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 $f \propto γ\exp[-β(H+Φ)]ζ(p_0)$, where $Φ$ is the mean field, $γ=dt/dτ$, $β$ is an inverse-temperature parameter, and $ζ$ encodes symmetry-induced degeneracy.

💡 Research Summary

The paper presents a comprehensive Hamiltonian formulation of relativistic kinetic theory on the cotangent bundle (T^{*}M) of a four‑dimensional Lorentzian manifold ((M,g)). Starting from the geodesic Hamiltonian \