Shock Wave Interactions and the Riemann-flat Condition: The Geometry behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity

TL;DR

This paper introduces the Riemann-flat condition as a geometric criterion to analyze the smoothness of gravitational metrics at shock waves in general relativity, addressing the existence of locally inertial frames and regularity singularities.

Contribution

It defines the Riemann-flat condition to determine metric regularity at shocks and proves the existence of locally inertial frames for Lipschitz metrics in spherically symmetric spacetimes.

Findings

The Riemann-flat condition characterizes when metrics are two derivatives smoother than curvature.

Locally inertial frames always exist for shock wave metrics in spherical symmetry.

Provides an explicit method to construct inertial coordinates for Lipschitz metrics.

Abstract

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.