Research Article LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field in Lyra s Geometry

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1 Advances in Mathematical Physics Volume 201, Article ID 89261, 5 pages Research Article LR Bianchi Type II Massive tring Cosmological Models with Magnetic Field in Lyra s Geometry Raj Bali, 1 Mahesh Kumar Yadav, 2 and Lokesh Kumar Gupta 1 1 Department of Mathematics, University of Rajasthan, Jaipur 02004, India 2 Department of Mathematics, Dr. H.. Gour Central University, agar 47000, India Correspondence should be addressed to Raj Bali; balir5@yahoo.co.in Received 10 May 201; Accepted 17 eptember 201 Academic Editor: hri Ram Copyright 201 Raj Bali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (φ i )intheframework of Lyra s geometry are investigated. The magnetic field is in YZ-plane. To get the deterministic solution, we have assumed that the shear (σ) is proportional to the expansion (θ). This leads to R= n,r and are metric potentials and n is a constant. We find that the models start with a big bang at initial singularity and expansion decreases due to lapse of time. The anisotropy is maintained throughout but the model isotropizes when n=1. The physical and geometrical aspects of the model in the presence and absence of magnetic field are also discussed. 1. Introduction Bianchi type II space time successfully explains the initial stage of evolution of universe. Asseo and ol [1] havegiven the importance of Bianchi type II space time for the study of universe. The string theory is useful to describe an event at the early stage of evolution of universe in a lucid way. Cosmic strings play a significant role in the structure formation and evolution of universe. The presence of string in the early universe has been explained by Kibble [2], Vilenkin [], and Zel dovich [4] using grand unified theories. These strings have stress energy and are classified as massive and geometric strings. The pioneer work in the formation of energy momentum tensor for classical massive strings is due to Letelier [5] who explained that the massive strings are formed by geometric strings (tachel [6]) with particle attached along its extension. Letelier [5] first used this idea in obtaining some cosmological solutions for massive string for Bianchi type I and Kantowski-achs space-times. Many authors namely, Banerjee et al. [7], Tikekar and Patel [8, 9], Wang [10], and Bali et al. [11 14], have investigated string cosmological models in different contexts. Einstein introduced general theory of relativity to describe gravitation in terms of geometry and it helped himtogeometrizeotherphysicalfields.motivatedbythe successful attempt of Einstein, Weyl [15] made one of the best attempts to generalize Riemannian geometry to unify gravitation and electromagnetism. Unfortunately Weyl s theory was not accepted due to nonintegrability of length. Lyra [16] proposed a modification in Riemannian geometry by introducing gauge function into the structureless manifold. This modification removed the main obstacle of the Weyl theory [15]. en [17] formulated a new scalar tensor theory ofgravitationandconstructedananalogueofeinsteinfield equations based on Lyra geometry. Halford [18] pointed out that the constant vector field (β) inlyrageometryplaysa similar role of cosmological constant(λ) in general theory of relativity. The scalar tensor theory of gravitation in Lyra geometry predicts the same effects within the observational limits as in the Einstein theory. The main difference between the cosmological theories based on Lyra geometry and Riemannian geometry lies in the fact that the constant displacement vector (β) arises naturally from the concept of gaugeinlyrageometryasthecosmologicalconstant (Λ) was introduced by Einstein in an ad hoc manner to find static solution of his field equations. Many authors, namely, Beesham [19], T. ingh and G. P. ingh [20], Chakraborty and Ghosh [21], Rahaman and Bera [22], Pradhan et al. [2, 24],

