Analytic treatment of the black-hole bomb

A bosonic field impinging on a rotating black hole can be amplified as it scatters off the hole, a phenomenon known as superradiant scattering. If in addition the field has a nonzero rest mass $\mu$, 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 field grows exponentially over time and is unstable. In this paper we study analytically for the first time the phenomenon of superradiant instability (the black-hole bomb mechanism) in the regime $M\mu =O(1)$ of greatest instability. We find a maximal instability growth rate of ${\tau }^{-1}=1.7\times {10}^{-3}{M}^{-1}$. This instability is 4 orders of magnitude stronger than has been previously estimated.

Figure 1

Superradiant instability for maximally rotating black holes $(a\simeq M)$. The results are for the $l=m=1$ mode, the mode with the greatest instability. The growth rate $M{\omega }_I$ is shown as a function of the dimensionless product $M\mu$. We display results for the direct solutions of both the exact resonance condition (19) and the polynomial approximation (20). The maximum growth rate is ${\tau }^{-1}\equiv {\omega }_I=1.7\times {10}^{-3}{M}^{-1}$, where $\tau$ is the $e$-folding time.