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

Abstract

Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (𝜙_𝑖) 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=𝑆^𝑛, 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.

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 𝑑𝑠²=𝜂_𝑎𝑏𝜃^𝑎𝜃^𝑏, (1) where 𝜂₁₁=𝜂₂₂=𝜂₃₃=1,𝜂₄₄=(),𝜃−1₁=𝑅𝑑𝑥,𝜃²,𝜃=𝑆𝑑𝑦−𝑥𝑑𝑧₃=𝑅𝑑𝑧,𝜃⁴=𝑑𝑡. (2) Thus the metric (1) leads to 𝑑𝑠²=−𝑑𝑡²+𝑅²𝑑𝑥²+𝑆²𝑑𝑦−𝑥𝑑𝑧²+𝑅²𝑑𝑧², (3) where 𝑅 and 𝑆 are functions of 𝑡 alone.

Energy momentum tensor 𝑇^𝑗_𝑖 for string dust in the presence of magnetic field is given by 𝑇^𝑗_𝑖=𝜌𝑣_𝑖𝑣^𝑗−𝜆𝑥_𝑖𝑥^𝑗+𝐸^𝑗_𝑖. (4) Einstein’s modified field equation in normal gauge for Lyra’s manifold obtained by Sen [17] is given by 𝑅^𝑗_𝑖−12𝑅𝑔^𝑗_𝑖+32𝜙_𝑖𝜙^𝑗−34𝜙_𝑘𝜙^𝑘𝑔^𝑗_𝑖=−𝑇^𝑗_𝑖ingeometrizedunits,where8𝜋𝐺=1,𝑐=1, (5) where 𝑣_𝑖=(0,0,0,−1); 𝑣^𝑖𝑣_𝑖=−1; 𝜙_𝑖=(0,0,0,𝛽(𝑡)); 𝑣₄=−1; 𝑣⁴=1⋅𝜌 is 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.

The electromagnetic field tensor 𝐸^𝑗_𝑖 given by Lichnerowicz [29] is given as 𝐸^𝑗_𝑖=𝜇|ℎ|²𝑣_𝑖𝑣^𝑗+12𝑔^𝑗_𝑖−ℎ_𝑖ℎ^𝑗, (6) with 𝜇 being the magnetic permeability and ℎ_𝑖 the magnetic flux vector defined by ℎ_𝑖=√−𝑔2𝜇∈_{𝑖𝑗𝑘ℓ}𝐹^𝑘ℓ𝑣^𝑗, (7) where 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 ℎ₁≠0, ℎ₂=0=ℎ₃=ℎ₄, and 𝐹₂₃ is the only nonvanishing component of 𝐹_𝑖𝑗. 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 𝐹_{𝑖𝑗;𝑘}+𝐹_{𝑗𝑘;𝑖}+𝐹_{𝑘𝑖;𝑗}=0 (8) leads to 𝜕𝐹₂₃𝜕t=0since𝐹₂₃istheonlynonvanishingcomponentand𝐹_𝑖𝑗=−𝐹_𝑗𝑖 (9) which leads to 𝐹₂₃=constant=𝐻say. (10) For 𝑖=1, (7) leads to ℎ₁=𝐻.𝜇𝑆 (11) Now the components of 𝐸^𝑗_𝑖 corresponding to the line element (3) are as follows: 𝐸¹₁𝐻=−²2𝜇𝑅²𝑆²=−𝐸²₂=−𝐸³₃=𝐸⁴₄. (12) Now the modified Einstein’s field equations (5) for the metric (3) lead to 𝑆²4𝑅⁴+𝑅₄𝑆₄+𝑅𝑅𝑆𝑅⁴⁴+𝑆₄₄𝑆+34𝛽²𝐾=𝜆+𝑅²𝑆²−3𝑆²4𝑅⁴+𝑅²₄𝑅²+2𝑅₄₄𝑅+34𝛽²𝐾=−𝑅²𝑆²−𝑆²4𝑅⁴+2𝑅₄𝑆₄+𝑅𝑅𝑆𝑅²⁴²−34𝛽²𝐾=𝜌+𝑅²𝑆², (13)(14)(15) 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 equation 𝑇^𝑗_𝑖;𝑗=0 leads to 𝜌₄+𝜌2𝑅₄𝑅+𝑆₄𝑆𝑅−𝜆⁴𝑅=0 (16) and conservation of left hand side of (5) leads to 𝑅^𝑗_𝑖−12𝑅𝑔^𝑗_𝑖;𝑗+32𝜙_𝑖𝜙^𝑗_;𝑗−34𝜙_𝑘𝜙^𝑘𝑔^𝑗_𝑖;𝑗=0 (17) which again leads to 32𝜙_𝑖𝜕𝜙^𝑗𝜕𝑥^𝑗+𝜙^ℓΓ^𝑗_ℓ𝑗+32𝜙^𝑗𝜕𝜙_𝑖𝜕𝑥^𝑗−𝜙_ℓΓ^ℓ_𝑖𝑗−34𝑔^𝑗_𝑖𝜙^𝑘𝜕𝜙_𝑘𝜕𝑥^𝑗+𝜙^ℓΓ^𝑗_ℓ𝑗−34𝑔^𝑗_𝑖𝜙_𝑘𝜕𝜙^𝑘𝜕𝑥^𝑗−𝜙_ℓΓ^ℓ_𝑖𝑗=0. (18) Equation (18) is automatically satisfied for 𝑖=1,2,3.

