The invariance of the equations of electrodynamics
We have found that the potentials $\phi$ and $\FLPA$ taken together form a four-vector which we call $A_\mu$, and that the wave equations—the full equations which determine the $A_\mu$ in terms of the $j_\mu$—can be written as in Eq. (25.22). This equation, together with the conservation of charge, Eq. (25.19), gives us the fundamental law of the electromagnetic field: \begin{equation} \label{Eq:II:25:29} \Box^2A_\mu=\frac{1}{\epsO}\,j_\mu,\quad \fournabla j_\mu=0. \end{equation} There, in one tiny space on the page, are all of the Maxwell equations—beautiful and simple. Did we learn anything from writing the equations this way, besides that they are beautiful and simple? In the first place, is it anything different from what we had before when we wrote everything out in all the various components? Can we from this equation deduce something that could not be deduced from the wave equations for the potentials in terms of the charges and currents? The answer is definitely no. The only thing we have been doing is changing the names of things—using a new notation. We have written a square symbol to represent the derivatives, but it still means nothing more nor less than the second derivative with respect to $t$, minus the second derivative with respect to $x$, minus the second derivative with respect to $y$, minus the second derivative with respect to $z$. And the $\mu$ means that we have four equations, one each for $\mu=t$, $x$, $y$, or $z$. What then is the significance of the fact that the equations can be written in this simple form? From the point of view of deducing anything directly, it doesn’t mean anything. Perhaps, though, the simplicity of the equations means that nature also has a certain simplicity.  Let us show you something interesting that we have recently discovered: All of the laws of physics can be contained in one equation. That equation is \begin{equation} \label{Eq:II:25:30} \mathsf{U}=0. \end{equation} What a simple equation! Of course, it is necessary to know what the symbol means. $\mathsf{U}$ is a physical quantity which we will call the “unworldliness” of the situation. And we have a formula for it. Here is how you calculate the unworldliness. You take all of the known physical laws and write them in a special form. For example, suppose you take the law of mechanics, $\FLPF=m\FLPa$, and rewrite it as $\FLPF-m\FLPa=\FLPzero$. Then you can call $(\FLPF-m\FLPa)$—which should, of course, be zero—the “mismatch” of mechanics. Next, you take the square of this mismatch and call it $\mathsf{U}_1$, which can be called the “unworldliness of mechanical effects.” In other words, you take \begin{equation} \label{Eq:II:25:31} \mathsf{U}_1=(\FLPF-m\FLPa)^2. \end{equation} Now you write another physical law, say, $\FLPdiv{\FLPE}=\rho/\epsO$ and define \begin{equation*} \mathsf{U}_2=\biggl(\FLPdiv{\FLPE}-\frac{\rho}{\epsO}\biggr)^2, \end{equation*} which you might call “the Gaussian unworldliness of electricity.” You continue to write $\mathsf{U}_3$, $\mathsf{U}_4$, and so on—one for every physical law there is.  Finally you call the total unworldliness $\mathsf{U}$ of the world the sum of the various unworldlinesses $\mathsf{U}_i$ from all the subphenomena that are involved; that is, $\mathsf{U}=\sum\mathsf{U}_i$. Then the great “law of nature” is \begin{equation} \label{Eq:II:25:32} \boxed{\mathsf{U}=0.} \end{equation} This “law” means, of course, that the sum of the squares of all the individual mismatches is zero, and the only way the sum of a lot of squares can be zero is for each one of the terms to be zero.  So the “beautifully simple” law in Eq. (25.32) is equivalent to the whole series of equations that you originally wrote down. It is therefore absolutely obvious that a simple notation that just hides the complexity in the definitions of symbols is not real simplicity. It is just a trick. The beauty that appears in Eq. (25.32)—just from the fact that several equations are hidden within it—is no more than a trick. When you unwrap the whole thing, you get back where you were before.  However, there is more to the simplicity of the laws of electromagnetism written in the form of Eq. (25.29). It means more, just as a theory of vector analysis means more. The fact that the electromagnetic equations can be written in a very particular notation which was designed for the four-dimensional geometry of the Lorentz transformations—in other words, as a vector equation in the four-space—means that it is invariant under the Lorentz transformations. It is because the Maxwell equations are invariant under those transformations that they can be written in a beautiful form.  It is no accident that the equations of electrodynamics can be written in the beautifully elegant form of Eq. (25.29). The theory of relativity was developed because it was found experimentally that the phenomena predicted by Maxwell’s equations were the same in all inertial systems. And it was precisely by studying the transformation properties of Maxwell’s equations that Lorentz discovered his transformation as the one which left the equations invariant.  There is, however, another reason for writing our equations this way. It has been discovered—after Einstein guessed that it might be so—that all of the laws of physics are invariant under the Lorentz transformation. That is the principle of relativity. Therefore, if we invent a notation which shows immediately when a law is written down whether it is invariant or not, we can be sure that in trying to make new theories we will write only equations which are consistent with the principle of relativity.  The fact that the Maxwell equations are simple in this particular notation is not a miracle, because the notation was invented with them in mind. But the interesting physical thing is that every law of physics—the propagation of meson waves or the behavior of neutrinos in beta decay, and so forth—must have this same invariance under the same transformation. Then when you are moving at a uniform velocity in a spaceship, all of the laws of nature transform together in such a way that no new phenomenon will show up. It is because the principle of relativity is a fact of nature that in the notation of four-dimensional vectors the equations of the world will look simple. 