The spinning Kerr-black-hole-mirror bomb: A lower bound on the radius of the reflecting mirror

TL;DR

This paper analytically establishes a lower bound on the mirror radius necessary for superradiant instabilities to occur in a Kerr black hole-mirror system, confirming previous numerical findings and advancing understanding of black-hole bombs.

Contribution

It provides the first analytical derivation of a lower bound on the mirror radius for superradiant instability in Kerr black-hole-mirror systems, aligning with numerical results.

Findings

Derived a necessary lower bound on mirror radius for instability.

Confirmed analytical bound matches numerical computations.

Enhanced understanding of superradiant instability conditions.

Abstract

The intriguing superradiant amplification phenomenon allows an orbiting scalar field to extract rotational energy from a spinning Kerr black hole. Interestingly, the energy extraction rate can grow exponentially in time if the black-hole-field system is placed inside a reflecting mirror which prevents the field from radiating its energy to infinity. This composed Kerr-black-hole-scalar-field-mirror system, first designed by Press and Teukolsky, has attracted the attention of physicists over the last four decades. Previous numerical studies of this spinning {\it black-hole bomb} have revealed the interesting fact that the superradiant instability shuts down if the reflecting mirror is placed too close to the black-hole horizon. In the present study we use analytical techniques to explore the superradiant instability regime of this composed Kerr-black-hole-linearized-scalar-field-mirror system. In particular, it is proved that the lower bound ${{r_{\text{m}}}\over{r_+}}>{1\over 2}\Big(\sqrt{1+{{8M}\over{r_-}}}-1\Big)$ provides a necessary condition for the development of the exponentially growing superradiant instabilities in this composed physical system, where $r_{\text{m}}$ is the radius of the confining mirror and $r_{\pm}$ are the horizon radii of the spinning Kerr black hole. We further show that, in the linearized regime, this {\it analytically} derived lower bound on the radius of the confining mirror agrees with direct {\it numerical} computations of the superradiant instability spectrum which characterizes the spinning black-hole-mirror bomb.