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We discuss a general method to study linear perturbations of slowly rotatingblack holes which is valid for any perturbation field, and particularlyadvantageous when the field equations are not separable. As an illustration ofthe method we investigate massive vector (Proca) perturbations in the Kerrmetric, which do not appear to be separable in the standard Teukolskyformalism. Working in a perturbative scheme, we discuss two important effectsinduced by rotation: a Zeeman-like shift of nonaxisymmetric quasinormal modesand bound states with different azimuthal number m, and the coupling betweenaxial and polar modes with different multipolar index l. We explicitly computethe perturbation equations up to second order in rotation, but in principle themethod can be extended to any order. Working at first order in rotation we showthat polar and axial Proca modes can be computed by solving two decoupled setsof equations, and we derive a single master equation describing axialperturbations of spin s=0 and s=+-1. By extending the calculation to secondorder we can study the superradiant regime of Proca perturbations in aself-consistent way. For the first time we show that Proca fields around Kerrblack holes exhibit a superradiant instability, which is significantly strongerthan for massive scalar fields. Because of this instability, astrophysicalobservations of spinning black holes provide the tightest upper limit on themass of the photon: mv<4x10^-20 eV under our most conservative assumptions.Spin measurements for the largest black holes could reduce this bound tomv<10^-22 eV or lower.

Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric