1. Introduction and Motivations

The inhomogeneous cosmological models play a significant role in understanding some essential features of the universe such as the formation of galaxies during the early stages of evolution and process of homogenization. The early attempts at the construction of such models have been done by Tolman [1] and Bondi [2] who considered spherically symmetric models. Inhomogeneous plane-symmetric models were considered by Taub [3, 4] and later by Tomimura [5], Szekeres [6], Collins and Szafron [7, 8], and Szafron and Collins [9]. Senovilla [10] obtained a new class of exact solutions of Einstein's equations without big bang singularity, representing a cylindrically symmetric, inhomogeneous cosmological model filled with perfect fluid which is smooth and regular everywhere satisfying energy and causality conditions. Later, Ruiz and Senovilla [11] have examined a fairly large class of singularity-free models through a comprehensive study of general cylindrically symmetric metric with separable function of and as metric coefficients. Dadhich et al. [12] have established a link between the FRW model and the singularity-free family by deducing the latter through a natural and simple in-homogenization and anisotropization of the former. Recently, Patel et al. [13] have presented a general class of inhomogeneous cosmological models filled with nonthermalized perfect fluid assuming that the background space-time admits two space-like commuting Killing vectors and has separable metric coefficients. Singh et al. [14] obtained inhomogeneous cosmological models of perfect fluid distribution with electromagnetic field. Recently, Pradhan et al. [15–18] have investigated cylindrically-symmetric inhomogeneous cosmological models in various contexts.

The occurrence of magnetic field on galactic scale is a well-established fact today, and its importance for a variety of astrophysical phenomena is generally acknowledged as pointed out by Zeldovich et al. [19]. Also Harrison [20] suggests that magnetic field could have a cosmological origin. As natural consequences, we should include magnetic fields in the energy-momentum tensor of the early universe. The choice of anisotropic cosmological models in Einstein system of field equations leads to the cosmological models more general than Robertson-Walker model [21]. The presence of primordial magnetic field in the early stages of the evolution of the universe is discussed by many [22–31]. Strong magnetic field can be created due to adiabatic compression in clusters of galaxies. Large-scale magnetic field gives rise to anisotropies in the universe. The anisotropic pressure created by the magnetic fields dominates the evolution of the shear anisotropy and decays slowly as compared to the case when the pressure is held isotropic [32, 33]. Such fields can be generated at the end of an inflationary epoch [34–38]. Anisotropic magnetic field models have significant contribution in the evolution of galaxies and stellar objects. Bali and Ali [39] obtained a magnetized cylindrically symmetric universe with an electrically neutral perfect fluid as the source of matter. Pradhan et al. [40–44] have investigated magnetized cosmological models in various contexts.

In 1917 Einstein introduced the cosmological constant into his field equations of general relativity in order to obtain a static cosmological model since, as is well known, without the cosmological term his field equations admit only nonstatic solutions. After the discovery of the red-shift of galaxies and explanation thereof Einstein regretted for the introduction of the cosmological constant. Recently, there has been much interest in the cosmological term in context of quantum field theories, quantum gravity, super-gravity theories, Kaluza-Klein theories and the inflationary-universe scenario. Shortly after Einstein's general theory of relativity Weyl [45] suggested the first so-called unified field theory based on a generalization of Riemannian geometry. With its backdrop, it would seem more appropriate to call Weyl's theory a geometrized theory of gravitation and electromagnetism (just as the general theory was a geometrized theory of gravitation only), instead of a unified field theory. It is not clear as to what extent the two fields have been unified, even though they acquire (different) geometrical significance in the same geometry. The theory was never taken seriously inasmuchas it was based on the concept of nonintegrability of length transfer; and, as pointed out by Einstein, this implies that spectral frequencies of atoms depend on their past histories and therefore have no absolute significance. Nevertheless, Weyl's geometry provides an interesting example of nonRiemannian connections, and recently Folland [46] has given a global formulation of Weyl manifolds clarifying considerably many of Weyl's basic ideas thereby.

