Retarded potential

Remarks

If $\rho$ and $J$ are twice continuously differentiable, then $\varphi$ and $A$ solve the inhomogeneous electromagnetic wave equations $\square \varphi = \rho / \varepsilon_0$ and $\square A = \mu_0 J$, with $\square = \frac{1}{c^2} \partial_t^2 - \Delta$. If moreover $\rho$ and $J$ satisfy the continuity equation $\partial_t \rho + \nabla \cdot J = 0$ of charge conservation, then they satisfy the Lorenz gauge condition, and $E = -\nabla \varphi - \partial_t A$ and $B = \nabla \times A$ solve Maxwell's equations. The kernel $\delta(t - \abs{r}/c) / (4 \pi \abs{r})$ behind these formulas is the retarded Green's function of $\square$.

Example: for a charge density $\rho$ that does not depend on time and $J = 0$, the delay has no effect. Then $\varphi$ is the electrostatic potential $\frac{1}{4 \pi \varepsilon_0} \int \rho(r') / \abs{r - r'} \, d^3 r'$ and $A = 0$.

Non-example: the advanced potentials use the time $t + \abs{r - r'}/c$ in place of $t_r$. They also solve the same wave equations, but they depend on the future of the sources, so they are not retarded potentials.