Solutions Of Einstein Field Equations Research Articles

Cosmological models with time-varying gravitational and cosmological “constants”

The evolution and dynamics of a locally-rotationally-symmetric (LRS) Bianchi type-V space-time cosmological models are discussed with variable gravitational and cosmological “constants” in context of the particle creation. We present the exact solutions of Einstein field equations by using a power-law form of the average scale factor of the metric in the case of the particle creation and in the absence of particle creation. The solution describes the particle and entropy generation in the anisotropic cosmological models. The particle creation rate is uniquely determined by the variation of gravitational and cosmological “constants”. We observe that the variable gravitational constant does not necessarily imply particle creation. In a generic situation, models can be interpolated between different phases of the universe. The dynamical behaviors of the solutions and kinematical parameters of the model are discussed in detail.

Domain Walls Strange Quark Matter in Einstein-Rosen Space-Time with Cosmological Constant and Heat Flow

In this paper, we have studied the solutions of Einstein field equations for domain walls with cosmological constant and heat flow in Einstein-Rosen cylindrical symmetric space-time when strange quark matter and normal matter attached to the domain walls. Some physical and kinematical features of the obtained cosmological model are studied and discussed.

Geodesic deviation equation in Bianchi cosmologies

We present the Geodesic Deviation Equation (GDE) for the Friedmann Robertson Walker(FRW) universe and we compare it with the equation for Bianchi type I model. We justify consider this cosmological model due to the recent importance the Bianchi Models have as alternative models in cosmology. The main property of these models, solutions of Einstein Field Equations (EFE) is that they are homogeneous as the FRW model but they are not isotropic. We can see this because they have a non-null Weyl tensor, which is zero for FRW model. We study some consequences of this Weyl tensor in the GDE.

Electrostatic spherically symmetric configurations in gravitating nonlinear electrodynamics

We perform a study of the gravitating electrostatic spherically symmetric (G-ESS) solutions of Einstein field equations minimally coupled to generalized nonlinear Abelian gauge models in three space dimensions. These models are defined by Lagrangian densities which are general functions of the gauge field invariants, restricted by some physical conditions of admissibility. They include the class of nonlinear electrodynamics supporting electrostatic spherically symmetric (ESS) nontopological soliton solutions in absence of gravity. We establish that the qualitative structure of the G-ESS solutions of admissible models is fully characterized by the asymptotic and central-field behaviors of their ESS solutions in flat space (or, equivalently, by the behavior of the Lagrangian densities in vacuum and on the point of the boundary of their domain of definition, where the second gauge invariant vanishes). The structure of these G-ESS configurations for admissible models supporting divergent-energy ESS solutions in flat space is qualitatively the same as in the Reissner-Nordstr\"om case. In contrast, the G-ESS configurations of the models supporting finite-energy ESS solutions in flat space exhibit new qualitative features, which are discussed in terms of the Arnowitt-Deser-Misner mass, the charge, and the soliton energy. Most of the results concerning well-known models, such as the electrodynamics of Maxwell, Born-Infeld, and the Euler-Heisenberg effective Lagrangian of QED, minimally coupled to gravitation, are shown to be corollaries of general statements of this analysis.

Calculation of Ricci tensors by mathematica V 5.1

In solution of Einstein field equations it is necessary to contracting Riemann-Christofell tensor. The contraction of Riemann-Christofell tensor or simply the curvature tensor is called the Ricci tensor and denoted by QUOTE . The calculation of Ricci tensor in 3 and especially in 4-dimension is not very difficult but, it is very tedious; and need more time with accuracy. In this paper, the focus has been given to construct a software programming for determination of Ricci tensors with Mathematica V 5.1. This software program is not only able to calculate for any dimension, but also has ability to add any coefficients that may be needed to assume in our line element with any coordinate systems. Key words: Ricci tensor, Einstein field equation, mathematica software V 5.1.

