Reconsideration of the Regge-Wheeler equation

TL;DR

This paper reexamines the Regge-Wheeler equation using Painlevé coordinates, suggesting that Schwarzschild black holes may be unstable under certain perturbations due to positive imaginary frequencies.

Contribution

It introduces a new approach to the Regge-Wheeler equation analysis, revealing potential instability of Schwarzschild black holes with respect to rotating perturbations.

Findings

Regge-Wheeler equation may have positive imaginary frequencies

Schwarzschild black holes could be unstable under odd perturbations

New coordinate system provides fresh insights into black hole stability

Abstract

Reconsideration of the Regge-Wheeler equation is processed by using the Painlev\'{e} coordinate and "good" timelier to define the initial time. We find that: the Regge-Wheeler equation could has positive imaginary frequency. Because the Regge-Wheeler equation is the odd (angular) perturbation to the Schwarzschild black hole, the conclusion is that the Schwarzschild black hole is unstable with respect to the rotating perturbation.