769 Expadg & Shearg ach ype-i No-Statc osmologcal Model Geeral Relatvty Reea Mathur, Gaedra P. Sgh * & tul yag Departmet of Mathematcs ad Statstcs, MohaLal Suhada Uversty, Udapur Departmet of Mathematcs, Geetaal Isttute of echcal Studes, Udapur rtcle bstract We have vestgated expadg, shearg ad o- statc ach type-i cosmologcal models Geeral Relatvty. o get determstc model, we have assumed that shear (σ) s proportoal to the expaso (θ) the model whch leads to = () where,, are metrc potetal ad s costat. Usg the codto, we foud a stff flud model. he varous physcal ad geometrcal aspects of model are also dscussed. Key Words: Expadg, shearg, electromagetc feld, cosmologcal Model. Itroducto he cosmologcal model presece of magetc feld, play a sgfcat role the evoluto of galaxes ad stellar bodes. Prmordal magetc of cosmologcal org s speculated by sseo ad Sol []. he brea dow of sotropy s also due to the magetc feld. olls [] gave a qualtatve aalyss of ach ype I model wth magetc feld. Jacobs [] vestgated ach ype I cosmologcal model wth barotropc flud presece of magetc feld. al ad yag [] have vestgated magetzed ach type I orthogoal cosmologcal model for perfect flud dstrbuto Geeral Relatvty. al ad Meea [5] have vestgated magetzed stff flud tlted uverse for perfect flud dstrbuto Geeral Relatvty. he presece of magetc feld galactc ad tergalactc spaces s evdet from recet observatos [6]. he large scale magetc felds ca detect by observg ther effects o the M radato. hese felds would ehace asotropes the M, sce the expaso rate wll be dfferet depedg o the drecto of the feld les [7, 8]. Matravers ad sagas [9] foud that teracto of cosmologcal magetc feld wth the space tme geometry could affect the expaso of the uverse. If the curvature s strog, the eve the wea magetc feld wll affect the evoluto of the uverse. he mageto-curvature couplg teds to accelerate the postvely curved regos whle t decelerates the egatvely curved regos [0]. Jacobs [] studed the spatally homogeeous ad asotropc ach type I cosmologcal model wth expaso ad shear but wthout rotato. e dscussed asotropy the temperature of M ad expaso both wth ad wthout magetc feld. It was cocluded that the prmordal magetc feld produce large expaso asotropes durg radato-domated * orrespodece: Gaedra P. Sgh, Departmet of Mathematcs, Geetaal Isttute of echcal Studes, Udapur. E-mal: gaedrasgh7@gmal.com
770 phase but t has eglgble effect durg the dust domated phase. Du ad upper [] dscussed propertes of ach type VI 0 models wth perfect flud ad magetc feld. Roy et.al [] explored the effects of cosmologcal costat ach type I ad VI 0 models wth perfect flud ad homogeeous magetc feld axal drecto. Several researchers le Zeldovch [], ertolm [5], Ozer ad aha [6], Frema ad Waga [7] ad Pradha et.al [8] vestgated more sgfcat cosmologcal model wth cosmologcal costat. ach type III cosmologcal models are studed by umbers of researcher dfferet cotext vz. ear ad Patel [9], al ad Dave [0], al ad Pradha []. Sgh ad yag [-] vestgated varous ach ype cosmologcal models wth varable cosmologcal ad gravtatoal costat presece ad absece of magetc feld. he perfect ad bul vscous fluds are cosdered as source of matter. Sgh et.al. [5] have vestgated LRS ach type III Massve Strg cosmologcal model wth Electromagetc feld. I ths paper, we have vestgated expadg, shearg ad o- statc ach type-i cosmologcal models Geeral Relatvty. o get determstc model, we have assumed that shear (σ) s proportoal to the expaso (θ) the model whch leads to = () where,, are metrc potetal ad s costat. It s observe that the model leads to stff flud case geeral. he physcal ad geometrcal mplcatos of models are also dscussed. he Metrc & Feld Equatos We cosder homogeeous asotropc ach type-i metrc the form of ds where, ad are fucto of t aloe. dx dt dy dz () I ths paper we have cosdered dstrbuto of matter s cosst of perfect flud wth a fte electrcal coductvty ad a magetc feld. he eergy-mometum tesor of the composte feld s assumed to be the sum of the correspodg eergy-mometum tesor. hus p v v pg E Were ρ s eergy desty, p s effectve pressure ad ρ () v the flow vector satsfyg the relato g v v () lso, E s the electromagetc feld gve by Lcherowcz [6] as E h vv g hh here μ s the magetc permeablty ad h s the magetc flux vector defe by where *F s the dual electromagetc feld tesor defe by Syge [7] as h F v () (5)
77 F g F s electromagetc feld tesor ad ε l lev-cevta tesor desty. ere, the co-movg coordates are tae to be v v v 0 ad v.we tae the cdet magetc feld to be the drecto of the x-axs so that h 0, h = 0 = h = h. O the assumpto of fte coductvty of the flud, we get F = 0 = F = F.he oly o vashg compoet of F s F. he frst set of Maxwell s equatos are satsfed by F l F ; F; F; 0 ad F; where the semcolo represet a covarat dfferetato. l 0 F = (costat) (7) (6) From equato (5) ad (7) we fd that h he Este s feld equato ( G c ) read as R Rg 8π (9) Este feld equato (9) together wth () ad () for metrc () lead to (8) 8π p 8π p 8π p 8π (0) () () ()
77 Solutos of the Feld Equatos Equatos (0), () ad () leads to d 8 () (5) Equatos (0) to () represet a system of four equatos fve uows,,, p ad. o get determate model we use a extra codto betwee metrc potetals as equatos () ad (6), lead to Puttg Equatos (5) ad (7) lead to d Equato (0) leads to K, Where where s costat of tegrato. (6) K 8 (7) ad (8) K 0 0 (9) (0) () Usg equato (9) ad (0), we get K () Equato () leads to ()
77 Equato (6), (8) ad () lead to K a L () where L s costat of tegrato. Equato (8), () ad () leads to Equatos (8) ad (5) leads to whch leads to where d K a L K a (5) L (6) K a t N (7) L K a t N (8) where β ad N are costat of Itegrato. From equatos (6), (8), (7) ad (8) we get t N (9) ece, the metrc reduces to the form ds t N (0) t N () α α t N dx d t t N α dy t N α dz () fter sutable trasformato of co-ordates, the metrc reduces to the form ds α α d α α dx dy dz () where t N, x X, y Y, z Z α.
