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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 251460, 9 pages
http://dx.doi.org/10.5402/2012/251460
Research Article

Bianchi Typeβ€”IX 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. BičÑk and D. 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 (𝜎) is proportional to expansion (πœƒ). This leads to π‘Ž=𝑏𝑛, where π‘Ž and 𝑏 are metric potentials and 𝑛 is a constant. To get the results in terms of cosmic time 𝑑, we have also considered a special case 𝛾=0 (dust filled universe) and 𝑛=2. We find that the model starts with a big bang initially and the displacement vector (𝛽) is initially large but decreases due to lapse of time. The models 𝑑𝑠2=βˆ’(𝑇6/[3𝑁2/20βˆ’((1βˆ’π›Ύ)/4(5𝛾+7))𝑇8βˆ’(𝛾/(5𝛾+4))𝑇6])𝑑𝑇2+𝑇4𝑑𝑋2+𝑇2π‘‘π‘Œ2+(𝑇2sin2π‘Œ+𝑇4cos2π‘Œ)𝑑𝑍2βˆ’2𝑇4cosπ‘Œπ‘‘π‘‹π‘‘π‘ and 𝑑𝑠2=βˆ’π‘‘πœ2√+[√21/5𝑁sin((2/7)𝜏)]𝑑π‘₯2√+[√21/5𝑁sin((2/7)𝜏)]1/2𝑑𝑦2+√([√21/5𝑁sin((2/7)𝜏)]1/2sin2βˆšπ‘¦+[√21/5𝑁sin((2/7)𝜏)]cos2𝑦)𝑑𝑧2βˆšβˆ’2[√21/5𝑁sin((2/7)𝜏)]cos𝑦𝑑π‘₯𝑑𝑧 have point-type singularity at 𝑇=0 and 𝜏=0, 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 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. [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 Weyl’s 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 Weyl’s theory [10] and made length of vector integrable under parallel displacement. Sen [12] investigated an analogue of Einstein’s field equation by introducing a new scalar theory of gravitation. Halford [13] pointed out that constant displacement vector (πœ™πœ‡) 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. [17–19], 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 (𝜎) is proportional to expansion (πœƒ). We have also considered the dust distribution (𝑝=0) 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 form𝑑𝑠2=βˆ’π‘‘π‘‘2+π‘Ž2𝑑π‘₯2+𝑏2𝑑𝑦2+𝑏2sin2𝑦+π‘Ž2cos2𝑦𝑑𝑧2βˆ’2π‘Ž2cos𝑦𝑑π‘₯𝑑𝑧,(2.1) where π‘Ž and 𝑏 are functions of 𝑑-alone.

The energy momentum tensor (𝑇𝑗𝑖) for perfect fluid distribution is given by 𝑇𝑗𝑖=(𝜌+𝑝)𝑣𝑖𝑣𝑗+𝑝𝑔𝑗𝑖.(2.2)

The modified Einstein’s field equation in normal gauge for Lyra’s manifold obtained by Sen [12] is given by π‘…π‘—π‘–βˆ’12𝑅𝑔𝑗𝑖+32πœ™π‘–πœ™π‘–βˆ’34πœ™π‘˜πœ™π‘˜π‘”π‘—π‘–=βˆ’π‘‡π‘—π‘–,(2.3) (in geometrized units where 8πœ‹πΊ=1 and 𝑐=1) where 𝑣𝑖=(0,0,0,βˆ’1);𝑣𝑖𝑣𝑖=βˆ’1,πœ™π‘–=(0,0,0,𝛽(𝑑)), 𝑝 is the isotropic pressure, 𝜌 the matter density, 𝑣𝑖 the fluid flow vector, and 𝛽 the gauge function.

The modified Einstein’s field equation (2.3) for the metric (2.1) leads to2𝑏44𝑏+𝑏24𝑏2+1𝑏2βˆ’34π‘Ž2𝑏4+34𝛽2π‘Ž=βˆ’π‘,(2.4)4𝑏4+π‘π‘Žπ‘44𝑏+π‘Ž44π‘Ž+π‘Ž24𝑏4+34𝛽2=βˆ’π‘,(2.5)2π‘Ž4𝑏4+π‘π‘Žπ‘24𝑏2+1𝑏2βˆ’π‘Ž24𝑏4βˆ’34𝛽2=𝜌.(2.6) Equations (2.5) and (2.6) after using barotropic condition 𝑝=π›ΎπœŒ lead toπ‘Ž(2𝛾+1)4𝑏4π‘π‘Žπ‘+𝛾24𝑏2+𝛾𝑏2+𝑏44𝑏+π‘Ž44π‘Žπ‘Ž+(1βˆ’π›Ύ)24𝑏4+34(1βˆ’π›Ύ)𝛽2=0.(2.7) The conservation equation 𝑇𝑗𝑖;𝑗=0 leads to32𝛽𝛽4+32𝛽2ξ‚΅π‘Ž4π‘Ž+2𝑏4𝑏=0,(2.8) which leads to𝑁𝛽=π‘Žπ‘2,(2.9)𝑁 being a constant of integration.

