This thesis examines the role of shear in inhomogeneous spherically symmetric spacetimes in the field of general relativity. The Einstein field equations are derived for a perfect fluid source in comoving coordinates. By assuming a barotropic equation of state, two classes of nonaccelerating solutions are obtained for the Einstein field equations. The first class has equation of state p = ⅓µ and the second class, with equation of state p = µ, generalises the models of Van den Bergh and Wils (1985). For a particular choice of a metric potential a new class of solutions is found which is expressible in terms of elliptic functions of the first and third kind in general. A class of nonexpanding cosmological models is briefly studied. The method of Lie symmetries of differential equations generates a self-similar variable which reduces the field and conservation equations to a system of ordinary differential equations. The behaviour of the gravitational field in this case is governed by a Riccati equation which is solved in general. Another class of solutions is obtained by making an ad hoc choice for one of the gravitational potentials. It is demonstrated that for a stiff fluid a particular case of the generalised Emden-Fowler equation arises. / Thesis (Ph.D.)-University of Natal, Durban, 1996.

In this thesis, we seek exact solutions for the interior of a radiating relativistic star undergoing gravitational collapse. The spherically symmetric interior spacetime, when matched with the exterior radiating Vaidya spacetime, at the boundary of the star, yields the governing equation describing the gravitational behaviour of the collapsing star. The investigation of the model hinges on the solution of the governing equation at the boundary. We first examine shear-free models which are conformally flat. The boundary condition is transformed to an Abel equation and several new solutions are generated. We then study collapse with shear in geodesic motion. Two classes of solutions are generated which are regular at the stellar centre. Our treatment extends the results of Naidu et al (2006) which had the undesirable feature of a singularity at the centre of the star. In an attempt to find more general models, we transform the fundamental equation to a Riccati equation. Two general classes of solution are found and are used to study the thermal evolution in the causal theory of thermodynamics. These solutions are shown to reduce to the Friedmann dust solution in the absence of heat flow. Furthermore, we obtain new categories of solutions for the case of gravitational collapse with expansion, shear and acceleration of the stellar fluid. This is achieved by transforming the boundary condition into a Riccati equation. In special cases the Bernoulli equation is regained. The solutions are given in terms of elementary functions and they permit the investigation of the physical features of radiative stellar collapse. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2007.

The study of exact solutions to the Einstein and Einstein-Maxwell field equations, by imposing a symmetry requirement on the manifold, has been the subject of much recent research. In this thesis we consider specifically conformal symmetries in static and nonstatic spherically symmetric spacetimes. We find conformally invariant solutions, for spherically symmetric vectors, to the Einstein-Maxwell field equations for static spacetimes. These solutions generalise results found previously and have the advantage of being regular in the interior of the sphere. The general solution to the conformal Killing vector equation for static spherically symmetric spacetimes is found. This solution is subject to integrability conditions that place restrictions on the metric functions. From the general solution we regain the special cases of Killing vectors, homothetic vectors and spherically symmetric vectors with a static conformal factor. Inheriting conformal vectors in static spacetimes are also identified. We find a new class of accelerating, expanding and shearing cosmological solutions in nonstatic spherically symmetric spacetimes. These solutions satisfy an equation of state which is a generalisation of the stiff equation of state. We also show that this solution admits a conformal Killing vector which is explicitly obtained. / Thesis (Ph.D.)-University of Natal, Durban, 1993.

We conduct a comprehensive investigative review of solution generating algorithms for the Einstein field equations governing the gravitational behaviour of an isolated neutral static spherical distribution of perfect fluid matter. Traditionally, the master field equation generated from the condition of pressure isotropy has been interpreted as a second order ordinary differential equation. However, since the pioneering work of Wyman (1949) it was observed that more success can be enjoyed by regarding the equation as a first order linear differential equation. There was a resurgence of the ideas of Wyman in 2000 and various researchers have been able to generate complete solutions to the field equations up to certain integrations. These have been accomplished by working in Schwarzschild (curvature) coordinates, isotropic coordinates, area coordinates and a coordinate system written in terms of the redshift parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a new algorithm. The algorithm is used to generate new classes of perfect fluid solutions as well as to regain familiar particular solutions reported in the literature. We find that our solution is well behaved according to elementary physical requirements. The pressure vanishes for a certain radius and this establishes the boundary of the distribution. Additionally the pressure and energy density are both positive inside the radius. The energy conditions are shown to be satisfied and it is particularly pleasing to have the causality criterion satisfied to ensure that the speed of light is not exceeded by the speed of sound. We also report some new solutions using the algorithms proposed by Lake (2006). / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

