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

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 $M\ensuremath{\mu}\ensuremath{\lesssim}0.5$. For a fast rotating hole ($a=0.99$) we find a maximum growth rate of ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\approx}1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}(GM/{c}^{3}{)}^{\ensuremath{-}1}$, at $M\ensuremath{\mu}\ensuremath{\approx}0.42$. The physical implications are discussed.