A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, June 2016. / This dissertation will focus on spin-3/2 perturbations on N-dimensional Schwarzschild black holes, with the aim of calculating the numerical values for the quasi-normal modes (QNMs) and absorption probabilities associated with these perturbations. We begin by determining the spinor-vector eigenmodes of our particles on an (N-2)-dimensional spherical background. This allows us to separate out the angular part and radial part on our N-dimensional Schwarzschild metric. We then determine the equations of motion and e ective potential of our particles near the N-dimensional black hole. Using techniques such as the Wentzel-Kramers-Brillouin and Improved Asymptotic Iterative Method we determine our QNMs and absorption probabilities. We see that higher dimensional black holes emit QNMs with larger real and imaginary values, this would imply they emit higher energy particles but that these particles are highly dampened and therefore would be di cult to detect. The results of the QNMs make sense if we also consider the e ective potential surrounding our black holes with the potential function increasing with increasing number of dimensions.

In this work, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. Our method employs Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For both portions, we derive Regge-Wheeler type equations for decoupled, gauge-invariant quantities at the level of perturbed connection coefficients. A general framework for the analysis of Regge-Wheeler type equations is presented, identifying sufficient conditions for decay estimates. With the choice of an appropriate gauge in each of the two portions, such decay estimates on these decoupled quantities are used to establish decay of the linearized metric coefficients, completing the proof of linear stability. The initial value problem is formulated on Cauchy data sets, complementing the work of Dafermos-Holzegel-Rodnianski [6], where the linear stability of Schwarzschild is established for characteristic initial data sets.

Thesis (Ph. D.)--University of Alberta, 2009. / Title from pdf file main screen (viewed on Jan. 5, 2010). "A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Physics, University of Alberta." Includes bibliographical references.

This thesis investigates the dynamics of a set of 8-600 Schwarzschild masses, randomly distributed inside cells which tessellate a 3-sphere. Furthermore the contents of each cell are mirror images of its neighbor. This symmetry give rise to a locally rotationally symmetric (LRS) curve, along which the Einstein field equations governing dynamics can be exactly integrated. The result is an irregular model consisting of discrete matter content, but where the dynamics is easy to calculate. We see that these local inhomogeneities will cause behavior deviating from the spherical dust-filled FLRW model. For instance, there are cases where configurations exhibit acceleration along the LRS curve, even though the content consists solely of ordinary matter with a vacuum filled exterior and no cosmological constant. / Denna avhandling undersöker konfigurationer av 8-600 Schwarzschild-massor, som är slumpmässigt utplacerade inom celler som tessellerar en 3-sfär. Utöver det är även innehållet i varje cell en spegelbild av granncellen. Denna symmetri ger upphov till en lokalt rotationssymmetrisk (LRS) kurva där Einsteins fältekvationer som beskriver dynamiken längs med är exakt integrerbara. Resultatet är en oregelbunden modell som består av diskreta massor, men vars dynamik är enkel att beräkna. Vi ser att dessa lokala inhomogeniteter ger upphov till beteenden som avviker från den sfäriska partikel-fyllda FLRW-modellen. Till exempel uppstår konfigurationer som uppvisar acceleration längs med LRS-kurvan, trots att innehållet består endast av ordinära massor med vakuum utanför och ingen kosmologisk konstant.

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)