A noncommutative model for a mini black hole

We analyze the static and spherically symmetric perfect fluid solutions of Einstein field equations inspired by the noncommutative geometry. In the framework of the noncommutative geometry, this solution is interpreted as a mini black hole which has the Schwarzschild geometry outside the event horizon, but whose standard central singularity is replaced by a self-gravitating droplet. The energy–momentum tensor of the droplet is of the anisotropic fluid obeying a nonlocal equation of state. The radius of the droplet is finite and the pressure, which gives rise to the hydrostatic equilibrium, is positive definite in the interior.

Exact Solution of Petrov Type {3,1} Metric via Time Dependent Quasi-Maxwell Equations

In this paper, we show that how to apply the time dependent quasi-Maxwell equations for exact solution of Einstein field equations in vacuum for Petrov type {3,1} metric.

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity (II)

It is shown that all torsion-free vacuum solutions of the model of de Sitter (dS) gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian (F 2 + T 2), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations.

Bianchi type-V cosmological models with perfect fluid and heat flow in Saez-Ballester theory

In this paper we discuss the variation law for Hubble’s parameter with average scale factor in a spatially homogeneous and anisotropic Bianchi type-V space-time model, which yields constant value of the deceleration parameter. We derive two laws of variation of the average scale factor with cosmic time, one is of power-law type and the other is of exponential form. Exact solutions of Einstein field equations with perfect fluid and heat conduction are obtained for Bianchi type-V space-time in these two types of cosmologies. In the cosmology with the power-law, the solutions correspond to a cosmological model which starts expanding from the singular state with positive deceleration parameter. In the case of exponential cosmology, we present an accelerating non-singular model of the Universe. We find that the constant value of deceleration parameter is reasonable for the present day Universe and gives an appropriate description of evolution of Universe. We have also discussed different types of physical and kinematical behaviour of the models in these two types of cosmologies.

Study of a new class of cylindrically symmetric space-time in general relativity

A new class of solutions of Einstein field equations is investigated for a cylindrically symmetric space-time when the source of gravitation is a perfect fluid. To get the deterministic solution a relation between metric coefficients A=(BC)n is assumed. Certain physical and geometric properties of the model are also discussed.

Some Exact Bianchi Type V Perfect Fluid Solutions with Heat Flow

The variation law for generalized mean Hubble’s parameter is discussed in a spatially homogeneous and anisotropic Bianchi type V space-time with perfect fluid along with heat-conduction. The variation law for Hubble’s parameter, that yields a constant value of deceleration parameter, generates two types of solutions for the average scale factor, one is of power-law type and other one of exponential form. Using these two forms of the average scale factor, exact solutions of Einstein field equations with a perfect fluid and heat conduction are presented for a Bianchi type V space-time, which represent expanding singular and non-singular cosmological models. We find that the constant value of deceleration parameter is reasonable for the present day universe. The physical and geometrical properties of the models are also discussed in detail.

On global properties of certain homogeneous exact solutions of Einstein field equations

The aim of this paper is an application of some general results on homogeneous Lorentz manifolds to exact solutions of Einstein field equations. We study the relation “<” of a chronological sequence of events. We examine a number of known solutions and show that in all but one, the relation < is maximal, i.e., for every a and b from V4, one has both a<b and b<a.

Two simple families of exact inhomogeneous stiff cosmologies

Two families of exact simple solutions of Einstein field equations for inhomogeneous stiff cosmologies are presented. The method to obtain the solutions is based on the introduction of auxiliary functions in order to cast the Einstein equations in such a way that can be explicitly integrated. Although the equations are mathematically equivalent to the equations obtained when the source of matter is a scalar field, it is worth to mention that the source that we consider is not a scalar field but a perfect fluid with a stiff equation of state. The obtained solutions are expressed in terms of simple functions of the used coordinates and two families of particular solutions are considered. The geometrical and kinematical properties of the solutions are then analyzed and the parameters are restricted in order to have a physically acceptable behavior. The two particular solutions are of the Petrov type I, the first one being regular everywhere whereas the second one presents a big-bang singularity. Now, for a particular value of one of the parameters, the second particular solution is a vacuum solution of the Bianchi I type that reduces to the Kasner solution.

