Art: Lenny, who is that dignified gentleman with the beard and wire-frame glasses?
Lenny: Ah, the Dutch uncle. That’s Hendrik. Would you like to meet him?
Art: Sure, is he a friend of yours?
Lenny: Art, they’re all friends of mine. Come on, I’ll introduce you.
Art Friedman, meet my friend Hendrik Lorentz.
Poor Art, he’s not quite prepared for this.
Art: Lorentz? Did you say Lorentz? Oh my God! Are you? Is he? Are you really, are you really the …”
Dignified as always, HL bows deeply.
Lorentz: Hendrik Antoon Lorentz, at your service.
Later a star-struck Art quietly asks, Lenny, is that really the Lorentz? The one who discovered Lorentz transformations?
Lenny: Sure he is, and a lot more than that. Bring me a napkin and a pen and I’ll tell you about his force law.
Of all the fundamental forces in nature, and there are many of them, few were known before the 1930s. Most are deeply hidden in the microscopic quantum world and only became observable with the advent of modern elementary particle physics. Most of the fundamental forces are what physicists call short-range. This means that they act only on objects that are separated by very small distances. The influence of a short-range force decreases so rapidly when the objects are separated that, for the most part, they are not noticed in the ordinary world. An example is the so-called nuclear force between nucleons (protons and neutrons). It’s a powerful force whose role is to bind these particles into nuclei. But as powerful as that force is, we don’t ordinarily notice it. The reason is that its effects disappear exponentially when the nucleons are separated by more than about 10−15 meters. The forces that we do notice are the long-range forces whose effects fade slowly with distance.
Of all the forces of nature, only three were known to the ancients—electric, magnetic, and gravitational. Thales of Miletos (600 BC) was said to have moved feathers with amber that had been rubbed with cat fur. At about the same time he mentioned lodestone, a naturally occurring magnetic material.
Aristotle, who was probably late on the scene, had a theory of gravity, even if it was completely wrong. These three were the only forces that were known until the 1930s.
What makes these easily observed forces special is that they are long-range. Long-range forces fade slowly with distance and can be seen between objects when they are well separated.
Gravitational force is by far the most obvious of the three, but surprisingly it is much weaker than electromagnetic force. The reason is interesting and worth a short digression. It goes back to Newton’s universal law of gravitational attraction: Everything attracts everything else. Every elementary particle in your body is attracted by every particle in the Earth. That’s a lot of particles, all attracting one another, and the result is a significant and noticeable gravitational attraction, but in fact the gravitational attraction between individual particles is far too small to measure.
The electric forces between charged particles is many orders of magnitude stronger than the gravitational force. But unlike gravity, electric force can be either attractive (between opposite charges) or repulsive (between like
charges). Both you and the Earth are composed of an equal number of positive charges (protons) and negative charges (electrons), and the result is that the forces cancel. If we imagined getting rid of all the electrons in both you and the Earth, the repulsive electric forces would easily overwhelm gravity and blast you from the Earth’s surface. In fact, it would be enough to blast the Earth and you into smithereens.
In any case, gravity is not the subject of these lectures, and the only other long-range forces are electromagnetic. Electric and magnetic forces are closely related to each other; in a sense they are a single thing, and the unifying link is relativity. As we will see, an electric force in one frame of reference becomes a magnetic force in another frame, and vice versa. To put it another way, electric and magnetic forces transform into each other under Lorentz transformation. The rest of these lectures are about electromagnetic forces and how they are unified into a single phenomenon through relativity. Going back to Pauli’s (fictitious) paraphrase of John Wheeler’s (real) slogan,
Fields tell charges how to move; charges tell fields how to vary.
We’ll begin with the first half—fields tell charges how to move. Or to put it more prosaically, fields determine the forces on charged particles.
An example that may be familiar to you—if not, it soon will be—is the electric field . Unlike the scalar field that we discussed in the last lecture, the electric field is a vector field—a 3-vector, to be precise. It has three components and points in some direction in space. It controls the electric force on a charged particle according to the equation
In this equation the symbol e represents the electric charge of the particle. It can be positive, in which case the force is in the same direction as the electric field; it can be negative, in which case the force and the field are in opposite directions; or, as in the case of a neutral atom, the force can be zero.
Magnetic forces were first discovered by their action on magnets or bits of lodestone, but they also act on electrically charged particles if the particles are in motion. The formula involves a magnetic field (also a 3-vector), the electric charge e, and the velocity of the particle We’ll derive it later from an
action principle, but jumping ahead, the force on a charged particle due to a magnetic field is
The symbol × represents the ordinary cross-product of vector algebra, which I assume you have seen before.2 One interesting property of magnetic forces is that they vanish for a particle at rest and increase as the particle velocity increases. If there happens to be both an electric and a magnetic field, the full force is the sum
Eq. 6.1 was discovered by Lorentz and is called the Lorentz force law.
We’ve already discussed scalar fields and the way they interact with particles. We showed how the same Lagrangian (and the same action) that tells the field how to influence the particle also tells the particle how to influence the field. Going forward, we’ll do the same thing for charged particles and the electromagnetic field. But before we do, I want to briefly review our notational scheme and extend it to include a new kind of object: tensors.
Tensors are a generalization of vectors and scalars and include them as special cases. As we will see, the electric and magnetic fields are not separate entities but combine together to form a relativistic tensor.
6.1 Extending Our Notation
Our basic building blocks are 4-vectors with upper and lower indices. In the context of special relativity, there’s little difference between the two types of indices. The only difference occurs in the time component (the component with index zero) of a 4-vector. For a given 4-vector, the time component with an upper index has the opposite sign of the time component with lower index. In symbols,
A0 = −A0.
It may seem like overkill to define a notation whose sole purpose is to keep track of sign changes for time components. However, this simple relationship
is a special case of a much broader geometric relationship based on the metric tensor. When we study general relativity, the relationship between upper and lower indices will become far more interesting. For now, upper and lower indices simply provide a convenient, elegant, and compact way to write equations.