ISRN Mathematical Physics

VolumeΒ 2012Β (2012), Article IDΒ 251460, 9 pages

http://dx.doi.org/10.5402/2012/251460

## Bianchi TypeβIX Barotropic Fluid Model with Time-Dependent Displacement Vector in Lyra Geometry

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

Received 17 September 2011; Accepted 19 October 2011

Academic Editors: J.Β BiΔΓ‘k and D.Β Singleton

Copyright Β© 2012 Raj Bali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], the de-Sitter universe [2], 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 [5] 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 Einsteinβs 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. [6], Chakraborty and Nandy [7], Chakraborty [8], and Bali and Upadhaya [9].

By geometrizing gravitation, Einstein derived the field equations of general relativity. Weyl [10] developed a theory to geometrize gravitation and electromagnetism inspired by the idea of geometrizing gravitation of Einstein. But Weylβs theory was discarded due to nonintegrability of length of vector under parallel displacement. Lyra [11] modified Riemannian geometry by introducing a gauge function into the structureless manifold. This step removed the main obstackle of Weylβs theory [10] and made length of vector integrable under parallel displacement. Sen [12] investigated an analogue of Einsteinβs field equation by introducing a new scalar theory of gravitation. Halford [13] 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 [14], Rahman and Bera [15], Rahman et al. [16], Pradhan et al. [17β19], Bali and Chandnani [20, 21], Ram et al. [22], and Bali et al. [23], 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 Einsteinβs field equation in normal gauge for Lyraβs manifold obtained by Sen [12] 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 Einsteinβs 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)β(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.

Equation (3.10) leads to Thus, the metric (2.1) can be written in the form where , , , , and cosmic time is given by

#### 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.

Thus, (2.1) takes the form The displacement vector is given byββ(2.9) The expansion is given by The shear is given by Thus, The matter density is given byββ(2.6)

#### 6. Discussion

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 (3.12) has point type singularity at (MacCallum [24]). The spatial volume increases as increases.

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 [24]).

#### References

- H. P. Robertson, βKinematics and World-Structure II,β
*Astrophysical Journal*, vol. 83, pp. 187β201, 1936. View at Google Scholar - W. De Sitter, βOn Einstein's theory of gravitation, and its astronomical consequences,β
*Monthly Notices of the Royal Astronomical Society*, vol. 78, pp. 3β28, 1917. View at Google Scholar - A. H. Taub, βEmpty space-times admitting a three parameter group of motions,β
*Annals of Mathematics. Second Series*, vol. 53, pp. 472β490, 1951. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. Newman, L. Tamburino, and T. Unti, βEmpty-space generalization of the Schwarzschild metric,β
*Journal of Mathematical Physics*, vol. 4, no. 7, pp. 915β923, 1963. View at Google Scholar Β· View at Scopus - P. C. Vaidya and L. K. Patel, βGravitational fields with space-times of Bianchi type IX,β
*Pramana*, vol. 27, no. 1-2, pp. 63β72, 1986. View at Publisher Β· View at Google Scholar Β· View at Scopus - K. D. Krori, T. Chaudhury, C. R. Mahanta, and A. Mazumdar, βSome exact solutions in string cosmology,β
*General Relativity and Gravitation*, vol. 22, no. 2, pp. 123β130, 1990. View at Publisher Β· View at Google Scholar Β· View at MathSciNet Β· View at Scopus - S. Chakraborty and G. C. Nandy, βCosmological studies in Bianchi II, VIII space-time,β
*Astrophysics and Space Science*, vol. 198, no. 2, pp. 299β308, 1992. View at Publisher Β· View at Google Scholar Β· View at Scopus - S. Chakraborty, βA study on Bianchi-IX cosmological model,β
*Astrophysics and Space Science*, vol. 180, no. 2, pp. 293β303, 1991. View at Publisher Β· View at Google Scholar Β· View at Scopus - R. Bali and R. D. Upadhaya, βBianchi type IX string dust cosmological model in general relativity,β
*Proceedings of the National Academy of Sciences. India. Section A*, vol. 73, no. 2, pp. 239β247, 2003. View at Google Scholar - H. Weyl, βGravitation and Elektricität,β
*Sber. Preuss. Akad. d. Wisson Chaften*, vol. 465, 1918. View at Google Scholar - G. Lyra, βÜber eine Modifikation der Riemannschen Geometrie,β
*Mathematische Zeitschrift*, vol. 54, pp. 52β64, 1951. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - D. K. Sen, βA static cosmological model,β
*Zeitschrift für Physik*, vol. 149, no. 3, pp. 311β323, 1957. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet Β· View at Scopus - W. D. Halford, βScalar-tensor theory of gravitation in a Lyra manifold,β
*Journal of Mathematical Physics*, vol. 13, no. 11, pp. 1699β1703, 1972. View at Google Scholar Β· View at Scopus - T. Singh and G. P. Singh, βBianchi type-I cosmological models in Lyra's geometry,β
*Journal of Mathematical Physics*, vol. 32, no. 9, pp. 2456β2458, 1991. View at Google Scholar - F. Rahaman and J. K. Bera, βHigher dimensional cosmological model in Lyra geometry,β
*International Journal of Modern Physics D*, vol. 10, no. 5, pp. 729β733, 2001. View at Publisher Β· View at Google Scholar - F. Rahaman, N. Begum, G. Bag, and B. C. Bhui, βCosmological models with negative constant deceleration parameter in Lyra geometry,β
*Astrophysics and Space Science*, vol. 299, no. 3, pp. 211β218, 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus - A. Pradhan, V. K. Yadav, and I. Chakrabarty, βBulk viscous FRW cosmology in Lyra geometry,β
*International Journal of Modern Physics D*, vol. 10, no. 3, pp. 339β349, 2001. View at Publisher Β· View at Google Scholar Β· View at Scopus - A. Pradhan, I. Aotemshi, and G. P. Singh, βPlane symmetric domain wall in Lyra geometry,β
*Astrophysics and Space Science*, vol. 288, no. 3, pp. 315β325, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus - A. Pradhan, L. Yadav, and A. K. Yadav, βIsotropic homogeneous universe with a bulk viscous fluid in Lyra geometry,β
*Astrophysics and Space Science*, vol. 299, no. 1, pp. 31β42, 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus - R. Bali and N. K. Chandnani, βBianchi type-I cosmological model for perfect fluid distribution in Lyra geometry,β
*Journal of Mathematical Physics*, vol. 49, no. 3, Article ID 032502, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - R. Bali and N. K. Chandnani, βBianchi type-III bulk viscous dust filled universe in Lyra geometry,β
*Astrophysics and Space Science*, vol. 318, no. 3-4, pp. 225β230, 2008. View at Publisher Β· View at Google Scholar Β· View at Scopus - S. Ram, M. Zeyauddin, and C. P. Singh, βBianchi type V cosmological models with perfect fluid and heat conduction in lyra's geometry,β
*International Journal of Modern Physics A*, vol. 23, no. 31, pp. 4991β5005, 2008. View at Publisher Β· View at Google Scholar Β· View at Scopus - R. Bali, N. K. Chandnani, and J. P. Dhanka, βBianchi type I magnetized dust filled universe in Lyra geometry,β
*International Journal of Modern Physics A*, vol. 25, no. 25, pp. 4839β4848, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus - M. A.H. MacCallum, βA class of homogeneous cosmological models III: asymptotic behaviour,β
*Communications in Mathematical Physics*, vol. 20, no. 1, pp. 57β84, 1971. View at Publisher Β· View at Google Scholar