Magnetized Dust Fluid Tilted Universe for Perfect. Fluid Distribution in General Relativity

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1 Adv. Studes Theor. Phys., Vol., 008, no. 7, 87-8 Mgnetzed Dust Flud Tlted Unverse for Perfect Flud Dstruton n Generl Reltvty Ghnshym Sngh Rthore Deprtment of Mthemtcs nd Sttstcs, Unversty ollege of Scence, M.L. Sukhd Unversty, Udpur-3300, Ind Ant gor* Deprtment of Mthemtcs, Seedlng Acdemy Jpur Ntonl Unversty, Jpur-3005, Ind nt_gor@yhoo.com mht_dve@yhoo.co.n Astrct. In ths pper, we hve nvestgted mgnetzed dust flud nch type-i nsotropc tlted cosmologcl model for perfect flud dstruton n generl reltvty. It hs een ssumed tht the expnson n the model s only n two drectons.e. one of the component of ule prmeter s zero. A A The physcl nd geometrcl spects of the model n the presence nd sence of the mgnetc feld together wth the sngulrty n the model re lso dscussed. Mthemtcs Suject lssfcton: 8350, 83F05 Keywords: Tlted, Mgnetzed, dust perfect flud, nch type-i. Introducton In recent yers, there hs een consderle nterest n nvestgtng sptlly homogeneous nd nsotropc cosmologcl models n whch mter does not move orthogonlly to the hyper surfce of homogenety. These types of models re clled tlted cosmologcl models. The generl dynmcs of these cosmologcl models hve een studed n detls y Kng nd Ells [], Ells nd Kng [], ollns nd Ells [3], Ells nd ldwn []. They hve shown tht we re lkely to e lvng n tlted unverse nd they hve ndcted tht how we my detect t. eeshm [5] derved tlted nch type V cosmologcl models n the sclr covrnt theory. *orrespondng Author

2 88 Gh. S. Rthore nd A. gor The nsotropc mgnetc flud models hve sgnfcnt contruton n the evoluton of glxes nd stellr odes. Prmordl mgnetc felds of cosmologcl orgn hve een speculted y Asseo nd Sol [6]. FRW models re pproxmtely vld s present dy mgnetc feld strength s very smll. In erly unverse, the strength must hve een pprecle. nch type-i mgnetzed orthogonl unverses hve een studed n detl y Thorne [7], Jcos [8, 9], Roy nd Prksh [0] nd nerjee nd Snyl []. l nd Tyg [] hve nvestgted mgnetc nch type-i stff perfect flud model n generl reltvty. In ths pper, we hve nvestgted mgnetzed dust flud nch type-i nsotropc tlted cosmologcl model for perfect flud dstruton n generl reltvty. The expnson n the model s long y nd z drecton. To get determnte soluton we hve ssumed tht the unverse s flled wth dust flud. The vrous physcl nd geometrcl spects of the models re dscussed n presence nd sence of the mgnetc feld.. The Feld Equton We consder the nch Type-I metrc n the form ds dt + dx + dy + dz, () where nd re functons of t lone. The energy momentum tensor for perfect flud dstruton wth het conducton s tken nto the form y Ells [3] s j j j j j j T ( + p)vv + pg + qv + vq + E, () together wth g j v v j, (3) q q j > 0, () q v 0. (5) j ere E s the electromgnetc feld gven y Lchnerowcz [] s j j j j E μ h (vv + g ) h h, (6) where μ s mgnetc permelty nd h s the mgnetc flow vector defned y ere h g kl j jkl F v. (7) μ F kl s the electromgnetc feld tensor nd jkl the Lev-vt tensor densty. From (7) we fnd tht h 0, h 0, h 3 0, h 0. Ths leds to F 0 F 3 y vrtue of (7). We lso fnd tht F 0 F due to the ssumpton of nfnte conductvty of the flud. We tke the ncdent mgnetc feld to e n the drecton of x-xs so tht the only non-vnshng component of F j s F 3. The frst set of Mxwell s equton

3 Mgnetzed dust flud tlted unverse 89 Now Snce F j;k + F jk;i + F k;j 0, leds to F 3 constnt (sy). h cosh λ, μ h snh λ. μ l h h l h h h + h h g (h ) + g (h ) cosh λ snh λ μ μ. μ Equton (6) leds to 3 E E E 3 E. (8) μ In the ove, p s the pressure, the densty, q the het conducton vector orthogonl to v. The flud flow vector v hs the components (snhλ, 0,0, coshλ) stsfyng (3) nd λ s the tlt ngle. The Ensten s feld equton j j j R Rg 8πT, (c G ) The feld equton for the lne element () leds to π ( + p) snh λ + p + qsnhλ μ () 8 π p + μ () 8 π p + μ (), (9), (0), () 8 π ( + p) cosh λ + p qsnhλ, μ () () snh λ ( + p) snhλ coshλ + q coshλ + q 0, coshλ (3) where the suffx stnds for ordnry dfferentton wth respect to the cosmc tme t lone.

