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.

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This work investigates how black holes can be described in terms of different definitions of horizons. Global definitions in terms of event horizons and Killing horizons are contrasted with local definitions in terms of trapping horizons and dynamical horizons. The discussion is framed in the context of the laws of black hole thermodynamics.

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by Liu Yuk Tung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 144-150) and index. / by Liu Yuk Tung. / Contents --- p.i / List of Figures --- p.v / List of Tables --- p.vii / Abstract --- p.viii / Acknowledgements --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Gravitational Wave Astronomy --- p.1 / Chapter 1.2 --- Quasinormal Modes of Black Holes --- p.2 / Chapter 1.3 --- Objective and Outline of this Thesis --- p.4 / Chapter Chapter 2. --- Perturbations of Schwarzschild Black Holes --- p.7 / Chapter 2.1 --- Introduction --- p.7 / Chapter 2.2 --- Weak Fields in the Schwarzschild Background --- p.8 / Chapter 2.3 --- Gravitational Perturbation of Schwarzschild Black Holes --- p.10 / Chapter 2.4 --- Scattering of Waves in a Schwarzschild Background --- p.12 / Chapter 2.5 --- Quasinormal Modes --- p.14 / Chapter Chapter 3. --- Green's Function Analysis --- p.16 / Chapter 3.1 --- Introduction --- p.16 / Chapter 3.2 --- Formalism --- p.17 / Chapter 3.3 --- The Signal at Early Time --- p.19 / Chapter 3.4 --- Quasinormal Ringings --- p.20 / Chapter 3.4.1 --- QNM spectrum of Schwarzschild Black Holes --- p.21 / Chapter 3.4.2 --- QNM spectrum of Kerr Black Holes --- p.23 / Chapter 3.5 --- Late Time Behavior --- p.24 / Chapter 3.6 --- Completeness of Quasinormal Modes --- p.25 / Chapter Chapter 4. --- Analytic Solutions of Regge-Wheeler Equation --- p.28 / Chapter 4.1 --- Introduction --- p.28 / Chapter 4.2 --- "Analytic Solutions for f(w, r) and g(w, r)" --- p.29 / Chapter 4.3 --- "Numerical Calculation of Leaver's series for(w, r)" --- p.32 / Chapter Chapter 5. --- Born Series --- p.36 / Chapter 5.1 --- Introduction --- p.36 / Chapter 5.2 --- Potentials with Exponential Tails --- p.37 / Chapter 5.2.1 --- Born Series Solution --- p.37 / Chapter 5.2.2 --- Poles in complex w plane --- p.38 / Chapter 5.3 --- Born Series Solution of Regge-Wheeler Potential --- p.39 / Chapter Chapter 6. --- Complex Integration --- p.44 / Chapter 6.1 --- Introduction --- p.44 / Chapter 6.2 --- Stokes and Anti-Stokes line --- p.45 / Chapter 6.3 --- Integration in the Complex Plane --- p.47 / Chapter 6.4 --- Stokes Phenomenon --- p.49 / Chapter 6.5 --- Integration of Regge-Wheeler Equation --- p.52 / Chapter Chapter 7. --- Semi-Analytic Method --- p.59 / Chapter 7.1 --- Introduction --- p.59 / Chapter 7.2 --- Application to Schwarzschild Black Holes --- p.60 / Chapter 7.3 --- Prospect of Application to Relativistic Stars --- p.63 / Chapter Chapter 8. --- Logarithmic Perturbation Theory --- p.65 / Chapter 8.1 --- Introduction --- p.65 / Chapter 8.2 --- Review on the Logarithmic Perturbation Theory --- p.67 / Chapter 8.3 --- General Properties of the Frequency Shift --- p.69 / Chapter 8.3.1 --- Open Systems in General --- p.69 / Chapter 8.3.2 --- Schwarzschild black holes --- p.72 / Chapter Chapter 9. --- The Shell Model of Dirty Black Holes --- p.78 / Chapter 9.1 --- Introduction --- p.78 / Chapter 9.2 --- The Master Equation --- p.79 / Chapter 9.3 --- Evaluation of Perturbation Formulas --- p.81 / Chapter 9.3.1 --- First Order Perturbation --- p.81 / Chapter 9.3.2 --- Second Order Perturbation --- p.84 / Chapter 9.4 --- Exact Calculation of QNMs of the Shell Model --- p.87 / Chapter 9.5 --- Comparison of Perturbation Calculation with Exact Result --- p.89 / Chapter 9.5.1 --- Dependence on μ and convergence --- p.89 / Chapter 9.5.2 --- Dependence on shell position --- p.91 / Chapter Chapter 10. --- Perturbations of Kerr Black Holes --- p.96 / Chapter 10.1 --- Introduction --- p.96 / Chapter 10.2 --- Teukolsky Equations --- p.96 / Chapter 10.3 --- The Radial Teukolsky Equation --- p.98 / Chapter 10.4 --- Superradiant Scattering --- p.100 / Chapter Chapter 11. --- Quasinormal Modes of Kerr Black Holes --- p.102 / Chapter 11.1 --- Introduction --- p.102 / Chapter 11.2 --- Angular Teukolsky Equation --- p.103 / Chapter 11.3 --- Born series solution --- p.104 / Chapter 11.4 --- Complex Integration of Teukolsky Equation --- p.106 / Chapter 11.5 --- The Semi-Analytic Method --- p.107 / Chapter Chapter 12. --- Conclusion --- p.114 / Chapter 12.1 --- Summary of Our Work --- p.114 / Chapter 12.2 --- Outlook --- p.116 / Appendix A. The Expansion Coefficients Vk for Black-Hole Potentials --- p.118 / Chapter A.1 --- Expansion of Regge-Wheeler Potential --- p.118 / Chapter A.2 --- Expansion of Teukolsky Potential --- p.120 / "Appendix B. Asymptotic Expression for g(w,r)" --- p.125 / Chapter B.l --- Regge-Wheeler Equation --- p.125 / Chapter B.2 --- Radial Teukolsky Equation --- p.126 / Appendix C. Numerical Derivatives and Root Searching Algorithm --- p.127 / Chapter C.1 --- Numerical Derivatives --- p.127 / Chapter C.2 --- Root Searching Algorithm --- p.130 / Appendix D. Derivation of the Equations for the Shell Model --- p.132 / Chapter D.1 --- The Metric of the Shell Model --- p.132 / Chapter D.2 --- The Master Equation for Scalar Waves --- p.134 / Chapter D.3 --- The Dominant Energy Condition for the Shell Model --- p.136 / Appendix E. Leaver's Analytic Solution of Teukolsky Equation --- p.139 / Chapter E.l --- Angular Equation --- p.139 / Chapter E.2 --- Radial Equation --- p.140 / Appendix F. Teukolsky-Starobinsky Identities --- p.142 / Bibliography --- p.144 / Index --- p.151

