Foundations of Physics (2019) 49:330–350 https://doi.org/10.1007/s10701-019-00251-5 Evaporating Black-Holes, Wormholes, and Vacuum Polarisation: Must they Always Conserve Charge? Jonathan Gratus1,2 · Paul Kinsler1,2 · Martin W. McCall3 Received: 17 May 2018 / Accepted: 28 February 2019 / Published online: 4 April 2019 © The Author(s) 2019 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 ϕ 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. Keywords Electromagnetism · Topology · Charge-conservation · Constitutive relations · Gauge freedom 1 Introduction It is not only a well established, but an extremely useful consequence of Maxwell’s equations, that charge is conserved [1]. However, this principle relies on some assump- tions, in particular those about the topology of the underlying spacetime, which are B Paul Kinsler
[email protected] Jonathan Gratus
[email protected] Martin W. McCall
[email protected] 1 Department of Physics, Lancaster University, Lancaster LA1 4YB, UK 2 The Cockcroft Institute, Sci-Tech Daresbury, Daresbury WA4 4AD, UK 3 Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK 123 Foundations of Physics (2019) 49:330–350 331 required for Stokes’ Theorem to hold.