Differential Forms and the Geometry of General Relativity

ISBN: 9781466510005 出版年：2014 页码：315 Dray, Tevian CRC Press

Spacetime Geometry Spacetime Line Elements Circle Trig Hyperbola Trig The Geometry of Special Relativity Symmetries Position and Velocity Geodesics Symmetries Example: Polar Coordinates Example: The Sphere Schwarzschild Geometry The Schwarzschild Metric Properties of the Schwarzschild Geometry Schwarzschild Geodesics Newtonian Motion Orbits Circular Orbits Null Orbits Radial Geodesics Rain Coordinates Schwarzschild Observers Rindler Geometry The Rindler Metric Properties of Rindler Geometry Rindler Geodesics Extending Rindler Geometry Black Holes Extending Schwarzschild Geometry Kruskal Geometry Penrose Diagrams Charged Black Holes Rotating Black Holes Problems General Relativity Warmup Differential Forms in a Nutshell Tensors The Physics of General Relativity Problems Geodesic Deviation Rain Coordinates II Tidal Forces Geodesic Deviation Schwarzschild Connection Tidal Forces Revisited Einstein's Equation Matter Dust First Guess at Einstein's Equation Conservation Laws The Einstein Tensor Einstein's Equation The Cosmological Constant Problems Cosmological Models Cosmology The Cosmological Principle Constant Curvature Robertson-Walker Metrics The Big Bang Friedmann Models Friedmann Vacuum Cosmologies Missing Matter The Standard Models Cosmological Redshift Problems Solar System Applications Bending of Light Perihelion Shift of Mercury Global Positioning Differential Forms Calculus Revisited Differentials Integrands Change of Variables Multiplying Differentials Vector Calculus Revisited A Review of Vector Calculus Differential Forms in Three Dimensions Multiplication of Differential Forms Relationships between Differential Forms Differentiation of Differential Forms The Algebra of Differential Forms Differential Forms Higher Rank Forms Polar Coordinates Linear Maps and Determinants The Cross Product The Dot Product Products of Differential Forms Pictures of Differential Forms Tensors Inner Products Polar Coordinates II Hodge Duality Bases for Differential Forms The Metric Tensor Signature Inner Products of Higher Rank Forms The Schwarz Inequality Orientation The Hodge Dual Hodge Dual in Minkowski 2-space Hodge Dual in Euclidean 2-space Hodge Dual in Polar Coordinates Dot and Cross Product Revisited Pseudovectors and Pseudoscalars The General Case Technical Note on the Hodge Dual Application: Decomposable Forms Problems Differentiation of Differential Forms Gradient Exterior Differentiation Divergence and Curl Laplacian in Polar Coordinates Properties of Exterior Differentiation Product Rules Maxwell's Equations I Maxwell's Equations II Maxwell's Equations III Orthogonal Coordinates Div, Grad, Curl in Orthogonal Coordinates Uniqueness of Exterior Differentiation Problems Integration of Differential Forms Vectors and Differential Forms Line and Surface Integrals Integrands Revisited Stokes' Theorem Calculus Theorems Integration by Parts Corollaries of Stokes' Theorem Problems Connections Polar Coordinates II Differential Forms which are also Vector Fields Exterior Derivatives of Vector Fields Properties of Differentiation Connections The Levi-Civita Connection Polar Coordinates III Uniqueness of the Levi-Civita Connection Tensor Algebra Commutators Problems Curvature Curves Surfaces Examples in Three Dimensions Curvature Curvature in Three Dimensions Components Bianchi Identities Geodesic Curvature Geodesic Triangles The Gauss-Bonnet Theorem The Torus Problems Geodesics Geodesics Geodesics in Three Dimensions Examples of Geodesics Solving the Geodesic Equation Geodesics in Polar Coordinates Geodesics on the Sphere Applications The Equivalence Problem Lagrangians Spinors Topology Integration on the Sphere Appendix A: Detailed Calculations Appendix B: Index Gymnastics Annotated Bibliography References