The Painlevé VI tau-function of Kerr-AdS5

Abstract

The aim of this thesis is to present the method of isomonodromic deformations to treat linear perturbations of matter fields propagating in a five dimensional Kerr-AdS black hole. Yet, the method applies to different space-times, as well as other physical systems. The Klein-Gordon equation leads to radial and angular second order ordinary differential equations with four regular singular points. The associated Fuchsian system can be deformed while preserving its monodromy data, where the isomonodromic equations reduce to the Painleve VI (PVI) equation, with a consistent definition of the PVI tau-function. By means of the tau-function we can reformulate the eigenvalue problem of the radial (angular) Heun equation into an initial value problem of the corresponding tau-function. An asymptotic expansion for the separation constant is computed in terms of the angular PVI tau-function for slowly rotating or near-equally rotating black hole, while the quasi-normal modes frequencies are found in the small radius limit. Scalar quasinormal-modes for the s-wave case and even orbital quantum number turn out to be stable for small black holes. Instead, modes with odd orbital quantum number do exhibit a regime of superradiance in the this limit. Furthermore, we consider vector perturbations in this background, where the separability of the Maxwell equations comes at the expense of the introduction of a new parameter mu, which can be associated to the apparent singularity of the isomonodromy method by a Mobius transformation. Finally, a numerical analysis is performed for QNMs in the ultraspinning limit.