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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010101117 | QC173.6 G76 2007 | Open Access Book | Book | Searching... |
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Summary
Summary
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.
Table of Contents
1 Introduction | p. 3 |
1.1 Introduction to Differential Geometry and General Relativity | p. 5 |
1.2 Introduction to the Einstein Equations | p. 11 |
1.3 t The Simplest Setting for Shock Waves p. 16 | |
1.4 A Covariant Glimm Scheme | p. 18 |
2 The Initial Value Problem in Special Relativity | p. 21 |
2.1 Shock Waves in Minkowski Spacetime | p. 21 |
2.2 The Relativistic Euler Equations as a System of Conservation Laws | p. 26 |
2.3 The Wave Speeds | p. 28 |
2.4 The Shock Curves | p. 33 |
2.5 Geometry of the Shock Curves | p. 37 |
2.6 The Rieman Problem | p. 43 |
2.7 The Initial Value Problem | p. 45 |
2.8 Appendix | p. 50 |
3 A Shock Wave Formulation of the Einstein Equations | p. 53 |
3.1 Introduction | p. 53 |
3.2 The Einstein Equations for a Perfect Fluid with Spherical Symmetry | p. 56 |
3.3 The Einstein Equations aks a System of Conservation Laws with Sources | p. 62 |
3.4 Statement of the General Problem | p. 67 |
3.5 Wave Speeds | p. 69 |
4 Existence and Consistency for the Initial Value Problem | p. 73 |
4.1 Introduction | p. 73 |
4.2 Preliminaries | p. 77 |
4.3 The Fractional Step Scheme | p. 83 |
4.4 The Riemann Problem Step | p. 91 |
4.5 The ODE Step | p. 99 |
4.6 Estimates for the ODE Step | p. 104 |
4.7 Analysis of the Approximate Solutions | p. 110 |
4.8 The Elimination of Assumptions | p. 124 |
4.9 Convergence | p. 135 |
4.10 Concluding Remarks | p. 144 |
References | p. 147 |
Index | p. 149 |