4 80 Gh. S. Rthore nd A. gor 3. Soluton of Feld Equton Equtons from (9) to (3) re fve equtons n sx unknown,,, p, q nd λ. For the complete determnton of these qunttes, we requre one more condton. We ssume tht the model s flled wth dust of perfect flud whch leds to p 0. () From equtons (0) nd (), we hve + 0, (5) whch leds to ν, ν μ (6) where μ, ν nd s constnt of ntegrton. Equtons (9) nd (), led to 8π + + 8π( p) + μ(). (7) y usng () n (7), we hve 8π + + 8π + μ(). (8) Whch leds to K + + 8π + (), (9) 8π where K. μ (0) Agn from the equton (0) nd () K + (). () Usng μ nd ν n equton (). Then t ecomes μ μ K +. μ μ μ μ () Ths gves f ( + K) ff, (3) μ μ where μ f(μ). Equton (3) leds to f + K +, () ( ) μ

5 Mgnetzed dust flud tlted unverse 8 where s constnt of ntegrton. From equtons (6) nd (), we hve μ + ( + K) + K + K ν, (5) μ + ( + K) + + K where s constnt of ntegrton. ence the metrc () reduces to the form ds dμ f + dx + μ μ + ( μ + ( + K) + K) + + K + K + K μ + ( + K) + + K μ + ( + K) + K y ntroducng the followng trnsformtons μ T, x X, y Y, z Z. The metrc (6) reduces to the form ds ( μ + K + dz. (6) dt + K) + T + T T + ( dx + T + K) + T + ( T + ( + K + K) + K) + dy + K + K + K + K + T dz. (7) T + ( + K) + K In the sence of the mgnetc feld.e. K 0, the model (7) reduces to T + ds dt + dx + T dy + + T T + + T + + T dz. (8) T + dy. Some Physcl nd Geometrcl Fetures The densty for the model (7) s gven y 8 π [ T K]. (9) T The tlt ngle λ s gven y T K cosh λ K, (30)

6 8 Gh. S. Rthore nd A. gor T cosh λ, K (3) T K snh λ. K (3) The relty condtons gven y Ells () + p > 0, () + 3p > 0, for the model (3) led to K T /3 >. (33) The sclr of expnson θ clculted for the flow vector v for the model (7) s gven y 3 ( + K) + T θ TK. (3) The flow vector v nd het conducton vector q for the model (7) re gven y v T K, (35) T v, K (36) q 6πT T K, (37) (T K) K q 3 / 6πT. (38) The non-vnshng components of sher tensor (σ j ) nd rotton tensor (ω j ) re gven y 3 / σ (3K ) T( + K + T), 6 K (39) ν T σ [3 + K + T], K (0) σ ν T K [ K T ] 33 + T σ ( + K + T)(T T K ω, () K), () + K + T, (3) T K + K + T σ. () K(T K) The rte of expnson n the drecton of x, y nd z - xes re gven y 0, (5)

7 Mgnetzed dust flud tlted unverse 83 [ + KT + T + ], (6) T 3 [ + KT + T ]. (7) T In the sence of mgnetc feld.e. K 0, the ove mentoned qunttes re gven y 3 T θ, coshλ, snhλ, ν, ν, q T, T 6πT q oncluson The model (7) strts wth g-ng t T 0 nd expnson n the model stops t T. The model n generl represents, sherng, rottng nd tlted unverse n presence of mgnetc feld. The model hs sngulrty t T 0. Rotton goes on decresng s tme ncrese. ut t T the rotton s present only due to the mgnetc feld. For K K T, energy densty 0, where s for T, the ngle λ0 nd model (7) reduces to non-tlted n nture nd t ths stge velocty components ν 0, ν nd het conducton vectors q q 0. ut fter ths stge the expnson n the model decreses s the tme ncreses nd t stops t T. Lt Snce σ 0, the model does not pproch sotropy for lrge vlue of T. T θ The x-component of ule prmeter s zero due to the ssumpton of metrc. owever densty 0, x, y nd z components to ule prmeters re zero t T, thus the metrc s symptotclly empty. In the sence of mgnetc feld, the model (8) hs sngulrty t T 0. Energy densty s nversely proportonl to tme. For ths cse the tlt ngle s gven y snhλ T. Relty condton show tht > 0. The velocty components T ν, ν, the het conducton vectors re gven y q T nd 6π q 0. T + Sher components tends to nfnty nd rotton s gven y ω. T Now s fr s the effect of the mgnetc feld s concerned, we cn conclude tht s the mgnetc feld ncreses, the tlt ngle λ, energy densty, velocty components ν nd ν decrese. et conducton vectors q nd q decrese n postve nd negtve drecton respectvely s mgnetc feld ncrese wheres rotton nd ule constnts ncrese wth the mgnetc feld.

8 8 Gh. S. Rthore nd A. gor References [] A.R. Kng, nd G.F.R. Ells, omm. Mth. Phys. 3 (973) 09. [] A.R. Kng, nd G.F.R. Ells, omm. Mth. Phys. 38 (97) 9. [3].. ollns nd G.F.R. Ells, Phys. Rep. 56 (979) 65. [] G.F.R. Ells nd J.E. ldwn, Monthly Notces. Roy. Astro. Soc. 06 (98) 377. [5] A. eeshm, Astrophys. Spce Sc. 5 (986) 99. [6] E. Assoe nd. Sol, Phys. Rep., 6 (987) 8. [7] K.S. Thorne, Phys. Rev., 38 (965) 35. [8] K.. Jcos, Astrophys. J. 53 (968) 66. [9] K.. Jcos, Astrophys. J. 55 (969) 339. [0] S.R. Roy nd S. Prksh, Ind. J. Phys. 5 (978) 7. [] A. nerjee nd A.K. Snyl, Gen. Reltv. Grvt. (U.S.A.), 8 (986),, 5. [] R. l nd A. Tyg, Int. J. Theor. Phys., 7 (988) 5. [3] G.F.R. Ells, Generl Reltvty nd osmology Acdemc Press, New York, (97) 77. [] A. Lchnerowcz, Reltvstc ydrodynmcs nd Mgneto ydrodynmcs, enjmn, New York, (967) 3. Receved: Aprl 0, 008