This thesis is concerned with the application of Lie's group theoretic method to the Einstein field equations in order to find new exact solutions. We analyse the nonlinear partial differential equation which arises in the study of non- static, non-conformally flat fluid plates of embedding class one. In order to find the group invariant solutions to the partial differential equation in a systematic and comprehensive manner we apply the method of optimal subgroups. We demonstrate that the model admits linear barotropic equations of state in several special cases. Secondly, we study a shear-free spherically symmetric cosmological model with heat flow. We review and extend a method of generating solutions developed by Deng. We use the method of Lie analysis as a systematic approach to generate new solutions to the master equation. Also, general classes of solution are found in which there is an explicit relationship between the gravitational potentials which is not present in earlier models. Using our systematic approach, we can recover known solutions. Thirdly, we study generalised shear-free spherically symmetric models with heat flow in higher dimensions. The method of Lie generates new solutions to the master equation. We obtain an implicit solution or we can reduce the governing equation to a Riccati equation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.

We present new exact solutions to the Einstein and Einstein-Maxwell field equations that model the interior of neutral, charged and radiating stars. Several new classes of solutions in static spherically symmetric interior spacetimes are found in the presence of charge. These correspond to isotropic matter with a specified electric field intensity. Our solutions are found by choosing different rational forms for one of the gravitational potentials and a particular form for the electric field. The models generated contain results found previously including Finch and Skea (1989) neutron stars, Durgapal and Bannerji (1983) dense stars, Tikekar (1990) superdense stars in the limit of vanishing charge. Then we study the general situation of a compact relativistic object with anisotropic pressures in the presence of the electromagnetic field. We assume the equation of state is linear so that the model may be applied to strange stars with quark matter and dark energy stars. Several new classes of exact solutions are found, and we show that the densities and masses are consistent with real stars. We regain as special cases the Lobo (2006) dark energy stars, the Sharma and Maharaj (2007) strange stars and the realistic isothermal universes of Saslaw et al (1996). In addition, we consider relativistic radiating stars undergoing gravitational collapse when the fluid particles are in geodesic motion. We transform the governing equation into Bernoulli, Riccati and confluent hypergeometric equations. These admit an infinite family of solutions in terms of simple elementary functions and special functions. Particular models contain the Minkowski spacetime and the Friedmann dust spacetime as limiting cases. Finally, we model the radiating star with shear, acceleration and expansion in the presence of anisotropic pressures. We obtain several classes of new solutions in terms of arbitrary functions in temporal and radial coordinates by rewriting the junction condition in the form of a Riccati equation. A brief physical analysis indicates that these models are physically reasonable. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2009.

In this thesis we study spherically symmetric spacetimes which are static with a perfect fluid source. The Einstein field equations, in a number of equivalent forms, are derived in detail. The physical properties of a relativistic star are briefly reviewed. We specify two particular choices for one of the gravitational potentials. The behaviour of the remaining gravitational potential is governed by a second order differential equation. This equation has solutions in terms of elementary functions for some cases. The differential equation, in other cases, may be expressed as Bessel, confluent hypergeometric and hypergeometric equations. In such instances the solution is given in terms of special functions. A number of solutions to the Einstein field equations are generated. We believe that these solutions may be used to model realistic stars. Many of the solutions found are new and have not been published previously. In some cases our solutions are generalisations of cases considered previously. For some choices of the gravitational potential our solutions are equivalent to well-known results documented in the literature; in these cases we explicitly relate our solutions to those published previously. We have utilised the computer package MATHEMATICA Version 2.0 (Wolfram 1991) to assist with calculations, and to produce figures to describe the gravitational field. In addition, we briefly investigate the approach of specifying an equation of state relating the energy density and the pressure. The solution of the Einstein field equations, for a linear equation of state, is reduced to integrating Abel's equation of the second kind. / Thesis (M.Sc.)-University of Natal, Durban, 1993.