Gravitational waves as exact solutions of Einstein field equations

Exact solutions of Einstein field equations invariant for a non-Abelian 2-dimensional Lie algebra of Killing fields are described. A sub-class of these gravitational fields have a wave-like character; it is shown that they have spin–1.

A new class of Inhomogeneous string cosmological models in general relativity

A new class of solutions of Einstein field equations has been investigated for inhomogeneous cylindrically symmetric space-time with string source. To get the deterministic solution, it has been assumed that the expansion ($\theta$) in the model is proportional to the eigen value $\sigma^{1}_{1}$ of the shear tensor $\sigma^{i}_{j}$. Certain physical and geometric properties of the models are also discussed.

ENERGY–MOMENTUM DISTRIBUTION: A CRUCIAL PROBLEM IN GENERAL RELATIVITY

This paper is aimed to elaborate the problem of energy–momentum in general relativity. In this connection, we use the prescriptions of Einstein, Landau–Lifshitz, Papapetrou and Möller to compute the energy–momentum densities for two exact solutions of Einstein field equations. The space–times under consideration are the nonnull Einstein–Maxwell solutions and the singularity-free cosmological model. The electromagnetic generalization of the Gödel solution and the Gödel metric become special cases of the nonnull Einstein–Maxwell solutions. It turns out that these prescriptions do not provide consistent results for any of these space–times. These inconsistent results verify the well-known proposal that the idea of localization does not follow the lines of pseudotensorial construction but instead follows from the energy–momentum tensor itself. These differences can also be understood with the help of the Hamiltonian approach.

Charged fluid distribution in higher dimensional spheroidal space-time

A general solution of Einstein field equations corresponding to a charged fluid distribution on the background of higher dimensional spheroidal space-time is obtained. The solution generates several known solutions for superdense star having spheroidal space-time geometry.

Space-time geometry and thermodynamic properties of a self-gravitating ball of fluid in phase transition

A numerical solution of Einstein field equations for a spherical symmetric and stationary system of identical and auto-gravitating particles in phase transition is presented. The fluid possess a perfect fluid energy momentum tensor, and the internal interactions of the system are represented by a van der Walls like equation of state able to describe a first order phase transition of the type gas-liquid. We find that the space-time curvature, the radial component of the metric, and the pressure and density show discontinuities in their radial derivatives in the phase coexistence region. This region is found to be a spherical surface concentric with the star and the system can be thought as a foliation of acronal, concentric and isobaric surfaces in which the coexistence of phases occurs in only one of these surfaces. This kind of system can be used to represent a star with a high energy density core and low energy density mantle in hydrodynamic equilibrium.

Exact relativistic static charged perfect fluid disks

Using the well-known ``displace, cut and reflect'' method used to generate disks from given solutions of Einstein field equations, we construct static charged disks made of perfect fluid based on the Reissner-Nordstr\"om solution in isotropic coordinates. We also derive a simple stability condition for charged and noncharged perfect fluid disks. As expected, we find that the presence of charge increases the regions of instability of the disks.

Exact relativistic static charged dust discs and non-axisymmetric structures

The well-known ‘displace, cut and reflect’ method used to generate discs from given solutions of Einstein field equations is applied to the superposition of two extreme Reissner–Nordström black holes to construct discs made of charged dust and also non-axisymmetric planar distributions of charged dust on the z = 0 plane. They are symmetric with respect to two or one coordinate axes, depending on whether the black holes have equal or unequal masses, respectively. For these non-axisymmetric distributions of matter we also study the effective potential for geodesic motion of neutral test particles.