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

2022/05/11 by Spyros Alexakis, Alexakis, Spyros, Nathan Thomas Carruth +1
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #35L15 #35L70 #83C40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Arctic and Antarctic ice dynamics #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · pdf · doi:10.48550/arxiv.2205.05526

openalex publication_date 2022/05/11 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

Abstract

We prove uniform finite-time existence of solutions to the vacuum Einstein equations in polarized U(1) symmetry which have uniformly positive incoming H¹ 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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