General relativity is the modern theory of the gravitational ?eld. It is a deep subject that couples ?uid dynamics to the geometry of spacetime through the Einstein equations. The subject has seen a resurgence of interest recently, partlybecauseofthespectacularsatellitedatathatcontinuestoshednewlight 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 ?eld equations, being the ?rst to describe a ?eld 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 ?uids, 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.
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.
