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

TL;DR

This paper formulates a Hamiltonian-based kinetic theory for self-gravitating particles in general relativity, deriving a collision operator and coupling it with Einstein's equations to study stationary states.

Contribution

It introduces a novel collision operator within a Hamiltonian framework for relativistic kinetic theory and couples it with Einstein's equations for self-gravitating systems.

Findings

Derived a Landau-type collision operator for relativistic particles.

Coupled the kinetic theory with Einstein's equations to form the Landau-Einstein system.

Identified stationary states under symmetry conditions.

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