The Geometry of Special Relativity

The Geometry of Special Relativity

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  • Presents special relativity in a geometric framework using basic trigonometry
  • Provides material suitable for both mathematicians and physicists
  • Covers hyperbolic triangle trigonometry
  • Emphasizes the equivalence between special and hyperbolic trigonometry


The Geometry of Special Relativity provides an introduction to special relativity that encourages readers to see beyond the formulas to the deeper geometric structure. The text treats the geometry of hyperbolas as the key to understanding special relativity. This approach replaces the ubiquitous ? symbol of most standard treatments with the appropriate hyperbolic trigonometric functions. In most cases, this not only simplifies the appearance of the formulas, but also emphasizes their geometric content in such a way as to make them almost obvious. Furthermore, many important relations, including the famous relativistic addition formula for velocities, follow directly from the appropriate trigonometric addition formulas.

The book first describes the basic physics of special relativity to set the stage for the geometric treatment that follows. It then reviews properties of ordinary two-dimensional Euclidean space, expressed in terms of the usual circular trigonometric functions, before presenting a similar treatment of two-dimensional Minkowski space, expressed in terms of hyperbolic trigonometric functions. After covering special relativity again from the geometric point of view, the text discusses standard paradoxes, applications to relativistic mechanics, the relativistic unification of electricity and magnetism, and further steps leading to Einstein’s general theory of relativity. The book also briefly describes the further steps leading to Einstein’s general theory of relativity and then explores applications of hyperbola geometry to non-Euclidean geometry and calculus, including a geometric construction of the derivatives of trigonometric functions and the exponential function.

Table of Contents

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)


Author Bio(s)

Editorial Reviews

"This short but remarkable book on special relativity develops the theory strictly from the geometry of hyperbolas, an approach unique within the textbook literature. … it gives an elementary, lively introduction to the theory. … The approximately 70 diagrams are not merely illustrations of the material given in the text of this well-thought-out representation but in most cases tools for the geometrical derivation of the results."
—Wolfgang Hasse, Mathematical Reviews, May 2013

"I enjoyed reading this book and certainly learned from it … it would serve best as a supplemental text for a course in special relativity … This is unquestionably a book that anybody who teaches special relativity will want to look at."
—Mark Hunacek, MAA Reviews, September 2012

"This book is very clearly and simply written. The treatment is mathematically and physically sound. The diagrams are especially good. Though there are many introductory books on special relativity, this book is unique in its emphasis on hyperbolic functions and geometry. The book can stand alone as an elementary introduction to relativity. Or it can serve well as a supplement to other books on relativity or electrodynamics. I strongly endorse it."
—David Hestenes, Professor Emeritus, Department of Physics, Arizona State University

"Einstein would have loved this book. It makes the beauty of his theory accessible to anybody familiar with just trigonometric functions and elementary calculus. But also experts may profit from its thoroughly geometric point of view. A great contribution to the literature on Special Relativity!"
—Jürgen Renn, Max Planck Institute for the History of Science and editor of Genesis of General Relativity

"This text successfully presents a geometrical pictorial view of special relativistic effects that cannot be found in any other introduction for non-scientists. Indeed, I wager that even a practicing relativist will encounter some surprisingly enriching perspectives."
—Don Salisbury, Austin College, Texas

"It is usually thought that spacetime geometry is part of the subject of general relativity, and that special relativity is all about clocks and trains and light. But special relativity has a geometry of its own: the Minkowskian geometry of spacetime, as opposed to the usual Euclidean geometry of space. Now The Geometry of Special Relativity by Tevian Dray comes along with a beautiful treatment of this much neglected approach. … The book is written in an extremely clear and engaging style. There are many examples as well as exercises for the reader. Anyone who wants to have a deep understanding of special relativity should read this book."
—David Garfinkle, Oakland University

"The approach to Special Relativity (SR) taken by the author is a novel, geometrical one, relying on the use of (elementary) hyperbolic geometry and employing lots of diagrams instead of the usual, more analytical, approach. … The author writes in a pleasing, informal and highly readable style and deals with time dilation, length contraction, the Lorentz transformations, the Doppler effect, etc. The chapter on the so-called ‘paradoxes’ of SR provides some welcome relief to any doubts which may arise in the mind of the reader and the three chapters of problems are most useful in the consolidation of the previous material. The author also includes a simple introduction to Minkowski’s unification of the electric and magnetic fields which arises as a consequence of SR. This book is a valuable and original addition to the literature on SR."
—G.S. Hall, Institute of Mathematics, University of Aberdeen

"Clear, beautiful, crystalline. Relativity is about hyperbolas in spacetime! The mathematically inclined will savor Tevian Dray’s friendly primer."
—Rudy Rucker, author of Geometry, Relativity, and the Fourth Dimension

"This book is essentially a grown-up version of the masterful Spacetime Physics by Taylor and Wheeler. Anyone who teaches or intends to teach special relativity needs to own a copy."
—Niall O’Murchadha, Department of Physics, National University of Ireland

"The answer to all the questions about special relativity that you didn’t know how to ask. A welcome addition to any physics library."
—A. Held, University of Bern

"Special relativity, which is fundamental to our understanding of the physical world, is best understood by enlarging our ideas of geometry to include both space and time. Tevian Dray’s book gives a thoroughly geometrical account of the theory. He clearly explains the interesting ways in which the geometry of space must be adapted to include time and develops the ideas of relativity in a purely geometrical form. The value of this geometrical approach is shown in a number of carefully worked examples, in which the reader is left to do some of the work and thereby acquire an intuitive understanding of the theory.
Tevian Dray is a respected researcher in general relativity and an experienced teacher of mathematical physics. He has written an original and valuable introduction to the concepts of special relativity."
—Tony Sudbery, Department of Mathematics, University of York

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