  We present authors' new theory of the RT-equations, nonlinear elliptic
partial differential equations which determine the coordinate transformations
which smooth connections $\Gamma$ to optimal regularity, one derivative
smoother than the Riemann curvature tensor ${\rm Riem}(\Gamma)$. As one
application we extend Uhlenbeck compactness from Riemannian to Lorentzian
geometry; and as another application we establish that regularity singularities
at GR shock waves can always be removed by coordinate transformation. This is
based on establishing a general multi-dimensional existence theory for the
RT-equations, by application of elliptic regularity theory in $L^p$ spaces. The
theory 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.
