Electromagnetic influences propagate at finite speed. In the Lorenz gauge, the scalar and vector potentials satisfy driven wave equations, $$\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi=-\frac{\rho}{\varepsilon_0},$$ $$\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\mathbf A=-\mu_0\mathbf J.$$ Their causal solutions are the retarded potentials: $$\phi(\mathbf r,t)=\frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf r',t_r)}{|\mathbf r-\mathbf r'|},d^3r',$$ $$\mathbf A(\mathbf r,t)=\frac{\mu_0}{4\pi}\int \frac{\mathbf J(\mathbf r',t_r)}{|\mathbf r-\mathbf r'|},d^3r',$$ where $$t_r=t-\frac{|\mathbf r-\mathbf r'|}{c}$$ is the retarded time.

The field observed at $(\mathbf r,t)$ therefore depends on what the source was doing earlier, at the time required for light to travel from the source point $\mathbf r'$ to the observation point.

For a static charge distribution, $ ho$ is time-independent and the retarded scalar potential reduces to the ordinary Coulomb potential. For rapidly changing currents, however, retardation is essential and generates electromagnetic radiation.

The retarded solution should not be interpreted as a source sending a continuously updated instruction to every point in space instantaneously. Changes in the source alter the field only after the causal light-travel delay.

Retarded potentials are the bridge between Maxwell's local differential equations and fields produced by time-dependent sources. They prepare the calculation of antenna radiation, dipole emission and relativistic electromagnetic interactions.