Bianchi Type I Bulk Viscous Fluid String Dust Cosmological Model with Magnetic Field in Bimetric Theory of Gravitation

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1 valable at ppl. ppl. Math. ISSN: Vol. 5, Issue (Jue 00) pp (Prevously, Vol. 5, No. ) pplcatos ad ppled Mathematcs: Iteratoal Joural (M) ach ype I ulk Vscous Flud Strg Dust osmologcal Model wth Magetc Feld metrc heory of Gravtato M. S. orkar Post Graduate Departmet of Mathematcs Mahatma Jotba Fule Educatoal ampus, mravat Road R.. M. Nagpur Uversty Nagpur-00, Ida borkar.mukud@redffmal.com S. S. hara Departmet of Mathematcs Late. Z. S. Scece ollege ramha (almeshwar) 50 Dst. Nagpur, Ida sschara@redffmal.com Receved: ugust 8, 009; ccepted: March 7, 00 bstract I the presece of both magetc feld ad bulk vscosty, ach ype I bulk vscous flud strg dust cosmologcal model Rose s bmetrc theory of gravtato have bee vestgated by usg the techque of Leteler ad Stachel. he ature of the model s dscussed the absece of both magetc feld ad bulk vscosty. o get a determate soluto, we have assumed the codto that σ s proportoal to θ ad ζθ = costat σ s the shear, θ s the expaso the model ad s the coeffcet of bulk vscosty. Further the physcal ad geometrcal sgfcace of the model are dscussed. Here, we compared betwee the case the presece of magetc feld ad bulk vscosty ad the case the absece of magetc feld ad bulk vscosty. eywords: metrc theory; bulk vscous; cosmc strg, magetc feld, ach type-i MS 000: 8 XX, 850, 8 F 05 96

2 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 97. Itroducto Several ew theores of gravtato have bee formulated whch are cosdered to be alteratves to Este s theory of gravtato. he most mportat amog them are Rose s bmetrc theory of gravtato ad Scalar-tesor theory of gravtato. he Rose s bmetrc theory s the theory of gravtato based o two metrcs, see Rose (97). Oe s the fudametal metrc tesor g descrbes the gravtatoal potetal ad the secod metrc γ refers to the flat space-tme ad descrbes the ertal forces assocated wth the accelerato of the frame of referece. he metrc tesor g determe the Remaa geometry of the curved space tme whch plays the same role as gve Este s geeral relatvty ad t teracts wth matter. he backgroud metrc refers to the geometry of the empty uverse (o matter but gravtato s there) ad descrbe the ertal forces. he metrc tesor has o drect physcal sgfcace but appears the feld equatos. herefore t teracts wth g but ot drectly wth matter. Oe ca regard as gvg the geometry that would exsts f there were o matter. I the absece of matter oe. would have g = Moreover, the bmetrc theory also satsfed the covarace ad equvalece prcples; the formato of geeral relatvty. he theory agrees wth the preset observatoal facts pertag to geeral relatvty [for detals oe may refer arade (980), atore ad Rae (006), ad Rose (97, 977)]. hus, at every pot of space-tme, there are two metrcs ds g dx dx () ad d dx dx. () he feld equatos of Rose (97) s bmetrc theory of gravtato are N N 8k (), pr s N g g s p r N N, ad g k together wth g det( g ) ad det( ). Here, the vertcal bar ( ) stads for -covarat dfferetato ad eergy-mometum tesor of matter felds. s the he several aspects of bmetrc theory of gravtato have bee studed by Rose (97), arade (980), Israelt (98), atore ad Rae (006), ad hadekar ad ade (007). I partcular Reddy ad N. V. Rao (998) have obtaed some ach type cosmologcal models bmetrc theory of gravtato. he purpose of Rose s bmetrc theory s to get rd of the sgulartes that

