Tilted Plane Symmetric Magnetized Cosmological Models

Transcription

1 Tlted Plne Symmetrc Mgnetzed Cosmologcl Models D. D. Pwr # *, V. J. & Y. S. Solnke & # School of Mthemtcl Scences, Swm Rmnnd Teerth Mrthwd Unversty, Vshnupur, Nnded-0, Dept. of Mthemtcs, Govt. College of Engneerng, Amrvt 0, Ind & Mungsj Mhrj Mhvdyly, Drwh,Ywtml Artcle Abstrct In ths pper we hve nvestgted tlted plne symmetrc cosmologcl models n presence nd bsence of mgnetc feld. To get the determnstc model, we hve ssumed the supplementry condtons p = 0 nd B = A n where A nd B re metrc potentls nd n s constnt. Some geometrc spects of the model re lso dscussed. Keywords: tlted models, plne symmetrc, dust flud.. Introducton An nsotropc cosmologcl model plys n mportnt role n the lrge scle behvor of the unverse. The mny reserchers workng on cosmology by usng reltvstc cosmologcl models hve not gven proper resons to beleve n regulr expnson for the descrpton of the erly stges of the unverse. There re some expermentl dt of CMR nd theoretcl rguments whch supports the exstence of nsotropc unverse (Verm et.l [], Chmento [], Msner [], Lnd et.l [], M. Demnsk [5], Chwl et l []). The consderble nterest hs been focused n nvestgtng sptlly homogeneous nd nsotropc unverses n whch the mtter does not move orthogonlly to the hyper surfce of homogenety n recent yers. These re clled tlted unverses. Kng nd Ells []; Ells nd Kng []; Collns nd Ells [] hve studed the generl dynmc of tlted cosmologcl models. Dunn nd Tupper [0,] hve been studed tlted Bnch type I cosmologcl model for perfect flud nd shown tht Bnch tltng unverse s possble when electromgnetc feld s present. Mny other reserchers lke Mtrvens et l. [], Bl nd Shrm [], Horwood et l. [], Hewt et l. [5], Aposotolopoulos [] hve studed dfferent spects of tlted cosmologcl models. Bnch type V tlted cosmologcl models n the Scle-covrnt theory derved by * Correspondence Author: D. D. Pwr, School of Mthemtcl Scences, Swm Rmnnd Teerth Mrthwd Unversty, Vshnupur, Nnded-0, (Ind). E-ml: dypwr@yhoo.com

2 Beeshm []. Bl nd Meen [] hve nvestgted tlted cosmologcl models flled wth dsordered rdton of perfect flud, het flow. Prdhn nd R [] hve obtned tlted Bnch type V cosmologcl models flled wth dsordered rdton n the presence of bulk vscous flud nd het flow. Bhwre et l. [0] studed tlted cosmologcl models wth vryng. Pwr et l. [] hve nvestgted tlted plne symmetrc cosmologcl models wth het conducton nd dsordered rdton. Pwr nd Dgwl [,] hve studed confrmlly flt tlted cosmologcl models nd recently two fluds tlted cosmologcl models n generl reltvty. Bnerjee et l. [] hve studed n xlly symmetrc Bnch type I strng dust cosmologcl model n presence nd bsence of mgnetc feld. LRS Bnch type strng dust-mgnetzed cosmologcl models hve been nvestgted by Bl nd Updhyy [5]. Sttonry dstrbuton of dust nd electromgnetc feld n generl reltvty hs been nvestgted by Bnerjee nd Bnerjee []. Pwr et l. [, ] hve studed bulk vscous flud wth plne symmetrc strng dust mgnetzed cosmologcl model n generl reltvty nd Lyr mnfold. Ptl et l. [, 0] obtned on thck domn wlls wth vscous feld coupled wth electromgnetc feld n generl reltvty nd Lyr geometry. Byskr et l. [] derved cosmologcl models of perfect flud nd mssless sclr feld wth electromgnetc feld. Bl et l. [] hve studed mgnetzed tlted unverse for perfect flud dstrbuton n generl reltvty. Bgor et l. [] hve nvestgted tlted Bnch type I dust flud mgnetzed cosmologcl model n generl reltvty.. Feld Equton We consder metrc n the form ds where A nd B re the functons of t lone. dx dy B dz dt A, () The Ensten s feld equton s j j j R Rg T e () The energy-momentum tensor for perfect flud dstrbuton wth het conducton s gven by Wth T j j j j j j p v v p g q v v q E. () j g v v, () j

