Self-Similar Solutions of the Kantowski-Sachs Model with a Perfect Fluid in General Relativity

Exact self-similar solutions to Einstein’s field equations for the Kantowski-Sachs space-time are determined. The self-similarity property is applied to determine the functional form of the unknown functions that define the gravitational model and to reduce the order of the field equations. The consequences of matter, described by the energy-momentum tensor, are investigated in the case of a perfect fluid. Some physical features and kinematical properties of the obtained model are studied.


Introduction
Space-time symmetries are important in identifying features of space-time that exhibit some kind of symmetry. The most important symmetries are those that simplify Einstein's field equations and provide a space-time classification based on the corresponding Li-algebra configuration. These symmetries preserve certain physical properties of space-time, such as metric, geodesic, curvature, Ricci scalar, and energy-momentum tensor. In the context of the theory of General Relativity (GR), symmetries have been studied based on Riemannian geometry and on Lyra geometry and in the framework of the theory of teleparallel gravity based on Weitzenböck geometry. Various types of symmetries, such as isometry, homothetic, conformal, Ricci and matter collineations, etc, have been thorough-studies in the literature [21] [22] [23] [24].
Self-similar solutions to Einstein's field equations play an important role in describing asymptotic properties of more general solutions and are therefore of great interest in physics [25]. These solutions are relevant to astrophysics and critical phenomena in gravitational collapse, see the references [26] [27] [28] [29] [30].
Einstein's field equations of GR are a system of nonlinear partial differential equations in some independent variables, depending on the matter distribution, determined by the stress-energy tensor, and unknown scale factors. This complicated system cannot generally be integrated without making simplistic assumptions on the variables to get their exact solutions. The exact solutions of Einstein's field equations are known as Lorentzian metrics obtained by solving these equations using the definition of energy-momentum tensor. Usually, there are two complementary approaches to get these exact solutions. For the first approach, one chooses a specific energy-momentum tensor model and studies the exact solutions corresponding to Einstein's field equations while assuming some physically acceptable properties on the scale factors. In the second approach, one focuses on some geometrical properties that are admitted by a space-time given by symmetries so as to simplify Einstein's field equations, and then searches for the matter source that depicts these properties. For the present investigation, we focus on a second approach, by assuming that a given space-time admits homothetic symmetry.
In the present work, we will focus our attention on the study of the homothetic symmetry of a Kantowski-Sachs space-time and solve Einstein's field equations without making assumptions, either on variables or on physical properties, as is common in the literature. We will only assume that a Kantowski-Sachs space-time understudy admits a homothetic vector.

Let ( )
, M g be a space-time, that is, a smooth, 4-dimensional para-compact manifold M with smooth metric g of Lorentzian signature ( ) , , , A global vector field ζ on M is said to be homothetic if in any coordinate domain of M one has ; ; ; where £ denotes a Lie derivative and semi-colon denotes a covariant derivative with respect to metric connection. The vector field ζ is said to be homothetic if ψ is constant on M (proper homothetic vector field if 0 ψ ≠ on M). If 0 ψ = on M, the vector field ζ is said to be a Killing vector field on M.
The paper is organized as follows. In the next section, we will give a brief overview of the space-time understudy and obtain its physical and geometric properties and obtain the homothetic vector field that this space-time admits. Section 7 concluding remarks are given.

Version of Model and Homothetic Vector Field
Consider a Kantowski-Sachs space-time which is a general measure of a homogeneous, anisotropic space-time with a spatial section of the topology 2 S ℜ× .
The standard representation of this space-time was given by [31] ( ) x θ = and 3 x φ = (two equivalent longitudinal directions) and the scale factors Ω and R are functions of t only.
From the geometrical point of view, the line element (2.1) admits a four-parameter continuous group of isometries that acts on space-like hypersurface and has no three-parameter subgroup that would be simply transitive on the orbits (see the references [31] and [32]). The energy-momentum distribution of this space-time has been studied in [33] [34] [35].
In the following, we define the physical and geometric parameters for use in discussing the physical and geometric properties of the self-similar solution obtained from the metric (2.1).
The average scale factor τ of the Kantowski-Sachs model (2.1) is given by and V represents a volume scale factor is defined as follows are the directional Hubble's parameters which measure the rate of expansion in the directions of , r θ and φ respectively. A dot denotes a derivative with respect to cosmic time t. From Equations (2.2)-(2.5), we obtain The deceleration parameter q of a Kantowski-Sachs model is an important observational quantity, which is given as follows The anisotropic parameter of expansion δ is defined by

Einstein's Field Equations in the Case of a Perfect Fluid
In this section, we will solve Einstein's field equations by assuming that the space-time under study admits a homothetic vector field and considering the matter is described by a perfect fluid.
Einstein's field equations are given by From the equation of stress-energy conservation ; 0

Solutions of Einstein's Field Equations
In this section, we'll solve Einstein's field equations for the space-time under study without making any assumption. We'll just assume that the space-time  Equations (4.8) (or (4.9)) and (4.11) give the dynamical variables of the obtained self-similar solution (2.31). In the next section, we give its kinematic variables.

Kinematic Variables
In this section we give the kinematic quantities of the obtained solution (2.31) as follows: The average scale factor τ and the volume scale factor V are given, respectively, by ( )

Barotropic Equation of State
As indicated in the references [8] [9] and [10] that if the matter is described by a  From the above discussion we obtained new classes of perfect fluid solutions whose matter energy density ρ and pressure p do not satisfy the barotropic equation of state (6.19).

Discussion and Conclusion
This work is devoted to studying the symmetries in particular self-similar symmetry of a Kantowski-Sachs model in the framework of Riemannian geometry. We have focussed on this kind of symmetry since space-time admitting it is stable from the dynamical system point and therefore is important from the physical one. For a Kantowski-Sachs space-time, we have solved the homothetic equations and found the homothetic vector field that this space-time admits. Moreover, the solution of homothetic equations helped us to get scale factors. We have used these scale factors in Einstein's field equations and found the dynamical variables, the energy density ρ and the pressure p, which depend on the cosmic time t. Using the values obtained for these variables, we found that the equation of state, which was indicated by Cahill and Taub [8] and Bicknell and Henriksen [9], was not satisfied in the three cases, namely, dust, radiation and stiff fluid. Therefore, the obtained self-similar solution can be considered as an addition to the rare perfect fluid solutions which do not satisfy any barotropic equation of state. We discussed the kinematical quantities of the obtained solution (2.31), we found that the model is not accelerated and expanding with time because its volume element increases as the time increases, which gives essentially empty universe for large time. The Hubble parameter, the scalar expansion and the shear scalar assume infinitely large values whereas with the growth of cosmic time they decrease to null values as t → ∞ . The behavior of the fluid is time-dependent and can be physically reasonable. For the obtained solution the limit of the ratio as t → ∞ is lim 0 , that is, the anisotropy of the model is maintained throughout.