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).

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)

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This thesis reviews the use of 2-dimensional conformal field theory applied to gravity, specifically calculating Bekenstein-Hawking entropy of black holes in (2+1) dimensions. A brief review of general relativity, Conformal Field Theory, energy extraction from black holes, and black hole thermodynamics will be given. The Cardy formula, which calculates the entropy of a black hole from the AdS/CFT duality, will be shown to calculate the correct Bekenstein-Hawking entropy of the static and rotating BTZ black holes. The first law of black hole thermodynamics of the static, rotating, and charged-rotating BTZ black holes will be verified. / vii, 119 leaves : ill. ; 29 cm