The reverse Burnett conjecture for null dusts

TL;DR

This paper proves the reverse Burnett conjecture for null dust solutions in general relativity by constructing vacuum solutions that approximate Einstein-null dust spacetimes using a multiphase geometric optics approach.

Contribution

It extends the reverse Burnett conjecture to a broad class of massless kinetic spacetimes without symmetry assumptions, using a novel geometric optics ansatz.

Findings

Constructed families of vacuum solutions approximating null dust spacetimes.

Controlled interactions of multiple oscillatory waves in Einstein vacuum equations.

Provided initial data satisfying vacuum constraints compatible with the ansatz.

Abstract

Given a regular solution $\mathbf{g}_0$ of the Einstein-null dusts system without restriction on the number of dusts, we construct families of solutions $(\mathbf{g}_\lambda)_{\lambda\in(0,1]}$ of the Einstein vacuum equations such that $\mathbf{g}_\lambda-\mathbf{g}_0$ and $\partial(\mathbf{g}_\lambda-\mathbf{g}_0)$ converges respectively strongly and weakly to 0 when $\lambda\to0$. Our construction, based on a multiphase geometric optics ansatz, thus extends the validity of the reverse Burnett conjecture without symmetry to a large class of massless kinetic spacetimes. In order to deal with the finite but arbitrary number of direction of oscillations we work in a generalised wave gauge and control precisely the self-interaction of each wave but also the interaction of waves propagating in different null directions, relying crucially on the non-linear structure of the Einstein vacuum equations. We also provide the construction of oscillating initial data solving the vacuum constraint equations and which are consistent with the spacetime ansatz.