The superradiant instability regime of the spinning Kerr black hole

TL;DR

This paper analytically derives an upper bound for the superradiant instability regime of spinning Kerr black holes with massive scalar fields, confirming it with numerical results and advancing understanding of black hole stability.

Contribution

It provides the first analytical upper bound on the superradiant instability regime of Kerr black holes, aligning with recent numerical findings.

Findings

Derived an explicit inequality bounding the instability regime

Confirmed the analytical bound with numerical resonance spectrum data

Enhanced understanding of the stability conditions for Kerr black holes

Abstract

Spinning Kerr black holes are known to be superradiantly unstable to massive scalar perturbations. We here prove that the instability regime of the composed Kerr-black-hole-massive-scalar-field system is bounded from above by the dimensionless inequality $M\mu < m \cdot \sqrt{{{2(1+\gamma) (1-\sqrt{1-\gamma^2}) - \gamma^2} \over {4\gamma^2}}}$, where $\{\mu,m\}$ are respectively the proper mass and azimuthal harmonic index of the scalar field and $\gamma\equiv r_-/r_+$ is the dimensionless ratio between the horizon radii of the black hole. It is further shown that this {\it analytically} derived upper bound on the superradiant instability regime of the spinning Kerr black hole agrees with recent {\it numerical} computations of the instability resonance spectrum.