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Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry
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abstract
We prove uniform finite-time existence of solutions to the vacuum Einstein equations in polarized U(1) symmetry which have uniformly positive incoming $H^1$ energy supported on an arbitrarily small set in the 2 + 1 spacetime obtained by quotienting by the U(1) symmetry. We also construct a subclass of solutions for which the energy remains concentrated (along a U(1) family of geodesics) throughout its evolution. These results rely on three innovations: a direct treatment of the 2 + 1 Einstein equations in a null geodesic gauge, a novel parabolic scaling of the Einstein equations in this gauge, and a new Klainerman-Sobolev inequality on rectangular strips.
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Cited by 1 Pith paper
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High-frequency backreaction for the Einstein equations under $\mathbb U(1)$ symmetry: from Einstein-dust to Einstein-Vlasov
Small U(1)-symmetric Einstein-massless Vlasov solutions are shown to be realizable as high-frequency limits of vacuum Einstein spacetimes, extending prior finite-null-dust constructions to general Vlasov fields.
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