Bianchi type I tilted cosmological model for barotropic perfect fluid distribution with heat conduction in general relativity

Bianchi type I tilted cosmological model for barotropic perfect fluid distribution with heat conduction in general relativity

Raj Bali^*; Pramila Kumawat

Department of Mathematics, University of Rajasthan, Jaipur-302004, India

ABSTRACT

Bianchi Type I tilted cosmological model for barotropic perfect fluid distribution with heat conduction is investigated.To get the deterministic solution, we have assumed barotropic condition p = γ ∈ ,0 < γ < 1, p being isotropic pressure, ∈ the matter density and a supplementary condition between metric potentials A, B, C as A = (BC)n where n is the constant. To get the model in terms of cosmic time, we have also discussed some special cases. The physical aspects of the model are also discussed.

Keywords: Tilted, barotropic perfect fluid, heat conduction, Bianchi I.

1. INTRODUCTION

Homogeneous and anisotropic cosmological models have been widely studied in the frame work of General Relativity in the search of realistic picture of the universe in the early stages of the evolution of universe. These models are of two types: (i) orthogonal models in which matter moves orthogonally to the hyper-surface of homogeneity (ii) the tilted models in which the fluid flow vector is not normal to the hyper-surface of homogeneity. The tilted models are more complicated than those of non-tilted one. The general dynamics of tilted cosmological models have been studied by King and Ellis [1], Ellis and King [2], Collins and Ellis [3]. Bradley and Sviestins [4] have investigated that heat flow is expected for tilted cosmological model. Mukherjee [5] has investigated tilted Bianchi Type I cosmological model with heat flux in General Relativity. He has shown that the universe assumes a Pan cake shape. The velocity vector is not geodesic and heat flux is comparable to the energy density. The cosmological models with heat flow have also been studied by number of researchers like Novello and Reboucas [6], Ray [7], Roy and Banerjee [8], Coley and Tupper [9], Deng [10]. Mukherjee [11], Banerjee and Santos [12], Coley [13], Roy and Prasad [14]. Bali and Sharma [15] have investigated tilted Bianchi Type I dust fluid cosmological model for perfect fluid distribution using the special condition A = Bⁿ between metric potential where n is the constant. Bali and Meena [16] have investigated Bianchi Type I tilted cosmological model for disordered radiation of perfect fluid using the supplementary condition A = (BC)n between metric potentials, n being a constant.

In this paper, we have investigated Bianchi type I tilted cosmological model for barotropic perfect fluid distribution (p = γ ∈ ) using the special condition A = (BC)ⁿ between metric potentials, n being a constant where p is the isotropic pressure, ∈ the matter density, 0 < γ < 1.

For complete solutions of equations (6) - (10),we need two extra conditions. An obvious one is equation of state p = γ ∈ (0 < γ < 1) given by (11), is general condition for barotropic equation of state, p being isotropic pressure and ∈ the matter density. This includes radiation for γ = , dust filled universe p = 0 (Friedmann model) for γ = 0, stiff fluid universe ∈ = p (Zel'dovich fluid) for γ = 1. These are physically valid conditions for the description of the universe.

The second condition A = (BC)ⁿ given by is obtained by assuming α θ for non-tilt model i.e. for λ = 0 where is the eigen value of shear tensor and θ the expansion in the model where

and

The motivation for assuming this condition is explained as : Referring to the Thorne [17], the observations of the velocity - redshift relation for extra galactic sources suggest that the Hubble expansion of the universe is isotropic today to within 30% [18,19]. More precisely, the redshift studies place the link

< 0.30 where σ is the shear and H is a Hubble constant. Collins et al. [20] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous hyper surface satisfies the condition = constant. The condition = constant for the metric leads to A = (BC)ⁿ where n is the constant.

Some special cases for different values of n and γ are discussed. The physical aspects of the model and singularities in the model are also discussed.

2. METRIC AND FIELD EQUATIONS

We consider the Bianchi Type I metric in the form

where A, B, C are functions of t alone.

The energy momentum tensor for perfect fluid distribution with heat conduction is taken into form given by Ellis [21] as

together with

where p is the isotropic pressure, ε the matter density and qi the heat conduction vector orthogonal to vi. The fluid flow vector vi has the components

satisfying (3), λ being the tilt angle.

