Differential Forms and the Geometry of General Relativity

Does the Past Still Exist?
Does the Past Still Exist?

1st Edition

Differential Forms and the Geometry of General Relativity

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


“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

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