3 0 Here q 0, (5) q j q 0. () v j E s the energy momentum tensor of electromgnetc feld gven by E j j j j h vv g hh, () where s mgnetc permeblty nd the mgnetc flux vector h s gven by k g k j h j k F v, () F s the electromgnetc feld tensor, j k s the Lev-Cvt tensor densty, P s pressure, s densty nd vector q s het conducton vector orthogonl to the flud flow vector v hs the components ngle. The ncdent mgnetc feld s tken on z-xs so tht The frst set of Mxwell s equton s Here v. The flud flow sn h 0,0,, cos h, stsfyng condton (); nd s the tlt B h 0, h 0, h 0, h 0. () F F F 0. (0) j; k j k; k ; j F constnt M(sy). () Here F F F 0, due to ssumpton of nfnte electrcl conductvty. The only non-vnshng component of Hence nd From () we hve, F j s F. BM h cos h A, () M h sn h () h E M hh. () A E M A E E The feld equton () for metrc () reduces to. (5)

4 A AB B p, A AB B A () A sn sn M p h p q, A A B A () A sn cos M p h p q, A AB B A () sn h p. () cos h B sn h cos h q cos h q 0 Here the dot (.) over feld vrble denotes the dfferentton wth respect to tme t.. Soluton of the Feld Equtons The set () () beng hghly non-lner contnng sx unknowns (A, B,,, p nd q ). So to obtn determnte soluton we hve to use two ddtonl constrnts. Let us frst ssume tht the model s flled wth dust whch leds to P = 0 (0) nd secondly we consder tht the sclr expnson θ s proportonl to the sher sclr σ leds to B = A n, () where n s constnt. Equton (), () led to Equtng () nd (0) we hve Equtng () nd () we hve where A A AB M. () A A AB A A AB B M. () A AB B A A The equton cn be rewrtten s n A N n A n A, () N M. (5)

5 where df da A N f, () n A n, () n nd A f A, () Equton () leds to A ff ' when da dt where n n N A C A df f '. () da, (0) n, & C s constnt of ntegrton. n The metrc () reduces to form dt n ds da da A dx dy A dz Equtng () we get Where ds N T C T dt A T, dx dx, dy dy, dz dz. T n dx dy T dz () (). Some Physcl nd Geometrcl Propertes The densty for the model () s gven by N C, n - () T T n n where, n n The tlt ngle s gven by cos h where n n 5n n n & n n n. n 5NT, () NT n n 5,, n

6 n n n n C, n nd n n nc., sn h 0NT, (5) NT where n 5n, n n n n C. 0 The expnson () clculted for the flow vector The flow vector v q v s gven by n N C NT 5, n - () T T T NT v nd het conducton vector v T 5NT NT T q for the metrc () re gven by 0NT, () n NT 0 NT NT, () 5NT NT nn nn n T 0 NT T C NT NT q nn nn C T T The non-vnshng components of sher tensor NT n N C 5, (). (0) nd rotton tensor j w re gven by, n () T T T The rte of expnson N C. () T T NT n N n nt n N C 0 NT T T nt T H n the drecton of x, y nd z-xs respectvely re gven by j

7 H N C H T T T, () n N C H. () T T T 5. Dscusson At the ntl moment energy densty s nfnte where s t vnshes when T s nfnte..e The model () strts wth bg bng t T = 0 nd stops t T.Thus model hs pont -type sngulrty t T = 0.The tlt ngles nd the flow vectors re cos h, sn h, v, v, when T = 0 provded n, 5 0 Wheres cos h, sn h, 5 v 0, v, when T provded n. Thus tlt ngles re constnt for both T = 0 nd T. From (), ntlly the rte of expnson s nfnte, t decreses when tme ncreses nd the expnson stop t T. At T = 0 sher sclr s nfnte provded n where s t vnshes when T s nfnte. Intlly drectonl Hubble prmeters re nfnte nd t vnshes for lrge vlue of T. Het conducton vector > nd n > nd q s nfnte t T = 0 nd vnshes t nfnte tme provded n q vnshes when T = 0 but t s nfnte for lrge vlue of T. Snce lm T 0 the model not pproch sotropy for lrge vlue of T. In ths cse, the models re expndng, sherng, rottng for tlted unverse. When the mgnetc feld s bsent ( N 0 ): k n k ds dt kt dx dy kt dz (5) n n where T ct d & k n The densty for the model (5) s gven by

8 5 c n k () kt Tlt ngles re gven by cos h k nk () sn h The flow vector k n5 nk v nd het conducton vector k n 5 v n k kt nk k v nk q q 5 ( kn) k nk kt c k k n kt nk c k n k 5 q for the metrc (5) re gven by () () (50) (5) (5) Sclr expnson (), the non-vnshng component of sher tensor nd rotton tensor j respectvely re gven by n k kt nk n c kt nc kt The rte of expnson k n k n 5 n k n k n k k H n the drecton of x, y nd z-xs respectvely re gven by j (5) (5) (55) c H H (5) kt

