Instability of the massive Klein-Gordon field on the Kerr spacetime

TL;DR

This paper studies the superradiant instability of massive scalar fields around rotating black holes, calculating the spectrum of bound states and identifying conditions for maximum growth rates.

Contribution

It introduces a continued fraction method to compute bound state spectra and analyzes the instability's dependence on black hole and field parameters.

Findings

Maximum growth rate occurs at Mμ ≈ 0.42 for a rapidly rotating black hole.

Instability is most significant for the l=1, m=1 state.

Growth rate of the instability is approximately 1.5 x 10^{-7} (GM/c^3)^{-1}.

Abstract

We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the $l = 1$, $m = 1$ state, for $M \mu \lesssim 0.5$. For a fast rotating hole ($a = 0.99$) we find a maximum growth rate of $\tau^{-1} \approx 1.5 \times 10^{-7} (GM/c^3)^{-1}$, at $M \mu \approx 0.42$. The physical implications are discussed.