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

Abstract

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 L p 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.

Strong Cosmic Censorship with bounded curvature

“…Theorem 3.1 in [33] applies to general (affine) connections on the tangent bundle of an n-dimensional manifold M with L p bounded curvature, and extends the classical optimal regularity result of Kazdan-DeTurck [19] from Riemannian to Lorentzian metrics and to affine connections. The proof of theorem 3.1 in [33] was a long time coming [29][30][31][32], motivated by earlier work on non-optimal Lorentzian metrics of shock wave solutions of the Einstein-Euler equations [16,18,27,28], all summarized in the RSPA article [35], (including the extension to vector bundles and Yang-Mills gauge theories in [34]). The main idea for establishing theorem 3.1 was to derive from the connection transformation law a non-invariant system of elliptic PDE's on the regularizing Jacobian J as an unknown, an idea motivated by the Riemann-flat 6 Our use here of Ω, as a chart on M, slightly differs from the use of Ω ⊂ R 4 in section 2, (where we identified Ω ≡ Ωx ⊂ R 4 ), but this ambiguity is irrelevant since our result and methods are local.…”