2 2 Advances in Mathematical Physics Bali and Chandnani [25, 26], and Ram et al. [27], have studied cosmological models in the frame work of Lyra s geometry. The present day magnitude of magnetic field is very small as compared to estimated matter density. It might not have been negligible during early stage of evolution of universe. Asseo and ol [1] speculated a primordial magnetic field of cosmological origin. Vilenkin [] haspointedoutthat cosmic strings may act as gravitational lensing. Therefore, it is interesting to discuss whether it is possible to construct an analogue of cosmic string in the presence of magnetic field in the frame work of Lyra s geometry. Recently, Bali et al. [28] investigated Bianchi type I string dust magnetized cosmological model in the frame work of Lyra s geometry. In this paper, we have investigated LR Bianchi type II massive string cosmological models with magnetic field in Lyra s geometry. We find that it is possible to construct an analogue of cosmic string solution in presence of magnetic field in the frame work of Lyra geometry. The physical and geometrical aspects of the model together with behavior of the model in the presence and absence of magnetic field are also discussed. 2. The Metric and Field Equations We consider Locally Rotationally ymmetric (LR) Bianchi type II metric as ds 2 =η ab θ a θ b, (1) η 11 =η 22 =η =1, η 44 = ( 1), θ 1 =Rdx, θ 2 = (dy x dz), θ =Rdz, θ 4 =dt. Thus the metric (1)leadsto ds 2 = dt 2 +R 2 dx (dy x dz) 2 +R 2 dz 2, R and are functions of t alone. Energy momentum tensor T j i for string dust in the presence of magnetic field is given by (2) () T j i =ρv i V j λx i x j +E j i. (4) Einstein s modified field equation in normal gauge for Lyra s manifold obtained by en [17]is given by R j i 1 2 Rgj i + 2 φ iφ j 4 φ kφ k g j i = T j i (in geometrized units, 8πG = 1, c = 1), V i = (0, 0, 0, 1); V i V i = 1; φ i = (0, 0, 0, β(t)); V 4 = 1; V 4 =1 ρis the matter density, λ the cloud string s tension density, V i the fluid flow vector, β the gauge function, E j i the electromagnetic field tensor and x a the space like 4-vectors representing the string s direction. (5) The electromagnetic field tensor E j i given by Lichnerowicz [29]isgivenas E j i = μ[ h 2 (V i V j gj i ) h ih j ], (6) with μ being the magnetic permeability and h i the magnetic fluxvectordefinedby h i = g 2μ ijklf kl V j, (7) F kl is the electromagnetic field tensor and ijkl the Levi-Civita tensor density. We assume that the current is flowing along the x-axis, so magnetic field is in yz-plane. Thus h 1 =0, h 2 = 0 = h = h 4,andF 2 is the only nonvanishing component of F ij. This leads to F 12 =0=F 1 by virtue of (7). We also find F 14 = 0 = F 24 = F 4 due to the assumption of infinite electrical conductivity of the fluid (Maartens [0]). A cosmological model which contains a global magnetic field is necessarily anisotropic since the magnetic vector specifies a preferred spatial direction (Bronnikov et al. [1]). The Maxwell s equation leads to F 2 F ij;k +F jk;i +F ki;j =0 (8) =0 t (since F 2 is the only nonvanishing component and F ij = F ji ) For i=1,(7)leadsto (9) F 2 = constant =H(say). (10) h 1 = H μ. (11) Now the components of E j i corresponding to the line element () are as follows: H2 E 1 1 = 2μR 2 2 = E2 2 = E =E4 4. (12) Now the modified Einstein s field equations (5)forthemetric ()leadto 2 4R 4 + R 4 4 R + R β2 =λ+ K R 2 2 (1) 2 4R 4 + R2 4 R 2 + 2R 44 R + 4 β2 = K R 2 2 (14) 2 4R 4 + 2R R2 4 R R 2 4 β2 =ρ+ K R 2 2, (15) K=H 2 /2μ and the direction of string is only along the x-axis so that x 1 x 1 =1, x 2 x 2 =0=x x.

3 Advances in Mathematical Physics The energy conservation equation T j i;j =0leads to ρ 4 +ρ( 2R 4 ) λr 4 =0 (16) R and conservation of left hand side of (5)leadsto (R j i 1 2 Rgj i ) + ;j 2 (φ iφ j ) ;j 4 (φ kφ k g j i ) =0 (17) ;j 2 φ i [ φj x j +φl Γ j lj ]+ 2 φj [ φ i x j φ lγ l ij ] Now we assume that Thus Therefore, (24) leadsto 4 =f(). (25) 44 =ff, (26) f = df d. (27) 4 gj i φk [ φ k x j +φl Γ j lj ] 4 gj i φ k [ φk x j φ lγ l ij ]=0. Equation (18) is automatically satisfied for i = 1, 2,. For i=j=4,(18)leadsto 2 β[ t (g44 φ 4 )+φ 4 Γ 4 44 ] + 2 g44 φ 4 [ φ 4 t φ 4Γ 4 44 ] 4 g4 4 φ 4 [ φ 4 t +φ4 Γ 4 44 ] 2 g4 4 g44 φ 4 [ φ 4 t φ4 Γ 4 44 ]=0 (18) (19) 2 ββ β2 ( 2R 4 )=0 (20). olution of Field Equations φ i = (0, 0, 0, β (t)). (21) For the complete determination of the model of the universe, we assume that the shear tensor (σ) isproportionaltothe expansion (θ) R= n. (22) From (20), we have β= α R 2, (2) with α beingconstantofintegration. Using (22) and(2)in(14), we have (n 2) 2 4 = 4n 4n+ α2 4n 4n 1 K n 2n 1. (24) df 2 d + n 2 f 2 = f 2 = 4n 4n+ α2 4n 4n 1 K n 2n 1 4n (2 n) 4 4n + Equation (29)leadsto α 2 K 4n (n+2) 4n n (n 2) 2n. (28) (29) f=( d dt )= Lτ 4 Nτ 2n +M τ 2n, (0) L = /(4n(2 n)), M=α 2 /(4n(n+2)), N = K/(n(n 2)), =τ,anewcoordinateisused,andα=0. By (22), we have R= n (1) R=τ n, (2) =τ. Using (0) and(2), the metric () leads to ds 2 = ds 2 = ( dt d ) 2 d 2 +R 2 dx (dy xdz) 2 +R 2 dz 2 () dτ 2 Lτ 4 4n Nτ 2n +Mτ 4n +τ 2n (dx 2 +dz 2 )+τ 2 (dy xdz) 2, the cosmic time t is defined as t= (4) dτ Lτ 4 4n Nτ 2n. (5) +Mτ 4n

