An analysis of linear perturbations and quasi-normal modes of Kerr black holes

This work is a thesis about linear perturbation theory of rotating black holes, so-called Kerr black holes (Kerr BH's). As such we start with a review of several analytical methods that have been developed in the last few decades. We use these methods to investigate the analytical properties of the retarded Green Function (GF) of the wave equation in this space-time. The properties of the GF that we are concerned with in this work are the branch cuts (BC's) and the poles of its Fourier modes in the complex-frequency plane. Each one of them is related to specific features of the BH, for instance the poles in the GF are related to its characteristic modes of vibration (the {\it quasi-normal modes}, (QNM). Another interesting subject of study is the BH (in)stability. When discussing mode stability of BH's, QNM's and BC's play a major role, since they can lead to instabilities depending on their localization on the complex-frequency plane. The novelty of our work is that our search for QNM modes is performed using the semi-analytical method called MST method, which allow us to construct solutions to the homogeneous wave equation and the retarded GF in term of series of hypergeometric functions. This method is explained in our review. We also used this method to search for BC's and to study the superradiance phenomenon. These topics were investigated first in the sub-extremal rotating case for which we were able to reproduce results found in the literature. The MST method was also applied to the extremal rotating BH and its use allowed us to access QNM frequencies in the extremal Kerr case. Since it is known that there is a new BC in the extremal case, we studied the formation of this new feature as extremality is approached. We hope that our work will contain some of the rst few steps towards the understanding of wave propagation around extremal Kerr BH's.