Perfect Fluid Cosmological Model in the Frame Work Lyra s Manifold

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1 Prespacetme Journal December 06 Volume 7 Issue 6 pp Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold 095 Artcle A. M. Pund * & G.. Avachar Dept. of Math., S.S.E.S. Amravat s Scence College, Congress Nagar, Nagpur, Inda Abstract In ths paper we have obtaned Perfect Flud cosmologcal model n the frame work Lyra s Manfold n uban s back ground. o get the determnstc solutons assume the relaton between metrc potental. Some physcal and knematcal propertes of the models are also cussed. Keywor: uban s metrc, perfect flud, Lyra s geometry.. Introducton Ensten proposed hs general theory of relatvty, n whch gravtaton s descrbed n terms of geometry and t motvated the geometrzaton of other physcal fel. One of the attempts n ths drecton was made by Weyl [6] who proposed a more general theory n whch gravtaton and electromagnetsm s also descrbed geometrcally. But ths theory was not accepted as t was based on non-ntegrablty of length transfer. Later Lyra [] ntroduced a gauge functon.e. placement vector n emannan space-tme whch removes the non-ntegrablty condton of a vector under parallel transport. hs modfed geometry was named as Lyra s geometry. Lyra [] proposed a modfcaton of emannan geometry by ntroducng gauge functon nto structure less manfold, as a result of whch the cosmologcal constant arses naturally from the geometry. In subsequent nvestgaton Sen [] and Sen and Dunn [5] formulated a new scalartensor theory of gravtaton and constructed an anologue of the Ensten feld equatons based on Lyra s geometry. eddy and Innah [0], eddy and venkateswarlu [] are some of the authors who have nvestgated varous aspects of the four dmensonal cosmologcal models n Lyra s manfold. Bhowmk and aput [] obtaned ansotropc Banch type cosmologcal models on the bass of Lyra s geometry. eddy DK [] examned plane symmetrc cosmc strngs n Lyra s manfold Mohanty et.al. [] has studed non-exstence of fve dmensonal perfect flud cosmologcal model n Lyra s manfold. * Correspondence Author: A. M. Pund, Dept. of Math., S.S.E.S. Amravat s Scence College, Congress Nagar, Nagpur, Inda. E-mal: ashokpund6@redffmal.com ISSN: 5-80 Prespacetme Journal Publshed by QuantumDream, Inc.

2 Prespacetme Journal December 06 Volume 7 Issue 6 pp Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold 096 Samanta et.al.[] cussed fve dmensonal Banch-I strng cosmologcal model n Lyra manfold. Nmkar et. al. [6] have been studed wet dark flud Cosmologcal Model n Lyra s Manfold. Also,recently Nmkar et.al.[7], Pund et.al.[9], Nmkar [8], and Mete et.al.[5] have studed dfferent cosmologcal model n uban s background. he purpose of the present work s to obtan Perfect flud uban s cosmologcal model n the frame work of Lyra s Manfold. Our paper s organzed as follows. In secton, we derve the feld equaton n presence Perfect flud wth the ad of uban s space tme. Secton, contan soluton of the feld Equatons, Secton s manly concerned wth the physcal and knematcal propertes of the model. he last secton contans some concluson.. he Metrc and feld equatons We consder the space- tme of uban s [] n the form where sn h ( y k y k sn y y snh y dt Q ( x, t dx ( t( dy h dz ( f f f k k 0 k and k s the curvature parameter of the homogeneous -spaces t and x constants. he functons Q and are free and wll be determned. he relatvstc feld equatons n normal gauge n Lyra s manfold are as k g g k ( where s a placement vector feld and the other symbols have ther usual meanng as n emmanan geometry. We now assume the vector placement feld to be the tme lke constant vector. where t s a functon of tme alone. he energy momentum tensor for Perfect flud s gven by 0,0,0, ( p u u pg where p and are the energy densty and pressure of the flud respectvely together wth co-movng coordnates u u where u 0,0,0,, ( ISSN: 5-80 Prespacetme Journal Publshed by QuantumDream, Inc.

