Bianchi Type-III String Cosmological Models in The Presence of Magnetic Field in General Relativity

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1 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 Bach ye-iii Strg Cosmologcal odels he Presece of agetc Feld Geeral Relatvty adalkar S.P., Samdurkar S.W. ad Gawade S.P. Deartmet of athematcs, Govt. Vdarbha Isttute of Scece & Humates, mravat (Ida). bstract I ths aer we have examed Bach tye-iii strg cosmologcal model the resece of magetc feld. o get determate solutos, the Este s feld equatos have bee solved for two cases () Reddy strg ad () Nambu strg. he hyscal ad geometrcal behavour of these models are dscussed. Idex erms Bach tye-iii model, Reddy strg, Nambu strg, agetzed. INRODUCION t very early stage of evoluto of uverse, t s assumed that durg the hase trasto, the symmetry of uverse s broke sotaeously. It ca gve rse to toologcal stable defects such as doma walls, strgs ad moooles (bble976).of all these cosmologcal structures, cosmc strg lay a vtal role structure formato cosmology (Zel dovch990).it s beleved that the vacuum strgs gve rse to desty fluctuatos suffcet formato of galaxes (Zel dovch990).he cosmc strgs have ther stress-eergy couled to the gravtatoal feld. herefore, the study of gravtatoal effects of such strgs wll be of terest. he reset day cofgurato of the uverse s ot cotradcted by large scale etwork of strgs the early uverse. he geeral relatvstc formato of cosmc strgs, are gve by Leteler (979 98) ad Stachel (980).I strg theory, the myrad of artcle tye s relaced by sgle fudametal buldg block, a strg. hese strgs ca be closed, lke a loo or oe, lke a har. s the strg moves through tme t tress out a tube or a sheet, accordg whether t s close or oe. Sce strgs are ot observed at reset tme of evoluto of uverse oe ca llumate strgs ad ed u wth cloud of artcles. Strg cosmologcal models have bee studed by may authors. umber of geeral relatvstc exact soluto were vestgated for Bach tye II,VI0,VIII & IX strg cosmologcal models by ror et al.(990). class of cosmologcal solutos of massve strgs for Bach tye VI0 sace tme has bee obtaed by Chakraborty (99). Roy ad Baeree (995) have vestgated some LRS Bach tye II strg cosmologcal models whch reresetgeometrcal ad massve strgs. Wag ( ) has vestgated ad dscussed some cosmologcal models ad ther hyscal mlcato some Bach tye sace tmes. he magetc feld has mortat role at the cosmologcal scale ad s reset galactc ad tergalactc saces. he mortace of the magetc feld for varous astrohyscal heomeo has bee studed may aers. elv (975) has oted out that durg the evoluto of uverse, the matter s hghly ozed state ad s smoothly couled wth the feld, subsequetly formg eutral matter as a result of uverse IJSER 0 exaso. herefore cosderg the resece of magetc feld strgs uverse s ot urealstc ad has bee vestgated by may authors (Baeree et al. 990;Shr Ram ad Sgh995; Sgh ad Sgh999; Balad Uadhaya00).Baeree et al.(990) vestgated some cosmologcal solutos of massve strgs for Bach tye I sace tme the reseces ad absece of magetc feld. kekar ad Patel (99) obtaed some exact solutos of massve strg of Bach tye III sace tme resece ad absece of magetc feld. Recetly Uadhaya ad Dave (008) have vestgated Bach tye III massve strg cosmologcal model the resece ad absece of magetc feld, uder the assumto that the exaso ( ) the model s roortoal to the shear ( ) whch lead ad have obtaed artcular soluto for. C I ths aer we have vestgated Bach tye III cosmologcal model the resece ad absece of magetc feld. o obtaed exact soluto feld equatos, we have cosdered Reddy strg ad Nambu strg. he hyscal behavour of models are also dscussed.. HE ERIC ND FIELD EQUIONS We cosder the Bach tye III sace tme the form where x dt B e dy C, B adc are fuctos of t oly ad s a costat. he eergy mometum tesor for a cloud of strg dust wth magetc feld alog the z-drecto of the strg s gve by where lm lm uu x x g FlFm gflmf () u ad x satsfy the codtos u u x x u x 0 () s the eergy desty for a cloud strg wth artcles at- tached to them, s the strg teso desty, u s the fourvelocty of the artcles, ad x s a ut sace-lke vector re- ()

