Bianchi Type V Magnetized String Dust Universe with Variable Magnetic Permeability

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1 EJTP 5, No. 19 (008) Electronic Journal of Theoretical Physics Bianchi Type V Magnetized String Dust Universe with Variable Magnetic Permeability Raj Bali Department of Mathematics, University of Rajasthan, Jaipur-30004, India Received 18 August 008, Accepted 0 September 008, Published 10 October 008 Abstract: Bianchi Type V magnetized string dust universe with variable magnetic permeability is investigated. The magnetic field is due to an electric current produced along x-axis. Thus F 3 is the only non-vanishing component of electro-magnetic field tensor F ij. Maxwell s equations F [ij;k] =0,F ij ;j = 0 are satisfied by F 3 = constant. The physical and geometrical aspects of the model with singularity in the model are discussed.the physical implications of the model are also explained. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Bianchi V, Magnetized, String Dust, Variable Magnetic Permeability PACS (008): z; Cq, q; k 1. Introduction Bianchi Type V universes are the natural generalization of FRW (Friedmann-Robertson- Walker) models with negative curvature. These open models are favoured by the available evidences for low density universes (Gott et al [1]). Bianchi Type V cosmological model where matter moves orthogonally to the hyper surface of homogeneity, has been studied by Heckmann and Schucking[]. Exact tilted solutions for the Bianchi Type V space-time are obtained by Hawking[3], Grishchuk et al. [4]. Ftaclas and Cohen[5] have investigated LRS (Locally Rotationally Symmetric) Bianchi Type V universes containing stiff matter with electromagnetic field. Lorentz[6] has investigated LRS Bianchi Type V tilted models with stiff perfect fluid and electromagnetic field. Roy and Singh [7] have investigated a Bianchi Type V universe with stiff fluid and a source free electromagnetic field. Banerjee and Sanyal [8] have investigated Bianchi Type V cosmological models with viscosity and heat flow. Coley [9] has investigated Biachi type V imperfect fluid cosmological models in General Relativity. Nayak and Sahoo [10] have investigated Bianchi type V models with balir5@yahoo.co.in

2 106 Electronic Journal of Theoretical Physics 5, No. 19 (008) matter distribution admitting anisotropic pressure and heat flow. Bali and Meena[11] have investigated Bianchi Type V tilted cosmological model for stiff perfect fluid distribution. Cosmic string play a significant role in the study of the early universe. These strings arise during the phase transitions after the big-bang explosion. Linde [1]conjectured that universe might have experienced a number of phase transitions after the big-bang explosions. The phase transitions produce vacuum domain structure such as domain walls, strings and monopoles(kibble[13], Zel dovich[14]). Cosmic strings create a considerable interest as these act as a gravitational lenses and give rise to density perturbations leading to the formation of galaxies (Vilekin[15]). These strings have stress energy and they can be classified as massive and geometrical strings. Each massive string is formed by geometric string with particles attached along its extension. This is the interesting situations wherein we have particles and strings together. The pioneering work in the formulation of the energy-momentum tensor for classical massive strings is due to Letelier[16] Who explained that the massive strings are formed by geometric string(stachel[17]) with particles attached along its extension. Letelier[18] first used this idea in finding some cosmological solutions for massive strings for Bianchi Type I and Kantowski-Sachs spacetime. Melvin[19] in his cosmological solution for dust and electromagnetic field suggested that during the evolution of the universe, the matter was in a highly ionized state and is smoothly coupled with the field. Hence the presence of magnetic field in string dust universe is not unrealistic. Banerjee et al.[0] have investigated an axially symmetric Bianchi Type I string dust cosmological model in presence and absence of magnetic field. A class of cosmological solutions of massive strings has been derived by Chakraborty[1] for Bianchi Type VI 0 space-time. Tikekar and Patel [,3]have investigated cosmological models in Biachi Type III and VI 0 space-times in presence and absence of magnetic field. Patel and Maharaj[4] investigated stationary rotating world model with magnetic field. Singh and Singh [5] have investigated string cosmological models with magnetic field in the context of space-time with G3 symmetry. Wang [6] has investigated massive string cosmological model in presence of magnetic field in the context of Bianchi Type III space-time.bali and Upadhaya [7]have investigated LRS(Locally Rotationally Symmetric)Bianchi Type I string dust magnetized cosmological models using the condition that σ (shear) is proportional to the expansion (θ). Bali and Anjali [8]have investigated Bianchi Type I magnetized string dust cosmological model using supplementary condition between metric potentials A, B, Cas A =(BC) n, n being a constant. Recently Bali et al.[9]have investigated Bianchi Type I massive string cosmological model with magnetic field for Barotropic perfect fluid distribution. In the above mentioned studies, the magnetic permeability where it is considered, is assumed as constant quantity. In this paper, we have investigated Bianchi Type V string dust universe in the presence of magnetic field with variable magnetic permeability. To get the deterministic model, we have assumed thatf 3 is the only non-vanishing component off ij. The physical implications of the model are also discussed.

