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The Geometry of Special Relativity

Tevian Dray
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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A student learning special relativity for the first time must contend not only with counter-intuitive concepts such as time dilation and length contraction but also with fairly cumbersome equations such as the Lorentz equations \[ x' = \gamma (x - vt)\] \[ t' = \gamma (t - (v/c^2)x)\]

(where \((\gamma = 1/\sqrt{1-v^2/c^2}\) and \(c\) is the speed of light), which relate the position and time of an observer O at rest and a moving observer O’. The novel idea of this slim, succinct book is to seek to replace reliance on these equations with geometric reasoning. Of course the “geometry” here is not the ordinary Euclidean geometry we all learned in high school, and which is based on circles in the sense that the set of all points at distance 1 from the origin is a circle. Here, instead, the geometry is based on hyperbolas: the squared distance of a point \((x, y)\) from the origin is defined to be \(x^2- y^2\), so the “unit circle” in this geometry is really a hyperbola. The ordinary trigonometric functions are then replaced by the hyperbolic functions sinh, cosh, and tanh, applied to angles that show up in spacetime diagrams. (Such a diagram plots the position \(x\) and time \(t\) of an object, with t running up the vertical axis and x moving along the horizontal one; thus, for example, a vertical line depicts an object that stays motionless as time passes.)

The book starts with two introductory chapters, the first of which (very) rapidly contrasts Newton’s and Einstein’s physics and the second of which functions as a quick overview of the physics of special relativity, including the derivation of the Lorentz equations from the two basic postulates of special relativity (that the laws of physics apply in all inertial reference frames and that the speed of light is the same for all inertial observers). This is followed by two chapters on Euclidean “circle geometry” and the non-Euclidean “hyperbola geometry” that will be used throughout the book, introducing the hyperbolic trigonometric functions from several different (but equivalent) points of view.

The geometry of special relativity begins in earnest with the next chapter, which introduces spacetime diagrams and a particular angle in them called the rapidity, which turns out to be the angle between a worldline and the vertical (ct) axis (ct, rather than t, because we want to measure time and space in the same units). The ideas developed here are exploited in two subsequent chapters (“Applications” and “Paradoxes”) which discuss the geometric ideas behind such topics as length contraction, time dilation, the twins paradox, and others.

There are then three (non-consecutive) chapters that struck me as the most demanding in the book: chapter 9, on relativistic mechanics, addresses the relationship between mass and energy and gives insight into the famous equation \(E = mc^2\); chapter 11, on relativistic electromagnetism (which, thanks to my dismal background in physics, I only dimly understood) unifies electricity and magnetism; chapter 13 provides a warp-speed look at general relativity.

The book concludes with two chapters that are really just straight mathematics, no physics: one chapter (“Hyperbolic Geometry”) discusses various models for hyperbolic geometry, and the other (“Calculus”) offers a geometric derivation of the trigonometric and exponential functions.

There are thirteen substantial worked-out examples in the book, all of which struck me as both interesting and illuminating; instead of scattering them throughout the text the author puts them all in three chapters (7, 10 and 12) devoted entirely to them. There are no exercises, but the author does occasionally explicitly leave something to the reader to ponder, such as the resolution of the “manhole cover” paradox in chapter 8.

Despite the fact that there are at least two other books (one quite recent) with titles that are very similar to this book’s, Dray’s approach really does appear to be novel. Dragon’s new book The Geometry of Special Relativity: A Concise Course, for example, seems (based on an admittedly cursory glance) to be addressed to a considerably more sophisticated audience than is this text, and while geometric ideas are certainly mentioned the ones that are discussed in Dray do not appear to take center stage in Dragon. There is also a book by Callahan with the similar-sounding title The Geometry of Spacetime, but that book is considerably different than this one; unlike this text, it covers both general and special relativity (so “geometry” as used in the title of that book really refers to differential geometry, which finds extensive use) and the discussion of special relativity in the first half of the book emphasizes linear algebra notation; matrix computations appear throughout. This is consistent with a primary idea in Dray’s book (that the Lorentz transformation is really just a hyperbolic rotation) but the manner of presentation is different. (Anybody planning to look at both books should also note that Callahan, in contravention of what I believe is now standard practice, uses the horizontal, rather than vertical, axis for time.) Finally, I should perhaps mention Taylor and Wheeler’s Spacetime Physics, endorsed by the author but unseen by me, which apparently contained a lot of this material in its first edition but left much out of it out of the second.

The question then arises, of course, whether “novel” necessarily means “better”. I found that the geometric discussions did shed some light on the underlying ideas, but I also thought at times that the calculations used in them were just disguised versions of the calculations used from a more traditional approach (which, I think, any serious student of special relativity should be familiar with, since these ideas appear so frequently in the literature). So, while I enjoyed reading this book and certainly learned from it, I tend to think that it would serve best as a supplemental text for a course in special relativity rather than as a main text. (The very succinct writing style, and the total lack of homework exercises, also influenced this opinion.) The author himself may think this, because he writes in the preface that the book “is not intended as a replacement for any of the excellent textbooks on special relativity” but is intended as an introduction “to a particularly beautiful way of looking at special relativity… encouraging students to see beyond the formulas to the deeper structure.” This goal, I think, has been met: on more than one occasion as I read this book I found myself looking at other texts to compare discussions, and I generally found that the process seemed to have a synergistic effect: I got more out of both books by doing this. This is unquestionably a book that anybody who teaches special relativity will want to look at.

Mark Hunacek ( teaches mathematics at Iowa State University.

Newton’s Relativity
Einstein’s Relativity

The Physics of Special Relativity
Observers and Measurement
The Postulates of Special Relativity
Time Dilation and Length Contraction
Lorentz Transformations
Addition of Velocities
The Interval

Circle Geometry
Triangle Trig
Addition Formulas

Hyperbola Geometry
Triangle Trig
Addition Formulas

The Geometry of Special Relativity
The Surveyors
Spacetime Diagrams
Lorentz Transformations
Space and Time
Dot Product

Drawing Spacetime Diagrams
Addition of Velocities
Length Contraction
Time Dilation
Doppler Shift

Problems I
The Getaway
Angles are not Invariant
Interstellar Travel
Cosmic Rays
Doppler Effect

Special Relativity Paradoxes
The Pole and Barn Paradox
The Twin Paradox
Manhole Covers

Relativistic Mechanics
Proper Time
Conservation Laws
Useful Formulas

Problems II
Mass isn’t Conserved
Colliding Oarticles I
Colliding Oarticles II
Colliding Oarticles III
Colliding Oarticles IV

Relativistic Electromagnetism
Magnetism from Electricity
Lorentz Transformations
The Electromagnetic Field
Maxwell’s Equations
The Unification of Special Relativity

Problems III
Electricity vs. Magnetism I
Electricity vs. Magnetism II

Beyond Special Relativity
Problems with Special Relativity
Tidal Effects
Differential Geometry
General Relativity
Uniform Acceleration and Black Holes

Hyperbolic Geometry
Non-Euclidean Geometry
The Hyperboloid
The Poincaré Disk
The Klein Disk
The Pseudosphere

Circle Trigonometry
Hyperbolic Trigonometry
Exponentials (and Logarithms)