Relativistic electrodynamics

“If [the mathematicians of the nineteenth century] had taken Maxwell’s equations to heart as Euler took Newton’s, they would have discovered...Einstein’s theory of special relativity...simply by exploring to the end the mathematical concepts to which Maxwell’s equations naturally lead.” Freeman Dyson

Special relativity on its own may seem deeply at odds with our everyday experience of the world. Yet electricity and magnetism is certainly familiar to us; could you go without a day without seeing a battery, a circuit, or some other electronic device? Surprisingly, the laws of physics tell us that these two are not unrelated theories of physics. Indeed, special relativity is implied by electromagnetic theory! So in this chapter, we will explore why special relativity and electromagnetism are deeply interrelated, and what it means when we say that electromagnetism is a relativistic field theory.

The tensor formulation of Maxwell’s equations¶

It is not immediately apparent that Maxwell’s equations have anything to do with special relativity, but this is more to due with a choice of notation than anything else. Typically, we express Maxwell’s equations using vector calculus notation, as follows:

This notation is all well and good for a variety of calculations, but it also obscures a lot of information; moreover, it is quite inelegant to have a system of four partial differential equations to express the dynamics of just two fields! When using tensor calculus to express Maxwell’s equations, the advantages of tensors becomes much more clear.

To start, it is simplest to work in terms of electromagnetic potentials instead of the fields themselves. Recall that the electric (scalar) potential $\phi$ and the magnetic (vector) potential $\mathbf{A}$ are defined as follows:

We may then define the electromagnetic 4-potential $A^\mu$ and the 4-current $J^\mu$ as follows:

Then, we define the Faraday tensor (also called the electromagnetic field tensor) as:

Using the Faraday tensor, we can then rewrite Maxwell’s equation using just 2 tensor equations, which, like any tensor equation, is coordinate-independent^[1]:

If we are willing to dispense with the fields and only consider the potentials, we can write the Maxwell equations in an even simpler form as:

Where $\square = \partial^\mu \partial_\mu = \eta^{\mu \nu} \partial_\nu \partial_\mu$ is known as the d’Alembert operator (also called the d’Alembertian), and is given by:

Relativistic solutions to the EM wave equation¶

The superposition principle tells us, among other things, that the general solution to an inhomogeneous differential equation in the form $\hat L u(\mathbf{r}) = f(\mathbf{r})$ (where $\hat L$ is a linear differential operator) may be obtained in the following manner:

1. First, solve the homogeneous version of the differential equation by setting $f(\mathbf{r}) = 0$. This yields a solution $u_0$ that satisfies $\hat L u_0 = 0$

2. Second, solve the inhomogeneous version of the differential equation for the given source function $f(\mathbf{r})$. This is given by $\hat L u_p = f(\mathbf{r})$, where $u_p$ is the particular solution.

   • Frequently, the source function is chosen to be a delta function, which physically represents a point source, since a superposition of delta functions can be used to describe any arbitrarily-complex source.

3. Then, the general solution to the differential equation $\hat L u = f(\mathbf{r})$ for a particular source function $f(\mathbf{r})$ is given by $u(\mathbf{r}) = u_0 + u_p$

Once the general solution is found to the inhomogeneous PDE, we can then solve specific initial/boundary-value problems by imposing the boundary condition(s) and initial condition(s) on the general solution.

Wave solutions¶

To start, let us consider the homogeneous Maxwell equations in 4-potential form, given by:

This expands out to two partial differential equations for the scalar and vector potentials:

Both of these PDEs are wave equations, meaning that their solutions are given by electromagnetic waves. The two most common wave solutions are plane waves and spherical waves. Both are idealized solutions that (for reasons we’ve already discussed) cannot exist in the real world.

It is also possible to write out a generalized solution in terms of a Fourier transform:

Liénard–Wiechert potentials¶

It can be shown that the inhomogeneous Maxwell equations have the particular solution (for arbitrary $J^\mu$) of $A^\mu_{(h)} = (\phi_p/c, \mathbf{A}_p)$, where:

Or alternatively, in 4-vector notation:

where $t_r$ is called the retarded time, given by:

In the case that we have a 4-current in the form of a delta function, corresponding to a point charge, we obtain the Liénard–Wiechert potentials.

Radiation by moving charges¶

Derive the Larmor formula for the radiated power of an accelerated charge, and its relativistic generalization.

The electromagnetic Lagrangian¶