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

TL;DR

The paper introduces the RT-equations, a new nonlinear elliptic PDE system, to optimize the regularity of connections in Lorentzian geometry, extending Uhlenbeck compactness and addressing regularity singularities in GR shock waves.

Contribution

It develops a general existence theory for the RT-equations in $L^p$ spaces, enabling optimal regularity smoothing of connections and removal of singularities in GR shock waves.

Findings

Extended Uhlenbeck compactness to Lorentzian geometry.

Proved regularity singularities at GR shock waves can be smoothed by coordinate transformations.

Established a multi-dimensional existence theory for the RT-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 ${\rm Riem}(\Gamma)$. 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.