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We present authors' new theory of the RT-equations, nonlinear ellipticpartial differential equations which determine the coordinate transformationswhich smooth connections $\Gamma$ to optimal regularity, one derivativesmoother than the Riemann curvature tensor ${\rm Riem}(\Gamma)$. As oneapplication we extend Uhlenbeck compactness from Riemannian to Lorentziangeometry; and as another application we establish that regularity singularitiesat GR shock waves can always be removed by coordinate transformation. This isbased on establishing a general multi-dimensional existence theory for theRT-equations, by application of elliptic regularity theory in $L^p$ spaces. Thetheory and results announced in this paper apply to arbitrary $L^\infty$connections on the tangent bundle $T\mathcal{M}$ of arbitrary manifolds$\mathcal{M}$, including Lorentzian manifolds of General Relativity.

How to smooth a crinkled map of spacetime: Uhlenbeck compactness for  $L^\infty$ connections and optimal regularity for general relativistic shock  waves by the Reintjes-Temple-equations