Reverse Burnett's conjecture on approximating Einstein–massless Vlasov solutions by vacuum spacetimes

Consider a sequence of $(3+1)$-dimensional spacetimes $\{(\mathcal M,g_n)\}_{n=1}^{+\infty}$ solving the Einstein vacuum equations, with $g_n\to g_\infty$ in $C^0_{loc}$ and derivatives converging weakly in $L^2_{loc}$. The limiting process is the weak high-frequency limit described above.

Reverse Burnett's conjecture. Any solution to the Einstein–massless Vlasov system arises as a limit of solutions to the Einstein vacuum equations in the sense described above.

This is the converse of Burnett's conjecture and asks whether every Einstein–massless Vlasov solution can be realized by a high-frequency vacuum approximation. It remains open in full generality, with some progress available under a $\mathbb U(1)$ symmetry assumption.