Complex scaling approach to quasinormal modes of Schwarzschild and Reissner--Nordstr\"om black holes

TL;DR

This paper introduces a complex scaling method (CSM) to compute black-hole quasinormal modes, transforming boundary conditions into an eigenvalue problem, and applies it to Schwarzschild and Reissner--Nordstr"om black holes.

Contribution

The paper develops and demonstrates a unified complex scaling approach for calculating quasinormal frequencies of black holes, including extremal Reissner--Nordstr"om cases.

Findings

CSM accurately computes Schwarzschild quasinormal modes.

The method extends to Reissner--Nordstr"om black holes, including extremal limits.

CSM offers a flexible spectral framework for black-hole perturbation analysis.

Abstract

We study black-hole quasinormal modes by applying the complex scaling method (CSM) to the perturbation equations of Schwarzschild and Reissner--Nordstr\"om black holes. The method converts the outgoing-wave boundary condition into a non-Hermitian eigenvalue problem, allowing quasinormal-mode frequencies to be computed within a common spectral framework. We first benchmark the method for the Schwarzschild Regge--Wheeler equation and then extend it to the Reissner--Nordstr\"om family, including the extremal limit. Our results show that CSM provides a unified and flexible approach to the computation of black-hole quasinormal frequencies.