Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole

TL;DR

This paper systematically analyzes the accretion of a collisionless relativistic gas into a Schwarzschild black hole, solving the Liouville equation and exploring steady-state solutions, revealing unique pressure behaviors and stability properties.

Contribution

It provides the first detailed solution of the relativistic Liouville equation on Schwarzschild spacetime and characterizes the properties and stability of collisionless gas accretion flows.

Findings

Tangential pressure at the horizon exceeds radial pressure in low-temperature limit.

Accretion rate is significantly lower than in hydrodynamic models.

Steady-state flows are asymptotically stable under broad initial conditions.

Abstract

We provide a systematic study for the accretion of a collisionless, relativistic kinetic gas into a nonrotating black hole. To this end, we first solve the relativistic Liouville equation on a Schwarzschild background spacetime. The most general solution for the distribution function is given in terms of appropriate symplectic coordinates on the cotangent bundle, and the associated observables, including the particle current density and stress energy-momentum tensor, are determined. Next, we explore the case where the flow is steady-state and spherically symmetric. Assuming that in the asymptotic region the gas is described by an equilibrium distribution function, we determine the relevant parameters of the accretion flow as a function of the particle density and the temperature of the gas at infinity. In particular, we find that in the low temperature limit the tangential pressure at the horizon is about an order of magnitude larger than the radial one, showing explicitly that a collisionless gas, despite exerting kinetic pressure, behaves very differently than an isotropic perfect fluid, and providing a partial explanation for the known fact that the accretion rate is much lower than in the hydrodynamic case of Bondi-Michel accretion. Finally, we establish the asymptotic stability of the steady-state spherical flows by proving pointwise convergence results which show that a large class of (possibly nonstationary and nonspherical) initial conditions for the distribution function lead to solutions of the Liouville equation which relax in time to a steady-state, spherically symmetric configuration.