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 isomon- odromic method) to the study of quasinormal modes, 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-Nordström). We show, using the method, that the QN 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 isomon- odromic method, we can 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 QN frequencies in the regime in which the methods used in the literature have generally convergence problems. Through the separation of variables, we show that the equations describing linear perturbations on both black holes can be rewritten in terms of second-order ordinary differential equations (ODEs), where for the cases in which both black holes are non- extremal and extremal, we have that such ODEs are the confluent and double-confluent Heun equations, respectively. In turn, we consider the matrix representation of the so- lutions of such ODEs and use the method of isomonodromic deformations, which is based on the existence of families of linear matrix systems with the same monodromy parameters that can be deformed isomonodromically. From the method, we derive con- ditions for the isomonodromic functions τV and τIII, which are strictly connected with isomonodromic deformations in the confluent and double-confluent Heun equations, respectively. By means of these conditions, we are able to perform the numerical analy- sis of the QN frequencies for both black holes, in the extremal or non-extremal regime. Subsequently, we show that it can be possible to reformulate, through the isomon- odromic method, the eigenvalue problem of the confluent and double-confluent Heun equations into an initial value problem for both τ-functions. Finally, for the case of the charged Reissner-Nordström black hole, following the same procedure applied to the Kerr black hole, we analyze the values of the QN frequencies for the extremal and non-extremal Reissner-Nordström black hole. For both cases, we present the results for the quasinormal frequencies associated with linear perturbations of charged scalar and spinorial fields.