Maxwell's equations are universal for locally conserved quantities
Classical Physics
2019-07-08 v1 Mathematical Physics
math.MP
Abstract
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 conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincar\'e lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.
Cite
@article{arxiv.1906.02675,
title = {Maxwell's equations are universal for locally conserved quantities},
author = {Lucas Burns},
journal= {arXiv preprint arXiv:1906.02675},
year = {2019}
}