4 4 Advances in Mathematical Physics 4. ome Physical and Geometrical Features Using (22), (2), (0), and (2)in(15), we have ρ=aτ 2 4n +Bτ 2n 2, (6) A = (n + 1)/(2 n) and B = 2nk/(2 n). imilarly from (15), the string tension density λ is given as λ=aτ 2 4n +bτ 2n 2 +dτ 4n 2 ρ p =ρ λ= (A a) τ (2 4n) Equation (2)gives + (B b) τ ( 2n 2) dτ ( 4n 2), a= (n+1)( 2n) K ( n), b = 2n (2 n) (n 2), d= α2 (n+4). (n+2) β= The expansion (θ)is given as θ= hear (σ)isgiven by σ = (7) (8) α. τ2n+1 (9) θ= 2R 4 (40) (2n + 1) τ 2n+1 Lτ 4 Nτ 2n +M. (41) σ= 1 (R 4 R 4 ) (42) (n 1) τ 2n+1 Lτ 4 Nτ 2n +M. (4) The deceleration parameter q is given by q= R 44/R R 2 4 /R2 (44) q=2+ 1 n [N2n (n+2) +2M L 4 N 2n +M ] =2+ 1 n [Nτ2n (n+2) +2M Lτ 4 Nτ 2n +M ]. (45) 5. Model in Absence of Magnetic Field To discuss the model in the absence of the magnetic field, we put K=0in (29)andhave f 2 =L 4 4n +M 4n, (46) L= 4n (2 n), M = α 2 4n (n+2). (47) Equation (45)leadsto ds dt = 4 = Lτ 4 +M τ 2n, (48) =τ,andanewcoordinateisused. By (22), we have Using (48)and(49)inmetric(), we get R=τ n. (49) ds 2 = ( dt d ) 2 d 2 +R 2 dx (dy xdz) 2 +R 2 dz 2 (50) ds 2 dτ 2 = Lτ 4 4n +Mτ 4n +τ2n dx 2 +τ 2 (dy xdz) 2 +τ 2n dz 2. (51) In this case, the energy density (ρ), the string tension density (λ), gauge function (β), the expansion (θ), shear (σ), and deceleration parameter (q)aregivenby ρ= n+1 2 n τ 4n 2 (2n )(n+1) λ= τ 4n+2 2n (n 2) ρ p =ρ λ β= α R 2 = α τ 2n+1 θ= 2R 4 (2n + 1) = Lτ 4 τ 2n+1 +M σ= 1 (R 4 R 4 ) = (n 1) Lτ 4 +M τ 2n+1 q= R 44/R R4 2/R2 =n 2 +n( M Lτ4 M+Lτ 4 ). (52)

5 Advances in Mathematical Physics 5 6. Discussion Model (4) in the presence of magnetic field starts with a big bang at τ = 0 and the expansion in the model decreases as τ increases. The spatial volume increases as τ increases. Thus inflationary scenario exists in the model. The model has point-type singularity at τ=0 n>0.inceσ/θ =0, hence anisotropy is maintained throughout. However, if n= 1, then the model isotropizes. The displacement vector β is initially large but decreases due to lapse of time 2n+1 > 0; however,β increases continuously when 2n + 1 < 0. The matter density ρ>0when 0<n<2. Model (51)startswithabigbangatτ=0whenn = 1/2 and the expansion in the model decreases as time increases. The displacement vector (β)isinitiallylargebutdecreasesdue to lapse of time. The model (51) has point-type singularity at τ = 0,n > 0.inceσ/θ =0, hence anisotropy is maintained throughout. However, if n=1, then the model isotropizes. 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