International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 251460 | 9 pages | https://doi.org/10.5402/2012/251460

Bianchi Type—IX Barotropic Fluid Model with Time-Dependent Displacement Vector in Lyra Geometry

Academic Editor: J. Bičák
Received17 Sep 2011
Accepted19 Oct 2011
Published02 Jan 2012

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 (ğœŽ) is proportional to expansion (𝜃). This leads to ğ‘Ž=𝑏𝑛, where ğ‘Ž and 𝑏 are metric potentials and 𝑛 is a constant. To get the results in terms of cosmic time 𝑡, 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=−(𝑇6/[3𝑁2/20−((1−𝛾)/4(5𝛾+7))𝑇8−(𝛾/(5𝛾+4))𝑇6])𝑑𝑇2+𝑇4𝑑𝑋2+𝑇2𝑑𝑌2+(𝑇2sin2𝑌+𝑇4cos2𝑌)𝑑𝑍2−2𝑇4cos𝑌𝑑𝑋𝑑𝑍 and 𝑑𝑠2=−𝑑𝜏2√+[√21/5𝑁sin((2/7)𝜏)]𝑑𝑥2√+[√21/5𝑁sin((2/7)𝜏)]1/2𝑑𝑦2+√([√21/5𝑁sin((2/7)𝜏)]1/2sin2√𝑦+[√21/5𝑁sin((2/7)𝜏)]cos2𝑦)𝑑𝑧2√−2[√21/5𝑁sin((2/7)𝜏)]cos𝑦𝑑𝑥𝑑𝑧 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 (𝜙𝜇) 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 (ğœŽ) is proportional to expansion (𝜃). 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=−𝑑𝑡2+ğ‘Ž2𝑑𝑥2+𝑏2𝑑𝑦2+𝑏2sin2𝑦+ğ‘Ž2cos2𝑦𝑑𝑧2−2ğ‘Ž2cos𝑦𝑑𝑥𝑑𝑧,(2.1) where ğ‘Ž and 𝑏 are functions of 𝑡-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 𝑅𝑗𝑖−12𝑅𝑔𝑗𝑖+32𝜙𝑖𝜙𝑖−34𝜙𝑘𝜙𝑘𝑔𝑗𝑖=−𝑇𝑗𝑖,(2.3) (in geometrized units where 8𝜋𝐺=1 and 𝑐=1) where 𝑣𝑖=(0,0,0,−1);𝑣𝑖𝑣𝑖=−1,𝜙𝑖=(0,0,0,𝛽(𝑡)), 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+1𝑏2−34ğ‘Ž2𝑏4+34𝛽2ğ‘Ž=−𝑝,(2.4)4𝑏4+ğ‘ğ‘Žğ‘44𝑏+ğ‘Ž44ğ‘Ž+ğ‘Ž24𝑏4+34𝛽2=−𝑝,(2.5)2ğ‘Ž4𝑏4+ğ‘ğ‘Žğ‘24𝑏2+1𝑏2âˆ’ğ‘Ž24𝑏4−34𝛽2=𝜌.(2.6) Equations (2.5) and (2.6) after using barotropic condition 𝑝=𝛾𝜌 lead toğ‘Ž(2𝛾+1)4𝑏4ğ‘ğ‘Žğ‘+𝛾24𝑏2+𝛾𝑏2+𝑏44𝑏+ğ‘Ž44ğ‘Žğ‘Ž+(1−𝛾)24𝑏4+34(1−𝛾)𝛽2=0.(2.7) The conservation equation 𝑇𝑗𝑖;𝑗=0 leads to32𝛽𝛽4+32𝛽2î‚µğ‘Ž4ğ‘Ž+2𝑏4𝑏=0,(2.8) which leads to𝑁𝛽=ğ‘Žğ‘2,(2.9)𝑁 being a constant of integration.

3. Solution of Field Equations

For deterministic model, we assume that the shear (ğœŽ) is proportional to the expansion (𝜃). This leads toğ‘Ž=𝑏𝑛,(3.1) where âˆšğœŽ=2/3(ğ‘Ž4/ğ‘Ž/−𝑏4/𝑏),𝜃=ğ‘Ž4/ğ‘Ž+2𝑏4/𝑏,𝑛 being a constant.

Equation (3.1) leads to ğ‘Ž4ğ‘Žğ‘=𝑛4𝑏,ğ‘Ž(3.2)44ğ‘Ž=𝑛2𝑏−𝑛4𝑏2𝑏+𝑛44𝑏.(3.3) Using (2.9)–(3.3) in (2.7), we have2𝑏44+22𝛾𝑛+𝑛2+𝛾𝑏(𝑛+1)24𝑏=−2𝛾−𝑏(𝑛+1)(1−𝛾)𝑏2𝑛−3−32(𝑛+1)2(1−𝛾)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=−2𝛾−3𝑏(1−𝛾)6𝑏−(1−𝛾)2𝑁2𝑏7.(3.5) To find the solution of (3.5), we assume 𝑏4=𝑓(𝑏).(3.6) Thus𝑏44=ğ‘“ğ‘“î…ž,(3.7) whereğ‘“î…ž=𝑑𝑓𝑑𝑏.(3.8) Therefore, (3.5) leads to𝑑𝑓2+2𝑑𝑏(5𝛾+4)𝑓3𝑏2=−2𝛾−3𝑏(1−𝛾)6𝑏−(1−𝛾)2𝑁2𝑏7(3.9) which again leads to 𝑓2𝛾=−−(5𝛾+4)(1−𝛾)𝑏4(5𝛾+7)2+3𝑁2𝑏20−6,(3.10) where constant of integration has been assumed zero.

