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