The Geometry of Spacetime. A Mathematical Introduction to Relativity Theory

Description

Preface Introduction Contents 1 Differentiable Manifolds 1.1 Charts and Atlases 1.2 Topologisation 1.3 Submanifolds of mathbbRm 2 Tangent Vectors 2.1 The Tangent Space 2.2 Generation of Tangent Vectors 2.3 Vector Fields 2.4 The Lie Bracket 3 Tensors 3.1 Introduction 3.2 Multilinear Forms 3.3 Components 3.4 Operations with Tensors 3.5 Tensors on Euclidean Spaces 4 Semi-Riemannian Manifolds 4.1 Tensor Fields 4.2 Riemannian Manifolds 4.3 Bilinear Forms 4.4 Orientation 4.5 Spacetime 5 Theory of Special Relativity 5.1 Kinematics 5.2 Dynamics 5.3 Electrodynamics 6 Differential Forms 6.1 p-forms 6.2 The Wedge Product 6.3 The Hodge-Star Operator 6.4 Outer Derivative 6.5 The Maxwell Equations in Vacuum 7 The Covariant Differentiation of Vector Fields 7.1 The Directional Derivative in mathbbRn 7.2 The Levi-Civita Connection 7.3 Christoffel Symbols 7.4 The Covariant Derivative on Hypersurfaces 7.5 The Covariant Derivative in the Schwarzschild Spacetime 8 Curvature 8.1 The Curvature Tensor 8.2 The Weingarten Map 8.3 The Ricci Tensor 8.4 The Curvature of the Schwarzschild Spacetime 8.5 Connection Forms and Curvature Forms 9 Matter 9.1 Mass 9.2 Energy and Momentum of a Flow 9.3 The Energy-Momentum Tensor 9.4 Charge 9.5 Energy and Momentum in the Electromagnetic Field 9.6 The Einstein Field Equation 9.7 Spherically Symmetric Solutions 9.8 Outer and Inner Schwarzschild Metric 10 Geodesics 10.1 Time 10.2 The Euler-Lagrange Equation 10.3 The Geodesic Equation 10.4 The Geodesic Deviation 10.5 Perihelion Precession 10.6 Light Deflection 10.7 Red Shift 11 Covariant Differentiation of Tensor Fields 11.1 Parallel Transport of Vectors 11.2 Parallel Transport of Tensors 11.3 Calculation Rules and Component Representation 11.4 The Second Bianchi Identity 11.5 Divergence 12 The Lie Derivative 12.1 The Flow and Its Tangents 12.2 Pull-Back and Push-Forward 12.3 Axiomatic Set Up 12.4 The Derivative Formula 12.5 Component Representation 12.6 Killing Vectors 12.7 The Lie Derivative of Differential Forms 13 Integration on Manifolds 13.1 Introduction 13.2 Partition of Unity 13.3 Integrals 13.4 Manifolds with Boundary 13.5 Integral Theorems 13.6 Extremal Principles 14 Nonrotating Black Holes 14.1 The Schwarzschild Half Plane 14.2 Optics of Black Holes 14.3 The Kruskal Plane 15 Cosmology 15.1 Spaces of Constant Curvature 15.2 The Robertson-Walker Metric 15.3 Universe Models 16 Rotating Black Holes 16.1 The Kerr Metric 16.2 Other Representations of the Kerr Metric 16.3 Causal Structure 16.4 Covariant Derivative and Curvature 16.5 Conservation Theorems 17 A Glimpse of String Theory 17.1 Quantum Theory Versus Relativity Theory 17.2 Elementary Particles as Strings 17.3 The Extremal Principle References Index