Teukolsky master equation and Painlevé transcendents: Numerics and extremal limit

We conduct an analysis of the quasinormal 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 τ-function for the Painlevé V transcendent. We show excellent 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 Painlevé V and III transcendents.

Paper https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.084051