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In 1989, Burnett conjectured that, under appropriate assumptions, the limitof highly oscillatory solutions to the Einstein vacuum equations is a solutionof the Einstein--massless Vlasov system. In a recent breakthrough, Huneau--Luk(arXiv:1907.10743) gave a proof of the conjecture in U(1)-symmetry and ellipticgauge. They also require control on up to fourth order derivatives of themetric components. In this paper, we give a streamlined proof of a strongerresult and, in the spirit of Burnett's original conjecture, we remove the needfor control on higher derivatives. Our methods also apply to general wave mapequations.

Oscillations in wave map systems and homogenization of the Einstein equations in symmetry