Maxwell's Equations from Vortex Mechanics: A Kelvin-Voigt Analogy

We establish a formal mechanical analogy for Maxwell's equations of electrodynamics, based on the dynamics of compact vortex structures in a Kelvin-Voigt viscoelastic medium. The central difficulty in hydrodynamic analogies of this type, that vorticity in a Newtonian fluid obeys a purely diffusive equation, incompatible with finite-speed wave propagation, is resolved by replacing the Newtonian constitutive model with the Kelvin-Voigt law. The elastic component of this law is physically motivated by the pressure-balance mechanism that stabilises compact Hill-type vortex cores: a small deformation of the core produces a restoring force, endowing the exterior vortex-ring medium with a finite shear modulus. The collective inter-vortex (Tkachenko) contribution to this modulus is derived exactly; the intra-core shape-mode contribution is computed via a pressure-jump method and shown to be of comparable magnitude. The Kelvin-Voigt constitutive law then yields, as an exact consequence, a damped shear-wave equation for the vector potential. Incompressibility of the exterior medium uniquely selects the Coulomb gauge condition without additional assumptions. All four Maxwell equations follow systematically: two are potential identities, one uses the Kelvin-Voigt wave closure, and one combines the gauge condition with the Poisson equation for the scalar potential. Electric charge is identified with the circulation invariant of the vortex core, conserved by Kelvin's theorem. The model reproduces Coulomb's law in the static limit, and all mapped quantities, including the conversion constant, identified as the Kelvin-Voigt mechanical relaxation time, have direct mechanical interpretations. Concrete experimental predictions are given for vortex interactions in superfluid systems such as Bose-Einstein condensates and liquid helium.