The extremal black hole bomb

TL;DR

This paper develops an analytical method to compute the superradiant instability spectrum of massive scalar fields around extremal Kerr black holes for generic mass parameters, improving understanding of black hole bomb phenomena.

Contribution

It introduces a new analytical approach to calculate the superradiant spectrum at intermediate mass regimes, avoiding unphysical poles present in previous approximations.

Findings

The matching at a single point removes unphysical poles.

The method agrees with numerical continued-fraction results.

Provides insights into the growth rates of superradiant instabilities.

Abstract

We analyze the spectrum of massive scalar bound states in the background of extremal Kerr black holes, focusing on modes in the superradiant regime, which grow exponentially in time and quickly deplete the black hole's mass and spin. Previous analytical estimates for the growth rate of this instability were limited to the $\mu M\ll1$ and $\mu M\gg1$ regimes, where $\mu$ and $M$ denote the scalar field and black hole masses, respectively. In this work, we discuss an analytical method to compute the superradiant spectrum for generic values of these parameters, namely in the phenomenologically interesting regime $\mu M\sim 1$. To do this, we solve the radial mode equation in two overlapping regions and match the solutions in their common domain of validity. We show that matching the functional forms of these functions involves approximations that are not valid for the whole range of scalar masses, exhibiting unphysical poles that produce a large enhancement of the growth rate. Alternatively, we match the functions at a single point and show that, despite the uncertainty in the choice of the match point, this method eliminates the spurious poles and agrees with previous numerical computations of the spectrum using a continued-fraction method.