# Maxwell's equations are universal for locally conserved quantities

## TL;DR

This paper proves that local conservation laws imply the existence of Maxwell-like fields, establishing a fundamental equivalence between conserved quantities and electromagnetic field equations in flat and curved spacetimes.

## Contribution

It introduces a strong Poincaré lemma and demonstrates that local conservation laws universally imply Maxwell's equations for any conserved quantity.

## Key findings

- Local charge conservation implies Maxwell's equations.
- A strong Poincaré lemma for divergence-free fields is established.
- Conditions for generalization to curved manifolds are provided.

## 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.