Starting from the covariant form of the Maxwell equations and making a suitable choice of gauge which you should specify, show that the 4-vector potential due to an arbitrary 4-current $J^{\mu}(x)$ obeys the wave equation,

$$\left(\nabla^{2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right) A^{\mu}=-\mu_{0} J^{\mu}$$

where $x^{\mu}=(c t, \mathbf{x})$.

Use the method of Green's functions to show that, for a localised current distribution, this equation is solved by

$$A^{\mu}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{J^{\mu}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$$

for some $t_{\text {ret }}$ that you should specify.

A point particle, of charge $q$, moving along a worldline $y^{\mu}(\tau)$ parameterised by proper time $\tau$, produces a 4 -vector potential

$$A^{\mu}(x)=\frac{\mu_{0} q c}{4 \pi} \frac{\dot{y}^{\mu}\left(\tau_{\star}\right)}{\left|R^{\nu}\left(\tau_{\star}\right) \dot{y}_{\nu}\left(\tau_{\star}\right)\right|}$$

where $R^{\mu}(\tau)=x^{\mu}-y^{\mu}(\tau)$. Define $\tau_{\star}(x)$ and draw a spacetime diagram to illustrate its physical significance.

Suppose the particle follows a circular trajectory,

$$\mathbf{y}(t)=(R \cos (\omega t), R \sin (\omega t), 0)$$

(with $y^{0}=c t$ ), in some inertial frame with coordinates $(c t, x, y, z)$. Evaluate the resulting 4 -vector potential at a point on the $z$-axis as a function of $z$ and $t$.