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A (3+1)-evolutionary method in the framework of Regge Calculus, known as "Parallelisable Implicit Evolutionary Scheme", is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolutionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in "Parallelisable Implicit Evolutionary Scheme" was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geometry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the "Reverse Triangle Inequality" which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Calculus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found.

Brown and York's gravitational microcanonical density of states is extended to general spacetime dimension and shown to be dependent upon features of the 4 dimensional gravitational action for its interpretation. Black hole entropy is calculated from the density of states path integral in general spacetime dimension, and the interpretation is shown to be likewise dependent upon the dimension of spacetime. The entropy of de Sitter and Rindler horizons are calculated using the black hole density of states and the notion of local horizon entropy density is shown to be supported. The applicability of the microcanonical ensemble to black hole mechanics is discussed at a fundamental level focussing on the absence of angular velocity as an external parameter in the gravitational Hamiltonian. The rotational ensemble and a new ensemble - the angular momentum ensemble - are introduced following Jaynes' information theory approach to statistical mechanics and proposed as more compelling candidates to calculate black hole entropy as a function of state. A program to calculate the density of states path integral non-perturbatively using causal dynamical triangulations is initiated. Regge calculus expressions for extrinsic curvature are extended to the case of Lorentzian hypersurfaces and used to derive Regge calculus expressions for quasilocal energy-momentum. The Regge version of the black hole density of states action is derived and specialised to the 3d and 4d spacetime constructions of causal dynamical triangulations. Finally, the recent suggestion that entropy is observer dependent is shown to be incompatible with the Tolman law for the equilibrium temperature in a gravitational field.

Kwong, Kwok Kun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 66-70) and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Master of Science / General Relativity (GR) is founded on the revolutionary idea that space and time are merely parts of a greater, unified whole: spacetime. Furthermore, the force we know as gravity results from the bending and stretching of the geometry of spacetime by its energetic contents. GR is notorious for its mathematical complexity and subtlety, meaning that an intuitive understanding of a spacetime is difficult. One of the best approaches to studying the properties of a given spacetime is to consider its geodesic structure—that is, to consider the motion of unaccelerated, “free-falling” particles. This report presents the results of such a study into two important spacetimes — the Kerr solution for a rotating black hole, and the Robertson-Walker solution for a homogeneous universe.

The present thesis deals with electromagnetic effectscreated by the motion or inhomogeneity of a dielectricmedium.In the first paper the quantum R\"ontgen effect isdiscussed. Here a rotating Bose-Einstein condensate -- oranother kind of quantum fluid -- is placed in a chargedcapacitor. The medium's rotation creates a magnetic field.Quantum media can only rotate in form of vortices, which leadsto a magnetic field corresponding to the field of a magneticmonopole. In the remaining part of the thesis the geometricalrepresentation of electromagnetic fields in moving andinhomogeneous media is discussed. It is shown that aninhomogeneously moving dielectric, e.g., a vortex, defines aspace-time metric and light rays follow null-geodesics definedby this metric. This means that light propagation in a movingmedium is analogous to light propagation in a gravitationalfield. The possibility of creating laboratory models ofastronomical objects, e.g., black holes is discussed. Theapplicability of the newly developed media with extremely lowgroup velocity for the actual creation of such an experiment isconsidered. Furthermore, a model for the case of the slowlymoving medium is discussed. Here the light propagation isanalogous to the motion of a charged particle propagatingthrough a magnetic field. The velocity of the flow correspondsto the vector potential. Consequently, light propagation in avortex corresponds to the Aharonov-Bohm effect. Finally, acomplete geometrical description of light in an inhomogeneousdielectric at rest is presented. It is shown that lighttrajectories are geodesics of a three-dimensional metricdefined by the medium. Here even the propagation of the fieldsis discussed in the language of differential geometry and it isshown that the field vectors are parallel transported along therays. These considerations can be generalized to thefour-dimensional case where the field-strength tensor isparallel transported along the ray. This emphasizes thefar-reaching analogy between light in moving media and light ingravitational fields.

In this thesis we first apply the 1+3 covariant description of general relativity to analyze n-fluid Friedmann-Lemaitre (FL) cosmologies; that is, homogeneous and isotropic cosmologies whose matter-energy content consists of n non-interacting fluids. We are motivated to study FL models of this type as observations suggest the physical universe is closely described by a FL model with a matter content consisting of radiation, dust and a cosmological constant. Secondly, we use the 1+3 covariant description to analyse scalar, vector and tensor perturbations of FL cosmologies containing a perfect fluid and a cosmological constant. In particular, we provide a thorough discussion of the behaviour of perturbations in the physically interesting cases of a dust or radiation background.