2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186164We 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 $M μ\lesssim 0.5$. For a fast rotating hole ($a = 0.99$) we find a maximum growth rate of $τ^{-1} \approx 1.5 \times 10^{-7} (GM/c^3)^{-1}$, at $M μ\approx 0.42$. The physical implications are discussed.Added references. 27 pages, 7 figuresGeneral Relativity and Quantum CosmologyHigh Energy Physics - Theorytext