Bulk Viscous Fluid Bianchi Type - I String Cosmological Model in General Relativity

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1 IOSR Joural o athematcs (IOSR-J) e-issn: , p-issn: 39-75X. Volume, Issue Ver. IV (ar. - pr. 0), PP -5 Bulk Vscous Flud Bach ype - I Strg Cosmologcal odel Geeral Relatvty Varu Humad, Harsh Nagar, Sapa Shrmal 3 Research Scholar o ewar Uversty, Chttorgarh (Rajastha), Ida. Departmet o athematcs, ewar Uversty, Chttorgarh (Rajastha), Ida. 3 Departmet o athematcs, Pacc cademy o Hgher Educato & Research Uversty, Udapur (Rajastha), Ida. bstract: I ths paper we have vestgated Bulk vscous lud Bach ype - I strg cosmologcal model geeral relatvty. o get a determstc model, t s assumed that ξθ = (costat), where ξ s the coecet o bulk vscosty s, θ s the scalar o expaso ad a relato betwee metrc potetal B =.he physcal ad geometrcal aspects o the model are also cussed. Key wor: Bach ype-i models, bulk vscosty, scalar o expaso. I. Itroducto It s a challegg problem to determe the exact physcal stuato at the very early stages o the ormato o our uverse. Strg cosmologcal models are wdely studed recet tmes because o ther prme role the descrpto o the evoluto o the early phase o uverse. hese arse durg the phase trasto ater the bg bag exploso as the temperature goes dow below some crtcal temperature as predcted by grad ued theores [,, 3]. It s beleved that cosmc strgs may act as gravtatoal leses ad these objects are cosdered as possble see or ormato o galaxes. he geeral relatvstc treatmet o strg was tated by Leteler[4] ad Stachel [5]. Leteler [4] has obtaed the soluto o Este s eld equato or a cloud strg wth sphercal, plae ad cyldrcal symmetry. he 983, he solved Este s eld equato or cloud massve strg ad obtas cosmologcal models Bach ype-i ad Katowsk-Sachs sphase tme. kekar ad patel [,7]. Bal ad yag [8] have obtaed a cyldrcally symmetrc homogeeous cosmologcal model wth electromagetc eld or perect lud trbuto. Beerjee et al. [9] have vestgated a axally symmetrc Bach ype I strg dust cosmologcal model presece ad absece o magetc eld. Bal et al.[0,,] have vestgated Bach ype I magetzed strg cosmologcal models. pawar et al. [3] have vestgated about deret aspects o plae symmetrc Bulk vscous lud strg dust magetzed cosmologcal model geeral relatvty. Saha et al. [4] ad Saha [5] have studed Bach ype I cosmologcal model presece o magetc lux deret cotexts. Recetly yag et al. [] vestgated Bulk vscous lud plae symmetrc strg dust magetzed cosmologcal model geeral relatvty. I ths paper, we have vestgated Bulk vscous lud Bach ype - I Strg cosmologcal model geeral relatvty. equatoξθ = (costat), where ξ s the coecet o bulk vscosty, θ s the scalar o expaso ad a relato betwee metrc potetals B = are assumed. he physcal ad geometrcal aspects o the model are also cussed. etrc d Feld Equato We cosder Bach ype - I metrc o the orm dt ( dx dy ) B dz... () where ad B are ucto o tme oly. he eergy mometum tesor or a cloud o strg wth bulk vscosty s u u x x u u g )... () j j ( j where s the rest eergy desty o the cloud o strg wth partcles attached to them wth s the rest eergy desty o partcle ad s the teso desty o the cloud o strgs, u ; s the scalar o expaso ad s the coecet o bulk vscosty. he vector u descrbes the low velocty vector ad x represet a drecto o asotropy.e. the drecto o strgs, satsy the stadard relato. u u x x,u x 0... (3) he expresso or scalar o expaso ad shear scalar are DOI: / Page

2 Bulk Vscous Flud Bach ype - I Strg Cosmologcal odel I Geeral Relatvty B u ; B... (4) B B... (5) he Este's eld equato ( gravtatoal uts c =, 8G = ) or a system o strg R Rg... () For the metrc (), Este's eld equato's ca be wrtte as B B B B B B... (7)... (8)... (9) Where a over dot sta or the rst ad double dot or the secod dervatve wth respect to cosmc tme t. Soluto O he Feld Equatos he eld equatos (7) to (9) are three equatos ve ukows parameters, B,, ad. I order to obta a determate soluto, we have assumed two codtos: ξθ = (costat) (0).e. the coecet o bulk vscosty s versely proportoal to expaso (θ). () B = () relato betwee metrc potetals ad B, Where s costat. From equato (7), (0) ad (), we get hs lea to Let us assume () d d /,... () / d d Usg (3) ad (4) equato (), we get d d ter tegrato, we get K Where K s the costat o tegrato. (4) (3)... (5) DOI: / Page

