You are here

Differential Forms and the Geometry of General Relativity

Tevian Dray
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Steven Deckelman
, on

This new book by Tevian Dray is truly remarkable and delightful to read. It introduces the basics of general relativity, an otherwise extraordinarily austere and rarefied topic, in the language of differential forms accessible to advanced undergraduates.

The book consists of three parts. The main sections are the one on general relativity and another on differential forms, with the ideas of differential geometry providing an underlying connective fiber between the two. Preceding these two parts, the book begins with a preview of what lies ahead, giving a glimpse of such ideas as the line element (or metric tensor), examples from Newtonian theory and how these relate to and fit with the metrics of Schwarzchild and Minkowski, Rindler geometry (a similar, but somewhat simpler geometry than Schwarzschild geometry) and finally black holes. While reading this first part, called Spacetime Geometry, it’s helpful to know that many of the ideas mentioned informally and in the context of specific examples are discussed more thoroughly later in the book and can be located in the index if necessary.

As a non-expert in general relativity, I found it a bit easier to read the third part on differential forms before tackling the second part on general relativity. Dray’s exposition of differential forms is more extensive than those often found in undergraduate texts. After defining differential forms and their standard properties he gives a very illuminating exposition of the chain of ideas leading from line elements to connections to curvature and to geodesics and Riemannian geometry. A key idea is that of a vector differential.

The author mentions in the beginning that it’s assumed the reader is somewhat familiar with the notions of connection and curvature forms, but these are explained in some detail later in the book. The book also includes an extensive bibliography and list of references. While reviewing this book, I found it useful to consult the classic 1962 book by Harley Flanders, Differential Forms with Applications to the Physical Sciences, for additional background. The author’s other book, The Geometry of Special Relativity, is also a good supplemental reference to have available. After reading the third part on differential forms, I found the second part much more accessible. It begins with the Einstein notation, discusses Einstein’s equation, cosmology and the Big Bang as well as solar system applications and the bending of light. As is known of course to most of us, a key idea is that gravity arises from the curvature of spacetime which in some sense can be identified with matter.

The book does have some exercises and, as the author says, could possibly be used as a textbook for an advanced undergraduate or beginning graduate course in mathematics or physics. But there are no answers in the back of the book. Perhaps more exercises could be added in subsequent revisions as well as a partial answer key. A list of symbols used would also be helpful. I believe this book will also be of interest to mathematicians who are nonspecialists in general relativity looking for an accessible entrée into the subject.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. His interests include complex analysis, mathematical biology and the history of mathematics.

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