Future global existence and asymptotic behaviour of solutions to the Einstein-Boltzmann system with Bianchi I symmetry

TL;DR

This paper extends previous results on the Einstein-Boltzmann system with Bianchi I symmetry, demonstrating future completeness, isotropization, and dust-like asymptotic behavior of solutions, generalizing to non-isotropic and Boltzmann cases.

Contribution

It generalizes prior isotropic and Vlasov results to non-isotropic Boltzmann systems, showing solutions are future complete and asymptotically Einstein-de Sitter.

Findings

Solutions are future geodesically complete.

Spacetimes isotropize at late times.

Solutions behave like Einstein-de Sitter universe asymptotically.

Abstract

In this paper we study the Einstein-Boltzmann system with Bianchi I symmetry. Isotropization of Bianchi I spacetimes was proved in [13] in the Vlasov case, and asymptotic behaviour of the relativistic Boltzmann equation was studied in [9] in the isotropic case. These results are generalized in the present paper to the non-isotropic and Boltzmann case. We assume that the initial data are small in a suitable sense, and show that solutions of the Einstein-Boltzmann system are future geodesically complete, that they isotropize and have a dust-like behaviour at late times. Detailed information about the metric and the matter terms is obtained. In short, the solution is asymptotically the Einstein-de Sitter solution.