3 Electronic Journal of Theoretical Physics 5, No. 19 (008) Formation of Field Equations We consider Bianchi Type V space-time in the form ds = dt + A dx + B e x dy + C e x dz (1) where A, B,C are functions of t. The energy-momentum tensor (T j i ) for a cloud of string is given by Letelier[16] T j i = ρv i v j λx i x j + E j i () where v i and x i satisfy the conditions v i v i = x i x i = 1,v i x i =0, x 1 0,x = x 3 = x 4 (3) ρ being the proper energy density for a cloud of string with particles attached to them,λ the string tension density, v i the four-velocity of the particles and x i is a unit space-like vector representing the direction of string. If the particle density of the configuration is denoted by ρ p then we have ρ = ρ p + λ (4) In a comoving coordinate system, we have E is electromagnetic field given by Lichnerowicz [30] as v i =(0, 0, 0, 1),x i =(1/A, 0, 0, 0) (5) E j i = μ [ h ( v i v j +1/g j i ) hi h j] (6) with g h i = μ ɛ ijklf kl v j (7) Where h i is the magnetic flux vector, ɛ ijkl the Levi-Civita tensor, F kl the electromagnetic field tensor, μ the magnetic permeability and h = h l h l, g ij the metric tensor. We assume that magnetic field is due to an electric current produced along x-axis. Thus F 3 is the only non-vanishing component of electromagnetic field tensorf ij and h 1 0,h =0=h 3 = h 4. Maxwell s equations F ij;k + F jk;i + F ki;j = o and F ij ;j = 0 are satisfied by F 3 = H(constant) (8) We also find that F 14 = 0 = F 4 = F 34 due to the assumption of infinite electrical conductivity(roy Maartens[31]). From equation (7), we find that h 1 = AHe x μbc (9)

4 108 Electronic Journal of Theoretical Physics 5, No. 19 (008) Using equation (9) in (6), we have The Einstein s field equations E 1 1 = H e 4x μb C = E = E 3 3 = E 4 4 (10) R j i 1/Rgj i = 8πT j i (11) for the line-element (1) with equations (), (5) and (10) lead to the following system of equations B 44 B + C 44 C + B 4C 4 BC 1 ( ) H A =8π B C + λ (1) A 44 A + C 44 C + A 4C 4 AC 1 ( ) H A = 8π (13) B C A 44 A + B 44 B + A 4B 4 AB 1 ( ) H A = 8π (14) B C A 4 B 4 AB + A 4C 4 AC + B 4C 4 BC 3 ) (ρ A =8π + H (15) B C A 4 A B 4 B C 4 C = 0 (16) where we have assumed that magnetic permeability is a variable quantity and assumed as μ = e 4x (17) Thus μ 0asx and μ = 1 when x 0, Zel dovich[14] in his investigation has explained that ρ s /ρ c where ρ s is the mass density and ρ c the critical density then strings frozen in plasma would change their density like a i.e. like t 1 in the radiation dominated universe where a is the radius of the universe. In this approximation, the strings would soon be dominant and the tension along the string (λ) is equal to its energy density (ρ) per unit length and the particle density (ρ p ) of the configuration is zero. Thus from equation (4) we have string dust condition as ρ = λ 3. Solution of Field Equations Equations (1) and (15) after using string dust condition ρ = λ lead to B 44 B + C 44 C A ( 4 B4 A B + C ) 4 + C A = 0 (18) Equation (16) leads to A 4 A = 1 ( B4 B + C ) 4 C (19)

