Table of Contents
ISRN Mathematical Physics
Volume 2012, Article ID 251460, 9 pages
http://dx.doi.org/10.5402/2012/251460
Research Article

Bianchi Type—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. Bičák and D. Singleton

Copyright © 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 = 𝑏 𝑛 , 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 𝑑 𝑠 2 = βˆ’ ( 𝑇 6 / [ 3 𝑁 2 / 2 0 βˆ’ ( ( 1 βˆ’ 𝛾 ) / 4 ( 5 𝛾 + 7 ) ) 𝑇 8 βˆ’ ( 𝛾 / ( 5 𝛾 + 4 ) ) 𝑇 6 ] ) 𝑑 𝑇 2 + 𝑇 4 𝑑 𝑋 2 + 𝑇 2 𝑑 π‘Œ 2 + ( 𝑇 2 s i n 2 π‘Œ + 𝑇 4 c o s 2 π‘Œ ) 𝑑 𝑍 2 βˆ’ 2 𝑇 4 c o s π‘Œ 𝑑 𝑋 𝑑 𝑍 and 𝑑 𝑠 2 = βˆ’ 𝑑 𝜏 2 √ + [ √ 2 1 / 5 𝑁 s i n ( ( 2 / 7 ) 𝜏 ) ] 𝑑 π‘₯ 2 √ + [ √ 2 1 / 5 𝑁 s i n ( ( 2 / 7 ) 𝜏 ) ] 1 / 2 𝑑 𝑦 2 + √ ( [ √ 2 1 / 5 𝑁 s i n ( ( 2 / 7 ) 𝜏 ) ] 1 / 2 s i n 2 √ 𝑦 + [ √ 2 1 / 5 𝑁 s i n ( ( 2 / 7 ) 𝜏 ) ] c o s 2 𝑦 ) 𝑑 𝑧 2 √ βˆ’ 2 [ √ 2 1 / 5 𝑁 s i n ( ( 2 / 7 ) 𝜏 ) ] c o s 𝑦 𝑑 π‘₯ 𝑑 𝑧 have point-type singularity at 𝑇 = 0 and 𝜏 = 0 , respectively. The physical and geometrical aspects of the models are also discussed.