dv. Studes Theor. Phys., Vol., 008, no. 19, 909-918 Magnetc Tlted omogeneous osmologcal Model wth Dsordered Radatons Ghanshyam Sngh Rathore Department of Mathematcs and Statstcs, Unversty ollege of Scence, M.L. Sukhada Unversty, Udapur-313001, Inda nta agora* Department of Mathematcs, Seedlng cademy, Japur Natonal Unversty, Japur-3005, Inda ddress : 1, Gayatr Nagar, adgaon, Udapur-313001, Raasthan anta_bagora@yahoo.com, mht_dave@yahoo.co.n Sushl Gandh Department of Mathematcs and Statstcs, Unversty ollege of Scence, M.L. Sukhada Unversty, Udapur-313001, Inda bstract. In ths paper, we have nvestgated a magnetzed homogeneous cosmologcal model of perfect flud dstrbuton havng dsordered radaton n the presence of magnetc feld n general relatvty. To get a determnate soluton, we have assumed that the unverse s flled wth dsordered radaton and a supplementary condton s. It has been shown that tlted nature of the model s preserved due to magnetc feld. The varous physcal and geometrcal aspects of the model are dscussed. The nature of the model n presence and absence of magnetc feld s also dscussed. Keywords : Tlted, Magnetzed, anch Type-I Mathematcs Subect lassfcaton : 8350, 83F05 *orrespondng uthor
910 Gh. Sngh Rathore,. agora, S. Gandh Introducton omogeneous and ansotropc cosmologcal models have been wdely studed n classcal general relatvty n the search for a relatvstc pcture of the unverse n ts early stages because they can explan a number of observed phenomena qute satsfactorly. It s well known that the magnetc feld has a sgnfcant role at the cosmologcal scale and ts present n galactc and ntergalactc spaces. The occurrence of magnetc felds on galactc scales and ther mportance for a varety of astrophyscal phenomena has been ponted out n [1,, 3]. Melvn [] has suggested that durng the evoluton of the unverse, matter was n a hghly onzed state, smoothly coupled wth the feld, subsequently formng neutral matter as a result of unverse expanson. anch Type I cosmologcal models wth perfect flud and magnetc feld dscussed by Roy et al. [5]. The dscusson of magnetc feld n spatally homogeneous unverse models appear to have been most extensvely covered by Jacobs and ughston [6] and by Macallum [7]. Interest n ths stream from the observaton of an ntergalactc magnetc feld [8] (whereas an upper lmt of order.10-8 gauss n gven). Klen [9] obtaned an approxmate soluton of Ensten s feld equaton n sphercal symmetry for a dstrbuton of dffuse radaton. Sngh and bdussattar [10] have obtaned Ensten s feld equaton for dsordered radaton of perfect flud, and to overcome the dffculty of nfnte densty at the centre. Roy and al [11] have nvestgated a statc cylndrcally symmetrc space-tme flled wth dsordered radaton for perfect flud dstrbuton. Texera, Wolk and Som [1] obtaned a cosmologcal model flled wth source free dsordered dstrbuton of electromagnetc feld radaton n general relatvty. Roy and Sngh [13] found a non-statc plane symmetrc spacetme flled wth dsordered radaton. new LRS perfect flud cosmologcal model wth equaton of state p(γ-1) ρ [1] has been derved by Senovlla [15]. anch Type I magnetzed orthogonal unverse have been studed n detal because of ther smplcty. The tlted cosmologcal models n whch the flud flow vector s not normal to the hypersurface of homogenety are more complcated. Kng and Ells [16] have nvestgated that there are no anch Type I tlted models f t has been obtaned under the assumpton that the matter taken the perfect flud form T ( p) pη, 1, >0, p>0, where s the velocty four vector and, p are the densty and pressure of the flud respectvely. owever, Dunn and Tupper [17] have shown than anch Tltng unverse are possble when an electromagnetc feld s present. Lorenz [18, 19] has nvestgated tlted electromagnetc anch Type I cosmologcal model n general relatvty. al and Meena [0] obtaned two tlted homogeneous cosmologcal models flled
