  Light bosonic scalars (e.g. axions) may form clouds around black holes via
superradiant instabilities, or via accretion if they form some component of the
dark matter. It has been suggested that their presence may lead to a
distinctive dephasing of the gravitational wave signal when a small compact
object spirals into a larger black hole. Motivated by this, we study
numerically the dynamical friction force on a black hole moving at relativistic
velocities in a background scalar field with an asymptotically homogeneous
energy density. We show that the relativistic scaling is analogous to that
found for supersonic collisional fluids, assuming an approximate expression for
the pressure correction which depends on the velocity and scalar mass. While we
focus on a complex scalar field, our results confirm the expectation that real
scalars would exert a force which oscillates between positive and negative
values in time with a frequency set by the scalar mass. The complex field
describes the time averaged value of this force, but in a real scalar the rapid
force oscillations could in principle leave an imprint on the trajectory. The
approximation we obtain can be used to inform estimates of dephasing in the
final stages of an extreme mass ratio inspiral.