5 Electronic Journal of Theoretical Physics 5, No. 19 (008) which on integration leads to where L is the constant of integration. Equation (19) leads to where A 44 A = B 44 B + C 44 C 1 From equations (13) and (14), we have A 44 A + B 44 B + C 44 C + A 4 A Using equations (0) and (1) in (), we have A = L BC (0) B 4 C 4 B 1 C + B 4C 4 BC ( B4 B + C 4 C (1) ) A = K () B C K =8πH (3) Let us assume B 44 B + C 44 C + B 4C 4 BC 1 L BC = K (4) B C BC = μ, B C = ν (5) Using equation (5)in (18)and (4), we have μ 44 μ μ 4 μ + ν 4 4ν + 1 L μ = 0 (6) and μ 44 μ μ 4 4μ + ν 4 4ν 1 L μ = K (7) μ Equations (6) and (7) lead to μ 44 +1/4μ 4 = 8 L K μ (8) which leads to f =( dμ dt ) = 4 L μ K + N μ where μ 4 = f(μ), μ 44 = ff, f = df dμ and N is the constant of integration. From equations (13), (14) and (19), we have B 44 B C 44 C + 1 ( B4 B + C 4 C which after using the condition (5)leads to ν 4 ν = (9) )( B4 B C ) 4 = 0 (30) C l Lμ 3/ (31)

6 110 Electronic Journal of Theoretical Physics 5, No. 19 (008) which again leads to dν ν = l dt dμ Lμ 3/ dμ (3) where l is the constant of integration. Equation (3) after using (9) leads to [ ] α + tan θ/ 4γ l/l αk L ν = M K α tan θ/+ 4γ L K (33) where tan θ/ is determined by (4T L K) tan θ/ = +(L 4 K +4NL ) 1 (4T L K) (34) and K = 1 L (35) Hence the metric (1) reduces to the form ( ) dt ds = dμ + L μdx + μνe x dy + μ dμ ν ex dz dt = + TdX + Tνe X 4T K + N L dy + T ν e X L dz L T where Lx = X, y = Y,z = Z, μ = T and ν is given by (33). In the absence of magnetic field i.e. when K 0 then the metric (36) reduces to the form ds = dt 4T L + N T + TdX + Tνe X L dy + T ν e X L dz where ν is determined by (33)in absence of magnetic field as [ ] l/α α + tan θ/ 4γ ν = M (38) α tan θ/+4γ and tan θ/ in the absence of magnetic field is given by (34) as 16T +4NL tan θ/ = 1 (39) 4T The energy density (ρ), the string tension density (λ),the scalar of expansion(θ) and the shear (σ) for the model (36) in the presence of magnetic field, are given by ( 3N 8πρ = 4 M ) 1 4 T K 3 T =8πλ (40) θ = A 4 A + B 4 B + C 4 C 4 =3 L T + N T K (41) 3 T l σ = (4) LT 3/ (36) (37)

