On the instability regime of the rotating Kerr spacetime to massive scalar perturbations

The instability of rotating Kerr black holes due to massive scalar perturbations is investigated. It is well known that a bosonic field impinging on a Kerr black hole can be amplified as it scatters off the hole. This superradiant scattering occurs for frequencies in the range $\omega<m\Omega$, where $\Omega$ is the angular frequency of the black hole and $m$ is the azimuthal harmonic index of the mode. If the incident field has a non-zero rest mass, $\mu$, then the mass term effectively works as a mirror, reflecting the scattered wave back towards the black hole. The wave may bounce back and forth between the black hole and some turning point amplifying itself each time. This may lead to a dynamical instability of the system, a phenomena known as a "black-hole bomb". In this work we provide a bound on the instability regime of rotating Kerr spacetimes. In particular, we show that Kerr black holes are stable to massive perturbations in the regime $\mu\geq\sqrt{2}m\Omega$.