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

Naoki Sato

TL;DR

This work develops a general relativistic kinetic theory on the cotangent bundle $T^{\ast}M$ by introducing a Hamiltonian framework and a Landau-type collision operator for binary, field-mediated interactions. The resulting Landau–Einstein system couples the kinetic equation to Einstein’s equations, yielding conservation laws and an $H$-theorem, and supports GR-specific equilibrium distributions that reduce to Synge/Jüttner or Maxwell–Boltzmann forms in appropriate limits. A covariant invariant measure with $J=\gamma$ provides a consistent entropy functional and enables a meaningful notion of relativistic temperature, $T_k$, linked to Tolman–Ehrenfest thermodynamics. The paper also analyzes how GR curvature and mean-field effects modify equilibrium structures and temperature transformations, with implications for self-gravitating astrophysical systems and curved-spacetime plasmas.

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

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

TL;DR

This work develops a general relativistic kinetic theory on the cotangent bundle by introducing a Hamiltonian framework and a Landau-type collision operator for binary, field-mediated interactions. The resulting Landau–Einstein system couples the kinetic equation to Einstein’s equations, yielding conservation laws and an -theorem, and supports GR-specific equilibrium distributions that reduce to Synge/Jüttner or Maxwell–Boltzmann forms in appropriate limits. A covariant invariant measure with provides a consistent entropy functional and enables a meaningful notion of relativistic temperature, , linked to Tolman–Ehrenfest thermodynamics. The paper also analyzes how GR curvature and mean-field effects modify equilibrium structures and temperature transformations, with implications for self-gravitating astrophysical systems and curved-spacetime plasmas.

Abstract

We develop a Hamiltonian framework for general relativistic kinetic theory on the cotangent bundle of a Lorentzian (pseudo-Riemannian) manifold. Starting from the geodesic Hamiltonian , we derive a Landau-type collision operator for self-gravitating particles undergoing binary interactions mediated by an arbitrary potential energy , 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 , where is the mean field, , is an inverse-temperature parameter, and encodes symmetry-induced degeneracy.

Paper Structure

This paper contains 21 sections, 3 theorems, 227 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structural differences between classical and general relativistic kinetic theory
  3. Thermodynamic equilibrium and temperature
  4. Energy and momentum
  5. Liouville measure, entropy, and thermodynamics
  6. Binary interactions in general relativistic spacetime
  7. A general relativistic collision operator
  8. Grazing scatterings
  9. Collision operator
  10. The Landau--Einstein system
  11. Energy-momentum tensor
  12. Conservation laws
  13. H-theorem
  14. Thermodynamic equilibrium
  15. Liouville equilibria
  16. ...and 6 more sections

Key Result

Proposition 5.1

The Landau–Einstein system LE with collision operator LECX preserves the total particle number and the total energy with respect to coordinate-time $t$.

Figures (2)

  • Figure 1: Schematic comparison between (a) the standard special-relativistic scattering process where the components $p_{\mu}$ of the four-momentum are additive invariants yielding the Synge–Jüttner distribution, and (b) the general-relativistic collision model developed here, where the geodesic Hamiltonian $\mathcal{H}$, including mean-field contributions, is an approximate additive invariant yielding the Landau–Einstein distribution.
  • Figure 2: Structure of general relativistic equilibria: the equilibrium distribution comprises a normalization factor $Z^{-1}$, the Lorentz factor (reflecting the bookkeeping time parameter $t$), a $\delta$-function enforcing the speed of light constraint, and an exponential weight determined by the additive invariants $\mathcal{H}^m$ with constant multipliers $\beta_m$, which are additively preserved by collisions and exactly preserved by the Poisson bracket, $\left\{{\mathcal{H}^m,\mathcal{H}}\right\}=0$. In addition, a function $\zeta$ depends on the exact invariants $\mathscr{P}_1, \mathscr{P}_2, \ldots$, which are preserved both by collisions and by the ideal part of the dynamics.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.1
  • Definition 3.1
  • Definition 4.1
  • Proposition 5.1
  • proof
  • Remark 5.1
  • Proposition 5.2
  • ...and 8 more