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

TL;DR

This paper demonstrates that a collisionless relativistic gas in the equatorial plane of a Kerr black hole relaxes to a stationary state through phase space mixing, with macroscopic observables reaching well-defined limits over time.

Contribution

It introduces the concept that phase space mixing causes relaxation of collisionless gases in Kerr black hole spacetimes, providing a predictive framework for the final equilibrium state.

Findings

Gas relaxes to a stationary, axisymmetric configuration.

Macroscopic observables have well-defined limits as time approaches infinity.

Final distribution depends only on constants of motion.

Abstract

It is shown that a collisionless, relativistic kinetic gas configuration propagating in the equatorial plane of a Kerr black hole undergoes a relaxation process and eventually settles down to a stationary, axisymmetric configuration surrounding the black hole. The underlying mechanism for this relaxation process is due to phase space mixing, which implies that although the one-particle distribution function $f$ satisfying the collisionless Boltzmann equation which describes the microscopic state of the gas is quasi-periodic in time, the associated macroscopic observables computed from averages over $f$ possess well-defined limits as time goes to infinity. The final state of the gas is described by an effective distribution function depending only on constants of motion which can be predicted from the initial distribution function.