Bianchi Type—IX Barotropic Fluid Model with Time-Dependent Displacement Vector in Lyra Geometry
Bianchi Type IX barotropic fluid cosmological model in the frame work of Lyra geometry is investigated. To get the deterministic model of the universe, it is assumed that shear () is proportional to expansion (). This leads to , where and are metric potentials and is a constant. To get the results in terms of cosmic time , we have also considered a special case (dust filled universe) and . We find that the model starts with a big bang initially and the displacement vector () is initially large but decreases due to lapse of time. The models and have point-type singularity at and , respectively. The physical and geometrical aspects of the models are also discussed.
Bianchi Type IX space-time is the generalization of FRW model with positive curvature. Bianchi Types cosmological models create more interest in the study, because familiar models like Robertson-Walker model , the de-Sitter universe , Taub-NUT [3, 4] space times are of Bianchi Type IX space-time. The solutions [3, 4] allow expansion, rotation, and shear. Vaidya and Patel  have obtained the solution for spatially homogeneous Bianchi Type IX space time and have given a general scheme for the derivation of exact solutions of Einstein’s field equations corresponding to a perfect fluid and pure radiation field. Bianchi Type IX space times are also studied by many research workers namely Krori et al. , Chakraborty and Nandy , Chakraborty , and Bali and Upadhaya .
By geometrizing gravitation, Einstein derived the field equations of general relativity. Weyl  developed a theory to geometrize gravitation and electromagnetism inspired by the idea of geometrizing gravitation of Einstein. But Weyl’s theory was discarded due to nonintegrability of length of vector under parallel displacement. Lyra  modified Riemannian geometry by introducing a gauge function into the structureless manifold. This step removed the main obstackle of Weyl’s theory  and made length of vector integrable under parallel displacement. Sen  investigated an analogue of Einstein’s field equation by introducing a new scalar theory of gravitation. Halford  pointed out that constant displacement vector in Lyra geometry plays the role of cosmological constant in General Relativity. A number of authors, namely, T. Singh and G. P. Singh , Rahman and Bera , Rahman et al. , Pradhan et al. [17–19], Bali and Chandnani [20, 21], Ram et al. , and Bali et al. , have investigated cosmological models for different Bianchi space time under different contexts in the frame work of Lyra geometry.
In this paper, we have investigated Bianchi Type IX barotropic fluid cosmological model in the frame work of Lyra geometry. To get the deterministic model, we have assumed that the shear is proportional to expansion . We have also considered the dust distribution model to get the result in terms of cosmic time. We find that the model starts with a big bang initially and expansion decreases as time increases. The displacement vector is initially large but decreases due to lapse of time. The physical and geometrical aspects of the models are also discussed.
2. The Metric and Field Equations
We consider Bianchi Type IX metric in the form where and are functions of -alone.
The energy momentum tensor for perfect fluid distribution is given by
The modified Einstein’s field equation in normal gauge for Lyra’s manifold obtained by Sen  is given by (in geometrized units where and ) where , is the isotropic pressure, the matter density, the fluid flow vector, and the gauge function.
The modified Einstein’s field equation (2.3) for the metric (2.1) leads to Equations (2.5) and (2.6) after using barotropic condition lead to The conservation equation leads to which leads to being a constant of integration.
3. Solution of Field Equations
For deterministic model, we assume that the shear is proportional to the expansion . This leads to where being a constant.
Equation (3.1) leads to Using (2.9)–(3.3) in (2.7), we have To get the simplified result, we assume , thus (3.4) leads to To find the solution of (3.5), we assume Thus where Therefore, (3.5) leads to which again leads to where constant of integration has been assumed zero.
4. Some Physical and Geometrical Properties
The displacement vector is given by (2.9) as The expansion is given by which leads to The shear is given by which leads to The matter density is given by which leads to which again leads to and the isotropic pressure is given by The spatial volume is given by
5. Special Case: Dust Model
To get the model of dust filled universe, we assume that , and using in (3.5), we get which leads to which after integration leads to where being constant of integration.
The model (3.12) starts with a big bang at , and the expansion in the model decreases as increases. The displacement vector is initially large but decreases due to lapse of time. Since , hence anisotropy is maintained throughout. The reality condition implies that the model exists during the span of time given by
The model (5.4) starts with a big bang at , and the expansion in the model decreases as increases. The displacement vector is initially large but decreases due to lapse of time. Since , hence anisotropy is maintained throughout. The reality condition implies that where .
The model has point type singularity at. (MacCallum ).
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