In 1951 Lyra [47] proposed a modification of Riemannian geometry by introducing a gauge function into the structureless manifold, as a result of which the cosmological constant arises naturally from the geometry. This bears a remarkable resemblance to Weyl's geometry. But in Lyra's geometry, unlike that of Weyl, the connection is metric preserving as in Remannian; in other words, length transfers are integrable. Lyra also introduced the notion of a gauge and in the “normal” gauge the curvature scalar in identical to that of Weyl. In consecutive investigations Sen [48], Sen and Dunn [49] proposed a new scalar-tensor theory of gravitation and constructed an analog of the Einstein field equations based on Lyra's geometry. It is, thus, possible [48] to construct a geometrized theory of gravitation and electromagnetism much along the lines of Weyl's “unified” field theory, however, without the inconvenience of nonintegrability length transfer.

Halford [50] has pointed out that the constant vector displacement field in Lyra's geometry plays the role of cosmological constant in the normal general relativistic treatment. It is shown by Halford [51] that the scalar-tensor treatment based on Lyra's geometry predicts the same effects within observational limits as the Einstein's theory. Several authors Sen and Vanstone [52], Bhamra [53], Karade and Borikar [54], Kalyanshetti and Waghmode [55], Reddy and Innaiah [56], Beesham [57], Reddy and Venkateswarlu [58], Soleng [59], have studied cosmological models based on Lyra's manifold with a constant displacement field vector. However, this restriction of the displacement field to be constant is merely one for convenience, and there is no a priori reason for it. Beesham [60] considered FRW models with time-dependent displacement field. He has shown that by assuming the energy density of the universe to be equal to its critical value, the models have the geometry. T. Singh and G. P. Singh [61–64], Singh and Desikan [65] have studied Bianchi-type I, III, Kantowaski-Sachs and a new class of cosmological models with time-dependent displacement field and have made a comparative study of Robertson-Walker models with constant deceleration parameter in Einstein's theory with cosmological term and in the cosmological theory based on Lyra's geometry. Soleng [59] has pointed out that the cosmologies based on Lyra's manifold with constant gauge vector will either include a creation field and be equal to Hoyle's creation field cosmology [66–68] or contain a special vacuum field, which together with the gauge vector term, may be considered as a cosmological term. In the latter case the solutions are equal to the general relativistic cosmologies with a cosmological term.

Recently, Pradhan et al. [69–73], Casana et al. [74], Rahaman et al. [75, 76], Bali and Chandnani [77], Kumar and Singh [78], Singh [79], Rao, Vinutha et al. [80] and Pradhan [81] have studied cosmological models based on Lyra's geometry in various contexts. Rahaman et al. [82, 83] have evaluated solutions for plane-symmetric thick domain wall in Lyra geometry by using the separable form for the metric coefficients. Rahaman [84–87] has also studied some topological defects within the framework of Lyra geometry. With these motivations, in this paper, we have obtained exact solutions of Einstein's modified field equations in cylindrically symmetric inhomogeneous space-time within the frame work of Lyra's geometry in the presence and absence of magnetic field for time varying displacement vector. This paper is organized as follows. In Section 1 the motivation for the present work is discussed. The metric and the field equations are presented in Section 2. In Section 3 the solutions of field equations are derived for time varying displacement field in presence of magnetic field. Section 4 contains the physical and geometric properties of the model in presence of magnetic field. The solutions in absence of magnetic field are given in Section 5. The physical and geometric properties of the model in absence of magnetic field are discussed in Section 6. Finally, in Section 7 discussion and concluding remarks are given.

2. The Metric and Field Equations

We consider the cylindrically symmetric metric in the form

where is the function of , alone and and are functions of and . The energy momentum tensor is taken as having the form

where and are, respectively, the energy density and pressure of the cosmic fluid, and is the fluid four-velocity vector satisfying the condition

In (2.2), is the electromagnetic field given by Lichnerowicz [88]

where is the magnetic permeability and the magnetic flux vector defined by

where the dual electromagnetic field tensor is defined by Synge [89]

Here is the electromagnetic field tensor and is the Levi-Civita tensor density.

The coordinates are considered to be comoving so that = = = and . If we consider that the current flows along the -axis, then is the only nonvanishing component of . The Maxwell's equations

require that is the function of -alone. We assume that the magnetic permeability is the functions of and both. Here the semicolon represents a covariant differentiation.

The field equations (in gravitational units , ), in normal gauge for Lyra's manifold, were obtained by Sen [48] as

where is the displacement field vector defined as

where other symbols have their usual meaning as in Riemannian geometry.

For the line-element (2.1), the field (2.8) with (2.2) and (2.9) leads to the following system of equations:

Here, and also in the following expressions, a dot and a dash indicate ordinary differentiation with, respect to and respectively.

