The Conformal Structure of Space-Time,9783540442806

The Conformal Structure of Space-Time

by ;
ISBN13: 9783540442806
ISBN10: 3540442804
Format: Hardcover
Pub. Date: 2002-11-01
Publisher(s): Springer Verlag
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Summary

Causal relations, and with them the underlying null cone or conformal structure, form a basic ingredient in all general analytical studies of asymptotically flat space-time. The present book reviews these aspects from the analytical, geometrical and numerical points of view. Care has been taken to present the material in a way that will also be accessible to postgraduate students and nonspecialist reseachers from related fields.

Table of Contents

1 Conformal Einstein Evolution
Helmut Friedrich
1(50)
1.1 Introduction
1(3)
1.2 The Conformal Field Equations
4(17)
1.2.1 Conformal Geometry
6(3)
1.2.2 Derivation of the Conformal Field Equations
9(20)
1.3 The Penrose Proposal
21(8)
1.4 Asymptotic Behaviour of Vacuum Fields with Vanishing Cosmological Constant
29(17)
1.4.1 The Hyperboloidal Initial Value Problem
30(1)
1.4.2 On the Existence of Asymptotically Simple Vacuum Solutions
31(2)
1.4.3 The Regular Finite Cauchy Problem
33(9)
1.4.4 Time-Like Infinity
42(23)
References
46(5)
2 Some Global Results for Asymptotically Simple Space-Times
Gregory J. Galloway
51(10)
2.1 Introduction
51(2)
2.2 The Null Splitting Theorem
53(2)
2.3 Proof of Theorem 2.1
55(3)
2.4 Concluding Remarks
58(1)
References
59(2)
3 Black Holes
Piotr T. Chrusciel
61(42)
3.1 Introduction
61(1)
3.2 Experimental Evidence
62(3)
3.3 Causality for Symmetric Hyperbolic Systems
65(6)
3.3.1 Dumb Holes
69(1)
3.3.2 Optical Holes
70(1)
3.3.3 Trapped Surfaces
70(1)
3.4 Standard Black Holes
71(6)
3.4.1 Scri Regularity Conditions, and the Area Theorem
75(8)
3.5 Horizons
77(2)
3.6 Apparent Horizons
79(1)
3.7 Classification of Stationary Solutions ("No Hair Theorems")
80(3)
3.8 Black Holes Without Scri
83(13)
3.8.1 Naive Black Holes
84(2)
3.8.2 Quasi-local Black Holes
86(4)
3.8.3 Finding Horizons
90(19)
References
96(7)
4 Conformal Geometry, Differential Equations and Associated Transformations
Simonetta Frittelli, Niky Kamran, Ezra T. Newman
103(10)
4.1 Introduction
103(2)
4.2 Example of Contact-Envelope Transformation
105(4)
4.3 Generalizations
109(2)
4.3.1 Three-Dimensional Conformal Lorentzian Geometries
109(1)
4.3.2 Four-Dimensional Conformal Lorentzian Geometries
110(21)
References
111(2)
5 Twistor Geometry of Conformal Infinity
Roger Penrose
113(10)
5.1 Non-linear Gravitons
113(2)
5.2 The Reasonableness of f+
115(1)
5.3 The Construction of Projective Twistor Space PT from I+
115(2)
5.4 The Construction of the Full Twistor Space T from I+
117(1)
5.5 The Local Structure of Twistor Space PT
118(1)
5.6 Present Status of the Role of T in Encoding Ricci-Flatness
119(1)
References
120(3)
6 Isotropic Cosmological Singularities
K. Paul Tod
123(12)
6.1 Introduction
123(2)
6.2 Formalism and Extensions
125(3)
6.3 Review of Polytropic Perfect Fluid Case
128(3)
6.4 Further Matter Models
131(2)
6.4.1 Massive Einstein-Vlasov
132(1)
6.4.2 Scalar Fields
132(1)
6.4.3 Einstein-Yang-Mills-Vlasov
132(1)
6.4.4 Einstein-Boltzmann
132(6)
6.5 Conclusion arnd Future Possibilities
133(1)
References
133(2)
7 Polyhomogeneous Expansions Close to Null and Spatial Infinity
Juan Antonio Valiente Kroon
135(26)
7.1 Introduction
135(1)
7.2 Minkowski Space-Time Close to Null and Spatial Infinity
136(2)
7.3 Linearised Gravity in the F-Gauge
138(4)
7.3.1 Initial Data for Linearised Gravity
140(5)
7.4 A Regularity Condition at Spatial Infinity
142(3)
7.5 Polyhomogencous Expansions
145(13)
7.5.1 A Substraction Argument
145(2)
7.5.2 An Investigation of Expansions Close to Null Infinity
147(10)
7.5.3 Concluding Remarks
157(16)
References
158(3)
8 Asymptotically Flat and Regular Cauchy Data
Sergio Mai
161(22)
8.1 Introduction
161(11)
8.2 Solution of thc Hamiltonian Constraint with Logarithmic Terms
172(1)
8.3 Explicit Solutions of thc Momentum Constraint
173(5)
8.3.1 The Momentum Constraint on Euclidean Space
173(3)
8.3.2 Axially Symmetric Initial Data
176(11)
8.4 Main Ideas in the Proof of Theorem 8.1
178(2)
8.5 Final Continents
180(1)
References
180(3)
9 Construction of Hyperboloidal Initial Data
Lars Andersson
183(12)
9.1 Introduction
183(1)
9.2 Preliminaries
184(1)
9.3 Conformal Rescalings of Minkowski Space
185(2)
