Tevian Dray - The Geometry of Special Relativity

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Second edition published 2021

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Library of Congress Cataloging-in-Publication Data

Names: Dray, Tevian, author.

Title: The geometry of special relativity / Tevian Dray, Department of Mathematics, Oregon State University.

Description: Second edition. | Boca Raton: Chapman & Hall/CRC Press, 2021. | Series: Advances in applied mathematics | Includes bibliographical references and index.

Identifiers: LCCN 2021000964 (print) | LCCN 2021000965 (ebook) | ISBN 9781032008202 (hardback) | ISBN 9781138063921 (paperback) | ISBN 9781351663212 (ebook)

Subjects: LCSH: Special relativity (Physics) | Space and timeMathematical models.

Classification: LCC QC173.65.D73 2021 (print) | LCC QC173.65 (ebook) | DDC 530.11dc23

LC record available at https://lccn.loc.gov/2021000964

LC ebook record available at https://lccn.loc.gov/2021000965

ISBN: 9781032008202 (hbk)

ISBN: 9781138063921 (pbk)

ISBN: 9781351663212 (ebk)

Lorentz transformations are just hyperbolic rotations.

Preface to the First Edition

The unification of space and time introduced by Einstein's special theory of relativity is one of the cornerstones of the modern scientific description of the universe. Yet the unification is counterintuitive because we perceive time very differently from space. Even in relativity, time is not just another dimension, it is one with different properties. Some authors have tried to unify the treatment of time and space, typically by replacing t by it, thus hiding some annoying minus signs. But these signs carry important information: Our universe, as described by relativity, is not Euclidean.

This short book treats the geometry of hyperbolas as the key to understanding special relativity. This approach can be summarized succinctly as the replacement of 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 emphasizes their geometric content in such a way as to make them almost obvious. Furthermore, many important relations, including but not limited to the famous relativistic addition formula for velocities, follow directly from the appropriate trigonometric addition formulas.

I am unaware of any other introductory book on special relativity that adopts this approach as fundamental. Many books point out the relationship between Lorentz transformations and hyperbolic rotations, but few actually make use of it. A pleasant exception is the original edition of Taylor and Wheeler's marvelous book [].

This book is not intended as a replacement for any of the excellent textbooks on special relativity. Rather, it is intended as an introduction to a particularly beautiful way of looking at special relativity, in hopes of encouraging students to see beyond the formulas to the deeper structure. Enough applications are included to get the basic idea, but these would probably need to be supplemented for a full course.

Much of the material presented can be understood by those familiar with the ordinary trigonometric functions, but occasional use is made of elementary differential calculus. In addition, the chapter on electricity and magnetism assumes the reader has seen Maxwell's equations and has at least a passing acquaintance with vector calculus. A prior course in calculus-based physics, up to and including electricity and magnetism, should provide the necessary background.

Preface to the Second Edition

The core of the book remains unchanged from the first edition, with its short, descriptive, accessible treatment of the material. However, in response to reader feedback, the initial treatment of Minkowski space and spacetime diagrams has been expanded slightly, with minor changes in notation. In addition, several new topics have been added, numerous small typos have been corrected, and more homework problems have been added.

The primary notational change made was to eliminate the use of for hyperbolic angles, in favor of , since other authors often use as shorthand for vc. Where already present, was changed to .

The new material includes a geometric derivation of Lorentz transformations in , and how it can be used to measure time and distance.

Overview

After a general introduction in .

Acknowledgments

This book started out as class notes [].