1st Edition

Differential Forms and the Geometry of General Relativity

By Tevian Dray Copyright 2015
    321 Pages 87 B/W Illustrations
    by A K Peters/CRC Press

    Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.

    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.

    Spacetime Geometry
    Line Elements
    Circle Trig
    Hyperbola Trig
    The Geometry of Special Relativity

    Position and Velocity
    Example: Polar Coordinates
    Example: The Sphere

    Schwarzschild Geometry
    The Schwarzschild Metric
    Properties of the Schwarzschild Geometry
    Schwarzschild Geodesics
    Newtonian Motion
    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

    General Relativity
    Differential Forms in a Nutshell
    The Physics of General Relativity

    Geodesic Deviation
    Rain Coordinates II
    Tidal Forces
    Geodesic Deviation
    Schwarzschild Connection
    Tidal Forces Revisited

    Einstein's Equation
    First Guess at Einstein's Equation
    Conservation Laws
    The Einstein Tensor
    Einstein's Equation
    The Cosmological Constant

    Cosmological Models
    The Cosmological Principle
    Constant Curvature
    Robertson-Walker Metrics
    The Big Bang
    Friedmann Models
    Friedmann Vacuum Cosmologies
    Missing Matter
    The Standard Models
    Cosmological Redshift

    Solar System Applications
    Bending of Light
    Perihelion Shift of Mercury
    Global Positioning

    Differential Forms
    Calculus Revisited
    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
    Inner Products
    Polar Coordinates II

    Hodge Duality
    Bases for Differential Forms
    The Metric Tensor
    Inner Products of Higher Rank Forms
    The Schwarz Inequality
    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

    Differentiation of Differential Forms
    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

    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

    Polar Coordinates II
    Differential Forms which are also Vector Fields
    Exterior Derivatives of Vector Fields
    Properties of Differentiation
    The Levi-Civita Connection
    Polar Coordinates III
    Uniqueness of the Levi-Civita Connection
    Tensor Algebra

    Examples in Three Dimensions
    Curvature in Three Dimensions
    Bianchi Identities
    Geodesic Curvature
    Geodesic Triangles
    The Gauss-Bonnet Theorem
    The Torus

    Geodesics in Three Dimensions
    Examples of Geodesics
    Solving the Geodesic Equation
    Geodesics in Polar Coordinates
    Geodesics on the Sphere

    The Equivalence Problem
    Integration on the Sphere

    Appendix A: Detailed Calculations
    Appendix B: Index Gymnastics

    Annotated Bibliography



    Tevian Dray

    "In this book, the author outlines an interesting path to relativity and shows its various stages on the way … The author inserts suggestive pictures and images, which make the book more attractive and easier to read. The book addresses not only specialists and graduate students, but even advanced undergraduates, due to its interactive structure containing questions and answers."
    Zentralblatt MATH 1315

    "…the presentation is very far from the ‘definition-theorem-proof-example’ style of a traditional mathematics text; rather, we meet important ideas several times, and they are developed further with each new exposure. This is a pedagogical decision which seems to me to be sound, as it allows the student’s understanding of the ideas to develop."
    —Robert J. Low, Mathematical Reviews, June 2015

    "This is a brilliant book. Dray has an extraordinary knack of conveying the key mathematics and concepts with an elegant economy that rivals the expositions of the legendary Paul Dirac. It is pure pleasure to see far-reaching results emerge effortlessly from easy-to-follow arguments, and for simple examples to morph into generalizations. It is so refreshing to find a book that does not obscure the basics with unnecessary technicalities, yet can develop sophisticated formalism from very modest mathematical investments."
    —Paul Davies, Regents’ Professor and Director, Beyond Center for Fundamental Concepts in Science; Co-Director, Cosmology Initiative; and Principal Investigator, Center for the Convergence of Physical Science and Cancer Biology, Arizona State University

    "It took Einstein eight years to create general relativity by carefully balancing his physical intuition and the rather tedious mathematical formalism at his disposal. Tevian Dray’s presentation of the geometry of general relativity in the elegant language of differential forms offers even beginners a novel and direct route to a deep understanding of the theory’s core concepts and applications, from the geometry of black holes to cosmological models."
    —Jürgen Renn, Director, Max Planck Institute for the History of Science, Berlin