Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in $L^p$-spaces

TL;DR

This paper proves that in General Relativity, the regularity of the gravitational metric can be improved through coordinate transformations by solving nonlinear elliptic RT-equations in $L^p$-spaces, establishing optimal regularity results.

Contribution

The paper introduces a new existence theory for the RT-equations, showing that connections with certain regularity can be transformed to higher regularity, resolving the optimal regularity problem in GR.

Findings

Existence of solutions to RT-equations in $W^{m,p}$ spaces.

Coordinate transformations can improve connection regularity by one order.

Optimal regularity of the metric is achieved via elliptic regularity theory.

Abstract

Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $\Gamma$ and curvature $Riem(\Gamma)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the condition that $Riem(\Gamma)$ has the same regularity as $\Gamma$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $\delta\Gamma$, thereby raising the regularity of the connection and the metric by one order--a subtle problem. We have now discovered, in a framework much more general than GR shock waves, that the regularization of non-optimal connections is determined by a nonlinear system of elliptic equations with matrix valued differential forms as unknowns, the Regularity Transformation equations, or RT-equations. In this paper we establish the first existence theory for the nonlinear RT-equations in the general case when $\Gamma, {\rm Riem}(\Gamma)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $\Gamma$ is any affine connection on an $n$-dimensional manifold. From this we conclude that for any such connection $\Gamma(x) \in W^{m,p}$ with ${\rm Riem}(\Gamma) \in W^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x\to y$ such that $\Gamma(y) \in W^{m+1,p}$. That is, $\Gamma$ exhibits optimal regularity in $y$-coordinates. The problem of optimal regularity for the hyperbolic Einstein equations is thus resolved by elliptic regularity theory in $L^p$-spaces applied to the RT-equations.