Maxwell's equations are universal for locally conserved quantities
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 Poincaré 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2019
- DOI:
- 10.48550/arXiv.1906.02675
- arXiv:
- arXiv:1906.02675
- Bibcode:
- 2019arXiv190602675B
- Keywords:
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- Physics - Classical Physics;
- Mathematical Physics
- E-Print:
- doi:10.1007/s00006-019-0979-7