4. Maxwell's theory of the aether

https://doi.org/10.48550/arXiv.2403.13981

I now summarize some of the reasoning that led Maxwell to his macroscopic theory of electromagnetism. My purpose is to show that almost all of his reasoning is inconsistent with what has since been learned about materials and spacetime and that, in hindsight, the existence of $\pp$ and $\D$ fields appears not to have a sound logical basis. The sources I have relied on most heavily are [Maxwell, 1865], [Maxwell, 1873], [Maxwell, 1892], [Heaviside, 1893], [Lorentz, 1916], and [Buchwald and Fox, 2013].

4.1 Maxwell's reasoning

In vacuum he regarded the aether as isotropic, homogeneous, and with properties characterised by only two scalar constants, such as its permittivity ${\epsilon_0}$ and its permeability ${\mu_0}$, or either one and the speed of light $c$. He believed that the aether’s properties were altered in the presence of matter, but that the effects of matter on electromagnetic phenomena were indirect and could, to a first approximation, be described by the changes ${\epsilon_0\mapsto\tilde{\epsilon}(\br)}$ and ${\mu_0\mapsto\tilde{\mu}(\br)}$ of the aether’s characteristic constants from uniform scalars to tensor fields.

Maxwell used the term electricity in an abstract or vague sense and he likened electricity in the aether to elasticity in a solid. He regarded this analogy as so compelling that, on the basis of it, he was willing to impute to the aether the minimal set of physical properties necessary to make his theory internally consistent. He reasoned that, just as a slack rope or an unstrained rod cannot transmit forces between its two ends, the aether must be in a state of mechanical stress when electric forces are transmitted through it. Therefore, just as an elastically-deformed solid has a density of stored energy at each point, which is released when the deforming force is removed and the body resumes its original shape, he speculated that an electric force field was always accompanied by a displacement field in the aether, which stored potential energy.

As an alternative to specifying a deformed state of an elastic solid with a vector field whose value at each point is the point’s displacement from equilibrium, one can describe it by a flux density vector field that is parallel to the direction of material flow at each point and has a magnitude equal to the total quantity of material, per unit area, that flowed through a small imaginary surface at the point during the deformation. Maxwell chose this latter approach to describe the state of the aether and the motion of electricity within it. One of his reasons was that certain fluxes, namely electric currents, were measureable, and measured fluxes were spatial averages, which could not be calculated using the former approach unless much more detailed information about the aether was available. For example, a rate of fluid flow cannot be calculated if one only knows the velocity of the fluid at each point; knowledge of its density is also required.

Maxwell defined the confusingly-named electric displacement field $\D$ as a flux density. It specified how much electricity passed through each point, and in which direction, as an applied field that caused and maintained this displacement was ‘switched on’. He regarded $\D$ as a specification of the electrically-deformed state of the aether, albeit one that was a spatial average of a more detailed microscopic flux density. He regarded the current density $\J$ as a rate of motion or a velocity of the aether, that was driven by the electric force $\E$, and which changed the aether’s displacement $\D$.

For reasons that remained mysterious to Maxwell, conductors lacked the restoring force that returned the $\D$ field in an electrically-deformed dielectric to its original state when the electric field supporting it was switched off. Therefore, instead of simply displacing, electricity flowed freely as a current. As it flowed, it dissipated some of the aether’s energy into heat within the material. Similarly, when a dielectric was placed in an electric field, a transient current $\bdot{\D}$ flowed and dissipated energy until the equilibrium displacement was reached.

For energy to be conserved the energy stored in the aether by the displacement field had to be lower in the presence of a dielectric than it was in free space. It followed that, for the same electric force $\E$, $\D$ was different within a dielectric to its value of ${\epsilon_0\E}$ in vacuum. Its value in a dielectric was ${\D=\epsilon_0\E+\pp}$, where $\pp$ was known as the electric polarization of the dielectric.

Since $\D$ was different in dielectrics, it must change abruptly at a dielectric’s boundaries. Maxwell viewed charge, not as a substance that can accumulate, but as a spatial discontinuity of $\D$. He did not understand a current to be a flow of charge but as a state of motion of the aether, which changed the $\D$ field, creating those discontinuities. So, although current did not transport charge, it caused it to exist.

4.2 Heaviside's concern

Heaviside made his concern about the challenges the theory faced clear in 1893, more than a decade after Maxwell’s death, when he wrote: "Whether Maxwell’s theory will last, as a sufficient and satisfactory primary theory upon which the numerous secondary developments required may be grafted, is a matter for the future to determine. Let it not be forgotten that Maxwell’s theory is only the first step towards a full theory of the aether ; and, moreover, that no theory of the aether can be complete that does not fully account for the omnipresent force of gravitation" [Heaviside, 1893].

Maxwell’s theory should not be expected to make sense conceptually as a theory of material response, because he developed it as a theory of the aether. However, because he ensured that it reproduced all of the empirically-known mathematical relationships between macroscale observables, its accuracy as a macroscale tool is undiminished by the historical peculiarities of its mathematical form.

It became confusing conceptually when it was stripped of its logical foundations by the concept of an aether becoming obsolete; and it fails as a conceptual scaffold when it is used beyond the macroscale domain for which it was constructed: At the microscale it conflicts with 20th century theories of material structure and composition. This is one of the reasons why there has been so much confusion and debate about how to calculate macroscopic fields from microscopic ones.