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.
Tilted; barotropic perfect fluid; heat conduction; Bianchi I
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, Jaipur302004, 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 hypersurface of homogeneity (ii) the tilted models in which the fluid flow vector is not normal to the hypersurface of homogeneity. The tilted models are more complicated than those of nontilted 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 = B^{n} 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 nontilt 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 lineelement (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 nontilt 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 toA = (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 bigbang 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 R^{3} = = ABC = T^{n+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 nontilted 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