Neo-Hertzian Wave Equation for Variable Detector Velocity

It has been argued previously [1-3] that a first-order invariant covering theory of Maxwell’s electromagnetism (EM) originally proposed by Hertz [4] provides not only a formally more comprehensive approach to EM description but one that is more competent to describe the weak-field (single photon) limit. This claim depends on a specific physical interpretation of the Galilean invariant formalism, whereby the Hertzian fields are operationally defined in a different way from the Maxwellian ones. That is, Maxwellian fields are measured at a field point stationary with respect to the inertial observer, by a field detector or “test charge” that is by definition also at rest at that field point; whereas Hertzian fields, likewise referred to a stationary field point, are measured by a detector or test particle that can move with arbitrary velocity d v with respect to that point (so that it momentarily passes through the field point at the instant of measurement). This introduces formally a new “detector velocity” parameter into EM theory, so that Hertzian theory parametrizes both source and sink motions, whereas Maxwellian theory leaves sink motions unparametrized. Mathematically, invariance in the Hertzian case and covariance in the Maxwellian case express the relativity principle with respect to inertial motions. The claim of Hertzian superiority in the weak-field limit arises from the fact that Hertz’s theory is adequately parametrized to describe from the viewpoints of n different inertial observers the relative motions of a single “public” instrument of field detection; whereas n Maxwellian observers require n different macro instruments, one comoving with each inertial observer (permanently fixed at his field point), in effect the “private property” of that observer. Both alternative ways of treating the relative motions of observers and detectors are equally valid if the signal to be detected is carried by a superabundance of field quanta. In this case all Maxwellian detectors, as their field points instantaneously coincide, receive enough quanta to register significant numbers—i.e., those quantifying covariantly related field components. But in the single-quantum limit it is obvious that at most a single macro