On the instability of the fundamental mode of the Regge-Wheeler effective potential

TL;DR

This paper refines the analysis of the fundamental mode stability in the Regge-Wheeler potential, showing that certain physical details significantly influence instability, while others like the spiral period remain largely unaffected.

Contribution

It provides a more nuanced understanding of the stability of black hole quasinormal modes by considering the effects of potential features and metric perturbations.

Findings

The imaginary part of the quasinormal frequency determines stability.

Features at spatial infinity significantly impact instability.

The spiral period is largely insensitive to perturbation details.

Abstract

It was recently pointed out that the fundamental mode of the Regge-Wheeler effective potential is unstable against an insignificant Gaussian metric perturbation, which, in turn, might substantially challenge the black hole spectroscopy. This intriguing result has been interpreted by some authors as arising from essentially replacing the black hole's effective potential and its perturbation with two disjoint potential barriers. We argue that such an analysis may have oversimplified the real physical scenario. To be more precise, a metric perturbation planted farther away from the black hole horizon might not always be appropriately approximated by a disjoint minor barrier. Particularly, for the perturbed P\"oschl-Teller potential, joint and disjoint metric perturbations might lead to drastically different stability properties for the low-lying modes. Following this line of thought, this study conducts a refined analysis of the stability of the fundamental mode of the Regge-Wheeler effective potential by closely examining a few physically relevant ingredients. While our analysis qualitatively confirms the main findings of previous studies, as the stability of the fundamental mode is primarily determined by the imaginary part of the quasinormal frequency, we show that specific features of both the effective potential at spatial infinity and the metric perturbation can have a sizable impact on the instability. In contrast, the spiral period, governed by the real part of the quasinormal frequency, appears largely insensitive to the details of the black hole metric or its perturbations. The analytic estimates are in reasonable agreement with the numerical results.