  We study the coupling of massive scalar fields to matter in orbit around
rotating black holes. It is generally expected that orbiting bodies will lose
energy in gravitational waves, slowly inspiralling into the black hole.
Instead, we show that the coupling of the field to matter leads to a surprising
effect: because of superradiance, matter can hover into "floating orbits" for
which the net gravitational energy loss at infinity is entirely provided by the
black hole's rotational energy. Orbiting bodies remain floating until they
extract sufficient angular momentum from the black hole, or until perturbations
or nonlinear effects disrupt the orbit. For slowly rotating and nonrotating
black holes floating orbits are unlikely to exist, but resonances at orbital
frequencies corresponding to quasibound states of the scalar field can speed up
the inspiral, so that the orbiting body "sinks". These effects could be a
smoking gun of deviations from general relativity.