For 𝑖=𝑗=4, (18) leads to 32𝛽𝜕𝑔𝜕𝑡⁴⁴𝜙₄+𝜙⁴Γ⁴₄₄+32𝑔⁴⁴𝜙₄𝜕𝜙₄𝜕𝑡−𝜙₄Γ⁴₄₄−34𝑔⁴₄𝜙₄𝜕𝜙₄𝜕𝑡+𝜙⁴Γ⁴₄₄−32𝑔⁴₄𝑔⁴⁴𝜙₄𝜕𝜙₄𝜕𝑡−𝜙⁴Γ⁴₄₄=0 (19) which again leads to 32𝛽𝛽₄+32𝛽²2𝑅₄𝑅+𝑆₄𝑆=0 (20) where 𝜙_𝑖=0,0,0,𝛽(𝑡). (21)

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 𝑅=𝑆^𝑛. (22) From (20), we have 𝛼𝛽=𝑅²𝑆, (23) with 𝛼 being constant of integration.

Using (22) and (23) in (14), we have 2𝑆₄₄+()𝑆3𝑛−2²⁴𝑆=3𝑆4𝑛^−4𝑛+3−3𝛼²𝑆4𝑛^−4𝑛−1−𝐾𝑛𝑆^−2𝑛−1. (24) Now we assume that 𝑆₄=𝑓(𝑆). (25) Thus 𝑆₄₄=𝑓𝑓^′, (26) where 𝑓^′=𝑑𝑓.𝑑𝑆 (27) Therefore, (24) leads to 𝑑𝑓²+𝑑𝑆3𝑛−2𝑆𝑓²=3𝑆4𝑛^−4𝑛+3−3𝛼²𝑆4𝑛^−4𝑛−1−𝐾𝑛𝑆^−2𝑛−1 (28) which again leads to 𝑓²=3()𝑆4𝑛2−𝑛^4−4𝑛+3𝛼²()𝑆4𝑛𝑛+2^−4𝑛−𝐾𝑛()𝑆𝑛−2^−2𝑛. (29) Equation (29) leads to 𝑓=𝑑𝑆=√𝑑𝑡𝐿𝜏⁴−𝑁𝜏^2𝑛+𝑀𝜏^2𝑛, (30) 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 𝑅=𝑆^𝑛 (31) which leads to 𝑅=𝜏^𝑛, (32) where 𝑆=𝜏.

Using (30) and (32), the metric (3) leads to 𝑑𝑠²=−𝑑𝑡𝑑𝑆²𝑑𝑆²+𝑅²𝑑𝑥²+𝑆²𝑑𝑦−𝑥𝑑𝑧²+𝑅²𝑑𝑧² (33) which again leads to 𝑑𝑠²=−𝑑𝜏²𝐿𝜏^4−4𝑛−𝑁𝜏^−2𝑛+𝑀𝜏^−4𝑛+𝜏^2𝑛𝑑𝑥²+𝑑𝑧²+𝜏²𝑑𝑦−𝑥𝑑𝑧², (34) where the cosmic time 𝑡 is defined as 𝑡=𝑑𝜏𝐿𝜏^4−4𝑛−𝑁𝜏^−2𝑛+𝑀𝜏^−4𝑛. (35)

4. Some Physical and Geometrical Features

Using (22), (23), (30), and (32) in (15), we have 𝜌=𝐴𝜏^2−4𝑛+𝐵𝜏^−2𝑛−2, (36) where  𝐴=(𝑛+1)/(2−𝑛) and   𝐵=2𝑛𝑘/(2−𝑛).

