  In this work we derive simple closed-form expressions for the dynamical
friction acting on black holes moving through ultralight (scalar field) dark
matter, covering both non-relativistic and relativistic black hole speeds. Our
derivation is based on long known scattering amplitudes in black hole
spacetimes, it includes the effect of black hole spin and can be easily
extended to vector and tensor light fields. Our results cover and complement
recent numerical and previous non-relativistic treatments of dynamical friction
in ultralight dark matter.