9.4 Conformal Constraint Equations
187(4)
9.4.1 Constraint Means Curvature Hypersurfaces
188(1)
9.4.2 Degenerate Elliptic Equations
189(1)
9.4.3 Regularity of Solutions to the Conformal Constraint Equations
190(1)
9.5 The Initial Value Problem
191(2)
9.5.1 Gauge Condition at OM
191(1)
9.5.2 Evolution it OM
192(5)
9.6 Discussion
193(1)
References
193(2)
10 Exploring the Conformal Constraint Equations
Adrian Butcher
195(28)
10.1 Introduction
195(2)
10.2 The Conformal Constraint Equations
197(9)
10.2.1 Deriving thc Equations
197(4)
10.2.2 Reduction to the Extended Constraint Equations
201(1)
10.2.3 Properties of the Extended Constraint Equations
202(4)
10.3 Asymptotically Flat Solutions of the Extended Constraint
Equations in the Time Symmetric Case
206(15)
10.3.1 Statement of the Main Theorem
206(2)
10.3.2 Formulating an Elliptic Problem
208(1)
10.3.3 Choosing the Banach Spaces
209(3)
10.3.4 First Attempt to Solve the Associated System
212(5)
10.3.5 Reestablishing Surjectivity and Solving the Associated System 215
10.3.6 Satisfying the Harmonic Coordinate Condition
217(23)
References
221(2)
11 Criteria for (In)finite Extent of Static Perfect Fluids
Walter Simon
223(16)
11.1 Introduction
223(9)
11.2 The Main Theorem 225
11.3 The Virial Theorem 230
11.4 Proof of the Main Theorem
232(3)
11.5 Discussion
235(2)
References
237(2)
12 Problems and Successes in the Numerical Approach to the Conformal Field Equations
Sascha Husa
239(22)
12.1 Introduction
239(1)
12.2 Algorithms
240(8)
12.2.1 Problem Overview
240(3)
12.2.2 Construction of "Extended" Hyperboloidal Initial Data
243(3)
12.2.3 Black Hole Initial Data
246(1)
12.2.4 Numerical Setup for Evolutions
247(1)
12.2.5 Physics Extraction
248(44)
12.3 Results for Weak Data
248(3)
12.4 Computational Aspects
251(4)
12.5 Discussion
255(2)
References
257(4)
13 Some Aspects of the Numerical Treatment of the Conformal Field Equations
J÷rg Frauendiener
261(22)
13.1 Introduction
261(2)
13.2 The Andersson-Chrusciel-Friedrich Procedure
263(3)
13.3 The LichnTrowicz-Yamabe-Equation
266(7)
13.4 Constructing Initial Data
273(5)
13.5 Conclusion
278(2)
References
280(3)
14 Data for the Numerical Calculation of the Kruskal Space-Time
Bernd G. Schmidt
283(14)
14.1 Introduction
283(1)
14.2 Conformal Extension of the Kruskal Space-Time
284(5)
14.3 A Space-Like Hypersurface
289(3)
Intersecting I+L and I+R
14.4 A Foliation Intersecting Both I
292(1)
14.5 Numerical Calculation of tbc Kruskal Space-Time
293(2)
References
295(2)
15 Numerics of the Characteristic Formulation in Bondi Variables. Where We Are and What Lies Ahead
Luis Lehner
297(16)
15.1 Introduction
297(1)
15.2 Characteristic Formulation of GR in Bomb Variables
298(6)
15.2.1 Initial Boundary Value Problem
299(2)
15.2.2 News
301(1)
15.2.3 Inertial Coordinates
302(4)
15.3 Numerical Details
304(1)
15.4 Applications
305(5)
15.4.1 Black Hole-Star Binaries
306(2)
15.4.2 Binary Black Hole Problem
308(28)
15.5 Final Comments
310(1)
References
311(2)
16 Numerical Experiments at Null Infinity
Robert A. Bartnik. Andrew H. Norton
313(14)
16.1 Introduction
313(2)
16.2 NQS Metric
315(2)
16.3 NQS Formal Asymptotics
317(4)
16.4 Genericity
321(1)
16.5 The NQS Code
322(1)
16.6 Numerical Results
323(2)
References
325(2)
17 Local Characteristic Algorithms for Relativistic Hydrodynamics
Jose A. Font
327(22)
17.1 Introduction
327(2)
17.2 Relativistic Hydrodynamic Equations
329(3)
17.3 High-Resolution Numerical Schemes
332(4)
17.4 Applications
336(7)
17.4.1 Shock Tube Test
336(2)
17.4.2 Gravitational Collapse of Supermassive Stars
338(2)
17.4.3 Null Cone Evolution of Relativisitic Stars
340(20)
17.5 Summary
343(1)
References
344(5)
18 Simulations of Generic Singularities in Harmonic Coordinates
David Garfinkle
349(10)
18.1 Introduction 349
351(1)
18.2 Equations and Numerical Methods
352(4)
18.3 Results
18.4 Discussion
356(1)
References
357(2)
19 Some Mathematical and Numerical Questions Connected with First and Second
Order Time-Dependent Systems of Partial Differential Equations
Heinz-O. Kreiss, Omar E. Ortiz
359(12)
19.1 Introduction
359(1)
19.2 Well Posed Problems
360(4)
19.2.1 First Order Systems
360(2)
19.2.2 Second Order Systems
362(2)
19.3 Second Order Initial Value Formulations for General Relativity
364(2)
19.4 Difference Approximations
366(2)
19.5 Constraints
368(2)
References
370(1)
Index 371

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