  Consider a sequence of $C^4$ Lorentzian metrics $\{h_n\}_{n=1}^{+\infty}$ on
a manifold $\mathcal M$ satisfying the Einstein vacuum equation
$\mathrm{Ric}(h_n)=0$. Suppose there exists a smooth Lorentzian metric $h_0$ on
$\mathcal M$ such that $h_n\to h_0$ uniformly on compact sets. Assume also that
on any compact set $K\subset \mathcal M$, there is a decreasing sequence of
positive numbers $\lambda_n \to 0$ such that $$\|\partial^{\alpha} (h_n -
h_0)\|_{L^{\infty}(K)} \lesssim \lambda_n^{1-|\alpha|},\quad |\alpha|\geq 4.$$
It is well-known that $h_0$, which represents a "high-frequency limit", is not
necessarily a solution to the Einstein vacuum equation. Nevertheless, Burnett
conjectured that $h_0$ must be isometric to a solution to the Einstein-massless
Vlasov system.
  In this paper, we prove Burnett's conjecture assuming that
$\{h_n\}_{n=1}^{+\infty}$ and $h_0$ in addition admit a $\mathbb U(1)$ symmetry
and obey an elliptic gauge condition. The proof uses microlocal defect measures
- we identify an appropriately defined microlocal defect measure to be the
Vlasov measure of the limit spacetime. In order to show that this measure
indeed obeys the Vlasov equation, we need some special cancellations which rely
on the precise structure of the Einstein equations. These cancellations are
related to a new "trilinear compensated compactness" phenomenon for solutions
to (semilinear) elliptic and (quasilinear) hyperbolic equations.
