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

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.

Phase space mixing in the equatorial plane of a Kerr black hole

“…In our case, the presence of the Killing vector field ∂ t induces a natural vector fieldk on phase space Γ bound defined as the complete lift of ∂ t (see Refs. [19,20] for details). This vector field, in turn, induces a flow ψ t on Γ bound with respect to which one may define the time-translated test function: 3…”

“…1. The validity of the determinant condition (20) will be analyzed towards the end of this section, following the proof of the theorem. As will be verified numerically, it is satisfied everywhere except for points lying on a certain curve in (E, L)-space, see Fig.…”

“…6. However, we stress that the theorem does not require condition (20) to hold everywhere; it is sufficient to hold for points lying in the support of ϕ, that is, in the vicinity where the observer performs the measurement.…”

“…Therefore, due to condition (20) the map W m is (at least) locally invertible on the support of ϕ. If not globally invertible, we may cover its domain with open subsets U i on which W m is invertible.…”

Relativistic kinetic gases as direct sources of gravity

“…2.3], or [4]. A geometric description of the kinetic gas and its dynamics on the tangent bundle has been investigated in [5] and was applied to the formation of accretion discs in [6]. Moreover first post Newtonian corrections to the behaviour of a gravitating gas in a fixed gravitational behaviour have been studied [7,8].…”