International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 2 August 2017

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1 Internatonal Journal of Mathematcs Trends and Technoloy (IJMTT) Volume 8 Number Auust 7 Ansotropc Cosmolocal Model of Cosmc Strn wth Bulk Vscosty n Lyra Geometry.N.Patra P.G. Department of Mathematcs, Berhampur Unversty, Odsha(Inda) Abstract A four dmensonal sphercally symmetrc cosmolocal model wth cosmc strn and bulk vscosty n Lyra Geometry has been consdered. The soluton for cloud strn modelρ + λ s = yelds that, the spatal volume decreases wth ncrease of tme,.e. the unverse s contractn wth a constant deceleraton and shows ansotropy throuhout the evoluton. Themodel s non-rotatn, not a recurrent space and also not a space of constant curvature n eneralzed curved space. Keywords: Sphercally Symmetrc, Lyra Geometry, Cosmc Strn, Bulk Vscosty, ecurrent Space, Space of Constant Curvature, Deceleraton Parameter.. Introducton: Ensten s dea of eometrzn ravtaton n eneral theory of relatvty motvated others to eometrze other physcal felds. Lyra [] proposed a modfcaton of emannan eometry n whch he ntroduced a aue functon to remove the non-nterablty of lenh of a vector under paralled transport. Varous Cosmolocal models n Lyra manfold has been constructed by ahaman[], snh [], Mohantry [] and eddy [5]. The study of strn theory s mportant n the early stae of evoluton of the unverse before the partcle creaton. Cosmc strns are consdered as two lnes of concentrated enery formn a tanled web permttn the entre unverse wth closed loops. Kbble [6] Vlenkn [7] beleved that strns may be one source of densty for lare scale structures of the unverse. The desrable thn for the strn models s that, ether the strns fade away at a certan epoch of cosmc evoluton or t has a partcle domnated future asymptote wth barely vsble strns. Ths fact attracted many researchers for further studes.. Feld Equaton and the Cosmolocal Model: Here we have consdered the four dmensonal sphercally symmetrc metrc n the form ds e dr r d r sn d e dt () Where and are functons of t only. Assumn the coordnates to be co-movn.e. u u u and u and the dsplacement vector n the form φ = (β,,, ) () ( s a constant), the enery momentum tensor T for cosmc strn wth bulk vscosty s T = ρ u u λ s x x ξθ(u u ), () ρ partcle densty ISSN: Pae 7

2 Internatonal Journal of Mathematcs Trends and Technoloy (IJMTT) Volume 8 Number Auust 7 λ s Strn tenson densty ξ Bulk vscous coeffcent u Four velocty vector covarant fundamental tensor x drecton of ansotropy of cosmc strn satsfyn u u x x and u x. () The Ensten s feld equaton based on Lyra s manfold s ven by m m T, (5) c Scalar. Usn equatons (),() and() the explct form of feld equaton (5) for the lne element () are obtaned as eλ μ λ e λ μ λ 8 μ β + eλ β = χλ s + χξθ e λ (6) r (7) r λ e μ + r λ e μ r 8 λ μ e μ β + r β = χξθr. (8) β cosec θ + r β = χξθr.(9) λ 8 μ e μ e λ + 9 r r β = χρ. (). Cosmolocal Soluton: From equaton (7), we et the value of λ as rt c, () c s the constant of nteraton. Addn equaton (6) and (), we et λ e λ μ e λ μ λ 8 μ λ 8 μ e μ e λ + 5 r β + r β e λ = χ λ s + ρ + χξθe λ () Specal Case: Cloud Strn Model : s Usn ths condton n equaton () and takn e λ common from both the sdes, we et ISSN: Pae 8

