The extremal black hole bomb

TL;DR

This paper analyzes the spectrum of superradiant bound states for a massive scalar field in an extremal Kerr black hole, focusing on the regime $μM\sim1$ where growth is most pronounced. It develops two analytic strategies: functional matching, which reduces to confluent hypergeometric solutions but produces unphysical poles that inflate the growth rate, and point matching, which matches in the overlap region at a single point to avoid these poles. The point-matching results align well with numerical continued-fraction computations, giving a maximum growth rate $\omega_I M\simeq 1.49\times10^{-7}$ at $l=m=1$, $μM\simeq 0.454$, and demonstrating that the problematic peaks in functional matching are artifacts of the approximation. The work clarifies the validity range of analytic methods for the Kerr superradiant spectrum and supports implications for astrophysical black holes and axion-like moduli in the string axiverse, including potential gravitational-wave and spin-down signatures.

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 $μM\ll1$ and $μM\gg1$ regimes, where $μ$ 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 $μM\sim 1$. 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.