Shock Wave Interactions in General Relativity
by Jeffrey Groah, Joel Smoller, B. Temple

Table of contents
  • Introduction
  • The Initial Value Problem in Special Relativity
  • A Shock Waves Formulation of Einstein Equations
  • Existence and Consistency for the Initial Value Problem
  • Bibliography

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Temple organized these notes from joint work with collaborators Smoller and Groah, and the purpose is to provide an exposition of the locally inertial Glimm Scheme that was first introduced by Groah and Temple.

"This monograph presents a self contained mathematical treatment of the initial value problem for shock wave solutions of the Einstein equations in General Relativity. The first two chapters provide background for the introduction of a locally intertial Glimm Scheme, a non-dissipative numerical scheme for approximating shock wave solutions of the Einstein equations in spherically symmetric spacetimes. What follows is a careful analysis of this scheme providing a proof of the existence of (shock wave) solutions of the spherically symmetric Einstein equations for a perfect fluid, starting from initial density and velocity profiles that are only locally of bounded total variation. The book covers the initial value problems for Einstein's gravitational field equations with fluid sources and shock wave initial data. It has a clearly outlined goal: proving a certain local existence theorem. Concluding remarks are added and commentary is provided throughout. The book will be useful to graduate students and researchers in mathematics and physics."