3 Volume Internatonal Journal of Mathematcs Trends and Technoloy (IJMTT) Volume 8 Number Auust 7 λ e μ λ 8 μ e μ λ 8 μ e μ λ e λ + 5 r β + r e λ + β = χξθ () Multplyn equaton () wth r and equatn ts LHS of equaton (9) and usn λ = from equaton (), we et λ e μ + r 8 λ μ e μ λ = 5 r β + e λ + e λ + β. () Puttn the value of λ from equaton () n equaton (), we et 6 r β e μ 6 r β μ e μ+ β r t c = 5 r β + 7 e β r t c + e β r t c + β (5) To avod the complcacy n fndn the soluton, let us take, μ = or μ = K(constant) (6) for a partcular caset = 7 ln 6 r β e K β 5 r β + c + r β. Usn equaton () and (6) the metrc () takes the form ds = e r β t+c dr r dθ r sn θ dφ + e K dt. (7). Physcal and Geometrcal Propertes:. Takn = constant, at ntal epoch t, the metrc () becomes flat. As t ncreases, the frst dmenson contracts, but the others don t chane.. Volume V = = e β r t+ K r snθ, K = c + K =constant. Here we observe that, the spatal volume of the unverse decreases wth the ncrease ofcosmc tme and may collapse shortly whch has shown n the V ~ traph Tme. The mean ansotropy parameter A = ansotropy throuh out the evoluton. = ΔH H =.e.a. Hence the model show ISSN: Pae 9

4 Hubble parameter Internatonal Journal of Mathematcs Trends and Technoloy (IJMTT) Volume 8 Number Auust 7. The model of the unverse s non rotatn snce the vertcty tensor w 5. It s found that. Hence the space of the strn model s not a Ensten s space. 6. the space of constant curvature s ven by hk h k hk k (8) hk r hk rx rx x h k rx r h k x x k rx rx ab a b hk ab a k b h = emannan curvature tensor. K s a constant known as the curvature of emannan manfold. We found that, the value of K after comparn the exstn values of L.H.S. and.h.s. of equaton (8) s not an unque constant. So our strn space tme n Lyra s manfold s not a space of constant curvature. 7. ecurrent Space: hk, l K e hk For our matrc, the covarant dervatves of the exstn components of emannan curvature tensor vanshes. Whch mples that, ultmately the correspondn values of k k, k, k,,, K l., Hence the matrcs s not a recurrent space. K l are also zero, 8.The Hubbleparameter (H) = H + H = raph shows the necessary varaton. λ + μ λ μ = β r β r t, whch s a functon of t. The H~t c Tme ISSN: Pae 5

5 Internatonal Journal of Mathematcs Trends and Technoloy (IJMTT) Volume 8 Number Auust 7 9. The scalar expanson θ = H = β r cosmc tme.. The deceleraton parameter q s defned as β r t, whch means that the volume expanson depend on the c q = d dt H =, as >, the model of the unverse s a deceleratn one, the spatal volume decreases wth the ncrease n cosmc tme. 5.Concluson:-We havestuded varous physcal and eometrcal propertes of a fourdmensonal cosmolocal model of cosmc strn wth bulk vscosty n Lyra eometry. We found the exact cosmolocal soluton for the value of λ but usn cloud strn model μ s found to be ndependent of cosmc tme. The vertcty tensor, w ndcates the non-rotatn nature of the unverse. The deceleraton parameter (q) s rater than zero shows that the spatal volume decreases unformly wth t and ultmately the unverse may collapse at some tme. The mean ansotropy parameter A s not equal to zero shows the ansotropy nature of the model. Also t s observed that the unverse s not a space of constant curvature, nether t s an Ensten s space nor a recurrent space. eferences:. G.Lyra, Math. Z., 5,(95), 5.. F.ahaman, S.Chakrabarty, N.Beum, M.Hossan and M.Kalam, FIZIKAB, (), 59.. G.P.Snh,.V. Deshpande and T.Snh, Pramana. J. Phys, 6, (), 97.. G.Mohanty, G.c. Samanta and K.L.Mahanta, Communcaton n Physcs, 7(7),. 5. D..K.eddy, Astrophys, Space Sc., (5), T.W.B. Kbble. Phys. A., 9(976), A.Vlenkn, Phys. ep.,, (985), 6. ISSN: Pae 5