The charge and current densities are given by $\rho(t, \mathbf{x}) \neq 0$ and $\mathbf{j}(t, \mathbf{x})$ respectively. The electromagnetic scalar and vector potentials are given by $\phi(t, \mathbf{x})$ and $\mathbf{A}(t, \mathbf{x})$ respectively. Explain how one can regard $j^{\mu}=(\rho, \mathbf{j})$ as a four-vector that obeys the current conservation rule $\partial_{\mu} j^{\mu}=0$.

In the Lorenz gauge $\partial_{\mu} A^{\mu}=0$, derive the wave equation that relates $A^{\mu}=(\phi, \mathbf{A})$ to $j^{\mu}$ and hence show that it is consistent to treat $A^{\mu}$ as a four-vector.

In the Lorenz gauge, with $j^{\mu}=0$, a plane wave solution for $A^{\mu}$ is given by

$$A^{\mu}=\epsilon^{\mu} \exp \left(i k_{\nu} x^{\nu}\right)$$

where $\epsilon^{\mu}, k^{\mu}$ and $x^{\mu}$ are four-vectors with

$$\epsilon^{\mu}=\left(\epsilon^{0}, \boldsymbol{\epsilon}\right), \quad k^{\mu}=\left(k^{0}, \mathbf{k}\right), \quad x^{\mu}=(t, \mathbf{x}) .$$

Show that $k_{\mu} k^{\mu}=k_{\mu} \epsilon^{\mu}=0$.

Interpret the components of $k^{\mu}$ in terms of the frequency and wavelength of the wave.

Find what residual gauge freedom there is and use it to show that it is possible to set $\epsilon^{0}=0$. What then is the physical meaning of the components of $\boldsymbol{\epsilon}$ ?

An observer at rest in a frame $S$ measures the angular frequency of a plane wave travelling parallel to the $z$-axis to be $\omega$. A second observer travelling at velocity $v$ in $S$ parallel to the $z$-axis measures the radiation to have frequency $\omega^{\prime}$. Express $\omega^{\prime}$ in terms of $\omega$.