Description
The Regularity Transformation (RT-)equations are an elliptic system of partial differential equations which determines coordinate and gauge transformations that remove apparent singularities in space- time by furnishing optimal metric regularity. The resulting gain of one derivative for connections above their Lp curvature suffices to establish Uhlenbeck compactness. By developing an existence theory for the RT-equations we prove optimal regularity and Uhlenbeck compactness in Lorentzian geometry, including general affine connections and connections on vector bundles with both compact and non-compact gauge groups. As an application in General Relativity, our optimal regularity result proves that the Lorentzian 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. It is currently an open problem whether the RT-equations could provide a general procedure for removing apparent singularities at black hole horizons.