Accretion of the relativistic Vlasov gas onto a moving Schwarzschild black hole: Exact solutions

We derive an exact, axially symmetric solution representing stationary accretion of the relativistic, collisionless Vlasov gas onto a moving Schwarzschild black hole. The gas is assumed to be in thermal equilibrium at infinity, where it obeys the Maxwell-J\"{u}ttner distribution. The Vlasov equation is solved analytically in terms of suitable action-angle variables. We provide explicit expressions for the particle current density and the accretion rate. In the limit of infinite asymptotic temperature of the gas, we recover the qualitative picture known form the relativistic Bondi-Hoyle-Lyttleton accretion of the perfect gas with the ultra-hard equation of state, in which the mass accretion is proportional to the Lorentz factor associated with the black-hole velocity. Surprisingly, for a finite asymptotic temperature, the mass accretion rate is not in general a monotonic function of the velocity of the black hole.