# LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field in Lyra's Geometry

Authors: Raj Bali; Mahesh Kumar Yadav; Lokesh Kumar Gupta
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (ϕi) in the frame work 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=Sn, where R and S 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.


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## Body

## 1. Introduction

Bianchi type II space time successfully explains the initial stage of evolution of universe. Asseo and Sol [1] have given 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 [3], 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 (Stachel [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-Sachs 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 him to geometrize other physical fields. Motivated by the 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]. Sen [17] formulated a new scalar tensor theory of gravitation and constructed an analogue of Einstein field equations based on Lyra geometry. Halford [18] pointed out that the constant vector field (β) in Lyra geometry plays a 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 gauge in Lyra geometry whereas the cosmological constant (Λ) was introduced by Einstein in an ad hoc manner to find static solution of his field equations. Many authors, namely, Beesham [19], T. Singh and G. P. Singh [20], Chakraborty and Ghosh [21], Rahaman and Bera [22], Pradhan et al. [23, 24], 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 Sol [1] speculated a primordial magnetic field of cosmological origin. Vilenkin [3] has pointed out that 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 LRS 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 Symmetric (LRS) Bianchi type II metric as(1)ds2=ηabθaθb,
where
(2)η11=η22=η33=1,η44=(-1),θ1=Rdx,θ2=S(dy-xdz),θ3=Rdz,θ4=dt.
Thus the metric (1) leads to
(3)ds2=-dt2+R2dx2+S2(dy-xdz)2+R2dz2,
where R and S are functions of t alone.Energy momentum tensorTij for string dust in the presence of magnetic field is given by
(4)Tij=ρvivj-λxixj+Eij.
Einstein’s modified field equation in normal gauge for Lyra’s manifold obtained by Sen [17] is given by
(5)Rij-12Rgij+32ϕiϕj-34ϕkϕkgij=-Tij(ingeometrizedunits,where8πG=1,c=1),
where vi=(0,0,0,-1); vivi=-1; ϕi=(0,0,0,β(t)); v4=-1; v4=1·ρ is the matter density, λ the cloud string’s tension density, vi the fluid flow vector, β the gauge function, Eij the electromagnetic field tensor and xa the space like 4-vectors representing the string’s direction.The electromagnetic field tensorEij given by Lichnerowicz [29] is given as
(6)Eij=μ-[|h|2(vivj+12gij)-hihj],
with μ- being the magnetic permeability and hi the magnetic flux vector defined by
(7)hi=-g2μ-∈ijkℓFkℓvj,
where Fkℓ 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 h1≠0, h2=0=h3=h4, and F23 is the only nonvanishing component of Fij. This leads to F12=0=F13 by virtue of (7). We also find F14=0=F24=F34 due to the assumption of infinite electrical conductivity of the fluid (Maartens [30]). A cosmological model which contains a global magnetic field is necessarily anisotropic since the magnetic vector specifies a preferred spatial direction (Bronnikov et al. [31]). The Maxwell’s equation
(8)Fij;k+Fjk;i+Fki;j=0
leads to
(9)∂F23∂t=0(Fij=-FjisinceF23istheonlynonvanishingcomponentandFij=-Fji)
which leads to
(10)F23=constant=H(say).
For i=1, (7) leads to
(11)h1=Hμ-S.
Now the components of Eij corresponding to the line element (3) are as follows:
(12)E11=-H22μ-R2S2=-E22=-E33=E44.
Now the modified Einstein’s field equations (5) for the metric (3) lead to
(13)S24R4+R4S4RS+R44R+S44S+34β2=λ+KR2S2(14)-3S24R4+R42R2+2R44R+34β2=-KR2S2(15)-S24R4+2R4S4RS+R42R2-34β2=ρ+KR2S2,
where K=H2/2μ- and the direction of string is only along the x-axis so that x1x1=1, x2x2=0=x3x3.The energy conservation equationTi;jj=0 leads to
(16)ρ4+ρ(2R4R+S4S)-λR4R=0
and conservation of left hand side of (5) leads to
(17)(Rij-12Rgij);j+32(ϕiϕj);j-34(ϕkϕkgij);j=0
which again leads to
(18)32ϕi[∂ϕj∂xj+ϕℓΓℓjj]+32ϕj[∂ϕi∂xj-ϕℓΓijℓ]-34gijϕk[∂ϕk∂xj+ϕℓΓℓjj]-34gijϕk[∂ϕk∂xj-ϕℓΓijℓ]=0.
Equation (18) is automatically satisfied for i=1,2,3.Fori=j=4, (18) leads to
(19)32β[∂∂t(g44ϕ4)+ϕ4Γ444]+32g44ϕ4[∂ϕ4∂t-ϕ4Γ444]-34g44ϕ4[∂ϕ4∂t+ϕ4Γ444]-32g44g44ϕ4[∂ϕ4∂t-ϕ4Γ444]=0
which again leads to
(20)32ββ4+32β2(2R4R+S4S)=0
where
(21)ϕi=(0,0,0,β(t)).


