Connecting scattering, monodromy, and MST's renormalized angular momentum for the Teukolsky equation in Kerr spacetime

TL;DR

This paper introduces a novel method to compute the renormalized angular momentum parameter in the Teukolsky equation using monodromy eigenvalues, linking complex analysis with black hole perturbation theory.

Contribution

It establishes a direct relation between monodromy data and the MST parameter $ u$, providing a new numerical scheme for its calculation in Kerr spacetime.

Findings

The monodromy-based method accurately computes $ u$ across parameter space.

The approach enhances understanding of scattering amplitudes in Kerr perturbations.

Numerical stability of the method is confirmed in various regimes.

Abstract

The Teukolsky equation describes perturbations of Kerr spacetime and is central to the study of rotating black holes and gravitational waves. In the frequency domain, the Teukolsky equation separates into radial and angular ordinary differential equations. Mano, Suzuki, and Takasugi (MST) found semi-analytic solutions to the homogeneous radial Teukolsky equation in terms of series of analytic special functions. The MST expansions hinge on an auxiliary parameter known as the renormalized angular momentum $\nu$, which one must calculate to ensure the convergence of these series solutions. In this work, we present a method for calculating $\nu$ via monodromy eigenvalues, which capture the behavior of ordinary differential equations and their solutions in the complex domain near their singular points. We directly relate the monodromy data of the radial Teukolsky equation to the parameter $\nu$ and provide a numerical scheme for calculating $\nu$ based on monodromy. With this method we evaluate $\nu$ in different regions of parameter space and analyze the numerical stability of this approach. We also highlight how, through $\nu$, monodromy data are linked to scattering amplitudes for generic (linear) perturbations of Kerr spacetime.