Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 36A / a / Solution

The electromagnetic four-potential is the field generated by the four-current $J^{\mu }$: every source volume element contributes with the inverse-distance Green-function factor. Electromagnetic influence propagates at speed $c$, so causality requires the source to be evaluated where its worldline intersects the field event's past light cone. ThusThe source time is exactly the retarded time $t_{\mathrm{ret}}=t-|x-x^{\prime }|/c$, rather than the observation time $t$.