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

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 ${\tau }^{-1}\approx 1.5\times {10}^{-7}(GM\mathrm{/}{c}^3{)}^{-1}$, at $M\mu \approx 0.42$. The physical implications are discussed.