Aspects of stability of the toroidal AdS Schwarzschild black hole

Abstract

In this thesis various aspects about the dynamical stability of the toroidally symmetric Schwarzschild AdS black hole are discussed and proven. The first chapter of the thesis is a literature review. This covers the key relevant results within the area and provides context for the results of the later chapters. The second chapter concerns the Klein-Gordon equation with Dirichlet, Neumann and Robin boundary conditions on the exterior of the toroidally symmetric Schwarzschild AdS black hole. Through the vector field method, energy estimates, and degenerating Morawetz estimates are proven. From these it is seen that the energy of the solutions on these spacetimes are bounded and decay polynomially in time. Furthermore, it is shown that there exist null geodesics on this spacetime remain exterior to the event horizon boundary for arbitrary coordinate time. Through a Gaussian beam argument, it follows that the degeneration in the Morawetz estimates is necessary. The third chapter proves the non-linear stability of the toroidally symmetric Schwarzschild AdS black hole as a solution to the AdS-Einstein--Klein-Gordon system within the class of square toroidal symmetries where the field satisfies Dirichlet or Neumann boundary conditions. This is done through establishing wellposedness of the system in a region near null infinity. Then for initial data `sufficiently small' it is shown through bootstrap arguments that the energy remains bounded by the initial data on the regular region exterior to the black hole. This is then used to establish the orbital stability of the spacetime. Then through the vector field method, exponential decay of the field on the regular region exterior to the black hole is established. From this the asymptotic stability follows. Finally, a vacuum stability result is established in the toroidal symmetry class where the periods of the torus are allowed to vary.