Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes

TL;DR

This paper employs confluent Heun functions to analytically compute quasinormal modes of nonrotating black holes, introducing a new epsilon-method for analyzing branch cuts and confirming the non-special nature of mode n=8.

Contribution

It presents an exact analytical approach using Heun functions to find quasinormal modes and introduces the epsilon-method for studying branch cuts in black hole perturbation theory.

Findings

Successfully computed the n=8 mode with high precision.

Validated frequencies against previous literature.

Demonstrated the epsilon-method's effectiveness in analyzing mode stability.

Abstract

Although finding numerically the quasinormal modes of a nonrotating black hole is a well-studied question, the physics of the problem is often hidden behind complicated numerical procedures aimed at avoiding the direct solution of the spectral system in this case. In this article, we use the exact analytical solutions of the Regge-Wheeler equation and the Teukolsky radial equation, written in terms of confluent Heun functions. In both cases, we obtain the quasinormal modes numerically from spectral condition written in terms of the Heun functions. The frequencies are compared with ones already published by Andersson and other authors. A new method of studying the branch cuts in the solutions is presented -- the epsilon-method. In particular, we prove that the mode $n=8$ is not algebraically special and find its value with more than 6 firm figures of precision for the first time. The stability of that mode is explored using the $\epsilon$ method, and the results show that this new method provides a natural way of studying the behavior of the modes around the branch cut points.