7 Electronic Journal of Theoretical Physics 5, No. 19 (008) The energy condition ρ 0 leads to 0 <T 3N M 64πK Conclusions The model (36) starts with a big-bang at T = 0 and the expansion in the model decreases as time increases. When T 0 then ρ and when T then ρ 0. Since σ 0 when T then the model isotropizes for large values of T. There is a Point type singularity in the model (36) at T = 0 (MacCallum[3]). The scale factor R is given by R 3 = ABCe x = Le x T 3/ Thus R increases as T increases. The deceleration parameter (q) is given by q = R/R Ṙ /R ( ) K 4N T = ( 4T ) (43) K + N L T The decelaration parameter approaches the value -1 as in de-sitter universe when 5N + 4T =4K T L In the absence of magnetic field i.e. when K 0,The energy density (ρ), the string tension density (λ),the scalar of expansion (θ) and the shear (σ) for the model (37), are given by ( 3N 8πρ = 4 M ) 1 4 T 3 =8πλ (44) 4 θ =3 L T + N (45) T 3 l σ = (46) LT 3/ The energy condition ρ 0 leads to 3N M. The model (37) in the absence of magnetic field, starts with a big-bang at T =0 and the expansion in the model decreases as time increases. Since σ 0 when T. Therefore the model isotropizes for large values of T.The scale factor R is given by R 3 = Le x T 3/ Thus R increases as T increases. The deceleration parameter (q) in the absence of magnetic field is given by q = R/R Ṙ /R = 4N/T 4T L + N T Thus the decelaration parameter approaches the value -1 as in de-sitter universe if 5NL + 4T =0.

8 11 Electronic Journal of Theoretical Physics 5, No. 19 (008) Acknowledgement The author is thankful to the Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune, India for providing facility and support where this work was carried out. References [1] Gott, J.R., Gunn, J.E., Schramn, D.N. and Tinsley,B.M Astrophys. J [] Heckmann,O. and Schucking, E.196, In Gravitation:An Introduction to Current Research ed.witten,l.(john Wiley, NewYork) [3] Hawking, S.W Mon. Not. R. Astron. Soc [4] Grishchuk, L.P., Doroshkevich, A.G. and Novikov, I.D Sov.Phys. JETP [5] Ftaclas, C. and Cohen, J.M Phys. Rev D [6] Lorentz, D Gen. Relat. Gravit. 13, 795 [7] Roy, S.R. and Singh, J.P Aust. J. Phys [8] Banerjee, A. and Sanyal,A.K Gen.Relati. Grav [9] Coley,A.A Gen.Relati.Grav. 3 [10] Nayak, B.K. and Sahoo, B.K Gen.Relati.Grav.8 51 [11] Bali, R. and Meena, B.L. 005 Proc. Nat. Acad. Sci. India, 75(A) IV 73 [1] Linde,A.B Rep.Prog.Phys. 4 5 [13] Kibble, T.W.B J. Phys. A.:Math. Gen [14] Zel dovich, Ya.B Mon.Not.Roy.Astron.Soc [15] Velenkin, A. 198 Phys. Rev D 4 08 [16] Letelier, P.S Phys. Rev D [17] Stachel, J Phys. Rev D [18] Letelier, P.S Phys. Rev D [19] Melvin, M.A Ann. New York Acad.Sci 6 53 [0] Banerjee, A.,Sanyal, A.K. and Chakravorty,S Pamana - J. Phys [1] Chakravorty, S Ind. J. Pure and Applied Phys [] Tikekar, R. and Patel, L.K. 199 Gen. Rel. Grav [3] Tikekar, R. and Patel, L.K Pramana - J. Phys [4] Patel, L.K. and Maharaj, S.D Pramana - J. Phys [5] Singh, G.P. and Singh, T Gen. Rel. Grav [6] Wang, X.X. 006 Chin. Phys. Lett [7] Bali,R. and Upadhaya, R.D. 003 Astrophys. and Space-Science 83 97

9 Electronic Journal of Theoretical Physics 5, No. 19 (008) [8] Bali, R. and Anjali 006 Astrophys. and Space-Science [9] Bali, R., Pareek, U.K. and Pradhan, A. 007 Chin.Phys. Lett [30] Lichnerowicz,A Relativistic Hydrodynamics and Magneto Hydrodynamics, Benjamin, NewYork, p.13 [31] Roy, Maartens 000 Pramana-J.Phys [3] MacCallum, M.A.H Comm. Math.Phys

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