Introduction

Dear readers and students of The Theoretical Minimum,

Hello there, and welcome back to Lenny & Art’s Excellent Adventure. We last left the intrepid pair recovering from a wild rollicking roller coaster ride through the quantum world of entanglement and uncertainty. They were ready for something sedate, something reliable and deterministic, something classical. But the ride continues in Volume III, and it’s no less wild. Contracting rods, time dilation, twin paradoxes, relative simultaneity, stretch limousines that do and don’t fit into Volkswagen-size garages. Lenny and Art are hardly finished with their madcap adventure. And at the end of the ride Lenny tricks Art with a fake monopole.

Well, maybe that is a bit overwrought, but to the beginner the relativistic world is a strange and wondrous fun house, full of dangerous puzzles and slippery paradoxes. But we’ll be there to hold your hand when the going gets tough. Some basic grounding in calculus and linear algebra should be good enough to get you through.

Our goal as always is to explain things in a completely serious way, without dumbing them down at all, but also without explaining more than is necessary to go to the next step. Depending on your preference, that could be either quantum field theory or general relativity.

It’s been a while since Art and I published Volume II on quantum mechanics. We’ve been tremendously gratified by the thousands of e-mails expressing appreciation for our efforts, thus far, to distill the most important theoretical principles of physics into TTM.

The first volume on classical mechanics was mostly about the general framework for classical physics that was set up in the nineteenth century by Lagrange, Hamilton, Poisson, and other greats. That framework has lasted, and provides the underpinning for all modern physics, even as it grew into quantum mechanics.

Quantum mechanics percolated into physics starting from the year 1900, when Max Planck discovered the limits of classical physics, until 1926 when Paul Dirac synthesized the ideas of Planck, Einstein, Bohr, de Broglie, Schrödinger, Heisenberg, and Born into a consistent mathematical theory. That great synthesis (which, by the way, was based on Hamilton’s and Poisson’s framework for classical mechanics) was the subject of TTM Volume II.

In Volume III we take a historical step back to the nineteenth century to the origins of modern field theory. I’m not a historian, but I think I am accurate in tracing the idea of a field to Michael Faraday. Faraday’s mathematics was rudimentary, but his powers of visualization were extraordinary and led him to the concepts of electromagnetic field, lines of force, and electromagnetic induction. In his intuitive way he understood most of what Maxwell later combined into his unified equations of electromagnetism. Faraday was lacking one element, namely that a changing electric field leads to effects similar to those of an electric current.

It was Maxwell who later discovered this so-called displacement current, sometime in the early 1860s, and then went on to construct the first true field theory: the theory of electromagnetism and electromagnetic radiation. But Maxwell’s theory was not without its own troubling confusions.

The problem with Maxwell’s theory was that it did not seem to be consistent with a basic principle, attributed to Galileo and clearly spelled out by Newton: All motion is relative. No (inertial) frame of reference is more entitled to be thought of as at rest than any other frame. However this principle was at odds with electromagnetic theory, which predicted that light moves with a distinct velocity c = 3 × 10⁸ meters per second. How could it be possible for light to have the same velocity in every frame of reference? How could it be that light travels with the same velocity in the rest frame of the train station, and also in the frame of the speeding train?

Maxwell and others knew about the clash, and resolved it the simplest way they knew how: by the expedient of tossing out Galileo’s principle of relative motion. They pictured the world as being filled with a peculiar substance—the ether—which, like an ordinary material, would have a rest frame in which it was not moving. That’s the only frame, according to the etherists, in which Maxwell’s equations were correct. In any other frame, moving with respect to the ether, the equations had to be adjusted.

This was the status until 1887 when Albert Michelson and Edward Morley did their famous experiment, attempting to measure the small changes in the motion of light due to the motion of Earth through the ether. No doubt most readers know what happened; they failed to find any. People tried to explain away Michelson and Morley’s result. The simplest idea was called ether drag, the idea being that the ether is dragged along with Earth so that the Michelson-Morley experiment was really at rest with respect to the ether. But no matter how you tried to rescue it, the ether theory was ugly and ungainly.

According to his own testimony, Einstein did not know about the Michelson-Morley experiment when in 1895 (at age sixteen), he began to think about the clash between electromagnetism and the relativity of motion. He simply felt intuitively that the clash somehow was not real. He based his thinking on two postulates that together seemed irreconcilable:

As uncomfortable as it probably seemed, the two principles together implied that light must move with the same velocity in all frames.

It took almost ten years, but by 1905 Einstein had reconciled the principles into what he called the special theory of relativity. It is interesting that the title of the 1905 paper did not contain the word relativity at all; it was “On the Electrodynamics of Moving Bodies.” Gone from physics was the ever more complicated ether; in its place was a new theory of space and time. However, to this day you will still find a residue of the ether theory in textbooks, where you will find the symbol ₀, the so-called dielectric constant of the vacuum, as if the vacuum were a substance with material properties. Students new to the subject often encounter a great deal of confusion originating from conventions and jargon that trace back to the ether theory. If I’ve done nothing else in these lectures, I tried to get rid of these confusions.

As in the other books in TTM I’ve kept the material to the minimum needed to move to the next step—depending on your preference, either quantum field theory or general relativity.

You’ve heard this before: Classical mechanics is intuitive; things move in predictable ways. An experienced ballplayer can take a quick look at a fly ball and from its location and its velocity know where to run in order to be there just in time to catch the ball. Of course, a sudden unexpected gust of wind might fool him, but that’s only because he didn’t take into account all the variables. There is an obvious reason why classical mechanics is intuitive: Humans, and animals before them, have been using it many times every day for survival.

In our quantum mechanics book, we explained in great detail why learning that subject requires us to forget our physical intuition and replace it with something entirely different. We had to learn new mathematical abstractions and a new way of connecting them to the physical world. But what about special relativity? While quantum mechanics explores the world of the VERY SMALL, special relativity takes us into the realm of the VERY FAST, and yes, it also forces us to bend our intuition. But here’s the good news: The mathematics of special relativity is far less abstract, and we don’t need brain surgery to connect those abstractions to the physical world. SR does stretch our intuition, but the stretch is far more gentle. In fact, SR is generally regarded as a branch of classical physics.

Special relativity requires us to rethink our notions of space, time, and especially simultaneity. Physicists did not make these revisions frivolously. As with any conceptual leap, SR was resisted by many. You could say that some physicists had to be dragged kicking and screaming to an acceptance of SR, and others never accepted it at all.¹ Why did most of them ultimately relent? Aside from the many experiments that confirmed the predictions made by SR, there was strong theoretical support. The classical theory of electromagnetism, perfected by Maxwell and others during the nineteenth century, quietly proclaimed that “the speed of light is the speed of light.” In other words, the speed of light is the same in every inertial (nonaccelerating) reference frame. While this conclusion was disturbing, it could not just be ignored—the theory of electromagnetism is far too successful to be brushed aside. In this book, we’ll explore SR’s deep connections to electromagnetic theory, as well as its many interesting predictions and paradoxes.

1 Notably Albert Michelson, the first American to win a Nobel Prize in physics, and his collaborator Edward Morley. Their precise measurements provided strong confirmation of SR.