How to smooth a crinkled map of space-time : Uhlenbeck compactness for L^∞ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equations

Abstract

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Γ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic 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 Lp spaces. The theory and results announced in this paper apply to arbitrary L∞ connections on the tangent bundle TM of arbitrary manifolds M, including Lorentzian manifolds of general relativity.