Teukolsky master equation and Painlev\'e transcendents: numerics and extremal limit

TL;DR

This paper analyzes Kerr black hole quasi-normal modes using isomonodromic methods and Painlevé transcendents, revealing their behavior in extremal limits and connecting them to monodromy properties.

Contribution

It introduces a novel approach using the Painlevé V transcendent and isomonodromic techniques to study black hole quasi-normal modes, especially in extremal conditions.

Findings

Good agreement with existing literature for non-extremal cases

Mode dependence on black hole temperature in extremal limit

Extremal modes derived from Painlevé V and III transcendents

Abstract

We conduct an analysis of the quasi-normal modes for generic spin perturbations of the Kerr black hole using the isomonodromic method. The strategy consists of solving the Riemann-Hilbert map relating the accessory parameters of the differential equations involved to monodromy properties of the solutions, using the $\tau$-function for the Painlev\'e V transcendent. We show good accordance of the method with the literature for generic rotation parameter $a<M$. In the extremal limit, we determined the dependence of the modes with the black hole temperature and establish that the extremal values of the modes are obtainable from the Painlev\'e V and III transcendents.