TRILINEAR COMPENSATED COMPACTNESS AND BURNETT’S CONJECTURE IN GENERAL RELATIVITY

Consider a sequence of C⁴ Lorentzian metrics {h_n}^+∞_n=1 on a manifold M satisfying the Einstein vacuum equation Ric(h_n) = 0. Suppose there exists a smooth Lorentzian metric h₀ on M such that h_n → h₀ uniformly on compact sets. Assume also that on any compact set K ⊂ M, there is a decreasing sequence of positive numbers λ_n → 0 such that (Formula presented). It is well-known that h₀, which represents a “high-frequency limit,” is not necessarily a solution to the Einstein vacuum equation. Nevertheless, Burnett conjectured that h₀ 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 and h₀ in addition admit a 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.