I will present our theory of the Regularity Transformation (RT-)equations, an elliptic system of partial differential equations which determines coordinate and gauge transformations that remove apparent singularities in spacetime by establishing optimal regularity for general connections. This gain of one derivative for the connections above their $L^p$ curvature then 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 implies 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 —, resolving a longstanding open problem in the field.