Magnetc tlted homogeneous cosmologcal model 911 wth dsordered radaton of perfect flud and heat flow. Mukheree [1] has nvestgated tlted homogeneous anch Type I unverse wth heat flux n general relatvty. e has shown that the unverse assumes a pancake shape. The velocty vector s not geodesc and heat flux s comparable to energy densty. al and Sharma [] obtaned tlted anch Type I cosmologcal model wth dust flud. onformally flat tlted anch Type V cosmologcal models n general relatvty has been derved by al and Meena [3]. In ths paper, we have nvestgated a magnetzed tlted homogeneous cosmologcal model of perfect flud flled wth dsordered radaton n general relatvty. The Feld Equaton We consder homogeneous metrc n the form ds dt dx dy dz, (1) where, and are functons of t alone. The energy-momentum tensor for perfect flud dstrbuton wth heat conducton s taken nto the form by Ells [] as T ( p)vv pg q v vq E, () together wth g v v 1, (3) q q > 0, () q v 0. (5) ere E s the electromagnetc feld gven by Lchnerowcz [5] as 1 E μ h v v g h h, (6) where μ s magnetc permeablty and h s the magnetc flux vector defned by h g kl kl F v. (7) μ F kl s the electromagnetc feld tensor and kl the Lev-vta tensor densty. From (7) we fnd that h 1 0, h 0, h 3 0, h 0. Ths leads to F 1 0 F 13 by vrtue of (7). We also fnd that F 1 0 F due to the assumpton of nfnte conductvty of the flud. We take the ncdent magnetc feld to be n the drecton of x-axs so that the only non-vanshng component of F s F 3. The frst set of Maxwell s equaton
91 Gh. Sngh Rathore,. agora, S. Gandh F ;k F k; F k; 0, leads to F 3 constant (say). From the equaton (7), we have h 1 cosh λ, μ h snhλ. μ Snce, l h h l h h 1 h 1 h h g 11 (h 1 ) g (h ) cosh λ snh λ μ μ. (8) μ Equaton (6) leads to 1 3 E 1 E E 3 E. μ In the above p s the pressure, the densty, q the heat conducton vector orthogonal to v. The flud flow vector snh λ has the components (, 0,0, coshλ) satsfyng (3). The Ensten s feld equaton R 1 Rg 8πT, ( G 1) (9) For the lne-element (1) leads to snhλ 8 π ( p) snh λ p q1 μ, (10) 8 π p μ, (11)
Magnetc tlted homogeneous cosmologcal model 913 8 π p μ, (1) snhλ 8 π ( p) cosh λ p q1 μ (13) snh λ ( p) snhλ coshλ q1 coshλ q1 0, (1) coshλ where the suffx stands for ordnary dfferentaton wth respect to the cosmc tme t alone. Soluton of Feld Equaton Equatons from (10) to (1) are fve equatons n seven unknown,,,, p, λ and q. For complete determnacy these quanttes, we need two extra condtons. () We assume the model s flled wth dsordered radaton whch leads to 3p. (15) () We also assume the condton between metrc potental as (). (16) Equatons (10) and (13) lead to 8π 8π ( p). (17) μ Usng (15) n (17), we have Or 8π 16πp, μ K 16πp, (18) () 8π where K. (19) μ Equatons (11) and (1) lead to
91 Gh. Sngh Rathore,. agora, S. Gandh 0, (0) a μ, (1) where a s constant of ntegraton and μ, From (11) and (1), we also have K 16πp (). () Equatons (18) and () lead to 0. (3) Equaton (3) leads to μ μ 3 μ μ 1 where μ. From (1) and (), we have 1 3 f 1 a 3 0., () ff, (5) μ μ where μ f(μ). Equaton (5) leads to 1 5 / f [a 5bμ ], (6) 5μ where b s constant of ntegraton. From (1), we have dμ log a 5, (7) 5 / μ a 5bμ ence the metrc (1) reduces to the form dμ μ ds f where s determned by (7). y ntroducng the followng transformatons μ dx μdy dz, (8)
