The Regularity Transformation Equations: An elliptic mechanism for smoothing gravitational metrics in General Relativity

TL;DR

This paper introduces the RT-equations, a set of nonlinear elliptic equations, to determine coordinate transformations that smooth gravitational metrics in General Relativity, resolving the existence of regularity singularities above certain smoothness thresholds.

Contribution

The paper proves the equivalence between smoothing gravitational metrics via coordinate transformations and solving the RT-equations, establishing the existence of such transformations above a smoothness threshold.

Findings

Existence of coordinate transformations smoothing metrics to optimal regularity.

No regularity singularities above the threshold of smoothness.

Framework applicable to connections and curvature in GR.

Abstract

A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime.