On the stability of the massive scalar field in Kerr space-time

TL;DR

This paper investigates the stability of a massive scalar field in Kerr space-time, providing new bounds for stability and reformulating the problem as a time-dependent wave equation with potential implications for future research.

Contribution

It offers an improved mass bound for stability of solutions and introduces new formulations of the reduced Klein-Gordon equation in Kerr space-time.

Findings

Derived an improved mass bound for scalar field stability.

Reformulated the Klein-Gordon equation as a time-dependent wave equation.

Showed that abstract properties alone do not guarantee solution stability.

Abstract

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass $\mu$. Among others, the paper provides an improved bound for $\mu$ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.