Similarly from (15), the string tension density 𝜆 is given as 𝜆=𝑎𝜏^2−4𝑛+𝑏𝜏^−2𝑛−2+𝑑𝜏^−4𝑛−2𝜌_𝑝=𝜌−𝜆=(𝐴−𝑎)𝜏^(2−4𝑛)+(𝐵−𝑏)𝜏^{(−2𝑛−2)}−𝑑𝜏^{(−4𝑛−2)}, (37) where ()𝑎=𝑛+1)(3−2𝑛()𝐾()2𝑛2−𝑛,𝑏=3−𝑛(),𝑛−2𝑑=3𝛼²(𝑛+4)().𝑛+2 (38) Equation (23) gives 𝛼𝛽=𝜏^2𝑛+1. (39) The expansion (𝜃) is given as 𝜃=2𝑅₄𝑅+𝑆₄𝑆 (40) which leads to ()𝜃=2𝑛+1𝜏^2𝑛+1√𝐿𝜏⁴−𝑁𝜏^2𝑛+𝑀. (41) Shear (𝜎) is given by 1𝜎=√3𝑅₄𝑅−𝑆₄𝑆 (42) which leads to ()𝜎=𝑛−1√3𝜏^2𝑛+1√𝐿𝜏⁴−𝑁𝜏^2𝑛+𝑀. (43) The deceleration parameter 𝑞 is given by 𝑅𝑞=−⁴⁴/𝑅𝑅²₄/𝑅² (44) which leads to 1𝑞=2+𝑛𝑁𝑆^2𝑛(𝑛+2)+2𝑀𝐿𝑆⁴−𝑁𝑆^2𝑛1+𝑀=2+𝑛𝑁𝜏^2𝑛(𝑛+2)+2𝑀𝐿𝜏⁴−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=0 in (29) and have 𝑓²=𝐿𝑆^4−4𝑛+𝑀𝑆^−4𝑛, (46) where 3𝐿=4𝑛(2−𝑛),𝑀=3𝛼²().4𝑛𝑛+2 (47) Equation (45) leads to 𝑑𝑠𝑑𝑡=𝑆₄=√𝐿𝜏⁴+𝑀𝜏^2𝑛, (48) where 𝑆=𝜏, and a new coordinate is used.

By (22), we have 𝑅=𝜏^𝑛. (49) Using (48) and (49) in metric (3), we get 𝑑𝑠²=−𝑑𝑡𝑑𝑆²𝑑𝑆²+𝑅²𝑑𝑥²+𝑆²𝑑𝑦−𝑥𝑑𝑧²+𝑅²𝑑𝑧² (50) which again leads to 𝑑𝑠²=−𝑑𝜏²𝐿𝜏^4−4𝑛+𝑀𝜏^−4𝑛+𝜏^2𝑛𝑑𝑥²+𝜏²𝑑𝑦−𝑥𝑑𝑧²+𝜏^2𝑛𝑑𝑧². (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) are given by 𝜌=𝑛+1𝜏2−𝑛^−4𝑛−2()𝜆=2𝑛−3)(𝑛+1()𝜏2𝑛𝑛−2^−4𝑛+2𝜌_𝑝𝛼=𝜌−𝜆𝛽=𝑅²𝑆=𝛼𝜏^2𝑛+1𝜃=2𝑅₄𝑅+𝑆₄𝑆=(2𝑛+1)𝜏^2𝑛+1√𝐿𝜏⁴1+𝑀𝜎=√3𝑅₄𝑅−𝑆₄𝑆=(𝑛−1)√3𝜏^2𝑛+1√𝐿𝜏⁴𝑅+𝑀𝑞=−⁴⁴/𝑅𝑅²₄/𝑅²=𝑛²+𝑛𝑀−𝐿𝜏⁴𝑀+𝐿𝜏⁴. (52)

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.