A Relatively Painless Guide to Special Relativity

By Dave Goldberg

Serious and accessible—finally the special relativity course book that both physics majors and lifelong learners deserve.
 
Special relativity challenges one’s physical intuition of space, time, matter, and energy in a way that few other topics in physics do. Yet the subject is often treated as an extra in undergraduate courses—something to be picked up in a few random lectures and presented as a combination of geometric and logical puzzles (seemingly with the premise of getting the novice student to concede that Einstein was a genius and that the universe is weird). But special relativity is absolutely fundamental to modern physics. It is the canvas on which electromagnetism, particle physics, field theory, and ultimately general relativity are based. For physics students, developing a relativistic intuition isn’t just a luxury: it’s a requirement.
 
Physicist and popular author Dave Goldberg provides a rigorous but conversational introduction to fill this void in spacetime education. Employing the standard calculus a sophomore or junior university student in science, engineering, or computer science will have encountered, Goldberg connects relativity to a student’s work ahead, acquainting them with topics like tensors, the development of new physical theories, and how relativity directly relates to other disciplines. But more than this, Goldberg welcomes lifelong learners who may have encountered special relativity in popular accounts, but are seeking a mathematical challenge to understand an elegant physical theory.

• Language: English
• Publisher: University of Chicago Press
• Release date: Jul 20, 2023
• ISBN: 9780226821863

Book preview

1

Space

Euclid of Alexandria circa 300 BCE. His Elements provided much of the motivation for Einstein’s early mathematical education, and his reasoning in space will form the foundation for much of our work in spacetime. Woodcut by André Thevet, 1584.

1.1 What Relativity Is

Even outside of physics circles, one equation has become so famous that it’s transcended the discipline:

E = mc².

This equation forms the basis for the power of fusion in the Sun, for nuclear weapons, and for the creation and annihilation of matter and antimatter in the early universe. It also made Einstein an international celebrity. We won’t start off by talking about mass and energy, but instead, we need to think about perspective.The relative in relativity comes from the challenge of converting one perspective to another.

At its heart, relativity is fundamentally about the study of measurement: how we measure the distance between two points in space, and ultimately, two points in space and time. In ordinary space, measuring distances seems straightforward: mark off two points on the ground and run a tape measure from one to the other.

Like many tasks in the mathematical sciences, it’s far easier to do a thing than to describe it with equations. The vibrations of a guitar string require a knowledge of Newton’s laws, tension, and differential equations to describe accurately, but even a little kid will know what happens if you pluck it. The same is true here. It’s easy to measure distances but somewhat trickier to describe what we mean by measuring distances. Fortunately, we have the benefit of physical intuition. We’ll know when we’re done whether our mathematics makes sense. (That same intuition will break down spectacularly shortly after we introduce the mechanics of spacetime.) With that in mind, we can make a concise statement about what relativity is. [7, 13]

Relativity is the idea that (within certain constraints) it does not matter who is making the measurement, how they are oriented, or how or whether they are moving, some measurements will be the same no matter how you look at them.

But Einstein himself urged caution [43]:

The meaning of relativity has been widely misunderstood. Philosophers play with the word, like a child with a doll. Relativity, as I see it, merely denotes that certain physical and mechanical facts, which have been regarded as positive and permanent, are relative with regard to certain other facts in the sphere of physics and mechanics. It does not mean that everything in life is relative and that we have the right to turn the whole world mischievously topsy-turvy.

It will turn out that, to a very significant degree, the study of relativity will be very much the study of symmetry. As the mathematician Hermann Weyl [45] put it:

A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.

Our central goal in relativity will be to show that we can stretch and rotate space and time and that some thing will remain as it was before. We’ll spend the first couple of chapters figuring out what that thing is.

1.2 Measuring Distances

When Einstein was a boy, one of his first inspirations was Euclid’s Elements [21], a book written around 300 BCE in which the author set out a few simple rules and from them—using only a straight edge and a compass—was able to derive almost all of the fundamental rules of geometry. As Einstein [6] put it:

Here were assertions … which—though by no means evident—could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon