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Illustrated Special Relativity Through Its Paradoxes: A Fusion of Linear Algebra, Graphics, and Reality

John dePillis and José Wudka
Mathematical Association of America
Publication Date: 
Number of Pages: 
Electronic Book
[Reviewed by
Mark Hunacek
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Most of us, I think, can recall doing something stupid during our college years; one of my moments of Youthful Idiocy concerned my refusal to take any physics courses at all. Instead, I fulfilled my science graduation requirement by taking, as I recall, something like earth science, a course whose value to me in the long run can be summed up by the observation that I cannot now recall one single thing about it. This decision, based on a bad experience with my physics lab in high school, wound up costing me greatly in the years to come, making me miss out on a lot of physical insight into the mathematics that I was learning, and also adversely affecting my early teaching experiences.

As a result of my weak physics background, I have, over the years, gone through periods of atonement, when I try to pick up a book on something physicsy and teach myself something about the subject. Since advanced texts are out of the question for this endeavor, I generally try to stick to something designed to be accessible to the hoi polloi. I enjoyed, for example, The Physics of Superheroes by Kakalios, but that book makes a determined effort to have almost no mathematical content at all. What I really enjoy reading, when I get in these atoning moods, is something that is casual, accessible to people who have not studied college physics, but yet not too dumbed down. That’s why I asked to review this book, which though advertised for a general audience (and which even comes equipped with cartoon-like illustrations drawn by the author) still maintains a reasonable level of sophistication.

This book delivered exactly what I was looking for. Presupposing only a modest background in physics, it takes the reader on a tour of special relativity, concentrating on half a dozen of the paradoxes of the subject. Some of these (such as the twin paradox) I had heard of before, but others (like the bug-rivet paradox) were new to me. In addition to special relativity, the book also covers electric and magnetic fields and Maxwell’s equations. The discussion throughout the book is clear and accessible, but does not flee from mathematics; some knowledge of calculus and matrices would certainly stand the reader in good stead, particularly in the discussion of Maxwell’s equations, where line integrals and the like are, not surprisingly, used. (An entire chapter is spent reviewing this material, but people who have never been exposed to these ideas before will likely be at something of a disadvantage.)

In addition to calculus, linear algebra is often used: Just as Dray’s The Geometry of Special Relativity emphasized a geometric approach to the subject, this book focuses on basic linear algebra. However, nothing much more sophisticated than 2 × 2 matrices is invoked here, and it’s really interesting to see how much use the authors make of them.

Thus, notwithstanding the cartoons, this is by no means a child’s book: it covers serious material, and (as best as this non-expert can tell) it does so honestly. So, the presence of light-hearted illustrations throughout the book should not lead a reader to think he is reading something with the same intellectual heft as a comic book; this is not really something to be read at the beach.

The book begins with an informal overview of some of the paradoxes of relativity theory, designed to whet one’s appetite for the more sophisticated analysis to follow. After some discussion, the book begins a more detailed analysis, starting with the concept of inertial frames and spacetime graphs. Linear algebra is exploited here because inertial frames are modeled as two-dimensional vector spaces consisting of a space and time coordinate. Einstein’s fundamental assumption that the speed of light is finite and constant for all observers, combined with linear algebra, leads to a derivation of the Lorentz equation that, I gather, is novel. The various paradoxes, whose discussion began the text, are then revisited and analyzed in more detail over the course of the next six chapters (one paradox per chapter).

Einstein’s famous formula E = mc2 is the subject of the next chapter, and is shown to be related to relativistic addition of speed and the “pea-shooter” paradox described earlier in the text. Of course, this formula assumes that the speed of light is a constant, independent of whether the observer is moving or not; this is one of Einstein’s basic axioms for relativity theory, and is explored in more detail in the next two chapters, which include, among other things, a nice historical account of early attempts to measure this constant.

There follows a fairly lengthy, multi-chapter, discussion of electricity and magnetism, leading up to a derivation of Maxwell’s equations. Maxwell’s work, acknowledged by Einstein himself in the opening paragraphs of his famous 1905 paper on special relativity, resulted in the derivation of the fact that electrical and magnetic waves traveled at the speed of light in a vacuum, independent of the observer; while Maxwell did not realize that light was a special case of electromagnetic waves, he believed there was a connection, and the invariance of the speed of light, as noted earlier, is one of Einstein’s essential assumptions in the theory of special relativity.

These final chapters constitute the most demanding portions of the book, but even a reader who never gets to them will be the beneficiary of an excellent overview of the subject of special relativity. In addition to the textual material, there are a number of exercises at the end of each chapter, suggesting a possible use for this book as a text for a college course, or perhaps as a supplementary text for one. That would be an interesting course: while I still don’t have (and may never have) the kind of intuition for relativity that I do for some parts of mathematics, I have more now than I did before picking this book up. I enjoyed this book, and so, I suspect, will students.

One final comment: it should be noted that there are several different versions of this book available. The subject of this review is an e-book published by the MAA. However, for those who, like me, prefer holding an actual book in their hands, rather than reading on a computer screen, there is (according to a paperback version, published not by the MAA but by a company called J. DePillis Illustrations, presumably created and owned by the first-listed author. As of this writing, this paperback sells for about twenty dollars more than the e-book. Then, there is the “deluxe edition” of the book, also published in paperback by J. DePillis Illustrations, which costs (again, as of this writing) about 25 dollars more than the “standard” paperback. Apparently the two paperbacks differ in the inclusion of about 80 pages of text. To make things even more interesting, the number of pages in my e-book matches the number of pages reported by amazon for the deluxe edition, so I suspect that’s the one that is published electronically by the MAA, even though the phrase “deluxe edition” does not appear on the cover.

Mark Hunacek ( teaches mathematics at Iowa State University.