E-Book, Englisch, 321 Seiten
ISBN: 978-1-4665-1032-6
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes some of the surprising implications of relativity without introducing more formalism than necessary. This nonstandard approach uses differential forms rather than tensor calculus and minimizes the use of "index gymnastics" as much as possible.
The second half of the book takes a more detailed look at the mathematics of differential forms. It covers the theory behind the mathematics used in the first half by emphasizing a conceptual understanding instead of formal proofs. The book provides a language to describe curvature, the key geometric idea in general relativity.
Zielgruppe
Undergraduate students in mathematics and physics; general readers of popular science.
Autoren/Hrsg.
Weitere Infos & Material
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
