Robert P. Crease applauds the third volume of a thrilling guide to a special pursuit.
Marchipatrick via Flickr/CC by 2.0
Negotiating the peaks and crevasses of the strange world of special relativity is akin to scaling the Matterhorn.
Since 2007, physicist Leonard Susskind has regularly delivered a lecture series called the Theoretical Minimum, on the foundations needed to study different areas of physics (http://theoreticalminimum.com/home). Companion volumes have emerged, the first on classical mechanics and the second on quantum mechanics. Special Relativity and Classical Field Theory is the third volume. Like the book on quantum mechanics, it is co-authored by Art Friedman and aimed, in Susskind's words, at “physics enthusiasts” or “people who know, or once knew, a bit of algebra and calculus, but are more or less beginners”.
The latest volume concerns the strange world that Albert Einstein discovered by combining James Clerk Maxwell's field theory with Isaac Newton's mechanics — a world in which moving fast makes time compress and lengths shorten. To understand it, challenging mathematical tools are required. In this volume, as in the others, you get the sense of being led up a legendary mountain by a trained guide. The guide knows you are an amateur, but wants you to get to the top on your own, without being airlifted over risky terrain. You do not hike through some of the hardest passes or peaks, nor past some of the most magnificent vistas. It's a peculiar route, and you encounter many sites in a different order from the one early explorers adopted, or in the way experts are used to teaching. But it's accessible. And you do get to the top. As an amateur myself, I found it thrilling.
The path starts with the first lecture, a discussion of reference frames. These are tools for labelling the positions of objects in your immediate space, and in spaces moving with respect to yours (such as on a train), that allow you to go back and forth between the spaces. Understanding them is an important skill for traversing rough spots ahead. Other essential tools include space-time, in which the reference frame includes time as well as space; proper time; and four-vectors, special kinds of paths and objects in space-time.
In a book on special relativity, you might expect to meet Einstein's mass–energy equivalence, E = mc2, close to the beginning. Yet you don't encounter it until Lecture 3, about 100 pages in, where it is refreshingly derived from first principles. You don't get the important Euler–Lagrange equation, which describes particle motion, until Lecture 4. Poisson's equation, for the electrostatic potential of a particle, and the Klein–Gordon equation, which describes a particle as a wave and relativistically, don't show up until Lecture 5. Gauge invariance, the basis of modern field theory, appears first in Lecture 7; Maxwell's equations, which provide the foundation of classical electromagnetism, materialize in Lecture 8; and the Poynting vector, which describes the flow of energy in electromagnetic waves, surfaces first in Lecture 11.
From a historian's point of view, therefore, the path is topsy-turvy. But Susskind's approach is to subject the novice to an ahistorical mathematical boot camp to make the path seem natural, and ultimately easier.
He appeals to the reader's evolving understanding to stay motivated, rather than airing his own expertise. Whenever you are puzzled by the famous conundrums of special relativity — the twin paradox, for instance, in which a sibling journeying on a light-speed rocket ages less than one who stays at home — he instructs you to “draw a spacetime diagram”. Such visual representations, he notes, make most of the weirdness in relativistic events go away.
Friedman pops up as the most vocal hiker on this at-times steep slope. He is not averse to making protests: “I don't recognize any of this. I thought you said we were going to get the Lorentz force law.” (“Lenny” replies: “Hang on, Art, we're getting there.”) Such jousts are infrequent, yet preserve the book's informal tone. In that vein, the narrative is rich in remarks at once witty and insightful. Modifying physicist John Wheeler's quote on relativity — “space-time tells matter how to move; matter tells space-time how to curve” — Susskind remarks, “Fields tell charges how to move; charges tell fields how to vary.”
Understanding the theoretical minimum in special relativity and classical field theory, however, itself demands a certain minimum of preparation and research. The book occasionally bumps up against this problem, referring the reader to earlier volumes; or Susskind might impatiently write, “If you don't know what a cross product is, please take the time to learn.”
The last few chapters are the steepest. You meet landmarks that would have been encountered much earlier in a historical approach, such as the laws of Maxwell, Charles de Coulomb, André-Marie Ampère and Michael Faraday — and even Maxwell's discovery that light is composed of electromagnetic waves, not mentioned until close to the end. But these conclusions fall right out of the tools you have been given in your intensive training — which Susskind calls the “cold shower” approach.
So why buy the book when the lectures are online? The online course consists of ten lectures, each anywhere up to two hours long, whereas the book is orderly and concise. You can go at your own pace, make notes and appreciate where Friedman — a former student of the course — becomes your stand-in and asks the questions that nag at you. You can refer back to something you read earlier and locate it quickly, rather than try to remember how far into the lecture it was and skip around until you find it. Finishing the book, you the physics enthusiast may not have a more profound view of any particular landmark in physics than before. But you will surely have a much more reliable map of the territory.