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 nonrelativistic 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 nonrelativistic treatments of dynamical friction in ultralight dark matter.
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