3 98 orkar ad hara occur geeral relatvty that was appearg the bg-bag cosmologcal models ad therefore recetly, there has bee a lot of terest cosmologcal model o the bass of Rose s bmetrc theory of gravtato. I bmetrc theory, the backgroud metrc tesor should ot be take as descrbg a empty uverse but t should rather be chose o the bass of cosmologcal cosderato. Hece Rose proposed that the metrc be take as the metrc tesor of a uverse whch perfect cosmologcal prcple holds. I accordace wth ths prcple, the large scale structure of uverse presets the same aspect from every space ad at all tmes. he fact, however, s that whle takg the matter actually preset the uverse, ths prcple s ot vald o small scale structure due to rregulartes the matter dstrbuto ad also ot vald o large scale structure due to the evoluto of the matter. herefore, we adopt the perfect cosmologcal prcple as the gudg prcple. It does ot apply to g ad the matter the uverse but to the metrc. Hece, descrbes a space-tme of costat curvature. I the cotext of geeral relatvty cosmc strgs do ot occur ach type models, see ror et al. (99). Some ach type cosmologcal models two four ad oe hgher dmesos- are studed by ror et al. (99). hey have show that the cosmc strgs do ot occur ach type V cosmology. al ad Dave (00); al ad Upadhaya (00), al ad Sgh (005), al ad Pareek (007) have vestgated ach type IX, I ad V strg cosmologcal models uder dfferet physcal codtos geeral relatvty. he magetc feld s due to a electrc curret produced alog x-axs. Ra al ad al (00) have vestgated ach ype I bulk vscous flud strg dust magetzed cosmologcal model geeral relatvty. hey have assumed that the egevalue ( ) of shear tesor ( ) s proportoal to the expaso ( ) whch s physcally plausble codto. he strg dust codto leads to =, s the rest eergy desty ad the strg teso desty. Recetly people lke al et al. (00), Pradha et al. (007), Pradha (009) ad Wag (00, 006) developed the models the feld of bulk vscous flud solutos ad ach type strg models whch are the most useful models geeral relatvty. I a attempt to acheve our bulk vscous model bmetrc theory of gravtato, we used the termology ad the otatos of al et al. (00). I ths paper, we have vestgated ach ype I bulk vscous flud strg dust cosmologcal model wth ad wthout magetc feld Rose s bmetrc theory of gravtato as there has bee a lot of terest cosmologcal model o the bass of Rose s bmetrc theory of gravtato. o get determate soluto we have assumed that σ s proportoal to θ ad ζθ = costat σ s shear, θ s the expaso the model ad s the coeffcet of bulk vscosty. lso the physcal ad geometrcal sgfcace of the model are dscussed. I our models, bulk vscosty plays mportat role the presece of magetc feld as well as the absece of magetc feld. hs agreed wth the fdgs of rpathy et al. (008) that the bulk vscosty plays role the evoluto of the uverse.

4 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 99 We cosder ach ype I metrc ds dt dx dy dz, (), ad are fuctos of t aloe. he flat metrc correspodg to metrc () s d dt dx dy dz. (5) he eergy mometum tesor for strg dust s gve by x x ; ( g ) E (6) wth ad x x (7) 0. (8) x I ths model, s the rest eergy desty for a cloud of strgs ad s gve by p p ad deote the partcle desty ad the strg teso desty of the system of strgs respectvely, x s the drecto of strgs ad ζ s the coeffcet of bulk vscosty. he electromagetc feld E s gve by Lcherowcz (967) E h g h h, (9) four velocty vector s gve by g (0) ad s the magetc permeablty ad the magetc flux vector h defed by h g kl kl F, () F kl s the electromagetc feld tesor ad kl s the Lev vta tesor desty.

5 00 orkar ad hara ssume the comovg coordates system, so that 0,. Further, we assume that the cdet magetc feld s take alog x-axs so that h 0 ad h 0. h h he frst set of Maxwell s equato F (), k 0 Yelds F = costat H (say). Due to the assumpto of fte electrcal coductvty, we have F F F 0. he oly o-vashg compoet of F s F. So that H h () ad h H. () From equato (9) we obta H E E E E. (5) From equato (6) we obta ;, ;, H H H. (6) Substtutg these values of [equato (6)] the Rose s feld equatos (), we wrte H 6 ; (7)

6 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 0 H 6 ; (8) H 6 ; (9) H 6, (0) d d d,,, etc. dt dt dt Equatos (7) to (0) are four equatos fve ukows,,, ad. herefore, to deduce a determate soluto; we assume a supplemetary codto 0, (), for whch the shear (σ) s proportoal to the scalar of expaso. he uverse s flled wth Zel dovch matter strg dust ad perfect flud ad therefore we are usg Zel dovch (980) codto () our model. From equatos (9) ad (0), we obta H 6 ;. () ddg equatos (7) ad () together ad usg the codto ε = λ, we get H 6 ;. () From equatos () ad (), we wrte ( ) ( ) ( ) ( ) 6 ;, (5)

7 0 orkar ad hara H. From equatos (8) ad (9), we obta. (6) O smplfyg above equato, we get whch o tegratg, yeld, (7) L, (8) L s the costat of tegrato. Usg assumptos ad v, equato (8) leads to v L v. (9) Now usg equato () ad the codto ad v, the equato (5) gves ;. (0) 6 pplyg the codto ζθ = costat to the above equato, we get 6, () ;, whch reduces to

8 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 0 d d 6 f ) f (, () f. he dfferetal equato () has soluto f ( ) P, () P s the costat of tegrato. From equato (9) we wrte logv ( ) L d ( ) P logb. () Usg f ad expresso (), the metrc () wll be ds ( ) ( ) d ( ) ( ) P dx vdy dz v, (5) v s determed by equato (). fter sutable trasformato of coordates.e., puttg, x X, y Y, z Z the above metrc (5) takes the form ds ( ) ( ) d ( ) ( ) P dx v dy v dz. (6) hs s the ach ype-i bulk vscous flud strg dust cosmologcal model wth magetc feld bmetrc theory of gravtato. I the absece of magetc feld.e., = 0, the metrc (6) have the form

