Quick and dirty methods for studying black-hole resonances

TL;DR

This paper introduces simple, efficient numerical methods based on phase-amplitude transformations for calculating black hole resonances, including quasinormal modes and Regge poles, with applications to Kerr black holes.

Contribution

It presents a novel, straightforward approach using phase-amplitude methods and Mueller's algorithm for computing black hole resonances, including new results for Kerr black holes.

Findings

Verified known Schwarzschild quasinormal modes and Regge poles

Produced new results for Kerr black hole resonances

Provided a method for estimating excitation coefficients of quasinormal modes

Abstract

We discuss simple integration methods for the calculation of rotating black hole scattering resonances both in the complex frequency plane (quasinormal modes) and the complex angular momentum plane (Regge poles). Our numerical schemes are based on variations of "phase-amplitude" methods. In particular, we discuss the Pruefer transformation, where the original (frequency domain) Teukolsky wave equation is replaced by a pair of first-order non-linear equations governing the introduced phase functions. Numerical integration of these equations, performed along the real coordinate axis, or along rotated contours in the complex coordinate plane, provides the required S-matrix element (the ratio of amplitudes of the outgoing and ingoing waves at infinity). Mueller's algorithm is then employed to conduct searches in the complex plane for the poles of this quantity (which are, by definition, the desired resonances). We have tested this method by verifying known results for Schwarzschild quasinormal modes and Regge poles, and provide new results for the Kerr black hole problem. We also describe a new method for estimating the "excitation coefficients" for quasinormal modes. The method is applied to scalar waves moving in the Kerr geometry, and the obtained results shed light on the long-lived quasinormal modes that exist for black holes rotating near the extreme Kerr limit.