Semi-Riemannian manifolds is a subject popular in physics, with applications particularly to modern gravitational theory and electrodynamics. Semi-Riemannian geometry is a branch of differential geometry, similar to Riemannian geometry. In fact, Riemannian geometry is a special case of semi-Riemannian geometry where the scalar product of nonzero vectors is only allowed to be positive. This essay approaches the subject from a mathematical perspective, proving some of the main theorems of semi-Riemannian geometry such as the existence and uniqueness of the covariant derivative of Levi-Civita connection, and some properties of the curvature tensor. Finally, this essay aims to deal with the physical applications of semi-Riemannian geometry. In it, two key theorems are proven - the equivalenceof the Einstein field equations, the foundation of modern gravitational physics, and the Schwarzschild solution to the Einstein field equations. Examples of applications of these theorems are presented.

In this dissertation we study radiating stars in strong gravitational elds. We generate new classes of exact solutions to the Einstein eld equations and the boundary condition applicable to radiating relativistic stars. The model of a radiating star in general relativity, matching to the Vaidya exterior spacetime, is reviewed. The boundary condition is converted to a Riccati equation and we consider both cases involving geodesic and non-geodesic particle trajectories. We present the metrics found previously. We rst solve the boundary condition for the geodesic case and nd the gravitational potentials which are expanding and shearing. This is a new result. Secondly the boundary condition is analysed for the non-geodesic case and we seek new gravitational potentials which are accelerating, expanding and shearing. We are able to identify only geodesic solutions for this second case; this appears to be a new class of models. The solutions found are presented in terms of elementary functions which are helpful in studying the physical properties. The new solutions found cannot be categorised in existing classes of known solutions; they are examples of a new generic class di erent from previous studies. The matter variables of the model are generated . / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.

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

Thesis (Ph.D.)--Baylor University, 2009. / Includes bibliographical references (p. 141-147).

DANTAS, Emanuel Wendell Damasceno. Gravitação Holográfica. 2014. 38 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-05-22T21:17:38Z No. of bitstreams: 1 2014_dis_ewddantas.pdf: 390097 bytes, checksum: b3647afae9886b14a926927fbe54c326 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-05-25T21:16:50Z (GMT) No. of bitstreams: 1 2014_dis_ewddantas.pdf: 390097 bytes, checksum: b3647afae9886b14a926927fbe54c326 (MD5) / Made available in DSpace on 2015-05-25T21:16:50Z (GMT). No. of bitstreams: 1 2014_dis_ewddantas.pdf: 390097 bytes, checksum: b3647afae9886b14a926927fbe54c326 (MD5) Previous issue date: 2014 / Neste trabalho iremos mostrar uma nova perspectiva para a gravidade, consequentemente para a dinâmica do espaço. Faremos uma exposição padrão do tratamento da ação gravitacional e as condições para que obtenhamos as equações de campo. Daí exploraremos uma característica notável da Lagrangeana gravitacional, que é o fato de ela dar origem à equações de segunda ordem mesmo que ela contenha termos de derivada segunda ordem. Em seguida mostraremos que, ao impor que a ação gravitacional seja invariante por uma certa transformação de difeomorfismo, nóos conseguimos obter as equações de campo gravitacional usando o termo de superfície, ao invés do termo de bulk como é padrão na literatura. Em seguida mostramos que isso só é possível para o caso da gravidade pura, e discutimos sua invalidade quando matéria é rigorosamente tratada.

M.Sc. / General Relativity, as defined by Einstein's equations, defines the geometry of the universe. In Numerical Relativity, Einstein's equations are solved with the aid of numerical methods and computers. This dissertation discusses the ADM formulation of Numerical Relativity via a Cauchy approach. (ADM refers to the initials of the discoverers of this method: Arnowitt, Deser and Misner.) When working within relativistic equations, a computer algebra code is very useful and such a code is described in this dissertation. In order to illustrate computational cost saving techniques, only spherically symmetric space-times are considered. Furthermore, we present and test a numerical code that implements the standard ADM approach in order to accurately evolve a single black hole space-time. Finally, we discuss the implementation of a maximal slicing gauge condition that refines the numerical code by giving it singularity avoidance properties.

The study of geodesics in space-time is essential to a comprehensive understanding of the physics of the field. Global properties, e.g. the singularity structure and completeness of space-time, can be related to the geodesic properties, thus it is through the solutions of the geodesic equation of motion that many of the global properties of space-time can be obtained in an easily interpretable form. However, it is usually very difficult to integrate the geodesic equations for the particle motion in the presence of a gravitational field (Introduction, p. 1)