Kerr Scattering Coefficients via Isomonodromy

TL;DR

This paper derives an exact formula for Kerr black hole scalar wave scattering coefficients using isomonodromy and Painlevé V functions, connecting black hole physics with integrable systems and conformal field theory.

Contribution

It introduces a novel approach to compute Kerr scattering coefficients via isomonodromy flow and Painlevé V tau-functions, linking black hole scattering to integrable models and conformal blocks.

Findings

Exact scattering coefficient formula in terms of monodromy parameter σ

Calculation of σ using Painlevé V tau-function

Application of techniques to angular eigenvalue problem

Abstract

We study the scattering of a massless scalar field in a generic Kerr background. Using a particular gauge choice based on the current conservation of the radial equation, we give a generic formula for the scattering coefficient in terms of the composite monodromy parameter $\sigma$ between the inner and the outer horizons. Using the isomonodromy flow, we calculate $\sigma$ exactly in terms of the Painlev\'e V $\tau$-function. We also show that the eigenvalue problem for the angular equation (spheroidal harmonics) can be calculated using the same techniques. We use recent developments relating the Painlev\'e V $\tau$-function to Liouville irregular conformal blocks to claim that this scattering problem is solved in the combinatorial sense, with known expressions for the $\tau$-function near the critical points.