View of HIGHER DIMENSIONAL COSMOLOGICAL MODEL WITH PERFECT FLUID MODEL AND MASSIVE SCALAR FIELD

115 HIGHER DIMENSIONAL COSMOLOGICAL MODEL WITH PERFECT FLUID MODEL AND

MASSIVE SCALAR FIELD Dr. Bhumika Panigrahi

Associate Professor, Department of Mathematics, Gopal Krishna College of Engineering & Technology, Gourahari Vihar, Raniput, Jeypore - 764 001, Orissa, INDIA

Abstract- In this article I have constructed five dimension cosmological models in presence of perfect fluid distribution with massive scalar field based on Lyra Geometry. Some geometrical properties of the models are discussed.

Keywords: Five Dimensional Cosmological Models, Lyra Manifold, Perfect Fluid Distribution, Massive Scalar Field.

1 INTRODUCTION

It is known that general theory of relativity serves as a basis to construct mathematical models of the universe.

Though the theory exists, it has many controversies. Thus several modifications of this theory have been proposed by researchers to unify gravitation and other effects in the universe. The unification of gravitational forces with other forces in nature is not possible in the usual four- dimensional space-time. So in higher dimensional quantum field theory this may be possible. Such cosmological models were investigated by Appelquist et al.,(1987), Weinberg (1986), Chodos and Detweler (1980) showed that, the extra dimensions are unobservable due to dynamical contraction to very small scale.

This idea is important in the field of cosmology, as we know that the universe was much smaller than today. So we predict that the present four-dimensional space-time of the universe could have proceeded by a higher dimensional space- time. The extra diminutions reduced to a volume of the order of the Planck length with the passage of time, which are not observable at the present stage of the universe.

The work reported in this chapter is communicated in the journal “Indian Academy of Mathematics”.

Freund (1982) explained the smallness of the extra diminutions of the universe through the dynamical evolution in 10 and 16 dimensional super gravity models. Rahaman et al., (2002a) and Singh et al., (2004) claimed thorough the solutions of the field equations that there is an expansion of four dimensional

Space-time fifth dimension contracts or remains constant. Further Guth (1981) and Alvarez (1983) observed that during contraction process extra dimensions produce large amount of entropy, which provides an alternative resolution to the flatness and horizon problem.

In this chapter we considered a five dimensional spherical symmetric space- time within the framework of Lyra’s geometry. We obtained exact solutions of the five dimensional vacuum field equations and stiff fluid distribution in presence of scalar meson field. Some geometrical as well as physical properties of the models are also discussed.

The analog of Einstein’s field equation based on Lyra’s geometry in normal gauge as obtained by Sen (1957) and Sen and Dunn (1971) are

 

 

  











 χ χ '

4 3 2 R 3 2

R

_ij

1 g

_ij



_i



_j

g

_ij



_m



^m

T

_ij

T

_ij

T

_ij

(1) where



_iis the displacement vector and other symbols have their usual meanings as in the Riemannian geometry.

2 THE METRIC AND THE FIELD EQUATION

Here the five dimensional spherically symmetric metric is considered which is in the form.



^{2 2 2 2 2 2}



²

2

2

dt e dr r d r sin d e dy

ds  

^

     

^

(2) where



and



are the function of time coordinate only.

We assume here that the coordinates to be commoving

116 i.e., u⁰=1 and u¹= u² = u³ = u⁴= 0.

(3) Further, we consider the displacement vector



_i in the form

  , 0 , 0 , 0 , 0 



_i



(4) where β is constant.

The energy momentum tensor Tij for perfect fluid distribution with massive scalar field is

( ) ¹ (

²

)

2

' k 2

ij i j ij i j ij k

T = p + r u u - pg + u, u, - g u, u - M u

(5) where



is the particle density, p is the isotropic pressure,

u

ⁱis the five velocity vector,

g

_ij is the covariant fundamental tensor,

x

ⁱis the direction of anisotropy satisfying

 1





_i ⁱ

i

i

u x x

u

(6) and

u

_i

x

ⁱ

 0

^.

(7) Further, the expansion scalar is given by

.

;  u



 

(8) Using equation (4), (5) and (6), the explicit form of field equation (1) for the line element (2) are formulated as

 

 



 







^{ }









 χ 2

4 3 4

3 4 λ

3

₂ ^{2 2}

2

u M

, (9)

3λ

χ

2

2 2

3

2

M u

4 2 2 4 4 2 p

   

 

     

 

 

          

 

(10) and

3 λ 3λ

χ

2

2 2

3

2

M u

2 2 4  2 p

  

 

       

 

_.

(11)

where over head dot denotes differentiation w.r.t. ‘t’. In the following section we intend to derive the exact solution of the field equation using β (constant) in order to overcome the difficulties due to non linear nature of the field equations.

3 COSMOLOGICAL SOLUTIONS

Here there are three unknowns viz.,λ^,



^,

and



involved in three field equations (7.9) – (7.11). In order to solve three field equations,

we take

  a 

(12) where a (≠ 0 ) is a parameter.

And the equation of state

1 m 0 , m

p    

_.

1 Case I: For

  cons tan t

Solving the equation (7.9) (7.10) (7.11) and (7.12) we get

exp

 exp _ ^ log 

₂



₂



^a23

_ ^

a

b t

 a

(13) and exp

 exp _ ^ log 

₂



₂



^a23

_ ^

a

b t

 a

. (14) from (7.9),we get

 

   ^ _ ^

 

  





 

2 4

3 3

1 2

3 χ

1

₂ ^{2 2}

2 2

2

M u

b t a a

a

a 



(15) 4 DISCUSSION

Some Physical and geometrical properties of the models

(a) The anisotropy

σ

is defined as (Raychaudhuri, 1955)

 

²2 2



2a a 3 a t b

   

^

 

₍₁₆₎

where

b

₂is an integration constant.

(b) The expansion scalar is obtained as

 

2 2



2

; a 3 at b

a 2 2

a 3 2

a u 3



 



 



 



 



 





^



^

117



2 2



2

b t a

a

 



(17)

At the initial epoch t = 0 then

  

and when

t  

_then

  0

_.

Hence there is finite expansion in the model.

a 3

2 2 λ

a 3

λ θ

σ

 

 

 



  





.

(18) The universe remains anisotropic thought-out the evolution.

(d) The spatial volume of the universe is



^{3λ 4 2}



¹²

12

θ e sin (-g)

V    e r

^

3 λ 2

2

sin θ

2

0, 0.

e r e Sin





   

₍₁₉₎

Here the volume of the universe increase with time but in a damped way.The result shows the universe reduces to infinity, i.e., the universe reduces to black hole.

5. CONCLUSION

In this study of five dimensional cosmological model in Lyra Manifold taking into consideration perfect fluid distribution and massive scalar field, we have arrive at the following conclusion.

At the initial epoch t = 0 then

  

and when

t  

_then

  0

_.

Hence there is finite expansion in the model.

The result shows the universe reduces to infinity i.e. the universe reduces to black hole.

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