Steady state relativistic stellar dynamics around a massive black hole

A massive black hole (MBH) consumes stars whose orbits evolve into the small phase-space volume of unstable orbits, the "loss-cone", which take them directly into the MBH, or close enough to interact strongly with it. The resulting phenomena: tidal heating and tidal disruption, binary capture and hyper-velocity star ejection, gravitational wave (GW) emission by inspiraling compact remnants, or hydrodynamical interactions with an accretion disk, are of interest as they can produce observable signatures and thereby reveal the existence of the MBH, affect its mass and spin evolution, probe strong gravity, and provide information on stars and gas near the MBH. The continuous loss of stars and the processes that resupply them shape the central stellar distribution. We investigate relativistic stellar dynamics near the loss-cone of a non-spinning MBH in steady-state analytically and by Monte Carlo simulations of the diffusion of the orbital parameters. These take into account Newtonian mass precession due to enclosed stellar mass, in-plane precession due to general relativity, dissipation by GW, uncorrelated two-body relaxation, correlated resonant relaxation (RR) and adiabatic invariance due to secular precession, using a rigorously derived description of correlated post-Newtonian dynamics in the diffusion limit. We argue that general maximal entropy considerations strongly constrain orbital diffusion in steady-state, irrespective of the relaxation mechanism. We identify the exact phase-space separatrix between plunges and inspirals, predict their steady-state rates, and verify they are robust under a wide range of assumptions. We derive the dependence of the rates on the mass of the MBH, show that the contribution of RR is small, and discuss special cases where unquenched RR in restricted volumes of phase-space may affect the steady-state substantially.