

come infinite for t = 0, which indicates that the universe starts at t=0. Hence the

model (21) admits initial singularity. The scalar of expansion 0 as t .

Since

lim

t

2 2

=

0, the model does not approach isotropy for large value of t.

The spatial volume V is zero when t = 0 and becomes infinite when t .

The deceleration parameter is negative. Therefore the model (21) is inflationary.

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