Isomonodromy Method and Black Holes Quasinormal Modes: numerical results and extremal limit analysis

Abstract: In this thesis, we present and apply the isomonodromy method (or isomonodromic method) to the study of quasinormal modes (QNMs), more precisely, we consider the analysis of modes that are associated with linear perturbations in two distinct four-dimensional black holes one with angular momentum (Kerr) and one with charge (Reissner-Nordstr\"om). We show, using the method, that the quasinormal mode frequencies for both black holes can be analyzed with high numerical accuracy and, for certain regimes, even analytically. We also explore, by means of the equations involved, the regime in which both black holes become extremal. We reveal for this case that through the isomonodromic method, it is possible to calculate with good accuracy the values for the quasinormal frequencies associated with gravitational, scalar, and electromagnetic perturbations in the black hole with angular momentum, as well as spinorial and scalar perturbations in the charged black hole. Extending thus the analysis of the QNMs in the regime in which the methods used in the literature have generally convergence problems.