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

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 $f\propto \gamma \exp [-\beta (H+\mathrm{\Phi })]\zeta (p_0)$, where $\mathrm{\Phi }$ is the mean field, $\gamma =dt\mathrm{/}d\tau$, $\beta$ is an inverse-temperature parameter, and $\zeta$ encodes symmetry-induced degeneracy.