Hertz vector

Remarks

The potentials satisfy the Lorenz gauge condition: $\nabla \cdot A + \frac{1}{c^2} \partial_t \varphi = \mu_0 \partial_t \nabla \cdot \Pi_e - \frac{1}{\varepsilon_0 c^2} \partial_t \nabla \cdot \Pi_e = 0$, since $\varepsilon_0 c^2 = 1 / \mu_0$. Applying $\square = \frac{1}{c^2} \partial_t^2 - \Delta$ gives $\square \varphi = -\frac{1}{\varepsilon_0} \nabla \cdot P = \rho / \varepsilon_0$ and $\square A = \mu_0 \partial_t P = \mu_0 J$, the inhomogeneous electromagnetic wave equations. In terms of $\Pi_e$ the fields are $E = \frac{1}{\varepsilon_0} \left( \nabla (\nabla \cdot \Pi_e) - \frac{1}{c^2} \partial_t^2 \Pi_e \right)$ and $B = \mu_0 \, \partial_t \nabla \times \Pi_e$. A magnetic Hertz vector is defined in the same way with the magnetization as source.

Example: for a point dipole $P = p(t) \, \delta(r)$ with a smooth moment $p \colon \R \to \R^3$, the retarded solution is $\Pi_e(r, t) = \frac{p(t - \abs{r}/c)}{4 \pi \abs{r}}$. Away from the origin each component has the form $f(t - \abs{r}/c) / \abs{r}$, an outgoing spherical wave, which solves the homogeneous wave equation.

Non-example: $\Pi_e(r, t) = (t^2, 0, 0)$ with $P = 0$ is not a Hertz vector for $P$, since $\Delta \Pi_e - \frac{1}{c^2} \partial_t^2 \Pi_e = (-2/c^2, 0, 0) \neq 0$. The defining equation, not the Lorenz condition, is what ties a Hertz vector to its source.