Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916283934572544 |
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| author | Alexakis, Spyros Carruth, Nathan Thomas |
| author_facet | Alexakis, Spyros Carruth, Nathan Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry Alexakis, Spyros Carruth, Nathan Thomas General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry 83C40, 35L15, 35L70 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. |
| title | Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry 83C40, 35L15, 35L70 |
| url | https://arxiv.org/abs/2205.05526 |