Frame Indifferent Formulation of Maxwell's Elastic Fluid and the Rational Continuum Mechanics of the Electromagnetic Field

TL;DR

This paper presents a frame indifferent formulation of Maxwell's equations by modeling the electromagnetic field as an elastic fluid, unifying electromagnetism with continuum mechanics and deriving classical laws from this perspective.

Contribution

It introduces a covariant, frame indifferent formulation of electromagnetism based on the analogy with elastic fluids, extending classical Maxwell equations with additional terms.

Findings

Maxwell equations are derivable from elastic medium dynamics

The formulation unifies electromagnetism with continuum mechanics

Classical laws like Biot–Savart and Lorentz force emerge naturally

Abstract

We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form' in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field' may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from' of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum.