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We study the superradiant instability of massive vector fields, i.e. Procafields, around spinning black holes in the test field limit. This is motivatedby the possibility that observations of astrophysical black holes can probe theexistence of ultralight bosons subject to this mechanism. By making use oftime-domain simulations, we characterize the growth rate, frequency, spatialdistribution, and other properties of the unstable modes, including in theregime where the black hole is rapidly spinning and the Compton wavelength ofthe Proca field is comparable to the black hole radius. We find thatrelativistic effects in this regime increase the range of Proca masses that areunstable, as well as the maximum instability rate. We also study thegravitational waves that can be sourced by such an instability, finding thatthey can be significantly stronger than in the massive scalar field case.

Superradiant instability of massive vector fields around spinning black holes in the relativistic regime