Preprint Article Version 2 This version is not peer-reviewed

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

Version 1 : Received: 14 September 2018 / Approved: 16 September 2018 / Online: 16 September 2018 (08:12:22 CEST)
Version 2 : Received: 5 March 2019 / Approved: 11 March 2019 / Online: 11 March 2019 (10:36:30 CET)

How to cite: Gratus, J.; Kinsler, P.; McCall, M.W. Maxwell's Equations, Stokes' Theorem, and the Conservation of Charge. Preprints 2018, 2018090278 (doi: 10.20944/preprints201809.0278.v2). Gratus, J.; Kinsler, P.; McCall, M.W. Maxwell's Equations, Stokes' Theorem, and the Conservation of Charge. Preprints 2018, 2018090278 (doi: 10.20944/preprints201809.0278.v2).

## Abstract

A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials $\varphi$ and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth.

## Subject Areas

electromagnetism; topology; charge-conservation; constitutive relations; gauge freedom