3 Prespacetme Journal December 06 Volume 7 Issue 6 pp Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold 097 Comovng co-ordnates for perfect flud s gven by p, and 0 for. (5 he feld equatons ( for the metrc ( wth the help of equaton (-(5 reduce to k p (6. Q Q Q Q p (7 Q k Q Here over head dot represent partal dfferentaton wth respect to t. (8. Soluton and the model Here we have three ndependent feld equatons (6-(8 connectng fve unknown, Q, p, and. herefore n order to obtan exact solutons, we must need one more relaton connectng the unknown quanttes. We assume the equaton of state p,where 0.Snce the feld equatons are hghly nonlnear, we also assume the relaton between metrc coeffcents Q x n n. Usng ths relaton, the feld equatons (6-(8 admt the exact soluton. M n Q x N n ( c t c n n ( c t c Usng equatons (9 and (0 the Metrc ( becomes dt N x n ( c t c n n dx M ( c t c n ( dy hs metrc can be transformed through a proper choce of coordnates to the form where ct c d N x n n n dx M n ( dy h dz h dz (9 (0 ( ( ISSN: 5-80 Prespacetme Journal Publshed by QuantumDream, Inc.

4 Prespacetme Journal December 06 Volume 7 Issue 6 pp Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold 098. he Physcal and Knematcal Propertes he energy densty and pressure p for the model ( are gven by Also. p c 5 n c n ( k ( n M n he spatal volume, he Scalar expanson, Shear scalar, n V x NM (5. Q Q. c (6 c (7 V. c Hubble parameter, H V (8 he model ( has no ntal sngulartes whle the energy densty and pressure gven by ( and gven by ( possess ntal sngulartes. However as ncreases these sngulartes vansh. Spatal volume of the model gven by (5 shows ansotropc expanson of the unverse ( wth tme. For the model the Scalar expanson and shear scalar tend to zero as 5. Concluson In ths paper, we have obtaned uban s cosmologcal model n presence of perfect flud n the frame work of Lyra Manfold. For fndng the exact soluton the relaton between the metrc potental and the equaton of state of are used. he model s free from ntal sngulartes and they are expandng, shearng and non- rotatng n the standard way. Our model throws some lght on the understandng of structure formaton of the unverse n Lyra s manfold. eceved November, 06; Accepted December 7, 06 ISSN: 5-80 Prespacetme Journal Publshed by QuantumDream, Inc.

5 Prespacetme Journal December 06 Volume 7 Issue 6 pp Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold 099 eferences [] Bhowmk, B. B. and aput, A.:Pramana J. Phys.6,6,87 (00. [] Lma J. A. S., Nobre M. A. S.: Class. Quantum Grav (990. [] Lyra, G. :math.z.5,5 (95. [] Mohanty,G.,Mahanta,k.L.,Sahoo,..:Astrophyscs space Scence, 06,69-7(006. [5] Mete,V.G.,Nmkar,A.S,Elkar,V.D.:Pramanna-J.Physcs,87,7(06. [6] Nmkar, A.S.,Ugale M.., Pund A.M.: - Internatonal Journal of Scentfc & Engneerng esearch, Volume 5, Issue, March (0. [7] Nmkar,A.S,Pund,A.M.:IOS ournal of Mathematcs, Vol.,Issue,7-50(05. [8] Nmkar,Harshal.A.:Prespace tme ournal,vol.6,issue0, 9-96(05. [9] Pund,A.M.,Nmkar,A.S.:Internatonal Mathematcal Archve,6(9, 8- (05. [0] eddy,d..k. and Innah, P.: Astrophys. Space Sc.,85 (985. [] eddy,d..k. and venkateswarlu.: Astrophys. Sc.9,87 (988. [] eddy,d..k.: Astrophys. Space Sc.00,8 (005. [] Samanta,G.C., Debata,S.:Journal of modern physcs,,80-8 (0. [] Sen, D.K.:Z.Phys.9,(957. [5] Sen,D.K.and Dunn,K.A.:J.Math.Phys.,578(97. [6] Weyl, Stzungsber press. Akad.Wss (98. ISSN: 5-80 Prespacetme Journal Publshed by QuantumDream, Inc.

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