Thesis (Ph. D.)--University of Texas at Austin, 2002. / Supervisors: Matthew William Choptuik and Philip J. Morrison. Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 62-63). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 62-63). Also available on the Internet.

One of the most important predictions of General Relativity, Einstein’s theory of gravity, is the existence of gravitational radiation. The strongest source of such radiation is expected to come from the merging of black holes. Upgrades to large ground based interferometric detectors (LIGO, VIRGO, GEO 600) have increased their sensitivity to the point that the first direct observation of a gravitational wave is expected to occur within the next few years. The chance of detection is greatly improved by the use of simulated waveforms which can be used as templates for signal processing. Recent advances in numerical relativity have allowed for long stable evolution of black hole mergers and the generation of expected waveforms. openGR is a modular, open framework black hole evolution code developed at The University of Texas at Austin Center for Relativity. Based on the BSSN (strongly hyperbolic) formulation of Einstein’s equations and the moving puncture method, we are able to model the evolution of a binary black hole system through the merger and extract the gravitational radiation produced. Although we are generally interested in binary interactions, openGR is capable of handling any number of black holes. This work serves as an overview of the capabilities of openGR and a demonstration of the physics it can be used to explore. / text

Conformal holonomy theory is the holonomy theory of the tractor connection on a conformal manifold. In this thesis, we present the first application of conformal holonomy theory to theoretical physics and determine the conformal holonomy groups/algebras of physically relevant spaces. After recalling some necessary background on conformal structures, tractor bundles and conformal holonomy theory in chapter 1, we begin in chapter 2 by discussing the role of conformal holonomy in the gauge-theoretic MacDowell-Mansouri formulation of general relativity. We show that the gauge algebra of this formulation is uniquely determined by the conformal structure of spacetime itself, in both Lorentzian and Riemannian metric signatures, through the conformal holonomy algebra. We then show that one may construct a MacDowell-Mansouri action functional for scale-invariant gravity, and we discuss a geometric interpretation for the scalar field therein. In chapter 3 we study a class of spacetimes relevant to Maldacena's AdS5=CFT4 correspondence in quantum gravity. It is well known that a Lie group coincidence lies at the heart of this correspondence: the proper isometry group of the bulk precisely matches the conformal group of the boundary. It has previously been proposed that the AdS5=CFT4 correspondence be extended to so-called Poincar e-Einstein spacetimes, which need not be as symmetric as anti-de Sitter space. We show that the conformal holonomy groups of the boundary and bulk furnish such a Lie group coincidence for 5-dimensional Poincar e-Einstein spacetimes in general. We completely characterise this boundary-bulk conformal holonomy matching for the Riemannian theory and present partial results for the Lorentzian theory. In chapter 4 we use the tools developed in the preceding chapters to further the classiification of the conformal holonomy groups of conformally Einstein spaces. Specifically, we determine the conformal holonomy groups of generic neutral signature conformally Einstein 4-manifolds subject to a condition on the conformal holonomy representation. Lastly, in chapter 5, we investigate the conformal holonomy reduction of the Fefferman conformal structures of residual twistor CR manifolds. A sufficient condition for reducible conformal holonomy is that the (Fefferman conformal structure of a) residual twistor CR manifold admit a parallel tractor. We show that this occurs if and only if the residual twistor CR manifold admits a Sasakian structure.

An investigation of Hawking radiation and a method for calculating particle creation in Schwarzschild spacetime using a quantum Langevin approach is presented in this thesis. In particular we shall show that an oscillator confined to a free-fall trajectory in Schwarzschild spacetime radiates as a result of such motions, and this radiation can be interpreted as Hawking radiation. In chapter 1 we present a literature review of the underlying concept: the Unruh effect. We also present some introductory material pertinent to the calculations. Chapter 2 is concerned with the case of a thin collapsing shell to form a black hole in Schwarzschild anti-de Sitter spacetime. We determine the temperature of the black hole to be T[subscript H] = h(r[subscript h])/4π = κ/2π where h(r[subscript h]) is the factorization of the conformal factor, r is the radial coordinate with the location of the horizon situated atr = r[subscript h], and κ the surface gravity. We also calculate the stress tensor at early and late spacetimes which allows us to calculate the renormalized stress-tensor {T[subscript μν]} which satisfies the semi-classical Einstien field equations. In chapter 3 we examine the case of a harmonic oscillator in 2D Schwarzschild spacetime and we show that the choice of trajectory is responsible for making the oscillator radiate. In chapter 4 we derive a quantum Langevin equation for the oscillator in the Heisenberg picture. By solving this equation using the Wigner-Weiskopff approximation we show that, in the case of an oscillator confined to a free fall trajectory in Schwarzschild spacetime, the oscillator radiates with respect to the Boulware vacuum. In agreement with Hawking[1] we obtain a temperature of the black hole as T = 1/8πM[subscript B]. In chapter 5 we present our conclusions and recommendations for further work.

This thesis establishes a precise mathematical connection between the Einstein equations of general relativity and the incompressible Navier-Stokes equation of fluid dynamics. We carry out a holographic analysis which relates solutions to the Einstein equations to the behaviour of a dual fluid living in one fewer dimensions. Gravitational systems are found to exhibit Navier-Stokes behaviour when we study the dynamics of the region near an event horizon. Thus, we find non-linear deformations of Einstein solutions which, after taking a suitable near horizon limit and imposing our particular choice of boundary conditions, turn out to be precisely characterised by solutions to the incompressible Navier-Stokes equation. In other words, for any solution to the Navier-Stokes equation, the set-up we present provides a solution to the Einstein equations near a horizon. We consider the cases of fluids flowing on the plane and on the sphere. Fluid dynamics on the plane is analysed foremost in the context of a flat background geometry whilst the spherical analysis is undertaken for Schwarzschild black holes and the static patch of four-dimensional de Sitter space. / Physics

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