  We study the superradiant instability of a massive boson around a spinning
black hole in full general relativity without assuming spatial symmetries. We
focus on the case of a rapidly spinning black hole in the presence of a vector
boson with a Compton wavelength comparable to the black hole radius, which is
the regime where relativistic effects are maximized. We follow the growth of
the boson cloud through superradiance into the nonlinear regime as it spins
down the black hole, reaches a maximum energy, and begins to dissipate through
the emission of gravitational waves. We find that the superradiant instability
can efficiently convert a significant fraction of a black hole's rotational
energy into gravitational radiation.
