Methods and Applications of Analysis

Volume 27 (2020)

Number 3

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

Pages: 199 – 242



Moritz Reintjes (Fachbereich für Mathematik und Statistik, Universität Konstanz, Germany)

Blake Temple (Department of Mathematics, University of California, Davis, Cal., U.S.A.)


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 $\mathit{Riem}(\Gamma)$ are both in $L^\infty$, the curvature being one derivative smoother than the curvature of a general Lipschitz metric. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. In fact, the Einstein equations naturally admit coordinates in which $\Gamma$ has the same regularity as $\mathit{Riem}(\Gamma)$ because the curvature transforms as a tensor, but the connection does not. Here we address the mathematical problem as to whether the condition that $\mathit{Riem}(\Gamma)$ has the same regularity as $\Gamma$, or equivalently the exterior derivatives $d \Gamma$ have the same regularity as $\Gamma$, is sufficient to allow 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 $\Gamma, \operatorname{Riem}(\Gamma) \in W^{m,p}, m \geq 1, n \lt p \lt \infty$, 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 $\operatorname{Riem}(\Gamma) \in W^{m,p}, m \geq 1, n \lt p \lt \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x \to y$ such that $\Gamma(y) \in W^{m+1,p}$. This implies all discontinuities in $m^\prime$th derivatives of $\delta \Gamma$ cancel out, the transformation $x \to y$ raises the connection regularity by one order, and $\Gamma$ 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.


optimal metric regularity, Lorentzian geometry, regularity singularities, general relativity, elliptic PDE’s, geometric analysis, shock waves

2010 Mathematics Subject Classification

Primary 83C75. Secondary 76L05.

M. Reintjes is currently supported by the German Research Foundation, DFG grant FR822/10-1, and was supported by FCT/Portugal through (GPSEinstein) PTDC/MATANA/ 1275/2014 and UID/MAT/04459/2013 from January 2017 until December 2018.

Received 8 May 2019

Accepted 7 October 2020

Published 13 August 2021