A simple test for the stability of a black hole by S-deformation

Abstract

We study a sufficient condition for proving the stability of a black hole when the master equation for linear perturbation takes the form of the Schrödinger equation. If the potential contains a small negative region, the S-deformation method is usually used to show the non-existence of an unstable mode. However, in some cases, it is hard to find an appropriate deformation function analytically because the only way found so far to find it is by trial-and-error. In this paper, we show that it is easy to find a regular deformation function by numerically solving the differential equation such that the deformed potential vanishes everywhere, when the spacetime is stable. Even if the spacetime is almost marginally stable, our method still works. We also discuss a simple toy model which can be solved analytically, and show that the condition for the non-existence of a bound state is the same as that for the existence of a regular solution for the differential equation in our method. From these results, we conjecture that our criteria is also a necessary condition.