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BIMSA General Relativity Seminar
Optimal Regularity and Uhlenbeck Compactness in Lorentzian Geometry and Beyond
Optimal Regularity and Uhlenbeck Compactness in Lorentzian Geometry and Beyond
Organizers
Speaker
Moritz Reintjes
Time
Monday, May 22, 2023 4:00 PM - 5:00 PM
Venue
A3-3-301
Online
Zoom 928 682 9093
(BIMSA)
Abstract
We prove that curvature alone controls the derivatives of a connection, including the gravitational metrics of General Relativity, and the Yang-Mills connections of Particle Physics. Specifically, we prove that the regularity of $L^p$ connections can be lifted by coordinate/gauge transformation to one derivative above their $L^p$ bounded Riemann curvature, (i.e., to optimal regularity), thereby removing apparent singularities in the underlying geometry. This extends the classical result of Kazdan-DeTurck for Riemannian metric connections. As an application to General Relativity, our optimal regularity result implies that the Lipschitz continuous metrics of shock wave solutions of the Einstein-Euler equations are non-singular, (geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense).
By the extra connection derivative, we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry, and from compact to non-compact gauge groups. The proofs are based on our discovery of, and existence theory for, a novel system of non-linear partial differential equations, (the RT-equations), non-invariant equations which are elliptic independent of metric signature, and which provide a general procedure for constructing coordinate and gauge transformations that regularize spacetime connections.