Semi-Riemannian Geometry With Applications to Relativity (Volume 103) (Pure and Applied Mathematics, Volume 103)

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... 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. SEMI-RIEMANNIAN GEOMETRY Copyright Page CONTENTS Preface Notation and Terminology CHAPTER 1. MANIFOLD THEORY Smooth Manifolds Smooth Mappings Tangent Vectors Differential Maps Curves Vector Fields One-Forms Submanifolds Immersions and Submersions Topology of Manifolds Some Special Manifolds Integral Curves CHAPTER 2. TENSORS Basic Algebra Tensor Fields Interpretations Tensors at a Point Tensor Components Contraction Covariant Tensors Tensor Derivations Symmetric Bilinear Forms Scalar Products CHAPTER 3. SEMI-RIEMANNIAN MANIFOLDS Isometries The Levi-Civita Connection Parallel Translation Geodesics The Exponential Map Curvature Sectional Curvature Semi-Riemannian Surfaces Type-Changing and Metric Contraction Frame Fields Some Differential Operators Ricci and Scalar Curvature Semi-Riemannian Product Manifolds Local Isometries Levels of Structure CHAPTER 4. SEMI-RIEMANNIAN SUBMANIFOLDS Tangents and Normals The Induced Connection Geodesics in Submanifolds Totally Geodesic Submanifolds Semi-Riemannian Hypersurfaces Hyperquadrics The Codazzi Equation Totally Umbilic Hypersurfaces The Normal Connection A Congruence Theorem Isometric Immersions Two-Parameter Maps CHAPTER 5. RIEMANNIAN AND LORENTZ GEOMETRY The Gauss Lemma Convex Open Sets Arc Length Riemannian Distance Riemannian Completeness Lorentz Causal Character Timecones Local Lorentz Geometry Geodesics in Hyperquadrics Geodesics in Surfaces Completeness and Extendibility CHAPTER 6. SPECIAL RELATIVITY Newtonian Space and Time Newtonian Space–Time Minkowski Spacetime Minkowski Geometry Particles Observed Some Relativistic Effects Lorentz–Fitzgerald Contraction Energy–Momentum Collisions An Accelerating Observer CHAPTER 7. CONSTRUCTIONS Deck Transformations Orbit Manifolds Orientability Semi-Riemannian Coverings Lorentz Time-Orientability Volume Elements Vector Bundles Local Isometries Matched Coverings Warped Products Warped Product Geodesics Curvature of Warped Products Semi-Riemannian Submersions CHAPTER 8. SYMMETRY AND CONSTANT CURVATURE Jacobi Fields Tidal Forces Locally Symmetric Manifolds Isometries of Normal Neighborhoods Symmetric Spaces Simply Connected Space Forms Transvections CHAPTER 9. ISOMETRIES Semiorthogonal Groups Some Isometry Groups Time-Orientability and Space-Orientability Linear Algebra Space Forms Killing Vector Fields The Lie Algebra i(M) I( M ) as Lie Group Homogeneous Spaces CHAPTER 10. CALCULUS OF VARIATIONS First Variation Second Variation The Index Form Conjugate Points Local Minima and Maxima Some Global Consequences The Endmanifold Case Focal Points Applications Variation of E Focal Points along Null Geodesics A Causality Theorem CHAPTER 11. HOMOGENEOUS AND SYMMETRIC SPACES More about Lie Groups Bi-Invariant Metrics Coset Manifolds Reductive Homogeneous Spaces Symmetric Spaces Riemannian Symmetric Spaces Duality Some Complex Geometry CHAPTER 12. GENERAL RELATIVITY; COSMOLOGY Foundations The Einstein Equation Perfect Fluids Robertson–Walker Spacetimes The Robertson–Walker Flow Robertson–Walker Cosmology Friedmann Models Geodesics and Redshift Observer Fields Static Spacetimes CHAPTER 13. SCHWARZSCHILD GEOMETRY Building the Model Geometry of N and B Schwarzschild Observers Schwarzschild Geodesics Free Fall Orbits Perihelion Advance Lightlike Orbits Stellar Collapse The Kruskal Plane Kruskal Spacetime Black Holes Kruskal Geodesics CHAPTER 14. CAUSALITY IN LORENTZ MANIFOLDS Causality Relations Quasi-Limits Causality Conditions Time Separation Achronal Sets Cauchy Hypersurfaces Warped Products Cauchy Developments Spacelike Hypersurfaces Cauchy Horizons Hawking’s Singularity Theorem Penrose’s Singularity Theorem APPENDIX A. FUNDAMENTAL GROUPS AND COVERING MANIFOLDS APPENDIX B. LIE GROUPS Lie Algebras Lie Exponential Map The Classical Groups APPENDIX C. NEWTONIAN GRAVITATION References Index "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."--FROM THE PUBLISHER 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.