77 Some Physcal & Geometrcal Features he pressure ad desty for the model () are gve by 8π p () 8π (5) he model has to satsfy the realty codtos gve by Ells [8] () p > 0 () p > 0 Leads to < 6 (6) he scalar of expaso θ calculated for the flow vector v s gve by θ (7) he scalar of shear s gve by 8 σ (8) Now, (9) he o vashg compoets of shear tesor are gve by (0) 6 () 6 () he rotato ω s detcally zero ad the o vashg compoets of coformal curvature tesor are gve by
775 6 () 6 () 6 (5) ocluso he models start wth bg bag at = 0 for > - ad the expaso the model decreases as tme creases. he expaso stops at = - or. he model represets a expadg, shearg ad o rotatg uverse. s, p 0 ad 0 provded + > 0. Sce σ 0.ece the model does t approach sotropy for large value of. pot type θ sgularty [9] s observed for > 0 ad > α as 0, g 0, g 0, g 0. Refereces. sseo, Estelle ad Sol, elee (987): Phys. Reports, 6, 8.. olls,. S. (97): omm. Maths. Phys. 7, 7.. Jacobs, K.. (968): strophys. J. 5, 66.. al, R ad yag,. (987): It. J. heor. Phys, 7, 5, 67. 5. al, R ad Meea,. L. (999): strophys. d Space Sc., 6, 89. 6. Grasso, D. ad Rubste,. R. (00): Phys. Rep. 8, 6; Maartes, R (000) : Pramaa 55, 575. 7. Madse, M. S. (989): Mo. Not. R. stro. Soc. 7, 09. 8. Kgs, E. J. ad oles, P. (007): lass. Quat. Gravt., 06. 9. Matravers, D. R. ad sagas,. G. (000): Phys. Rev. D 6, 0, 59. 0. sagas,. G. ad arrow, J. D. (997): lass. Quat. Gravt., 59.. Jacobs, K.. (969): strophys. J. 55, 79.. Du, K.. ad upper,. O. J. (976): strophys. J. 0,.. Roy, S. R., Sgh, J. P. ad Nar, S. (985): strophys. Space Sc., 89; bd. ust. J. Phys. (985) 8, 9.. Zeldovch, Ya.. (968): Sov. Phys.Uspeh, 8. 5. ertolam, O. (986): Nuovo. meto. 9, 6. 6. Ozer, M. ad aha, M. O. (987): Nucl. Phys. 87, 776. 7. Frema, J.. ad Waga. I. (998): Phys. Rev. D57, 6. 8. Pradha,., Yadav, V. K. ad harabarty, I. (00): It. J. Mod. Phys. D, 89. 9. ear, R. ad Patel, L. K. (99) : Ge. Rel. Gravt., 97. 0. al, R. ad Dave, S. (00): strophys. Space Sc. 8, 6.. al, R. & Pradha,. (007): h. Phys. Lett., 585.. yag,. ad Sgh, G. P. (00): Joural of Ultra Scetst of Physcal Sceces, hopal, Vol. No. (M).
776. yag,. ad Sgh, G. P., (0): Joural of Raastha cademy of Physcal Sceces, Japur,.0(),07-6.. Sgh, G.P. ad yag,.(0): Joural of Ultra Scetst of Physcal Sceces, hopal,, (), 7-5,. 5. umad, V., Srmal, S. ad Sgh, G. P., (0) Ultra Scetst of Physcal Sceces, hopal, Vol. 6(), 7. 6. Lcherowcz,. (967): Relatvstc ydrodyamcs ad Mageto-hydrodyamcs, W.. eame, Ic., New Yor- msterdam., 9. 7 Syge, J. L. (960): Relatvty: he Geeral heory, North-ollad Publ., msterdam, p. 56. 8 Ells, G. F. R. (97): Geeral Relatvty ad osmology ed., Sachs, cademc Press, Lodo P. 7. 9 Mac allum, M... (97): omm. Math. Phys. 0, 57.