Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods

Christian Klein (auth.)354028589X, 9783540285892

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.

Table of contents :
Introduction….Pages 1-15
The Ernst Equation….Pages 17-42
Riemann–Hilbert Problem and Fay’s Identity….Pages 43-77
Analyticity Properties and Limiting Cases….Pages 79-96
Boundary Value Problems and Solutions….Pages 97-121
Hyperelliptic Theta Functions and Spectral Methods….Pages 123-146
Physical Properties….Pages 147-171
Open Problems….Pages 173-189
References….Pages 237-245