Global stability of Minkowski space for the Einstein--Vlasov system in the harmonic gauge

TL;DR

This paper proves the global stability of Minkowski space as a solution to the Einstein--Vlasov system using a harmonic gauge, quasilinear wave equations, and vector field methods, simplifying previous results in the vacuum case.

Contribution

It establishes the first global stability result for Minkowski space coupled with massive Vlasov matter in harmonic gauge, introducing adapted vector fields for controlling the distribution function.

Findings

Minkowski space is globally stable under the Einstein--Vlasov system.

The proof simplifies previous vacuum stability results.

A new method for controlling the Vlasov distribution function is developed.

Abstract

Minkowski space is shown to be globally stable as a solution to the massive Einstein--Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used to control the particle distribution function, a function of both spacetime and momentum variables. The vector fields are derived using a general procedure, are adapted to the geometry of the solution and reduce to the generators of the symmetries of Minkowski space when restricted to acting on spacetime functions. Moreover, when specialising to the case of vacuum, the proof provides a simplification of previous stability works.