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

Abstract: We discuss a construction of polarized U(1) space-times of a “fixed size”, where the initial data contains a given amount η>0 of incoming gravitational energy which is allowed to be supported at arbitrarily small scales, without affecting the time of existence. More specialized examples can be constructed where any chosen fraction of this initial incoming energy can stay focused along a null geodesic, in the U(1) picture. We discuss the relationship of this result with one direction of Thorne’s hoop conjecture. The work relies on a special structure of the Einstein equations in this symmetry class, a new re-scaling of the equations and large-time control, along with a new local Klainerman-Sobolev estimate. Time permitting, some comparisons of the result and the methods with large-data constructions for the Einstein equations over the last decade will also be presented.

general relativity and quantum cosmologymathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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JoMaReC - Joint Online Mathematical Relativity Colloquium

Series comments: This monthly online colloquium is meant to be accessible to and informative for mathematicians and mathematical physicists with a background in General Relativity, widely interpreted to include Lorentzian Geometry, and Geometric Analysis of various Partial Differential Equations related to General Relativity.

It is aimed to present motivation and applications of particular results and/or introduce specific subfields, while refraining from too much technicalities.

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