Burnett's conjecture in generalized wave coordinates

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 $d\ge 3$) 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.

28 pages