The Einstein-Boltzmann system and positivity

TL;DR

This paper investigates the Einstein-Boltzmann system, introducing a new parametrization to ensure non-negativity of solutions, and addresses the mathematical challenges of extending the theory to physically relevant scattering kernels.

Contribution

A new parametrization of post-collisional momenta simplifies conditions on collision cross-sections and advances the understanding of solution non-negativity in the Einstein-Boltzmann system.

Findings

Established non-negativity for a class of scattering kernels.

Simplified collision cross-section conditions.

Extended existing non-negativity results to curved spacetime.

Abstract

The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.