3 Bulk Vscous Flud Bach ype - I Strg Cosmologcal odel I Geeral Relatvty DOI: / Page K... () K... (7) Equato (7) lea to d K dt Hece the model () s reduce to dz dy dx d K... (8) ter usg a sutable trasormato o coordates the model (8) reduce to dz dy dx d K... (9) Where =, x = X, y = Y, z = Z Some Physcal d Geometrcal spects O he odel For the model o equato (9), the other physcal ad geometrcal parameters ca be easly obtaed. he eergy desty, the strg teso desty, the scalar o expaso, the coecet o bulk vscosty, ad the shear scalar are respectvely gve by K... (0) k... () K... () K... (3) K... (4)

4 Bulk Vscous Flud Bach ype - I Strg Cosmologcal odel I Geeral Relatvty DOI: / Page cosat... (5) I the absece o bulk vscosty.e. whe 0,the le elemet () reduce to dz dy dx d K () he eergy desty, the strg teso desty, the scalar o expaso, the coecet o bulk vscosty, ad the shear scalar are respectvely gve by K... (7) k... (8) K... (9) 0 (30) K... (3) cosat... (3) II. Cocluso he eergy codto ρ 0 the presece o bulk vscous lud lea to 0 K (33) hus the model (9) exsts durg the tme spa gve by (33). Whe, ad due to presece o bulk vscous lud. Sce 0 Lm Hece model does ot sotropze or large value o. s 0 the ρ, θ, the model starts wth a bg bag at =0. he eergy codto ρ 0 or the model () lea to

5 Bulk Vscous Flud Bach ype - I Strg Cosmologcal odel I Geeral Relatvty K 0 (34) hus the model () exsts durg the tme spa gve by (34). he model starts bg bag at =0 ad the Lm 0 expaso the model decreases as tme creases,hece model does ot sotropze or large value o. Reereces []. Kbble,.W.B., Some mplcatos a cosmologcal phase-trasto, Phys. Rep. 7, 83 (980). []. Zel sdovch, Ya. B., Kokzarev, I. Yu ad Oku. L. B. Cosmologcal cosequeces o a spotaeous breakdow o a crete symmetry, Z.Sov.phys. JEP.40, -5 (975). [3]. Vlek,.,Cosmc strgs Phys. Rev. D 4, 08 (983). [4]. Leteler, P.S., Strg cosmologes, Phys. Rev. D 0, 94 (979). [5]. Stachel, J., hckeg the strg I the strg perect dust, Phys. Rev. D, 7 (980). []. kekar, R. ad Patel, L.K., Some exact solutos o strg cosmology Bach III space tme, Ge. Rel. Grav. 4, 397 (99). [7]. kekar, R. ad Patel, L.K.,: Some exact solutos Bach VIo strg cosmology, Pramaa. J. Phys. 4, 483 (994). [8]. Bal, R. ad yag,., Some exact solutos Bach VIo strg cosmology, Ge. Re. Grav., (989). [9]. Baerjee,., Sayal,.K. ad Chakraborty, S., Strg cosmology Bach I space tme, Pramaa-J. Phys. 34, (990). [0]. Bal, R. ad Upadhaya, R.D., LRS Bach type I strg dust magetzed cosmologcal model geeral relatvty, strophys. Space Sc. 83, 97 (003). []. Bal, R. ad jal,: Bach type I magetzed strg cosmologcal model geeral relatvty, strophys. Space Sc. 30, 0 (00). []. Bal, R., Pareek, U.K. ad Pradha,., Bach type I massve strg magetzed barotropc perect cosmologcal model geeral relatvty,ch. Phys. Lett. 4, 585 (007). [3]. Pawar, D.D., Bhaware, S.W. ad Deshmukh,.G., Bulk Vscous Flud plae symmetrc strg dust magetzed cosmologcal model geeral relatvty, It. J. heor. Phys. 47, 599 (008). [4]. Saha, B. ad Vsescu,., Strg cosmologcal model the presece o a magetc lux, strophys. Space Sc. 35, 99, (008). [5]. Saha, B., strophys. Iteractg scalar ad Spor el Bach type I uverse eld wth mageto- lud, Space Sc. 99, 49, (005). []. yag,. ad Sharma, K., Bulk Vscous lud plae symmetrc strg dust magetzed cosmologcal model geeral relatvty, Id. cad. o athematcs. 3,, (00). DOI: / Page