LNP-685: Ernst Equation And Riemann Surfaces: Analytical And Numerical MethodsErnst方程与黎曼曲面:分析与数值法
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作者： Christian Klein，Olaf Richter著
出 版 社： 北京燕山出版社
出版时间： 2005-12-1字数：版次：页数： 249印刷时间： 2005/12/01开本： 16开印次：纸张： 胶版纸I S B N ： 9783540285892包装： 精装内容简介
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.
目录
1Introduction
1.1 General Remarks on Integrability
1.2 The Korteweg de Vries Equation
1.3 The Ernst Equation
1.4 Outline of the Content of the Book
2The Ernst Equation
2.1 Dimensional Reduction and Group Structure
2.2 The Stationary Axisymmetric Case
2.3 Bianchi Surfaces
2.4 The Yang Equation
2.5 Multi-Monopoles of the Yang Mills-Higgs Equations
3Riemann-Hilbert Problem and Fay's Identity
3.1 Linear System of the Ernst Equation
3.2 Solutions to the Ernst Equation via Riemann-Hilbert Problems
3.2.1 Riemann-Hilbert Problems on the Complex Plane and the Riemann Sphere
3.2.2 Gauge Transformations of the Riemann-Hilbert Problem
3.2.3 The Non-compact Case
3.2.4 The Compact Case
3.3 Hyperelliptic Solutions of the Ernst Equation
3.4 Finite Gap Solutions and Picard-F~chs Equations
3.5 Theta-functional Solutions to the KdV and KP Equation
3.5.1 Hyperelliptic and Solitonic Solutions
3.6 Ernst Equation, Fay Identities and Variational Formulas on Hyperelliptic Surfaces
3.6.1 First Derivatives of the Ernst Potential
3.6.2 Action of the Laplace Operator on the Ernst Potential and Ernst Equation
3.6.3 Metric Functions for the Stationary Axisymmetric Vacuum
3.6.4 Relation to the Previous Form of the Solutions
4Analyticity Properties and Limiting Cases
4.1 The Singular Structure of the Ernst Potential
4.1.1 Zeros of the Denominator
4.1.2 Essential Singularities
4.1.3 Contours
4.1.4 Axis
4.1.5 Asymptotic Behavior
4.1.6 Real Branch Points
4.1.7 Non-real Branch Points
4.2 Equatorial Symmetry
4.2.1 Reduction of the Ernst Potential
4.3 Solitonic Limit
5 Boundary Value Problems and Solutions
5.1 Newtonian Dust Disks
5.2 Boundary Conditions for Counter-rotating Dust Disks
5.3 Axis Relations
5.4 Differential Relations in the Whole Spacetime
5.5 Counter-rotating Disks of Genus 2
5.5.1 Newtonian Limit
5.5.2 Explicit Solution for Constant Angular Velocity and Constant Relative Density
5.5.3 Global Regularity
6 Hyperelliptic Theta Functions and Spectral Methods
6.1 Numerical Implementations
6.1.1 Spectral Approximation
6.1.2 Implementation of the Square-root
6.1.3 Numerical Treatment of the Periods
6.1.4 Numerical Treatment of the Line Integrals
6.1.5 Theta Functions
6.2 Integral Identities
6.2.1 Mass Equalities
6.2.2 Virial-type Identities
6.3 Testing LORENE
7 Physical Properties
7.1 Metric Functions
7.2 Physical Properties of the Counter-rotating Dust Disk
7.2.1 The Physical Parameters
7.2.2 Mass and Angular Momentum
……
8 Open Problems
A Riemann Surfaces and Theta Functions
B Ernst Equation and Twistor Theory
References
Index