Magnetc tlted homogeneous cosmologcal model 915 μ T, x X, y Y, z Z. The metrc (8) reduces to the form ds 5T 5bT dt T dx Tdy dz, (9) 5/ a where dt N exp a 5 5 /, (30) T a 5bT N s constant of ntegraton. T Some Physcal and Geometrcal Features The pressure and densty for the model (9) are gven by 1 3/ 8 π p (5b 8KT ), (31) 11/ 16T 3 3/ 8 π (5b 8KT ). (3) 11/ 16T The tlt angle λ s gven by cosh λ 5b 16KT 30b, (33) snh λ 5b 16KT 30b. (3) The realty condtons () p > 0, () 3p > 0, leads to 1 11/ T 16 > 0. (35) The scalar of expansons θ calculated for the flow vector s gven by
916 Gh. Sngh Rathore,. agora, S. Gandh 5 / 1 a 5bT θ. (36) 10T 6b(5b 16KT ) The non-vanshng components of flud flow vector and heat conducton vector q for the model (9) are gven by 3/ 1 1 5b 16KT, (37) T 30b 3/ 5b 16KT, (38) 30b q q (5b 16KT ) 5b 16KT, (39) 15 / 18πT 30b 1 (5b 16KT ) 5b 16KT. (0) 11/ 18πT 30b The non-vanshng components of shear tensor (σ )and rotaton tensor (ω ) are gven by 5 / (5b 8KT ) (a 5bT )(5b 16KT ) σ 11, (1) 50b 6b σ σ 30T (a 5bT 5 / )(5b 16KT 6b, 5 / 1 30T (a 5bT )(5b 16KT 6b ) a ) 3a 5 3/ 1KT 5bT (5b 16KT ) () 3a 5 1KT 5 / a 5bT (5b 16KT ) (3) 5 / 33, 1 (a 5bT 5 / ) σ (5b 16KT )(5b 8KT ) 50T (5b 16KT ), () 5 / 1 a 5bT σ 3 1 100Kb T 150b 160bK T 5 / (30b) 5( 5b 16KT ), (5)
Magnetc tlted homogeneous cosmologcal model 917 ω 1 (5b 16KT 300bT ) a 5bT 6b 5 / 3/ 3/ 7 / (35b 16KT ) 6(55b 16KT )KT (5b 16KT The rates of expanson n the drecton of x, y and z axes are gven by 3/ ) T. (6) 1 5 / 1 a 5bT, T 5 1 5 / a 5bT a 5 5T, 1 5 / a 5bT a 5 5T 3. (7) (8) (9) oncluson The model (9) has an ntal sngularty at T 0. The model starts expandng wth a bg-bang at T 0 and the expanson n the model stops at T. The model n general represents shearng and rotatng and tlted unverse n presence of magnetc feld. The ubble components at T 0. The magnetc feld posses the expanson n the model. References [1]. aneree,.k. Sanyal, and S. hakraborty, Pramana J. Phys. 3 (1990) 1. [] S. hakraborty, Ind. J. Pure ppl. Phys. 9 (1991) 31. [3] R. Tkekar and L.K. Patel, Gen. Rela. Gravt. (199) 397. [] M.. Melvn, nn. N.Y. cad. Sc. 6 (1975) 53. [5] S.R. Roy, S. Naran and J.P. Sngh, ust. J. Phys. 38 (1985) 39.
918 Gh. Sngh Rathore,. agora, S. Gandh [6] L.P. ughston and K.. Jacob, strophys. J. 160 (1970) 17. [7] M... Macallum, Ph.D. thess (1970) Unversty of ambrdge. [8] K. Fumoto Kawabata, Y.M. Sufune and M. Fuku, str. Soc. Japan 1 (1969) 93. [9] O. Klen, rk. Mat. str. Phys. 3 (197) 19, 10. [10] K.P. Sngh and bdussattar, Ind. J. Pure and ppl. Math. (1973), 68. [11] S.R. Roy and R. al, J. Sc. Research,..U. (Inda) 8 (1977). [1].F. Da.F. Texera, I. Wolk and M.M. Som, IL Nuovo m.della Socenta, Italan dffsca 1 (1977), 387. [13] S.R. Roy and P.N. Sngh, J. Phys.. Math. and Gen. 10 (1977) 1, 9. [1] D. Kramer,. Macallum Stephan, M... and E. erlt, Exact soluton of Ensten s feld equaton (1980) ambrdge Unv. Press. [15] M.M. Senovlla, Jose, lass. Quantum Gravty, (1987) 19. [16].R. Kng and G.F.R. Ells, omm. Math. Phys. 38 (1973) 119. [17] K.. Dunn and.o.j. Tupper, strophys. J. 35 (1980) 307. [18] D. Lorenz, Phys. Lett. 83 (1981a) 155. [19] D. Lorenz, Gen. Rela. Gravt. 13 (1981b) 795. [0] R. al and.l. Meena, strophys. and Space Sc. 81 (00) 565. [1] G. Mukheree, J. strophys. stron. 7 (1986) 59. [] R. al and K. Sharma, Pramana J. of Phys. 58 (001) 3, 57. [3] R. al and.l. Meena, Pramana J. of Phys. 6 (00) 5, 1007. [] G.F.R. Ells, General Relatvty and osmology, cademc Press, New York, (1971) 177. [5]. Lchnerowcz, Relatvstc ydrodynamcs and Magneto ydrodynamcs enamn, New York, (1967) 13. Receved: September 0, 008