# Superradiant stability of the Kerr black holes

## TL;DR

This paper analytically demonstrates the superradiant stability of Kerr black holes with massive scalar perturbations in a previously unconfirmed parameter region, expanding understanding of black hole stability conditions.

## Contribution

It provides an analytical proof of superradiant stability for Kerr black holes with scalar fields in a new parameter regime, complementing prior numerical results.

## Key findings

- Proves superradiant stability when <<rac{\u03bc}{\u221a{2}} and /r_+<0.802.
- Identifies simple inequalities that guarantee stability in the complementary parameter region.
- Extends the stability analysis of Kerr black holes beyond previously known conditions.

## Abstract

We study the superradiant stability of the system of a Kerr black hole and a massive scalar perturbation. It was proved previously that this system is superradiantly stable when $\mu\geq \sqrt{2}m\Omega_H$, where $\mu$ is the proper mass of the scalar, $m$ is the azimuthal number of the scalar mode, and $\Omega_H$ is the angular velocity of the Kerr black hole horizon. Our study is a complementary work of this result. We analytically prove that in the complementary parameter region $\mu<\sqrt{2}m\Omega_H$, when the parameters of scalar perturbation and Kerr black hole satisfy two simple inequalities, $\omega<\frac{\mu}{\sqrt{2}}$,~ $\frac{r_-}{r_+}<0.802$, the system is also superradiantly stable.