The Einstein's field equation , (In the generalized unit where c = 1, G = 1 and taking Λ = 0)

for the line-element (1) leads to

where the subscript '4' denotes the ordinary differentiation with respect to 't'.

3. SOLUTION OF FIELD EQUATIONS

Equations from (6) to (10) are five equations in seven unknowns, A, B, C, ε, p, λ and q1. For the complete determination of these quantities, we assume that the model is filled with barotropic perfect fluid which leads to

where 0 < γ < 1

To get the deterministic solution, we also assume a supplementary condition between metric potentials A, B and C as

where n is the constant.

For complete solutions of equations (6) – (10),we need two extra conditions. An obvious one is equation of state p = γ ∈ (0 < γ < 1) given by (11), is general condition for barotropic equation of state, p being isotropic pressure and ∈ the matter density. This includes radiation for γ = , dust filled universe p = 0 (Friedmann model) for γ = 0, stiff fluid universe ∈ = p (Zel'dovich fluid) for γ = 1. These are physically valid conditions for the description of the universe.

The second condition A = (BC)ⁿ given by (12) is obtained by assuming α θ for non-tilt model i.e. for λ = 0 where is the eigen value of shear tensor and θ the expansion in the model where

and

The motivation for assuming this condition is explained as : Referring to the Thorne [17], the observations of the velocity - redshift relation for extra galactic sources suggest that the Hubble expansion of the universe is isotropic today to within 30% [18,19]. More precisely, the redshift studies place the link

< 0.30 where σ is the shear and H is a Hubble constant. Collins et al. [20] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous hyper surface satisfies the condition = constant. The condition = constant for the metric (1) leads to

A = (BC)ⁿ

where n is the constant.

Equations (6) and (9) lead to

Using the barotropoic condition p = γε given by (11) in (13), we have

Using (8) in (14), we have

Equations (7) and (8) lead to

Equation (12) leads to

Thus equation (16) becomes

which on integration leads to

where b is constant of integration.

Let

Using (20) in (19), we have

Using the assumptions (20) and (12) in (15), we have

Equations (21) and (22) leads to

where

Let us assume that

Thus

Therefore equation (23) leads to

Equation (28) leads to

where L is constant of integration and

Using (31) in (29), we have

Equation (21) leads to

Thus, we have

Hence the metric (1) reduces to the form

which leads to

where ν is determined by (33) and µ = T.

4. SOME PHYSICAL AND GEOMETRICAL FEATURES

The isotropic pressure (p), the matter density (ε), the expansion (θ), cosh λ, v1, v4, q1, q4, σ11, σ14 are given by

which leads to

Now

Similarly

5. DISCUSSION

The reality conditions

(i) ρ + p > 0 (ii) ρ + 3p > 0

given by Ellis [22], lead to

The matter density ∈ → ∞ when T → 0 and ∈ → 0 when T → ∞. The model (35) starts with a big-bang at T = 0 and the expansion in the model decreases as time increases. v1 → 0 at T= 0 when n < 0. q1 = 0, q4 = 0 when L = 0. cosh λ > 1 implies that

are satisfied as shown in (44) and (45) where σij and wij are shear tensor and vorticity tensor respectively. Since ≠ 0. Hence the model does not approach isotropy for large values of T. There is a Point Type singularity in the model (35) at T = 0 (MacCallum [23] ). The spatial volume R³ = = ABC = Tⁿ⁺¹ . Thus spatial (R3) increases as time T increases where n+1 > 0.

Special Cases

We have also investigated the following cases:

(i) n = 1, γ = , a = lead to barotropic perfect fluid non-tilted cosmological model as in this case cosh λ = which is not defined as cosh λ > 1 for tilted model.

(ii) n = , a = 0 leads to γ =(disordered radiation condition) and cosh λ = 1.

(iii) n = -, a = - leads to stiff fluid case γ = 1 and cosh λ = 1.

(iv) n =, a = - leads to γ = 0 (dust distribution) but cosh λ is not defined.

Thus in all the above mentioned cases, no tilted cosmological models are possible because for tilted model cosh λ > 1.

(Received on 29 January, 2008)

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Publication Dates

• Publication in this collection
  27 Sept 2010
• Date of issue
  Sept 2010

History

• Received
  29 Jan 2008