Equation (3.10) leads to 𝑑𝑏𝑑𝑡2=𝑏−63𝑁2−20(1−𝛾)𝑏4(5𝛾+7)8−𝛾𝑏5𝛾+46.(3.11) Thus, the metric (2.1) can be written in the form𝑑𝑠2𝑇=−63𝑁2/20−((1−𝛾)/4(5𝛾+7))𝑇8−(𝛾/(5𝛾+4))𝑇6𝑑𝑇2+𝑇4𝑑𝑋2+𝑇2𝑑𝑌2+𝑇2sin2𝑌+𝑇4cos2𝑌𝑑𝑍2−2𝑇4cos𝑌𝑑𝑋𝑑𝑍,(3.12) where 𝑇=𝑏, 𝑥=𝑋, 𝑦=𝑌, 𝑧=𝑍, and cosmic time 𝑡 is given by𝑇𝑡=33𝑁2/20−((1−𝛾)/4(5𝛾+7))𝑇8−(𝛾/(5𝛾+4))𝑇61/2𝑑𝑇.(3.13)

4. Some Physical and Geometrical Properties

The displacement vector (𝛽) is given by (2.9) as𝑁𝛽=ğ‘Žğ‘2=𝑁𝑇4.(4.1) The expansion (𝜃) is given byğ‘Žğœƒ=4ğ‘Ž+2𝑏4𝑏,(4.2) which leads to4𝜃=𝑇43𝑁2−20(1−𝛾)𝑇4(5𝛾+7)8−𝛾𝑇65𝛾+41/2.(4.3) The shear (ğœŽ) is given byğœŽ2=23î‚µğ‘Ž4ğ‘Žâˆ’ğ‘4𝑏2,(4.4) which leads toğœŽ2=23𝑇83𝑁2−20(1−𝛾)𝑇4(5𝛾+7)8−𝛾𝑇65𝛾+4.(4.5) The matter density (𝜌) is given by5𝜌=𝑇83𝑁2−20(1−𝛾)𝑇4(5𝛾+7)8−𝛾𝑇6+15𝛾+4𝑇2−14−34𝑁2𝑇8,(4.6) which leads to5𝜌=−𝑇8(1−𝛾)𝑇4(5𝛾+7)8+𝛾𝑇6+1(5𝛾+4)𝑇2−14,(4.7) which again leads to4𝜌=𝑇2−3(5𝛾+4)5𝛾+7,(4.8) and the isotropic pressure is given by4𝑝=𝛾𝜌=𝛾𝑇2−3(5𝛾+4)5𝛾+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=−6𝑁𝑏−22𝑏7,(5.1) which leads to𝑑𝑏𝑑𝑡2=3𝑁2120𝑏6−1𝑏282,(5.2) which after integration leads to𝑏2=2152𝑁sin√7𝜏1/2,(5.3) where 𝑡+ℓ=𝜏,ℓ being constant of integration.

Thus, (2.1) takes the form𝑑𝑠2=−𝑑𝜏2+2152𝑁sin√7𝜏𝑑𝑥2+2152𝑁sin√7𝜏1/2𝑑𝑦2+⎛⎜⎜⎝2152𝑁sin√7𝜏1/2sin2𝑦+2152𝑁sin√7𝜏cos2ğ‘¦âŽžâŽŸâŽŸâŽ ğ‘‘ğ‘§2−22152𝑁sin√7𝜏cos𝑦𝑑𝑥𝑑𝑧.(5.4) The displacement vector (𝛽) is given by  (2.9)𝑁𝛽=ğ‘Žğ‘2=5221cosec√7𝜏.(5.5) The expansion (𝜃) is given byğ‘Žğœƒ=4ğ‘Ž+2𝑏4𝑏=2√72cot√7𝜏.(5.6) The shear (ğœŽ) is given byî‚™ğœŽ=23î‚µğ‘Ž4ğ‘Žâˆ’ğ‘4𝑏.(5.7) Thus,1ğœŽ=√42cot𝜏.(5.8) The matter density (𝜌) is given by  (2.6)1𝜌=√√21/5𝑁sin2/7𝜏1/2−37.(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 ğœŽ/𝜃≠0, hence anisotropy is maintained throughout. The reality condition 𝜌>0 implies that the model exists during the span of time given byîƒŽğ‘‡<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 ğœŽ/𝜃≠0, hence anisotropy is maintained throughout. The reality condition 𝜌>0 implies that2sin√7𝜏<7√35√9𝑁3,(6.2) where 0<𝑁<1.

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

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