Burnett's conjecture in generalized wave coordinates

TL;DR

This paper proves Burnett's conjecture in general relativity for metrics satisfying a generalized wave coordinate condition, showing that weak limits of such metrics solve the Einstein--massless Vlasov system, with the Vlasov field derived from microlocal defect measures.

Contribution

It establishes Burnett's conjecture under generalized wave coordinate conditions and introduces a microlocal defect measure approach for the Vlasov field in the limit.

Findings

Weak convergence of metrics implies Einstein--massless Vlasov system in the limit.

The Vlasov field can be obtained as a microlocal defect measure.

The proof leverages a compensation phenomenon based on Einstein equations' structure.

Abstract

We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose $\{g_n\}_{n=1}^\infty$ is a sequence of Lorentzian metrics (in arbitrary dimensions $d \geq 3$) satisfying a generalized wave coordinate condition and such that $g_n\to g$ in a suitably weak and "high-frequency" manner, then the limit metric $g$ satisfies the Einstein--massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations.