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# 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

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n3.a1

#### Authors

#### 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 $\mathit{Riem}(\Gamma)$ 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.

#### Keywords

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