Defining Electromagnetic Fields from Continuity and Divergence-Free Structure
Preferred Frame Writing — January 2026
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One-Sentence Summary
Continuity and divergence-free structure force electromagnetic energy to be described by directional, circulating degrees of freedom; however, energy density and flux alone do not uniquely determine the electric and magnetic fields without additional relational data.
Summary
We clarify the conceptual bridge between a field-only ontology of electromagnetic energy flow and the usual electric and magnetic field variables. We show rigorously that a single scalar energy density does not define propagation, but that the relation between scalar configurations at different times, constrained by continuity, forces the introduction of directional transport. We prove a reconstruction result: given an energy density u and an energy flux S satisfying |S| <= c u, one can always construct at least one pair of fields E and B that reproduce u and S. The non-uniqueness of this reconstruction matches the true physical degrees of freedom identified as polarization. We conclude with a precise statement of what is implied by continuity and what requires the full Maxwell dynamical law.
Keywords
Keywords: Maxwell theory, continuity equation, divergence-free fields, Poynting vector, energy density, field reconstruction, circulation, topology