In this thesis we study spherically symmetric spacetimes related to the Einstein field equations. We consider only neutral matter and apply the Einstein field equations with isotropic pressures. Our object is to model relativistic stellar systems. We express the Einstein field equations and the condition of pressure isotropy in terms of Schwarzschild coordinates and isotropic coordinates. For Schwarzschild coordinates we consider the transformations due to Buchdahl (1959), Durgapal and Bannerji (1983), Fodor (2000) and Tewari and Pant (2010). The condition of pressure isotropy is integrated and new exact solutions of the field equations are obtained utilizing the transformations of Buchdahl (1959) and Tewari and Pant (2010). These exact solutions are given in terms of elementary functions. For isotropic coordinates we can express the condition of pressure isotropy as a Riccati equation or a linear equation. An algorithm is developed that produces a new solution if a particular solution is known. The transformations reduce to a nonlinear Bernoulli equation in most instances. There are fundamentally three new classes of solutions to the condition of pressure isotropy. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models. / Thesis (M.Sc.)-University of Natal, Durban, 1998.

The study of embeddings is applicable and signicant to higher dimensional theories of our universe, high-energy physics and classical general relativity. In this thesis we investigate local and global isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Theorems have been established that guarantee the existence of such embeddings. However, most known explicit results concern embedded spaces with relatively simple Ricci curvature. We consider the four-dimensional gravitational field of a global monopole, a simple non-vacuum space with a more complicated Ricci tensor, which is of theoretical interest in its own right, and occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain an exact solution for its embedding into Minkowski space. Our local embedding space can be used to construct global embedding spaces, including a globally at space and several types of cosmic strings. We present an analysis of the result and comment on its signicance in the context of induced matter theory and the Einstein-Gauss-Bonnet gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black hole. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate which embedded spaces admit bulks of a specific type. We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space with a particular metric form, and we discuss their five-dimensional solutions. Furthermore, we determine that the only spherically symmetric spacetime in retarded time coordinates that can be embedded into a particular Einstein bulk is the general Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to the general embedding problem. We also consider the conformal Killing geometry of a five-dimensional Einstein space that embeds a static spherically symmetric spacetime, and we show how the Killing geometry of the embedded space is inherited by its bulk. The study of embedding properties such as these enables a deeper mathematical understanding of higher dimensional cosmological models and is also of physical interest as conformal symmetries encode conservation laws. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2012.

In this thesis we study classes of static spherically symmetric spacetimes admitting a perfect fluid source, electromagnetic fields and anisotropic pressures. Our intention is to generate exact solutions that model the interior of dense, relativistic stars. We find a sufficient condition for the existence of series solutions to the condition of pressure isotropy for neutral isolated spheres. The existence of a series solution is demonstrated by the method of Frobenius. With the help of MATHEMATICA (Wolfram 1991) we recovered the Tolman VII model for a quadratic gravitational potential, but failed to obtain other known classes of solution. This establishes the weakness, in certain instances, of symbolic manipulation software to extract series solutions from differential equations. For a cubic potential, we obtained a new series solution to the Einstein field equations describing neutral stars. The gravitational and thermodynamic variables are non-singular and continuous. This model also satisfies the important barotropic equation of state p = p(p). Two new exact solutions to the Einstein-Maxwell system, that generalise previous results for uncharged stars, were also found. The first of these generalises the solution of Maharaj and Mkhwanazi (1996), and has well-behaved matter and curvature variables. The second solution reduces to the Durgapal and Bannerji (1983) model in the uncharged limit; this new result may only serve as a toy model for quark stars because of negative energy densities. In both examples we observe that the solutions may be expressed in terms of hypergeometric and elementary functions; this indicates the possibility of unifying isolated solutions under the hypergeometric equation. We also briefly study compact stars with spheroidal geometry, that may be charged or admit anisotropic pressure distributions. The adapted forms of the pressure isotropy condition can be written as a harmonic oscillator equation. Two simple examples are presented. / Thesis (M.Sc.)-University of Natal, Durban, 2002.

Errata inserted opposite p.177. / Bibliography: p. 173-177. / x, 177 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / The classical dynamical structure of cosomological models in which the matter content of the universe consists of a scalar field with arbitrary non-negative potential is analyzed in full. (abstract) / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1996?

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Richard Matzner. Vita. Includes bibliographical references. Available also from UMI company.