3. Solution of Field Equations

For deterministic model, we assume that the shear (𝜎) is proportional to the expansion (πœƒ). This leads toπ‘Ž=𝑏𝑛,(3.1) where √𝜎=2/3(π‘Ž4/π‘Ž/βˆ’π‘4/𝑏),πœƒ=π‘Ž4/π‘Ž+2𝑏4/𝑏,𝑛 being a constant.

Equation (3.1) leads to π‘Ž4π‘Žπ‘=𝑛4𝑏,π‘Ž(3.2)44π‘Ž=𝑛2ξ€Έξ‚΅π‘βˆ’π‘›4𝑏2𝑏+𝑛44𝑏.(3.3) Using (2.9)–(3.3) in (2.7), we have2𝑏44+2ξ€·2𝛾𝑛+𝑛2ξ€Έ+𝛾𝑏(𝑛+1)24𝑏=βˆ’2π›Ύβˆ’π‘(𝑛+1)(1βˆ’π›Ύ)𝑏2π‘›βˆ’3βˆ’32(𝑛+1)2(1βˆ’π›Ύ)𝑁(𝑛+1)2𝑏2𝑛+3.(3.4) To get the simplified result, we assume 𝑛=2, thus (3.4) leads to2𝑏44+2(5𝛾+4)3𝑏24𝑏=βˆ’2π›Ύβˆ’3𝑏(1βˆ’π›Ύ)6π‘βˆ’(1βˆ’π›Ύ)2𝑁2𝑏7.(3.5) To find the solution of (3.5), we assume 𝑏4=𝑓(𝑏).(3.6) Thus𝑏44=π‘“π‘“ξ…ž,(3.7) whereπ‘“ξ…ž=𝑑𝑓𝑑𝑏.(3.8) Therefore, (3.5) leads to𝑑𝑓2+2𝑑𝑏(5𝛾+4)𝑓3𝑏2=βˆ’2π›Ύβˆ’3𝑏(1βˆ’π›Ύ)6π‘βˆ’(1βˆ’π›Ύ)2𝑁2𝑏7(3.9) which again leads to 𝑓2𝛾=βˆ’βˆ’(5𝛾+4)(1βˆ’π›Ύ)𝑏4(5𝛾+7)2+3𝑁2𝑏20βˆ’6,(3.10) where constant of integration has been assumed zero.

Equation (3.10) leads to 𝑑𝑏𝑑𝑑2=π‘βˆ’6ξ‚Έ3𝑁2βˆ’20(1βˆ’π›Ύ)𝑏4(5𝛾+7)8βˆ’π›Ύπ‘5𝛾+46ξ‚Ή.(3.11) Thus, the metric (2.1) can be written in the form𝑑𝑠2𝑇=βˆ’6ξ€Ί3𝑁2/20βˆ’((1βˆ’π›Ύ)/4(5𝛾+7))𝑇8βˆ’(𝛾/(5𝛾+4))𝑇6𝑑𝑇2+𝑇4𝑑𝑋2+𝑇2π‘‘π‘Œ2+𝑇2sin2π‘Œ+𝑇4cos2π‘Œξ€Έπ‘‘π‘2βˆ’2𝑇4cosπ‘Œπ‘‘π‘‹π‘‘π‘,(3.12) where 𝑇=𝑏, π‘₯=𝑋, 𝑦=π‘Œ, 𝑧=𝑍, and cosmic time 𝑑 is given byξ€œπ‘‡π‘‘=3ξ€Ί3𝑁2/20βˆ’((1βˆ’π›Ύ)/4(5𝛾+7))𝑇8βˆ’(𝛾/(5𝛾+4))𝑇6ξ€»1/2𝑑𝑇.(3.13)