The energy conservation equation leads to

Equation (2.16) leads to Equation (2.17) is identically satisfied for . For , (2.17) reduces to

3. Solution of Field Equations in Presence of Magnetic Field

Equations (2.10)–(2.14) are five independent equations in seven unknowns , , , , , and . For the complete determinacy of the system, we need two extra conditions which are narrated hereinafter. The research on exact solutions is based on some physically reasonable restrictions used to simplify the field equations.

To get determinate solution, we assume that the expansion in the model is proportional to the shear . This condition leads to

where is a constant. The motive behind assuming this condition is explained with reference to Thorne [90]; the observations of the velocity-red-shift relation for extragalactic sources suggest that Hubble expansion of the universe is isotropic today within percent [91, 92]. To put more precisely, red-shift studies place the limit

on the ratio of shear, , to Hubble constant, , in the neighbourhood of our Galaxy today. Collins et al. [93] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous expansion satisfies that the condition is constant.

From (2.15)–(2.17), we have

We also assume that

Using (3.1) and (3.5) in (2.14) and (3.3) leads to

Equation (3.6) leads to

where and is the constant of integration. From (3.7) and (3.9), we have

where

Equation (3.8) leads to

where is an integrating constant. Equation (3.10) leads to

where and , are integrating constants. Hence from (3.9) and (3.13), we have

Therefore, we obtain

where , being a constant of integration.

After using suitable transformation of the coordinates, the model (2.1) reduces to the form

where , , , and .

4. Some Physical and Geometric Properties of the Model in Presence of Magnetic Field

Equation (2.18) gives

which leads to

Equation (4.2) on integration gives

where is a constant of integration and

Using (3.15) and (4.3) in (2.10) and (2.13), the expressions for pressure and density for the model (3.16) are given by

where

From (3.4) the nonvanishing component of the electromagnetic field tensor is obtained as

From the above, equation it is observed that the electromagnetic field tensor increases with time.

The reality conditions (Ellis [94])

lead to

respectively.

The dominant energy conditions (Hawking and Ellis [95])

lead to

respectively. The conditions (4.10) and (4.12) impose a restriction on displacement vector .

The expressions for the expansion , Hubble parameter , shear scalar , deceleration parameter and proper volume for the model (3.16) are given by

From (4.14) and (4.15), we obtain

The rotation is identically zero.

The rate of expansion in the direction of , and is given by

Generally the model (3.16) represents an expanding, shearing, and nonrotating universe in which the flow vector is geodetic. The model (3.16) starts expanding at and goes on expanding indefinitely when . Since = constant, the model does not approach isotropy. As increases the proper volume also increases. The physical quantities and decrease as increases. However, if , the process of contraction starts at and at the expansion stops. The electromagnetic field tensor does not vanish when , and . It is observed from (4.16) that when and which implies an accelerating model of the universe. Recent observations of type Ia supernovae [96–100] reveal that the present universe is in accelerating phase and deceleration parameter lies somewhere in the range . It follows that our models of the universe are consistent with recent observations. Either when or , the deceleration parameter approaches the value as in the case of de-Sitter universe.

5. Field Equations and Their Solution in Absence of Magnetic Field

In absence of magnetic field, the field (2.8) with (2.2) and (2.9) for metric (2.1) reads as

Equations (5.2) and (5.3) lead to

Equations (3.5) and (5.6) lead to

Equations (3.9) and (5.7) lead to

which on integration gives

where and are constants of integration. Hence from (3.9) and (5.9), we have

In this case (3.8) also leads to the same as (3.12).

Therefore, in absence of magnetic field, we have

where is already defined in previous section.

After using suitable transformation of the coordinates, the metric (2.1) reduces to the form where , , , and .

6. Some Physical and Geometric Properties of the Model in Absence of Magnetic Field

With the use of (5.11), equation (2.18) leads to

which upon integration leads to

where is an integrating constant.

Using (5.11) and (6.2) in (5.1) and (5.4), the expressions for pressure and density for the model (5.12) are given by

The dominant energy conditions (Hawking and Ellis [95])

lead to

respectively.

The reality conditions (Ellis [94])

lead to

The conditions (6.5) and (6.9) impose a restriction on .

The expressions for the expansion , Hubble parameter , shear scalar , deceleration parameter and proper volume for the model (5.12) in absence of magnetic field are given by