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Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods 2005 ed. [Kietas viršelis]

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  • Formatas: Hardback, 249 pages, aukštis x plotis: 235x155 mm, weight: 1210 g, X, 249 p., 1 Hardback
  • Serija: Lecture Notes in Physics 685
  • Išleidimo metai: 18-Nov-2005
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 354028589X
  • ISBN-13: 9783540285892
Kitos knygos pagal šią temą:
Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods 2005 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 249 pages, aukštis x plotis: 235x155 mm, weight: 1210 g, X, 249 p., 1 Hardback
  • Serija: Lecture Notes in Physics 685
  • Išleidimo metai: 18-Nov-2005
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 354028589X
  • ISBN-13: 9783540285892
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783540285892.html
Raktažodžiai:
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.

Recenzijos

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 reviewers 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)

Introduction
1(16)
General Remarks on Integrability
1(5)
The Korteweg-de Vries Equation
6(2)
The Ernst Equation
8(5)
Outline of the Content of the Book
13(4)
The Ernst Equation
17(26)
Dimensional Reduction and Group Structure
18(4)
The Stationary Axisymmetric Case
22(4)
Bianchi Surfaces
26(7)
The Yang Equation
33(7)
Multi-Monopoles of the Yang--Mills--Higgs Equations
40(3)
Riemann--Hilbert Problem and Fay's Identity
43(36)
Linear System of the Ernst Equation
44(4)
Solutions to the Ernst Equation via Riemann--Hilbert Problems
48(7)
Riemann--Hilbert Problems on the Complex Plane and the Riemann Sphere
48(2)
Gauge Transformations of the Riemann--Hilbert Problem
50(2)
The Non-compact Case
52(2)
The Compact Case
54(1)
Hyperelliptic Solutions of the Ernst Equation
55(5)
Finite Gap Solutions and Picard--Fuchs Equations
60(2)
Theta-functional Solutions to the KdV and KP Equation
62(4)
Hyperelliptic and Solitonic Solutions
63(3)
Ernst Equation, Fay Identities and Variational Formulas on Hyperelliptic Surfaces
66(13)
First Derivatives of the Ernst Potential
70(1)
Action of the Laplace Operator on the Ernst Potential and Ernst Equation
71(2)
Metric Functions for the Stationary Axisymmetric Vacuum
73(3)
Relation to the Previous Form of the Solutions
76(3)
Analyticity Properties and Limiting Cases
79(18)
The Singular Structure of the Ernst Potential
79(8)
Zeros of the Denominator
80(1)
Essential Singularities
80(1)
Contours
81(1)
Axis
82(2)
Asymptotic Behavior
84(1)
Real Branch Points
84(2)
Non-real Branch Points
86(1)
Equatorial Symmetry
87(5)
Reduction of the Ernst Potential
89(3)
Solitonic Limit
92(5)
Boundary Value Problems and Solutions
97(26)
Newtonian Dust Disks
99(3)
Boundary Conditions for Counter-rotating Dust Disks
102(4)
Axis Relations
106(4)
Differential Relations in the Whole Spacetime
110(3)
Counter-rotating Disks of Genus 2
113(10)
Newtonian Limit
114(2)
Explicit Solution for Constant Angular Velocity and Constant Relative Density
116(3)
Global Regularity
119(4)
Hyperelliptic Theta Functions and Spectral Methods
123(24)
Numerical Implementations
124(13)
Spectral Approximation
125(2)
Implementation of the Square-root
127(2)
Numerical Treatment of the Periods
129(4)
Numerical Treatment of the Line Integrals
133(2)
Theta Functions
135(2)
Integral Identities
137(4)
Mass Equalities
138(2)
Virial-type Identities
140(1)
Testing Lorene
141(6)
Physical Properties
147(26)
Metric Functions
148(5)
Physical Properties of the Counter-rotating Dust Disk
153(13)
The Physical Parameters
153(3)
Mass and Angular Momentum
156(1)
Energy-momentum Tensor
157(8)
Ergospheres
165(1)
Ultrarelativistic Limit
166(7)
Ultrarelativistic Limit of the Static Disks
166(2)
Ultrarelativistic Limit for 0 < γ < 1
168(1)
Ultrarelativistic Limit of the One-component Disks
169(2)
Over-extreme Region
171(2)
Open Problems
173(18)
Integrated version of the Picard--Fuchs system
175(2)
Black-hole Disk Systems
177(5)
Newtonian Case
177(1)
Relativistic Case
178(2)
The Case g = 0
180(2)
Einstein--Maxwell Equations
182(9)
Harrison Transformations
185(1)
Asymptotic Behavior of the Harrison Transformed Solutions
186(1)
The Stationary Axisymmetric Case
187(4)
A. Riemann Surfaces and Theta Functions
191(18)
Riemann Surfaces and Algebraic Curves
191(3)
Differentiation and Integration on Riemann Surfaces
194(5)
Divisors and the Theorems of Abel and Jacobi
199(3)
Theta Functions of Riemann Surfaces
202(2)
Elliptic Surfaces
204(1)
The Trisecant Identity for Theta Functions on Riemann Surfaces
204(3)
Rauch's Formulas and Root Functions
207(2)
B. Ernst Equation and Twistor Theory
209(28)
The Quaternionic Hopf Bundle and the Twistor Transform
209(3)
Symmetry Reductions of the Penrose--Ward Transform
212(13)
The Reduced Twistor Space
212(6)
Holomorphic Vector Bundles over the Reduced Twistor Space
218(7)
Transition Matrices for the Holomorphic Vector Bundles
225(9)
The Covering of the Reduced Twistor Space
225(2)
Patching Matrices for Real, Symmetric Framed Bundles
227(3)
The Axis-simple Case
230(4)
Patching Matrices for the Class of Hyperelliptic Solutions
234(3)
References 237(10)
Index 247