2 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 resetg the drecto of strg. I a co-movg coordate system, we have u (0,0,0,), x (0,0,,0), () C he artcle desty of the cofgurato s gve by (5) he electromagetc feld tesor F has oly the o-zero comoet Fbecause the magetc feld s assumed to be alog the z-drecto. Subsequetly axwell equato F, k Fk, Fk, 0 ad F ( g) 0 lead to F e x (6) where s a costat so the magetc feld dee uo the sace coordate x oly. From (), () ad (6), t follows that F 0. he Este s feld equato R ; k Rg 8 (7) for the metrc () lead to the followg system of equatos: B BC C 8 B BC C B C BC 0 B C BC C C 0 C C B B 8 B B B 0 B (8) (9) (0) () () where a dot (.) over a varable deotes ordary dfferetato wth resect to tme t. From equato (), we have two cases 0 ad B.Whe 0, the metrc () degeerates to Bach tye I. Sce we have cosdered Bach III symmetry, we assume that s o-zero ad B. hus the feld equatos (8)-() reduces as C 8 C 8 C C 0 C C () () (5) SOLUIONS OF HE FIELD EQUIONS he feld equatos () -(5) are a system of three equatos wth four ukow arameters,, ad C.Oe addtoal costrat relatg these arameter s requred to obta exlct solutos of the system. We assume that the exaso ( ) s roortoal to the shear ( ). hs codto lead to C (6) where s a costat. o obta exact solutos, we solve the feld equatos for the followg two cases.. Case I: Reddy Strg I ths case 0 (7) From (), () ad (7) we obta C C 0 Usg (6), the above equato reduces to ( ) Let f () whch mlesthat f f df f. d Hece (9) ca be wrtte as d d ( ) f f By tegratg (0) we fd f ( ), where where s tegrato costat. herefore, we have d dt ( ) hus the metrc () reduces to the form dt d d d ( ) x e dy where, x X, y Y, z Z e x I the absece of magetc feld.e. whe 0 the the metrc () reduces to the form dy (8) (9) (0) () () () () IJSER 0

3 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 d x e dy ( ).. Geometrcal ad hyscal sgfcace of model he eergy desty ) desty ) (5) (, the strg teso (), the artcle, the scalar of exaso ( ) ad the shear ( ) ( for the model () are gve by 8 ( ) ( ) ( ) ( (6) ) 8 ( ) ( ) ( ) ( (7) ) 8 ( ) ( ) ( ) ( (8) ) ( ) ( ) (9) ( ) ( ) ( ( ) ) ( ) cost ( ) (0) he decelerato arameter s gve by R / R q R / R ( ) ( ) ( ) ( ( ) From the (), we observe that ) () () q 0 f ( ) ( ) ( ) ( ) he eergy codto 0 mles that ( ) ( ) ( ) ad teso desty ) ( he artcle desty ) () (5) ( of the cloud strg vash asymtotcally geeral f ( ) 0.he exaso the model stos whe.he model starts exadg wth a bg bag at 0 ad the exaso the model decreases as tme creases f ( ) 0.he model () has sgularty at 0.he hyscal arameters,, are fte at the sgularty 0 ad decreases as.he eergy desty ad exaso the model decreases radly the resece of magetc feld. Sce lm 0. Hece the model does ot aroach sotroy for large value of.for the codto ( ) ( ) ( ) ( ),the soluto gves acceleratg model of the uverse ad for the codto ( ) ( ) ( ) ( ), our soluto reresets deceleratg model of the uverse ad q aroaches at ( ) value - whe ( ).Further uder the costrat, ad, the eergy desty () ad artcle desty ( ) s ostve for ad 0 (Fg. ad Fg.). q 0 f ad ( ) ( ) ( ) ( ) () IJSER 0

4 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 ( ) ( ) (0) ( ) ( ) ( ( ) ( ) cost ( ) () ) () I the absece of magetc feld, the artcle desty ( ) ad teso desty () vash whe.he exaso the model decreases as tme creases f 0 excet. Sce lm 0.he the model does ot aroach sotroy for large values of. Further uder the costrat, ( ad artcle ad, the eergy desty ) desty ( ) s ostve for ad. I the absece of magetc feld, the eergy codto 0 lead to ( ) ( ) ad the hyscal arameters 8 ( ),,, ad are gve ( ) ( ) (6) (7) 8 ( ) ( ) ( (8) ) 8 ( ) ( ) ( (9) ) IJSER 0

