Stability study of a model for the Klein-Gordon equation in Kerr spacetime

TL;DR

This paper investigates the stability of scalar fields governed by the Klein-Gordon equation in Kerr spacetime, deriving a model that suggests potential instabilities near near-extremal black hole rotation.

Contribution

It introduces a model problem for the Klein-Gordon equation in Kerr spacetime that indicates possible instabilities for high rotational parameters, extending previous stability results.

Findings

Rigorous proof of stability for large scalar field masses.

Numerical evidence of potential instability near extremal rotation.

Model supports instability down to a/M ≈ 0.97.

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 of mass $\mu$ in the background of a rotating black hole. Rigorous results proof the stability of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, field for sufficiently large masses. Some, but not all, numerical investigations find instability of the reduced field for rotational parameters $a$ extremely close to 1. Among others, the paper derives a model problem for the equation which supports the instability of the field down to $a/M \approx 0.97$.