9 H nc (5) kt At the ntl moment energy densty nd sclr expnson re nfnte where s energy densty nd sclr expnson vnsh when T s nfnte. The tlt ngles re () nd () whch re the sme t ny nstnt. The flow vectors The flow vectors q v s nfnte t T= 0 nd vnshes when tme s nfnte. v s constnt for both T = 0 ndt. Intlly het conducton vector & q re nfnte nd t vnshes for lrge vlue of T. At T = 0 sher sclr s nfnte where s t vnshes when T s nfnte. Intlly drectonl Hubble prmeters re nfnte nd t vnshes for lrge vlue of T. lm Snce T 0 the model not pproch sotropy for lrge vlue of T.. Concluson In the present pper we hve constructed tlted plne symmetrc dust flud cosmologcl model n presence nd bsence of the mgnetc feld. We hve obtned determnte soluton by ssumng the condtons tht model s flled wth dust (whch leds to the zero pressure) nd the expnson sclr ө s proportonl to the sher sclr σ. We hve dscussed the physcl behvor of the models n presence nd bsence of the mgnetc feld. Energy densty, expnson sclr nd sher sclr hve the smlr behvor n presence nd bsence of mgnetc feld. At the ntl epoch ll these prmeters re nfnte nd decreses wth ncrese n cosmc tme. In the presence nd bsence of the mgnetc feld the models re expndng, sherng, rottng nd tlted. Only vrtons n tlt ngles re found. In the presence of the mgnetc feld tlt ngles re the functons of tme nd t tends to fnte number when tme s nfnte nd they re nfnte when tme zero. Wheres tlted ngles re constnt (very smll) throughout the evoluton of the unverse n bsence of mgnetc feld. The model hs rel sngulrty t T=0 nd t strt wth bg bng nd stops when cosmc tme s nfnte, n other word t becomes symptotclly empty. The present model does not pproch sotropy n presence nd bsence of mgnetc feld snce lm 0. T References ) M. K. Verm et.l (0). Rom-Journ. Phys., 5,-,- ) L.P. Chmento (00).phy. Rev. D..5. ) C.W. Msner ().Astrophys.J.5,.

10 ) K. Lnd, J. Mquejo (005).Phys. Revlelt.5, 00. 5) M. Demnsk () Phy. Rev. D.,, -. ) Chnchl Chwl, R. K. Mshr, Anrudh Prdhn (0): rxv 0.0v [physcs.gen-ph]. ) A. R. Kng, G. G. R. Ells (). Comm. Mth. Phys.,. 0. ) G. G. R. Ells, A. R. Kng (). Comm. Mth. Phys.,,. ) C. B. Collns, G. G. R. Ells (). Phys. Rep., 5, 5. 0) K. A. Dunn, B. O. J. Tpper (). Astrophys. J., 05. ) K. A. Dunn, B. O. J. Tupper (0). Astrophys. J., 5, 0. ) D. R. Mtrvers, M. s. Mdsen, D. L. Vogel (5). Astrophys. Spce Sc.,,. ) R. Bl, K. Shrm (00). Prmn J. Phys., 5, 5. ) J. T. Horwood, M. J. Hncock, D. The Wnwrght (00). Preprnt gr-c/000. 5) C. G. Hewtt, R. Brdson, J. Wnstrght (00). Gen. Reltv. Grvtton,, 5. ) P. S. Apostolopoulos (00). Preprnt gr-c/000. ) A. Beeshm (). Astrophys. J., 5,. ) R. Bl, B. L. Meen (00). Astrophys. J.,, 55. ) A. Prdhn, A. R (00). Astrophys. Spce Sc.,,. 0) S. W. Bhwre, D. D. Pwr, A. G. Deshmukh (00), JVR 5,, -5. ) D. D. Pwr, S. W. Bhwre, A. G. Deshmukh (00). Room. J. Phys., 5, -. ) D. D. Pwr, V. J. Dgwl (00). Bulg. J. Phys.,, 5-5. ) D. D. Pwr, V. J. Dgwl (0) Int. J. Theor. Phys DOI:0.00/s ) A. Bnerjee, A. K. Snyl nd S. Chkrbort (0). Prmn J. Phys.,,. 5) R. Bl, R. D. Updhyy (00). Astrophys. Spce Sc.,,. ) A. Bnerjee, S. Bnerjee (). J. Phys. A. Proc. Phys. Soc. (Gen) (GB), :. ) D. D. Pwr, S. W. Bhwre, A. G. Deshmukh (00). Int. J. Theor. Phys., 5 05 ) D. D. Pwr, V. R. Ptl (0), Vol. -, pp.-0. ) V. R. Ptl, D. D. Pwr, A. G. Deshmukh (00) Rom. Rep. Phys. (), 0. 0) V. R. Ptl, D. D. Pwr, G. U. Khpekr (0) Int J Theor Phys 5:0 0. ) S. N. Byskr, D. D. Pwr, A. G. Deshmukh (00) Rom. J. Phys. Vol. 5, Nos. -, 0. ) R. Bl, K. Shrm (00). Astrophys. J.,,. ) A. Bgor, G. S. Rthore, P. Bgor (00). Turk. J. Phys.,, -.