Maxwell’s equations are universal for locally conserved quantities

A fundamental result of classical electromagnetism is that Maxwell’s equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell’s equations. This holds true for any quantity satisfying a continuity equation.

As a first application, we find that pressure and velocity fields of acoustic waves, which satisfy a continuity equation, possess a pair of vector potentials satisfying Maxwell’s equations. A Lagrangian expressed in terms of these potentials is shown to yield the usual acoustic wave equation, with the important difference that its canonical angular momentum tensor contains a non-vanishing spin part, despite acoustic waves being purely longitudinal. This spin term agrees with an expression for acoustic spin density proposed in a recent string of publications.