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arXiv · 2205.05526

Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry

Spyros Alexakis·Nathan Thomas Carruth
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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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Spyros Alexakis, Nathan Thomas Carruth. 2022-05-11. Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry. https://arxiv.org/abs/2205.05526

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