Optimal regularity and Uhlenbeck compactness for general relativity and Yang-Mills theory

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Original languageEnglish
Article number20220444
Journal / PublicationProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2271
Online published1 Mar 2023
Publication statusPublished - 29 Mar 2023


We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of general relativity (GR). Compactness is the essential tool of mathematical analysis for establishing the validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with Lp curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature (i.e. to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative, we extend Uhlenbeck compactness from Uhlenbeck's setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a 'geometric' improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence.

Research Area(s)

  • shock waves, Yang-Mills theories, general relativity, apparent singularities, compensated compactness, SHOCK-WAVE SOLUTIONS, GRAVITATIONAL COLLAPSE, EINSTEIN EQUATIONS, EXISTENCE, METRICS

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