Onset of superradiant instabilities in the composed Kerr-black-hole-mirror bomb

TL;DR

This paper investigates the critical mirror radius in Kerr-black-hole-mirror systems that determines the onset of superradiant instabilities, revealing it approaches the horizon radius in rapidly rotating black holes and scales with temperature.

Contribution

It provides a detailed analysis of the stationary mirror radius boundary, linking it to black hole rotation and temperature, and clarifies the conditions for superradiant instability onset.

Findings

Critical mirror radius approaches the horizon in extremal black holes.

The critical radius scales linearly with black hole temperature.

Stable and unstable configurations are separated by this critical radius.

Abstract

It was first pointed out by Press and Teukolsky that a system composed of a spinning Kerr black hole surrounded by a reflecting mirror may develop instabilities. The physical mechanism responsible for the development of these exponentially growing instabilities is the superradiant amplification of bosonic fields confined between the black hole and the mirror. A remarkable feature of this composed black-hole-mirror-field system is the existence of a critical mirror radius, $r^{\text{stat}}_{\text{m}}$, which supports {\it stationary} (marginally-stable) field configurations. This critical (`stationary') mirror radius marks the boundary between stable and unstable black-hole-mirror-field configurations: composed systems whose confining mirror is situated in the region $r_{\text{m}}<r^{\text{stat}}_{\text{m}}$ are stable (that is, all modes of the confined field decay in time), whereas composed systems whose confining mirror is situated in the region $r_{\text{m}}>r^{\text{stat}}_{\text{m}}$ are unstable (that is, there are confined field modes which grow exponentially over time). In the present paper we explore this critical (marginally-stable) boundary between stable and explosive black-hole-mirror-field configurations. It is shown that the innermost ({\it smallest}) radius of the confining mirror which allows the extraction of rotational energy from a spinning Kerr black hole approaches the black-hole horizon radius in the extremal limit of rapidly-rotating black holes. We find, in particular, that this critical mirror radius (which marks the onset of superradiant instabilities in the composed system) scales linearly with the black-hole temperature.