Select causal solutions with retarded Green functions

In vacuum, introduce scalar potential phi in volts and vector potential A in volt seconds per metre. Their derivatives give the electric and magnetic fields. The Lorenz condition couples their divergence and time derivative so that each component satisfies a sourced wave equation. This choice simplifies solving Maxwell's equations; it is not an additional physical field equation. Consistency requires charge conservation, linking the time derivative of charge density to the divergence of current density.

Use the operator with time derivatives divided by speed squared and a negative Laplacian. Its Green function has a positive spatial delta source under this convention. Integrating that function against a physical source produces the particular solution. If instead the operator were time derivative squared minus speed squared times the Laplacian, its Green function would differ by a factor of speed squared. Carrying a remembered formula between those definitions silently gives incorrect SI potentials.

For a point impulse, the outgoing response spreads over spheres. The inverse distance factor supplies the three-dimensional geometric spreading, while the time delta restricts the response to the light cone. Its integral over source time evaluates the source at observation time minus travel time. The retarded kernel vanishes for negative elapsed time. An advanced kernel solves the same local equation with the opposite temporal support; selecting the retarded one supplies physical information about preparation and incoming waves.

For an extended source, each source position has its own distance and therefore its own retarded time. One common delay is only an approximation for a compact source viewed far away, and even then directional phase differences may matter. A static source recovers the Coulomb integral because delaying a constant changes nothing. This limit is a normalization check, not evidence that electrostatic rearrangements propagate instantly. Establishing a new static configuration still involves electromagnetic transients.

The source integral solves the inhomogeneous problem with the chosen causal prescription. One may still add a homogeneous Maxwell solution describing incident radiation or specified initial fields. Thus retarded potentials do not imply that every experiment starts with no electromagnetic field. Interfaces, conductors, and cavities require boundary conditions and often a different Green function. The free-space kernel cannot enforce a metal wall merely because the source was placed near one.

For a sinusoidal current element, a delay translates into phase. Doubling distance also reduces its vector-potential amplitude in the point-source approximation. These are separate effects: phase involves travel time, amplitude involves geometric spreading. The wave lab can isolate the addition of two equal-frequency contributions with a specified phase difference. It cannot establish causal support, create a switched pulse, or solve the three-dimensional Maxwell initial-boundary problem; those claims belong to the kernel and its assumptions.