O'Neill Semi-Riemannian Geometry With Applications to Relativity

1. Auflage 1983
ISBN: 978-0-08-057057-0
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 468 Seiten

This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Autoren/Hrsg.

O'Neill, Barrett

Inhaltsverzeichnis

1;Front Cover;1
2;SEMI-RIEMANNIAN GEOMETRY;4
3;Copyright Page;5
4;CONTENTS;6
5;Preface;12
6;Notation and Terminology;14
7;CHAPTER 1. MANIFOLD THEORY;16
7.1;Smooth Manifolds;16
7.2;Smooth Mappings;19
7.3;Tangent Vectors;21
7.4;Differential Maps;24
7.5;Curves;25
7.6;Vector Fields;27
7.7;One-Forms;29
7.8;Submanifolds;30
7.9;Immersions and Submersions;34
7.10;Topology of Manifolds;36
7.11;Some Special Manifolds;39
7.12;Integral Curves;42
8;CHAPTER 2. TENSORS;49
8.1;Basic Algebra;49
8.2;Tensor Fields;50
8.3;Interpretations;51
8.4;Tensors at a Point;52
8.5;Tensor Components;54
8.6;Contraction;55
8.7;Covariant Tensors;57
8.8;Tensor Derivations;58
8.9;Symmetric Bilinear Forms;61
8.10;Scalar Products;62
9;CHAPTER 3. SEMI-RIEMANNIAN MANIFOLDS;69
9.1;Isometries;73
9.2;The Levi-Civita Connection;74
9.3;Parallel Translation;80
9.4;Geodesics;82
9.5;The Exponential Map;85
9.6;Curvature;89
9.7;Sectional Curvature;92
9.8;Semi-Riemannian Surfaces;95
9.9;Type-Changing and Metric Contraction;96
9.10;Frame Fields;99
9.11;Some Differential Operators;100
9.12;Ricci and Scalar Curvature;102
9.13;Semi-Riemannian Product Manifolds;104
9.14;Local Isometries;105
9.15;Levels of Structure;108
10;CHAPTER 4. SEMI-RIEMANNIAN SUBMANIFOLDS;112
10.1;Tangents and Normals;112
10.2;The Induced Connection;113
10.3;Geodesics in Submanifolds;117
10.4;Totally Geodesic Submanifolds;119
10.5;Semi-Riemannian Hypersurfaces;121
10.6;Hyperquadrics;123
10.7;The Codazzi Equation;129
10.8;Totally Umbilic Hypersurfaces;131
10.9;The Normal Connection;133
10.10;A Congruence Theorem;135
10.11;Isometric Immersions;136
10.12;Two-Parameter Maps;137
11;CHAPTER 5. RIEMANNIAN AND LORENTZ GEOMETRY;141
11.1;The Gauss Lemma;141
11.2;Convex Open Sets;144
11.3;Arc Length;146
11.4;Riemannian Distance;147
11.5;Riemannian Completeness;153
11.6;Lorentz Causal Character;155
11.7;Timecones;158
11.8;Local Lorentz Geometry;161
11.9;Geodesics in Hyperquadrics;164
11.10;Geodesics in Surfaces;165
11.11;Completeness and Extendibility;169
12;CHAPTER 6. SPECIAL RELATIVITY;173
12.1;Newtonian Space and Time;173
12.2;Newtonian Space–Time;175
12.3;Minkowski Spacetime;178
12.4;Minkowski Geometry;179
12.5;Particles Observed;182
12.6;Some Relativistic Effects;186
12.7;Lorentz–Fitzgerald Contraction;189
12.8;Energy–Momentum;191
12.9;Collisions;194
12.10;An Accelerating Observer;196
13;CHAPTER 7. CONSTRUCTIONS;200
13.1;Deck Transformations;200
13.2;Orbit Manifolds;202
13.3;Orientability;204
13.4;Semi-Riemannian Coverings;206
13.5;Lorentz Time-Orientability;209
13.6;Volume Elements;209
