Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2022
- DOI:
- 10.48550/arXiv.2205.05526
- arXiv:
- arXiv:2205.05526
- Bibcode:
- 2022arXiv220505526A
- Keywords:
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- General Relativity and Quantum Cosmology;
- Mathematics - Analysis of PDEs;
- Mathematics - Differential Geometry;
- 83C40;
- 35L15;
- 35L70
- E-Print:
- 102 pages, 8 figures