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Massive bosonic fields of arbitrary spin are predicted by general extensionsof the Standard Model. It has been recently shown that there exists a family ofbimetric theories of gravity - including massive gravity - which are free ofBoulware-Deser ghosts at the nonlinear level. This opens up the possibility todescribe consistently the dynamics of massive spin-2 particles in agravitational field. Within this context, we develop the study of massivespin-2 fluctuations - including massive gravitons - around Schwarzschild andslowly-rotating Kerr black holes. Our work has two important outcomes. First,we show that the Schwarzschild geometry is linearly unstable for small tensormasses, against a spherically symmetric mode. Second, we provide solid evidencethat the Kerr geometry is also generically unstable, both against the sphericalmode and against long-lived superradiant modes. In the absence of nonlineareffects, the observation of spinning black holes bounds the graviton mass to besmaller than 5x10^{-23} eV.

Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass