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

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

“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