Shock Wave Interactions in General Relativity A Locally Inertial Gli
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 Gl
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Jeffrey Groah Joel Smoller Blake Temple
Shock Wave Interactions in General Relativity A Locally Inertial Glimm Scheme for Spherically Symmetric Spacetimes
Jeffrey Groah Department of Mathematics Montgomery College Conroe, TX 77384 USA
Joel Smoller Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA
Blake Temple Institute of Theoretical Dynamics University of California, Davis Davis, CA 95616 USA
Mathematics Subject Classification (2000): 35L65, 35L67, 83C05 Library of Congress Control Number: 2006932412 ISBN-10: 0-387-35073-X ISBN-13: 978-0-387-35073-8
e-ISBN-10: 0-387-44602-8 e-ISBN-13: 978-0-387-44602-8
Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring St, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com
Preface
General relativity is the modern theory of the gravitational field. It is a deep subject that couples fluid dynamics to the geometry of spacetime through the Einstein equations. The subject has seen a resurgence of interest recently, partly because of the spectacular satellite data that continues to shed new light on the nature of the universe...Einstein’s theory of gravity is still the basic theory we have to describe the expanding universe of galaxies. But the Einstein equations are of great physical, mathematical and intellectual interest in their own right. They are the granddaddy of all modern field equations, being the first to describe a field by curvature, an idea that has impacted all of physics, and that revolutionized the modern theory of elementary particles. In these notes we describe a mathematical theory of shock wave propagation in general relativity. Shock waves are strong fronts that propagate in fluids, and across which there is a rapid change in density, pressure and velocity, and they can be described mathematically by discontinuities across which mass, momentum and energy are conserved. In general relativity, shock waves carry with them a discontinuity in spacetime curvature. The main object of these notes is to introduce and analyze a practical method for numerically computing shock waves in spherically symmetric spacetimes. The method is locally inertial in the sense that the curvature is set equal to zero in each local grid cell. Although it formally appears that the method introduces singularities at shocks, the arguments demonstrate that thi
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