5 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 5. Case II: Nambu Strg I ths case () From (), () ad () we obta C C 0 Usg (6), the above equato reduces to 0 whch o tegrato gves ) ( at b) () (5) ( (6) where a ad b are costats of tegrato. Hece we obta B C ) ( at b) ( (7) ) ( at b) ( (8) hus the metrc () reduces to the form dt ( ) ( at b) x ( ) ( at b) e dy ( ) ( at b) (9) fter a sutable trasformato of coordate, the metrc (9) takes the form d a ( ) dx x ( ) e dy ( ) dz x X, y Y, z Z where ax b, (50) cost he decelerato arameter s gve by R R q R R From (56), we observe that q 0 ad q 0 (55) (56) f (57) f (58) he eergy codto 0 mles that ( ) a ( ) ( ) ( ) (59) he eergy desty ad strg teso are fte at sgularty 0 ad decreasg as.he exaso the model stos whe.he model starts exadg wth a bg bag at ad the exaso the model decreases as tme creases 0.Sce lm 0.Hece the model does ot aroach sotroy for large value of.he model (50) has sgularty at 0. For the codto the soluto gves acceleratg model of the uverse ad for the codto, ours soluto reresets deceleratg model of the uverse ad q aroaches at value - whe.further uder the costratad desty s ostve for (Fg.5)., the eergy ( ), the scalar of exaso ).. Geometrcal ad hyscal sgfcace of model he eergy desty (), the strg teso (), the artcle desty ( ad the shear ( ) for the model (50) are gve by ( ) a 8 ( ) ( ) ( ) (5) 0 (5) a( ) ( ) (5) ( ) a ( ) (5) IJSER 0

6 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 6 I absece of magetc feld, the eergy codto 0 lead to ( ) a ( ) ( ),,, ad are gve by ad the hyscal arameters ( ) a 8 ( ) ( ) 0 (60) (6) (6) a( ) ( ) (6) ( ) a ( ) (6) cost (65) I the absece of magetc feld, the artcle desty ad teso desty vash whe 0. he exaso the model decreases as tme creases. Sce lm 0. Hece the model does ot aroach sotroy for large value of Sce lm 0.Hece the model does ot aroach sotroy for large value of.further uder the costratad, the eergy desty () s ostve for (Fg.6). Cocluso We have dscussed Bach tye III cosmologcal model for two cases () Reddy strg ad () Nambu strg. It s foud that both the cases, the model always rereset acceleratg ad deceleratg uverse uder the codtos (), () ad (57), (58). It has bee show that the case of Reddy strg ad Nambu strg, the models are ot free from sgulartes. It s reasoable to say that cosmologcal model s requred to exla accelerato reset uverse. herefore, our theoretcal models are agreemet wth recet observatos (Perlmutter et al.999; Garavch et al.998; Resset al.998; Schmdt et al.998). Further uder the costrat,, for Reddy strg, the resece of maget, the artcle desty s egatve 0 ad IJSER 0

7 Iteratoal Joural of Scetfc & Egeerg Research, Volume, Issue, November-0 7 the absece of maget (Fg. ad Fg. ). For,, the res- ad (Fg. 7 ad Fg. 8). Nambu strg uder the costrat ece of maget, the eergy desty s egatve the absece of maget Refereces [] Wag, X.X.:Ch. Phys. Lett., 70 (006). [] Zel dovch, Ya.B. o Not. R.stro. Soc.9, 66 (990). [] Bal, R., Uadhaya, R.D.: strohyscs ad Sace-Scece 8, 97 (00) [] Baeree,., Sayual,.., Chakraborty, S.: Pramaa J.Phys., (990) [] Chakraborty, S.: Id. J. Pure adl.phys.9, (99) [] Garavch, P..: strohys. J. 9, L5 (998). [5] bble,.w.b., J.Phys.: ath.ge.9, 87 (976). [6] ror,.d., Chaudhary,., ahata, C.R., azumdar,.: Ge. Relatvty ad Grav., (990) [7] Leteler, P.S.: Phys. Rev.D 0, 9 (979) [8] Leteler, P.S.: Phys Rev.D 8, (98) [9] elv,..:.new York cad.sc.6,5 (975) [0] Perlmutter, S.: strohys. J. 57, 565 (999) [] Ress,.G.: stro. J. 6, 009 (998) [] Roy, S.R., Baeree, S..: Class Quatum Grav., 9 (995) [] Schmdt, B.P.: strohys. J. 507, 6 (998) [] Shr Ram, Sgh,..: Ge. Relat. Gravt. 7 (), 07 (995) [5] Sgh, G.P., Sgh,..: Ge. Relat. Gravt. (), 7 (999) [6] Stachel, J.: Phys. Rev.D, 7 (980). [7]kekar, R., Patel, L..: Ge. Relatvty & Grav., 97 (99). [8] Uadhaya, R.D., Dave, S.: Brazla Joural of Physcs, 8, (00) [9] Wag, X.X.: strohyscs ad Sace-Scece 9, 9 (000). [0] Wag, X.X.: Ch. Phys. Lett., 05 (00). [] Wag, X.X.: Ch. Phys. Lett., 9 (005). IJSER 0