Analytic treatment of the black-hole bomb

TL;DR

This paper analytically investigates the superradiant instability of a massive bosonic field around a rotating black hole in the regime where the instability is strongest, providing new estimates of the growth rate.

Contribution

It presents the first analytical study of the black-hole bomb instability in the regime where the product of black hole and field masses is order one, revealing a much stronger instability than previously estimated.

Findings

Instability growth rate: .0 imes 10^{-3} M^{-1}

Instability is four orders of magnitude stronger than earlier estimates

Analytical results match numerical findings in the regime of greatest instability

Abstract

A bosonic field impinging on a rotating black hole can be amplified as it scatters off the hole, a phenomena known as superradiant scattering. If in addition the field has a non-zero rest mass then the mass term effectively works as a mirror, reflecting the scattered wave back towards the black hole. In this physical system, known as a black-hole bomb, the wave may bounce back and forth between the black hole and some turning point amplifying itself each time. Consequently, the massive field grows exponentially over time and is unstable. Former analytical estimations of the timescale associated with the instability were restricted to the regimes $M\mu>>1$ and $M\mu<<1$, where $M$ and $\mu$ are the masses of the black hole and the field, respectively. In these two limits the growth rate of the field was found to be extremely weak. However, subsequent numerical investigations have indicated that the instability is actually greatest in the regime $M\mu=O(1)$, where the previous analytical approximations break down. Thus, a new analytical study of the instability timescale for the case $M\mu=O(1)$ is physically well motivated. In this Letter we study analytically for the first time the phenomena of superradiant instability (the black-hole bomb mechanism) in this physically interesting regime -- the regime of greatest instability. We find an instability growth rate of $\tau^{-1}=\omega_I=1.7\times 10^{-3}M^{-1}$ for the fastest growing mode. This instability is four orders of magnitude stronger than has been previously estimated.