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

  • Moritz ReintjesEmail author
  • Blake Temple
Article
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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 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 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.

Notes

Acknowledgements

We thank Heinrich Freistühler for a careful reading of our manuscript and very helpful suggestions.

Compliance with Ethical Standards

Funding

M. Reintjes was funded through CAPES-Brazil as a Post-Doctorate at IMPA (Rio de Janeiro) in 2015 and 2016. In 2017 and 2018, M. Reintjes was supported by FCT/Portugal through (GPSEinstein) PTDC/MAT-ANA/1275/2014 and UID/MAT/04459/2013. B. Temple was supported by NSF Applied Mathematics Grant No. DMS-010-2493.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fachbereich für MathematikUniversität KonstanzConstanceGermany
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

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