9 0 orkar ad hara ds P d ( ) ( ) dx v dy v dz. (7) I the absece of vscosty,.e., 0, the metrc (6) takes the form ds d ( ) ( ) P dx v dy v dz. (8). Some Physcal ad Geometrcal Features he desty (), the strg teso desty (), for the model (6) s gve by (5 ) ( ). (9) ( ) 6 ( ) Now, the expaso s gve by ( ) ( ) ( ) P, whch has the value ( ). (0) ( ) ( ) he compoets of shear tesor are gve by f, or or ( ) ( ) P. () ( ) ( ) Lkewse, we obta the other compoets of as ( ) ( ) L P 6 ( ) ( ), ()

10 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 05 ad hus, ( ) ( ) L P 6 ( ) ( ), () 0. () = ( ) ( ) L P ( ) ( ) ( ), (5) ad the spatal volume s R, (6). f ( ) ( ) P ( ) ( ) From these results, t s leared that the presece of magetc feld ad bulk vscosty, the rest eergy desty ad strg teso desty both are fte tally, as both are depeds o vscosty coeffcet at fte tme. he expaso the model creases as the coeffcet of vscosty decreases ad t becomes maxmum for = 0. For very very large value of, the expaso, the shear ad the spatal volume R, are fte ad our model (6) does ot approach sotropy.. Dscusso From the results of earler secto-, t s see that presece of magetc feld ad bulk vscosty, our model (6) s expadg, whe the coeffcet of vscosty decreasg, ad the maxmum expaso s P. he rest eergy desty ad strg teso desty both are fte tally, as both are depeds o vscosty coeffcet at fte tme. he expaso the model creases as the coeffcet of vscosty decreases ad t becomes maxmum for = 0. For very very

11 06 orkar ad hara large value of, the expaso, the shear ad the spatal volume model (6) does ot approach sotropy. R are fte ad our L For, there s shear values of.. Hece, the model (6) does ot approaches sotropy for large I the absece of magetc feld,, the equato (9) leads to, (7) 6 from whch we coclude that the rest eergy desty, strg teso desty o vscosty coeffcet. he expressos for, ad R, the absece of magetc feld are gve by ad P depeds oly sp, (8) P, (9) L P 6, (50) L P 6, (5) 0, (5) L P, (5) R, (5)

12 M: Iter. J., Vol. 5, Issue (Jue 00) [Prevously, Vol. 5, No. ] 07 P. I the absece of vscosty, the rest eergy desty, strg teso desty for the model (6) s gve by 5, (55) ad t becomes zero tally ad fte for very very large value of. he expressos for, ad R, the absece of vscosty, are gve by P, P, (56) 6 L P, (57) 6 L P, (58) 0, (59) ) ( L P, (60) ad R, (6)

13 08 orkar ad hara. P I the absece of magetc feld, our model (6) cotractg as ad creases, ad t s expadg the absece of bulk vscosty. he model does ot approaches sotropy for large values of. he spatal volume of our model s zero tally ad t s fte for very very large value of. We compared betwee the case the presece of magetc feld ad bulk vscosty ad the case the absece of magetc feld ad bulk vscosty ad t s realzed that our model s expadg as well as shearg presece of magetc feld ad bulk vscosty (for decreasg ), as t s ether expadg or shearg the absece of both magetc feld ad bulk vscosty. ckowledgemet he authors would lke to covey ther scere thaks ad grattude to the referees for ther useful commets ad suggestos for the mprovemet of the mauscrpt of ths paper. REFERENES al, R. ad al (00). ach ype I bulk vscous flud strg dust magetzed cosmologcal model geeral relatvty, Pramaa- J. of Physcs, 6 (), al, R., Dave, S. (00). ach ype -IX strg cosmologcal models wth bulk vscous flud geeral relatvty, strophys. Space Sc., 88, al, R., Pareek, U.. (007). ach ype I strg dust cosmologcal model wth magetc feld geeral relatvty, strophys. Space Sc.,, al, R., Pareek, U.. ad Pradha,. (007). ach ype I massve strg magetzed barotropc perfect flud cosmologcal model Geeral Relatvty, hese Phys.Lett., (8), al, R., Pradha. (007). ach ype III strg cosmologcal models wth tme depedet bulk vscosty Geeral Relatvty, hese Phys.Lett., (), al, R., Pradha,., ad mrhashch, H. (008). ach ype VI0 magetzed barotropc bulk vscous flud massve strg uverse Geeral Relatvty, It.J. heor. Phys., 7(7-8), al, R., Sgh, D. (005). ach type V bulk vscous flud strg dust cosmologcal model geeral relatvty, strophys. Space Sc., 00, al, R. ad Upadhaya, R. D. (00). LRS ach ype I strg dust magetzed cosmologcal models, strophys. Space Sc., 8,

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