4. Some Physical and Geometrical Properties

The displacement vector (𝛽) is given by (2.9) as𝑁𝛽=π‘Žπ‘2=𝑁𝑇4.(4.1) The expansion (πœƒ) is given byπ‘Žπœƒ=4π‘Ž+2𝑏4𝑏,(4.2) which leads to4πœƒ=𝑇4ξ‚Έ3𝑁2βˆ’20(1βˆ’π›Ύ)𝑇4(5𝛾+7)8βˆ’π›Ύπ‘‡6ξ‚Ή5𝛾+41/2.(4.3) The shear (𝜎) is given by𝜎2=23ξ‚΅π‘Ž4π‘Žβˆ’π‘4𝑏2,(4.4) which leads to𝜎2=23𝑇8ξ‚Έ3𝑁2βˆ’20(1βˆ’π›Ύ)𝑇4(5𝛾+7)8βˆ’π›Ύπ‘‡6ξ‚Ή5𝛾+4.(4.5) The matter density (𝜌) is given by5𝜌=𝑇8ξ‚Έ3𝑁2βˆ’20(1βˆ’π›Ύ)𝑇4(5𝛾+7)8βˆ’π›Ύπ‘‡6ξ‚Ή+15𝛾+4𝑇2βˆ’14βˆ’34𝑁2𝑇8,(4.6) which leads to5𝜌=βˆ’π‘‡8ξ‚Έ(1βˆ’π›Ύ)𝑇4(5𝛾+7)8+𝛾𝑇6ξ‚Ή+1(5𝛾+4)𝑇2βˆ’14,(4.7) which again leads to4𝜌=𝑇2βˆ’3(5𝛾+4)5𝛾+7,(4.8) and the isotropic pressure is given byξ‚Έ4𝑝=π›ΎπœŒ=𝛾𝑇2βˆ’3(5𝛾+4)ξ‚Ή5𝛾+7.(4.9) The spatial volume (𝑉3) is given by𝑉3=𝑇4.(4.10)

5. Special Case: Dust Model (𝑝=0)

To get the model of dust filled universe, we assume that 𝑛=2, and using 𝛾=0 in (3.5), we get2𝑏44+83𝑏24𝑏1=βˆ’6π‘π‘βˆ’22𝑏7,(5.1) which leads to𝑑𝑏𝑑𝑑2=3𝑁2120𝑏6βˆ’1𝑏282,(5.2) which after integration leads to𝑏2=2152𝑁sin√7𝜏ξƒͺξƒ­1/2,(5.3) where 𝑑+β„“=𝜏,β„“ being constant of integration.

Thus, (2.1) takes the form𝑑𝑠2=βˆ’π‘‘πœ2+2152𝑁sin√7𝜏ξƒͺ𝑑π‘₯2+2152𝑁sin√7𝜏ξƒͺξƒ­1/2𝑑𝑦2+βŽ›βŽœβŽœβŽξƒ¬ξ‚™2152𝑁sin√7𝜏ξƒͺξƒ­1/2sin2𝑦+2152𝑁sin√7𝜏ξƒͺξƒ­cos2π‘¦βŽžβŽŸβŽŸβŽ π‘‘π‘§2ξƒ¬ξ‚™βˆ’22152𝑁sin√7𝜏ξƒͺξƒ­cos𝑦𝑑π‘₯𝑑𝑧.(5.4) The displacement vector (𝛽) is given by  (2.9)𝑁𝛽=π‘Žπ‘2=ξ‚™5221cosec√7𝜏ξƒͺ.(5.5) The expansion (πœƒ) is given byπ‘Žπœƒ=4π‘Ž+2𝑏4𝑏=2√72cot√7𝜏ξƒͺ.(5.6) The shear (𝜎) is given byξ‚™πœŽ=23ξ‚΅π‘Ž4π‘Žβˆ’π‘4𝑏.(5.7) Thus,1𝜎=√42cot𝜏.(5.8) The matter density (𝜌) is given by  (2.6)1𝜌=ξ‚ƒβˆšβˆš21/5𝑁sinξ‚€ξ‚€2/7ξ‚πœξ‚ξ‚„1/2βˆ’37.(5.9)

6. Discussion

The model (3.12) starts with a big bang at 𝑇=0, and the expansion in the model decreases as 𝑇 increases. The displacement vector (𝛽) is initially large but decreases due to lapse of time. Since 𝜎/πœƒβ‰ 0, hence anisotropy is maintained throughout. The reality condition 𝜌>0 implies that the model exists during the span of time given byξƒŽπ‘‡<4(5𝛾+7)3(5𝛾+4).(6.1)

The model (3.12) has point type singularity at 𝑇=0 (MacCallum [24]). The spatial volume increases as 𝑇 increases.

The model (5.4) starts with a big bang at 𝜏=0, and the expansion in the model decreases as 𝜏 increases. The displacement vector (𝛽) is initially large but decreases due to lapse of time. Since 𝜎/πœƒβ‰ 0, hence anisotropy is maintained throughout. The reality condition 𝜌>0 implies that2sin√7𝜏ξƒͺ<7√35√9𝑁3,(6.2) where 0<𝑁<1.

The model has point type singularity at𝜏=0. (MacCallum [24]).

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