[Paper Review] Shock Wave Interactions in General Relativity: The Geometry behind Metric Smoothing and the Existence of Locally Inertial Frames

Moritz Reintjes, Blake Temple|arXiv (Cornell University)|Oct 7, 2016

Cosmology and Gravitation Theories8 references4 citations

TL;DR

This paper establishes a necessary and sufficient condition for smoothing a $C^{0,1}$ (Lipschitz continuous) spacetime metric to $C^{1,1}$ regularity via coordinate transformations, showing that such smoothing is possible if and only if the singular part of the metric connection extends to a Riemann-flat connection. It proves that locally inertial frames exist in spherically symmetric shock wave spacetimes, even when the metric cannot be smoothed, offering a new regularity result for the Glimm scheme.

ABSTRACT

We prove a necessary and sufficient condition for determining the essential smoothness of weak solutions of the Einstein equations at apparent singularities where the gravitational metric tensor is only Lipschitz continuous, but the tensor is in $L^{\infty}$, a regularity so low that locally inertial frames might not exist. Namely, we prove that the question whether there exists a coordinate transformation which smooths a metric from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of a point is equivalent to the condition that the singular part of the metric connection can be extended to a Riemann flat connection in that neighborhood. This applies to shock-wave solutions in General Relativity, and the framework leads to a definition of the curvature of the singular part of a connection, (i.e., the of the shock-set), and we prove that this vanishes if and only if the spacetime metric can be smoothed within the $C^{1,1}$ atlas. As an application of our method we prove that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, a new regularity result for the Glimm scheme, independent of whether the metric itself can be smoothed in a neighborhood.

Motivation & Objective

• To determine the geometric conditions under which a $C^{0,1}$ metric in general relativity can be smoothed to $C^{1,1}$ via coordinate transformations.
• To define and analyze the curvature of the singular part of a connection, particularly at shock fronts in spacetime.
• To establish the existence of locally inertial frames in spherically symmetric shock wave solutions, independent of metric smoothing.
• To provide a new regularity result for the Glimm scheme in the context of shock wave interactions in general relativity.

Proposed method

• The authors analyze the structure of the metric connection at points of $C^{0,1}$ regularity, focusing on the singular part of the connection.
• They introduce a geometric condition: the singular part of the connection extends to a Riemann-flat connection in a neighborhood if and only if the metric can be smoothed to $C^{1,1}$.
• The curvature of the singular connection is rigorously defined and shown to vanish precisely when smoothing is possible.
• The method applies to shock wave solutions in general relativity, particularly in spherically symmetric spacetimes.
• The framework uses tools from differential geometry and weak solutions of the Einstein equations, focusing on $L^∞$-regular metrics.
• The analysis relies on the equivalence between the existence of a $C^{1,1}$ coordinate atlas and the Riemann-flatness of the singular connection.

Experimental results

Research questions

• RQ1 Under what geometric conditions can a $C^{0,1}$ spacetime metric be smoothed to $C^{1,1}$ via coordinate transformations?
• RQ2 Can the curvature of the singular part of a connection be meaningfully defined in the context of weak solutions of the Einstein equations?
• RQ3 Does the existence of locally inertial frames in shock wave spacetimes depend on the smoothability of the metric?
• RQ4 Is there a regularity result for the Glimm scheme applicable to shock wave solutions in spherically symmetric spacetimes?

Key findings

• A $C^{0,1}$ metric can be smoothed to $C^{1,1}$ in a neighborhood if and only if the singular part of the metric connection extends to a Riemann-flat connection in that neighborhood.
• The curvature of the singular connection vanishes if and only if the metric can be smoothed within the $C^{1,1}$ atlas.
• Locally inertial frames exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, even when the metric is not $C^{1,1}$.
• The result provides a new regularity guarantee for the Glimm scheme, independent of metric smoothability.
• The framework offers a geometric criterion for metric smoothing that is both necessary and sufficient, applicable to weak solutions in general relativity.