Shock Wave Interactions in General Relativity: A Locally Inertial Glimm Scheme for Spherically Symmetric Spacetimes / Edition 1

by Jeffrey Groah, Joel Smoller, Blake Temple

Overview

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, partlybecauseof thespectacularsatellitedatathatcontinuestoshednewlight on the nature of the universe. . . Einstein’s theory of gravity is still the basic theorywehavetodescribetheexpandinguniverseofgalaxies. ButtheEinstein 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 noteswedescribeamathematicaltheoryofshockwavepropagationingeneral 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 bedescribedmathematicallybydiscontinuitiesacrosswhichmass,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 thesensethatthecurvatureissetequaltozeroineachlocalgridcell. Although it formally appears that the method introduces singularities at shocks, the arguments demonstrate that this is not the case. The third author would like to dedicate these notes to his father, Paul Blake Temple, who piqued the author’s interest in Einstein’s theory when he was a young boy, and whose interest and encouragement has been an inspirationthroughout his adult life.

Product Details

ISBN-13:	9781441922465
Publisher:	Springer New York
Publication date:	11/19/2010
Series:	Springer Monographs in Mathematics
Edition description:	Softcover reprint of hardcover 1st ed. 2007
Pages:	152
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

The Initial Value Problem in Special Relativity.- A Shock Wave Formulation of the Einstein Equations.- Existence and Consistency for the Initial Value Problem.