Analytical and Numerical Approaches to Mathematical Relativity

Paul E. Ehrlich (auth.), Jörg Frauendiener, Domenico J.W. Giulini, Volker Perlick (eds.)3-540-31027-4, 978-3-540-31027-3

Today, general relativity rates among the most accurately tested fundamental theories in all of physics. However, deficiencies in our mathematical and conceptual understanding still exist, and these partly hamper further progress. For this reason alone, but no less important from the point of view that a theory-based prediction should be regarded as no better than one’s own structural understanding of the underlying theory, one should undertake serious investigations into the corresponding mathematical issues. This book contains a representative collection of surveys by experts in mathematical relativity writing about the current status of, and problems in, their fields. There are four contributions for each of the following mathematical areas: differential geometry and differential topology, analytical methods and differential equations, and numerical methods. This book addresses graduate students and specialist researchers alike.

Table of contents :
Front Matter….Pages I-XVII
A Personal Perspective on Global Lorentzian Geometry….Pages 3-34
The Space of Null Geodesics (and a New Causal Boundary)….Pages 35-50
Some Variational Problems in Semi-Riemannian Geometry….Pages 51-77
On the Geometry of pp-Wave Type Spacetimes….Pages 79-98
Front Matter….Pages I-XVII
Concepts of Hyperbolicity and Relativistic Continuum Mechanics….Pages 101-116
Elliptic Systems….Pages 117-139
Mathematical Properties of Cosmological Models with Accelerated Expansion….Pages 141-155
The Poincaré Structure and the Centre-of-Mass of Asymptotically Flat Spacetimes….Pages 157-184
Front Matter….Pages I-XVII
Computer Simulation – a Tool for Mathematical Relativity – and Vice Versa….Pages 187-203
On Boundary Conditions for the Einstein Equations….Pages 205-222
Recent Analytical and Numerical Techniques Applied to the Einstein Equations….Pages 223-249
Some Mathematical Problems in Numerical Relativity….Pages 251-274