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

Abstract

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 $\mathrm{Riem}(\Gamma )$. 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.

This paper summarises the theory developed in arXiv:1805.01004, arXiv:1808.06455 and arXiv:1912.12997. V5: Improved wording and minor corrections in Section 7. V4: Cor 3.1 added. V3: Thms 2.6 - 2.8, addressing Uhlenbeck compactness and $L^{\infty }$ connections, added. Improved discussion section and introduction. V2: Improved presentation; Curvature bound added to Thm 2.4 and 2.5