## 3. Solution of Field Equations

For the complete determination of the model of the universe, we assume that the shear tensor (σ) is proportional to the expansion (θ) which leads to
(22)R=Sn.
From (20), we have
(23)β=αR2S,
with α being constant of integration.Using (22) and (23) in (14), we have
(24)2S44+(3n-2)S42S=34nS-4n+3-3α24nS-4n-1-KnS-2n-1.
Now we assume that
(25)S4=f(S).
Thus
(26)S44=ff′,
where
(27)f′=dfdS.
Therefore, (24) leads to
(28)df2dS+3n-2Sf2=34nS-4n+3-3α24nS-4n-1-KnS-2n-1
which again leads to
(29)f2=34n(2-n)S4-4n+3α24n(n+2)S-4n-Kn(n-2)S-2n.
Equation (29) leads to
(30)f=(dSdt)=Lτ4-Nτ2n+Mτ2n,
where L=3/(4n(2-n)),  M=3α2/(4n(n+2)),  N=K/(n(n-2)), S=τ, a new coordinate is used, and α≠0.By (22), we have
(31)R=Sn
which leads to
(32)R=τn,
where S=τ.Using (30) and (32), the metric (3) leads to
(33)ds2=-(dtdS)2dS2+R2dx2+S2(dy-xdz)2+R2dz2
which again leads to
(34)ds2=-dτ2Lτ4-4n-Nτ-2n+Mτ-4n+τ2n(dx2+dz2)+τ2(dy-xdz)2,
where the cosmic time t is defined as
(35)t=∫dτLτ4-4n-Nτ-2n+Mτ-4n.


## 4. Some Physical and Geometrical Features

Using (22), (23), (30), and (32) in (15), we have
(36)ρ=Aτ2-4n+Bτ-2n-2,
where  A=(n+1)/(2-n) andB=2nk/(2-n).Similarly from (15), the string tension density λ is given as
(37)λ=aτ2-4n+bτ-2n-2+dτ-4n-2ρp=ρ-λ=(A-a)τ(2-4n)+(B-b)τ(-2n-2)-dτ(-4n-2),
where
(38)a=(n+1)(3-2n)2n(2-n),b=K(3-n)(n-2),d=3α2(n+4)(n+2).
Equation (23) gives
(39)β=ατ2n+1.
The expansion (θ) is given as
(40)θ=2R4R+S4S
which leads to
(41)θ=(2n+1)τ2n+1Lτ4-Nτ2n+M.
Shear (σ) is given by
(42)σ=13(R4R-S4S)
which leads to
(43)σ=(n-1)3τ2n+1Lτ4-Nτ2n+M.
The deceleration parameter q is given by
(44)q=-R44/RR42/R2
which leads to
(45)q=2+1n[NS2n(n+2)+2MLS4-NS2n+M]=2+1n[Nτ2n(n+2)+2MLτ4-Nτ2n+M].


## 5. Model in Absence of Magnetic Field

To discuss the model in the absence of the magnetic field, we putK=0 in (29) and have
(46)f2=LS4-4n+MS-4n,
where
(47)L=34n(2-n),M=3α24n(n+2).
Equation (45) leads to
(48)dsdt=S4=Lτ4+Mτ2n,
where S=τ, and a new coordinate is used.By (22), we have
(49)R=τn.
Using (48) and (49) in metric (3), we get
(50)ds2=-(dtdS)2dS2+R2dx2+S2(dy-xdz)2+R2dz2
which again leads to
(51)ds2=-dτ2Lτ4-4n+Mτ-4n+τ2ndx2+τ2(dy-xdz)2+τ2ndz2.
In this case, the energy density (ρ), the string tension density (λ), gauge function (β), the expansion (θ), shear (σ), and deceleration parameter (q) are given by
(52)ρ=n+12-nτ-4n-2λ=(2n-3)(n+1)2n(n-2)τ-4n+2ρp=ρ-λβ=αR2S=ατ2n+1θ=2R4R+S4S=(2n+1)τ2n+1Lτ4+Mσ=13(R4R-S4S)=(n-1)3τ2n+1Lτ4+Mq=-R44/RR42/R2=n2+n(M-Lτ4M+Lτ4).


## 6. Discussion

Model (34) 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 where n>0. Since σ/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 where 2n+1>0; however, β increases continuously when 2n+1<0. The matter density ρ>0 when 0<n<2.Model (51) starts with a big bang at τ=0 when n=-1/2 and the expansion in the model decreases as time increases. The displacement vector (β) is initially large but decreases due to lapse of time. The model (51) has point-type singularity at τ=0, where n>0. Since σ/θ≠0, hence anisotropy is maintained throughout. However, if n=1, then the model isotropizes.Thus it is possible to construct globally regular Bianchi type II solutions with displacement vector (β) using geometric condition shear which is proportional to expansion.



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*Source: 892361-2013-10-30.xml*