A Magnetic Tilted Homogeneous Cosmological. Model with Disordered Radiations

Transcription

1 dv. Studes Theor. Phys., Vol., 008, no. 19, 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 , Inda nta agora* Department of Mathematcs, Seedlng cademy, Japur Natonal Unversty, Japur-3005, Inda ddress : 1, Gayatr Nagar, adgaon, Udapur , 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 , 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

2 910 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

3 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

4 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)

5 Magnetc tlted homogeneous cosmologcal model 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

6 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)

7 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

8 916 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 Kb T 150b 160bK T 5 / (30b) 5( 5b 16KT ), (5)

9 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 / 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.

10 918 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, [] 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