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This thesis examines exact solutions to gauged and ungauged supergravity theories in space-time dimensions D⩾5 as well as various instabilities of such solutions. I begin by using two solution generating techniques for five dimensional minimal ungauged supergravity, the first of which exploits the existence of a Killing spinor to generate supersymmetric solutions, which are time-like fibrations over four dimensional hyper-Kähler base spaces. I use this technique to construct a supersymmetric solution with the Atiyah-Hitchin metric as the base space. This solution has three independent parameters and possesses mass, angular momentum, electric charge and magnetic charge. Via an analysis of a congruence of null geodesics, I determine that the solution contains a region with naked closed time-like curves. The centre of the space-time is a conically singular pseudo-horizon that repels geodesics, otherwise known as a repulson. The region exterior to the closed time-like curves is outwardly geodesically complete and possesses an asymptotic region free of pathologies provided the angular momentum is chosen appropriately. The second solution generating technique exploits a hidden G2 symmetry in five dimensional minimal supergravity. I use this hidden symmetry approach to construct the most general black string solution in five dimensions, which is endowed with mass, angular momentum, linear momentum, electric charge and magnetic charge. This general black string satisfies the first law of thermodynamics, with the Bekenstein-Hawking entropy being reproduced via a microstate counting in terms of free M-branes in the decoupling limit. Furthermore it reduces to all previously known black string solutions in its various limits. A phase diagram for extremal black strings is produced to draw conclusions about extremal black rings, in particular why supersymmetric black rings exhibit a lower bound on the electric charge. The same phase diagram further suggests the existence of a new class of supersymmetric black rings, which are completely disconnected from the previously known class. A particular limit of this general black string is the magnetically charged black string, whose thermodynamic phase behaviour and perturbative stability were previously studied but not very well understood. I construct magnetically charged topological solitons, which I then show play an important role in the phase structure of these black strings. Topological solitons in Einstein-Maxwell gravity, however, were previously believed to generically correspond to unstable "bubbles of nothing" which expand to destroy the space-time. I show that the addition of a topological magnetic charge changes the stability properties of these Kaluza-Klein bubbles and that there exist perturbatively stable, static, magnetically charged bubbles which are the local vacuum and the end-point of Hawking evaporation of magnetic black strings. In gauged supergravity theories, bubbles of nothing are stabilised by the positive energy theorem for asymptotically anti-de Sitter space-times. For orbifold anti-de Sitter space-times in odd dimensions, a local vacuum state of the theory is just such a bubble, known as the Eguchi-Hanson soliton. I study the phase behaviour of orbifold Schwarzschild-anti-de Sitter black holes, thermal orbifold anti-de Sitter space-times, and thermal Eguchi-Hanson solitons from a gravitational perspective; general agreement is found between this analysis and the previous analysis from the gauge theory perspective via the AdS/CFT correspondence. I show that the usual Hawking-Page phase structure is recovered and that the main effect of the soliton in the phase space is to widen the range of large black holes that are unstable to decay despite the positivity of their specific heat. Furthermore, using topological arguments I show that the soliton and orbifold AdS geometry correspond to a confinement phase in the boundary gauge theory while the black hole corresponds to a deconfinement phase. An important instability for rotating asymptotically anti-de Sitter black holes is the superradiant instability. Motivated by arguments that the physical end point of this instability should describe a clump of scalar field co-rotating with the black hole, I construct asymptotically anti-de Sitter black hole solutions with scalar hair. Perturbative results, i.e. low amplitude boson stars and small radius black holes with low amplitude scalar hair, are presented in odd dimensions relevant to gauged supergravity theories, namely D=5,7. These solutions are neither stationary nor axisymmetric, allowing them to evade the rigidity theorem; instead the space-time plus matter fields are invariant under only a single helical Killing vector. These hairy black holes are argued to be stable within their class of scalar field perturbations but are ultimately unstable to higher order perturbative modes.

Thesis (Ph. D.)--California Institute of Technology, 1971. UM #72-00,482. / Advisor names found in the Acknowledgments pages of the thesis. Title from home page. Viewed 02/11/2010. Includes bibliographical references.

Academisch proefschrift--Universiteit van Amsterdam, 2008. / Description based on print version record. Includes bibliographical references (p. [131]-144).