Electrodynamics in four-dimensional notation
We have already encountered the d’Alembertian operator, without giving it that name, in Section 18-6; the differential equations we found there for the potentials can be written in the new notations as: \begin{equation} \label{Eq:II:25:21} \Box^2\phi=\frac{\rho}{\epsO},\quad \Box^2\FLPA=\frac{\FLPj}{\epsO}. \end{equation} The four quantities on the right-hand side of the two equations in (25.21) are $\rho$, $j_x$, $j_y$, $j_z$ divided by $\epsO$, which is a universal constant which will be the same in all coordinate systems if the same unit of charge is used in all frames. So the four quantities $\rho/\epsO$, $j_x/\epsO$, $j_y/\epsO$, $j_z/\epsO$ also transform as a four-vector. We can write them as $j_\mu/\epsO$. The d’Alembertian doesn’t change when the coordinate system is changed, so the quantities $\phi$, $A_x$, $A_y$, $A_z$ must also transform like a four-vector—which means that they are the components of a four-vector. In short, \begin{equation*} A_\mu=(\phi,\FLPA) \end{equation*} is a four-vector. What we call the scalar and vector potentials are really different aspects of the same physical thing. They belong together. And if they are kept together the relativistic invariance of the world is obvious. We call $A_\mu$ the four-potential.  In the four-vector notation Eqs. (25.21) become simply \begin{equation} \label{Eq:II:25:22} \Box^2A_\mu=\frac{j_\mu}{\epsO}, \end{equation} The physics of this equation is just the same as Maxwell’s equations. But there is some pleasure in being able to rewrite them in an elegant form. The pretty form is also meaningful; it shows directly the invariance of electrodynamics under the Lorentz transformation.  Remember that Eqs. (25.21) could be deduced from Maxwell’s equations only if we imposed the gauge condition \begin{equation} \label{Eq:II:25:23} \ddp{\phi}{t}+\FLPdiv{\FLPA}=0, \end{equation} which just says $\fournabla A_\mu=0$; the gauge condition says that the divergence of the four-vector $A_\mu$ is zero. This condition is called the Lorenz condition. It is very convenient because it is an invariant condition and therefore Maxwell’s equations stay in the form of Eq. (25.22) for all frames. 