Using Lorentz gauge, $A_{, a}^{a}=0$, Maxwell's equations for a current distribution $J^{a}$ can be reduced to $\square A^{a}(x)=\mu_{0} J^{a}(x)$. The retarded solution is

$$A^{a}(x)=\frac{\mu_{0}}{2 \pi} \int d^{4} y \theta\left(z^{0}\right) \delta\left(z_{c} z^{c}\right) J^{a}(y)$$

where $z^{a}=x^{a}-y^{a}$. Explain, heuristically, the rôle of the $\delta$-function and Heaviside step function $\theta$ in this formula.

The current distribution is produced by a point particle of charge $q$ moving on a world line $r^{a}(s)$, where $s$ is the particle's proper time, so that

$$J^{a}(y)=q \int d s V^{a}(s) \delta^{(4)}(y-r(s))$$

where $V^{a}=\dot{r}^{a}(s)=d r^{a} / d s$. Show that

$$A^{a}(x)=\frac{\mu_{0} q}{2 \pi} \int d s \theta\left(X^{0}\right) \delta\left(X_{c} X^{c}\right) V^{a}(s),$$

where $X^{a}=x^{a}-r^{a}(s)$, and further that, setting $\alpha=X_{c} V^{c}$,

$$A^{a}(x)=\frac{\mu_{0} q}{4 \pi}\left[\frac{V^{a}}{\alpha}\right]_{s=s^{*}}$$

where $s^{*}$ should be defined. Verify that

$$s^{*}, a=\left[\frac{X_{a}}{\alpha}\right]_{s=s^{*}} .$$

Evaluating quantities at $s=s^{*}$ show that

$$\left[\frac{V^{a}}{\alpha}\right]_{, b}=\frac{1}{\alpha^{2}}\left[-V^{a} V_{b}+S^{a} X_{b}\right]$$

where $S^{a}=\dot{V}^{a}+V^{a}\left(1-X_{c} \dot{V}^{c}\right) / \alpha$. Hence verify that $A^{a}{ }_{, a}(x)=0$ and

$$F_{a b}=\frac{\mu_{0} q}{4 \pi \alpha^{2}}\left(S_{a} X_{b}-S_{b} X_{a}\right) .$$

Verify this formula for a stationary point charge at the origin.

[Hint: If $f(s)$ has simple zeros at $s_{i}, i=1,2, \ldots$ then

$$\delta(f(s))=\sum_{i} \frac{\delta\left(s_{i}\right)}{\left|f^{\prime}\left(s_{i}\right)\right|}$$