13.7;Vector Bundles;212
13.8;Local Isometries;215
13.9;Matched Coverings;218
13.10;Warped Products;219
13.11;Warped Product Geodesics;222
13.12;Curvature of Warped Products;224
13.13;Semi-Riemannian Submersions;227
14;CHAPTER 8. SYMMETRY AND CONSTANT CURVATURE;230
14.1;Jacobi Fields;230
14.2;Tidal Forces;233
14.3;Locally Symmetric Manifolds;234
14.4;Isometries of Normal Neighborhoods;236
14.5;Symmetric Spaces;239
14.6;Simply Connected Space Forms;242
14.7;Transvections;246
15;CHAPTER 9. ISOMETRIES;248
15.1;Semiorthogonal Groups;248
15.2;Some Isometry Groups;254
15.3;Time-Orientability and Space-Orientability;255
15.4;Linear Algebra;257
15.5;Space Forms;258
15.6;Killing Vector Fields;264
15.7;The Lie Algebra i(M);267
15.8;I( M ) as Lie Group;269
15.9;Homogeneous Spaces;272
16;CHAPTER 10. CALCULUS OF VARIATIONS;278
16.1;First Variation;278
16.2;Second Variation;281
16.3;The Index Form;283
16.4;Conjugate Points;285
16.5;Local Minima and Maxima;287
16.6;Some Global Consequences;292
16.7;The Endmanifold Case;295
16.8;Focal Points;296
16.9;Applications;301
16.10;Variation of E;303
16.11;Focal Points along Null Geodesics;305
16.12;A Causality Theorem;308
17;CHAPTER 11. HOMOGENEOUS AND SYMMETRIC SPACES;315
17.1;More about Lie Groups;315
17.2;Bi-Invariant Metrics;319
17.3;Coset Manifolds;321
17.4;Reductive Homogeneous Spaces;325
17.5;Symmetric Spaces;330
17.6;Riemannian Symmetric Spaces;334
17.7;Duality;336
17.8;Some Complex Geometry;338
18;CHAPTER 12. GENERAL RELATIVITY; COSMOLOGY;347
18.1;Foundations;347
18.2;The Einstein Equation;351
18.3;Perfect Fluids;352
18.4;Robertson–Walker Spacetimes;356
18.5;The Robertson–Walker Flow;360
18.6;Robertson–Walker Cosmology;362
18.7;Friedmann Models;365
18.8;Geodesics and Redshift;368
18.9;Observer Fields;373
18.10;Static Spacetimes;375
19;CHAPTER 13. SCHWARZSCHILD GEOMETRY;379
19.1;Building the Model;379
19.2;Geometry of N and B;383
19.3;Schwarzschild Observers;386
19.4;Schwarzschild Geodesics;387
19.5;Free Fall Orbits;389
19.6;Perihelion Advance;393
19.7;Lightlike Orbits;395
19.8;Stellar Collapse;399
19.9;The Kruskal Plane;401
19.10;Kruskal Spacetime;404
19.11;Black Holes;407
19.12;Kruskal Geodesics;410
20;CHAPTER 14. CAUSALITY IN LORENTZ MANIFOLDS;416
20.1;Causality Relations;417
20.2;Quasi-Limits;419
20.3;Causality Conditions;422
20.4;Time Separation;424
20.5;Achronal Sets;428
20.6;Cauchy Hypersurfaces;430
20.7;Warped Products;432
20.8;Cauchy Developments;434
20.9;Spacelike Hypersurfaces;440
20.10;Cauchy Horizons;443
20.11;Hawking’s Singularity Theorem;446
20.12;Penrose’s Singularity Theorem;449
21;APPENDIX A. FUNDAMENTAL GROUPS AND COVERING MANIFOLDS;456
22;APPENDIX B. LIE GROUPS;461
22.1;Lie Algebras;462
22.2;Lie Exponential Map;464
22.3;The Classical Groups;465
23;APPENDIX C. NEWTONIAN GRAVITATION;468
24;References;471
25;Index;474