Robustness of the $S$-deformation method for black hole stability analysis

TL;DR

This paper provides a general proof for the existence of a regular S-deformation in stable black hole spacetimes, clarifying its construction and boundary conditions, thus enhancing the method's reliability for stability analysis.

Contribution

It establishes the existence of a regular S-deformation under stability assumptions and characterizes its boundary conditions, advancing the theoretical foundation of the method.

Findings

Existence of a regular S-deformation is proven under stability assumptions.

The boundary condition for the differential equation has a finite one-parameter degree of freedom.

No fine-tuning is needed to find the S-deformation numerically.

Abstract

The $S$-deformation method is a useful way to show the linear mode stability of a black hole when the perturbed field equation takes the form of the Schr\"odinger equation. While previous works where many explicit examples are studied suggest that this method works well, general discussion is not given yet explicitly. In this paper, we show the existence of a regular $S$-deformation when a black hole spacetime is stable under some reasonable assumptions. This $S$-deformation is constructed from a solution of a differential equation. We also show that the boundary condition for the differential equation which corresponds to a regular $S$-deformation has a one-parameter degree of freedom with a finite range. This is the reason why any fine-tune technique is not needed to find $S$-deformation numerically.