Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods

by Christian Klein, Olaf Richter

Overview

Exact solutions to Einstein‘s equations have been useful for the understanding of general relativity in many respects. They have led to physical concepts as black holes and event horizons and helped to visualize interesting features of the theory. In addition they have been used to test the quality of various approximation methods and numerical codes. The most powerful solution generation methods are due to the theory of Integrable Systems. In the case of axisymmetric stationary spacetimes the Einstein equations are equivalent to the completely integrable Ernst equation. In this volume the solutions to the Ernst equation associated to Riemann surfaces are studied in detail and physical and mathematical aspects of this class are discussed both analytically and numerically.

Product Details

ISBN-13:	9783642066771
Publisher:	Springer Berlin Heidelberg
Publication date:	12/15/2010
Series:	Lecture Notes in Physics , #685
Edition description:	Softcover reprint of hardcover 1st ed. 2005
Pages:	249
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

Introduction.- The Ernst Equation.- Riemann-Hilbert Problem and Fay's Identity.- Analyticity Properties and Limiting Cases.- Boundary Value Problems and Solutions.- Hyperelliptic Theta Functions and Spectral Methods.- Physical Properties.- Open Problems.- Riemann Surfaces and Theta Functions.- Ernst Equation and Twister Theory.- Index.

Editorial Reviews

From the reviews:

"This book covers these areas – the reduction of the Einstein vacuum equations to the Ernst equation, the reinterpretation of the Ernst equation as an integrable system and the use of techniques of integrable systems … . This book provides an excellent exposition of these ideas; as well as providing a sound introduction … . This is an excellently written monograph with an encyclopedic list of references and it should be of interest to a wide range of people … ." (Ian A. B. Strachan, Mathematical Reviews, Issue 2006 k)

"What the present book describes are some of the heroic efforts that have been undertaken to construct physically significant spacetimes by solving the vacuum Ernst equation. … It is the reviewer’s opinion that the resulting book will be more useful as a resource for those who are already well versed in the subject of integrable systems … ." (Frederick J Ernst, Classical and Quantum Gravity, Vol. 24, 2007)

From the Publisher