A Collision Operator for Field-Mediated Interactions in General Relativistic Kinetic Theory

Abstract

We develop a Hamiltonian framework for general relativistic kinetic theory on the cotangent bundle $T^{\ast }M$ of a Lorentzian (pseudo-Riemannian) manifold. Starting from the geodesic Hamiltonian $H$, we derive a Landau-type collision operator for self-gravitating particles undergoing binary interactions mediated by an arbitrary potential energy $V$, and couple the resulting kinetic stress-energy to the Einstein field equations to obtain the Landau-Einstein system. In the presence of a coordinate-time Killing symmetry we find a family of stationary states of the form $f\propto {\hat{I}}^3\exp [-{\hat{I}}^2(H+\hat{I}¦)]\hat{I}\P (p_0)$, where $\hat{I}¦$ is the mean field, ${\hat{I}}^3=dt/d\ddot{I}„$, ${\hat{I}}^2$ is an inverse-temperature parameter, and $\hat{I}\P$ encodes symmetry-induced degeneracy.

40 pages, 2 figures