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

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.

V5: Improved presentation, in particular, to Section 6. Results unchanged. V4: We shortened the presentation, added Def 3.1 and removed last section of previous version. V3: We extended results from connections of bounded variation to connections bounded in $L^{\infty }$, otherwise main results remain identical. V2: Result of Theorem 1.5 was extended from 2-D to spherically symmetric space-times

References in corpus (2)

• The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
• No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families

Cited by in corpus (1)

• Optimal Regularity and Uhlenbeck Compactness for General Relativity and Yang-Mills Theory