## Special Relativity

T.M. Helliwell Harvey Mudd College

THE SPECIAL THEORY OF RELATIVITY is a superb place to begin a serious study of physics, because it illustrates the fact that nature is stranger than we could have imagined, while also being the most accessible of the twentieth-century revolutions. It is likewise a superb subject to learn at any level, because it is fundamental physics with many important applications, while also being surprising, counterintuitive, and fun! Anyone who knows the Pythagorean theorem and a bit of algebra can understand its fundamentals. In this book we exploit the theory’s accessibility by emphasizing its physical content. Numerous illustrations and examples are discussed at the outset, while the concise mathematical description is postponed until after the reader has had the opportunity to build up some physical intuition for what is going on. The principal challenges of special relativity are in fact conceptual, not mathematical.

The plan of the book was fixed partly by experience in trying various orders of presentation, and partly by the desire to make it useful in a variety of situations. It can serve as part of an introductory first-year course or incorporated within a second-year “modern physics” course. It can also be used in a special-topics or advance placement course, or even as a supplement in advanced undergraduate courses such as theoretical mechanics, electromagnetism, or particle physics.

The main narrative of the book is as follows. Chapter I offers a brief review of classical mechanics–students with a good high-school course can skip this chapter or refer back to it as needed. Chapter II describes two of the important experiments leading up to the special theory. The core of the book (the physical description of what is called “relativistic kinematics”) is contained in Chapters III-VI. Here very little mathematics is used, because very little is needed. It is the physical ideas and the apparently paradoxical results that can be challenging, as exemplified particularly in Chapter VII. These chapters culminate in the Lorentz transformation of Chapter VIII. Thus the mathematical outcome of Einstein’s postulates is postponed until after time dilation, length contraction, and the relativity of simultaneity have already been deduced. Four-dimensional spacetime is constructed in Chapter IX; this spacetime arena then allows us to introduce four-scalars and four-vectors. Chapters X and XI show that momentum and energy join together to form the components of a four-vector, and that mass is a form of energy. Chapter XII takes up several applications, including the important topic of binding energy, especially in the context of nuclear physics, and then goes on to describe many examples of particle collisions and decays and how special relativity is used to predict outcomes, and ends up with an analysis of “photon rockets”. Energy and momentum transformations are central to Chapter XIII, with the aberration of light, the Doppler effect, threshold energies, and colliding-beam experiments as important applications. Finally, Chapter XIV provides a brief introduction to relativistic gravitation, introducing the Principle of Equivalence, gravitational red-shifts, and the effect of gravity on clocks. The Global Positioning System (GPS) is shown to be strongly influenced by relativistic effects.

In addition to the main narrative, the book has ten appendices, one or more of which can be taken up as interest and time allow. These are partly for fun and partly for more specialized topics. They include The Binomial Approximation, describing the often unsung but extraordinarily useful approximation for doing real physics calculations; The “Paradox” of Light Spheres, providing an interesting and thorough workout in the “three rules” of special relativity; The Appearance of Moving Objects, distinguishing between “optical illusions” and reality; The Twin Paradox Revisited, examining this (in)famous paradox in greater depth than there is room for in Chapter VII; The “Cosmic Speed Limit,” describing causal paradoxes and why signals are not supposed to exceed the speed of light; “Relativistic Mass” and Relativistic Forces, debating the pros and cons of believing that mass increases with velocity (and coming down hard on one side of this issue), and going on to explore the effect of forces in special relativity; The Ultimate Relativistic Spaceflight, recounting a truly fantastical voyage; Nuclear Decays, Fission, and Fusion, illustrating the interplay between relativity and nuclear physics, particularly in fusion reactions in the Sun and laboratory, and fission and fusion in nuclear weapons; Some Particles, listing mass and lifetime data for the lighter fundamental particles; and (finally) Relativity and Electromagnetism, giving a very brief example of the close relationship between relativity and the electric and magnetic fields around a wire and their effects upon a nearby electric charge.

This book grew out of an earlier book, ** Introduction to Special Relativity,** published by the author in 1966. In one incarnation or another, it has been used ever since to teach relativity to nearly all first-year students at Harvey Mudd College, whether they intend to major in physics, engineering, biology, chemistry, computer science, or mathematics. For most of that time the subject was taught in the spring, after an introduction to college-level classical mechanics. This was very successful, but more recently we have taken a different approach that has been even more successful: We now begin first-year physics with special relativity. It has been exciting to start off with a subject new to nearly everyone, and this newness, together with the subject’s mathematical simplicity, is a great leveler, in that students with stronger backgrounds have little advantage over those less well-prepared. Special relativity is also an excellent training ground for careful, logical thought, a useful skill to anyone, no matter what his or her primary interests may be.

A word about how much prior mathematics and physics is needed to use this text. Students familiar with elementary algebra and high-school level classical mechanics can skip the first two chapters, and start right in with Chapter III (perhaps with a brief glance back at Chapter II), and then take up Chapters IV, V, VI, and VII, which form the physical heart of relativistic kinematics. No calculus or background in college-level physics is needed to understand this material. A beginning knowledge of differential calculus is needed in Chaper VIII and in several subsequent chapters and appendices, and some familiarity with vectors is needed in Chapters X and beyond. Integral calculus is used only in Appendices D and G . The binomial approximation, which is based on the Taylor expansion of differential calculus, is explained in Appendix A, and used in a few of the problems throughout the book. One can of course use the approximation without necessarily understanding its derivation.

The author is grateful to literally thousands of students for their enthusiasm and help. This book really grew out of the process of teaching one another this fascinating subject. Many faculty colleagues offered valuable suggestions and support, including especially Profs. E. Wicher, D. Petersen, P. Saeta, G. Lyzenga, J. Townsend, C-Y Chen, P. Sparks, A. Esin, and V. Sahakian. My research colleague D. A. Konkowski read the entire manuscript and offered a great many important and detailed suggestions. Several reviewers for University Science Books made very valuable recommendations that have been incorporated. I am also greatly indebted to my wife Bonnie for her insights and for her unflagging patience and support.

A rather important person in all of this was a citizen of the world who signed his papers “A. Einstein.” His ten-year effort to understand the motion of light, beginning at age 16, led him to present the bulk of what is discussed here (along with a good deal else) in two of his five breathtakingly original papers of 1905. Not even Einstein could have foreseen that his ideas would be so intertwined not only with the science of physics, but also with the nature of warfare, the global economy, and even the history of nations.

The author hopes the reader enjoys learning special relativity, and that he or she is sometimes very confused by it! There is little chance of truly understanding this often counterintuitive theory without getting confused and working your way out of the confusion.