Maxwell's Equations, Stokes' Theorem, and the Conservation of Charge

Gratus, Jonathan and Kinsler, Paul and McCall, Martin (2020) Maxwell's Equations, Stokes' Theorem, and the Conservation of Charge. arXiv. ISSN 2331-8422

Abstract

We construct an analytic solution to Maxwell's equations that violates global charge conservation, by building on the possibilities demonstrated in [Foundations of Physics 49, 330 (2019)]. The construction is valid for a spacetime containing a temporary singularity and a Maxwellian electrodynamics containing a proposed "topological" axion field. As such, it demonstrates that although "physics breaks down" at a singularity, the physical laws on the spacetime still impose constraints on what can happen. Further, the concepts of transformation optics can be applied to show that our specific mathematical solution has a much wider applicability.