On Hertz's Invariant Form of Maxwell's Equations
Thomas E. Phipps
Keywords: Hertzian electromagnetism, Galilean invariance, Maxwell's equations, covariance, total time derivatives
The failure of Maxwell's equations to exhibit invariance under the Galilean transformation was corrected by Hertz through a simple, but today largely forgotten, mathematical trick. This involves substituting total (convective) time derivatives for partial time derivatives wherever the latter appear in Maxwell's equations. By this means Hertz derived a formally Galilean-invariant covering theory of Maxwell's vacuum electrodynamics - which, however, was not space-time symmetrical (in view of his tampering with the time but not space derivatives). Had Hertz's mathematical accom-plishment received wider recognition, his invariant covering theory of Maxwell's could have furnished the formal key (almost two decades before Minkowski's "covariance") to unification of the "relativistic" properties of electrodynamics and Newtonian mechanics, explanation of the Michelson-Morley result, etc. The task of finding a viable physical interpretation of the Hertzian convective velocity parameter - which Hertz himself did not live to accomplish - remains for continuing research. We discuss this and related matters and give an explicit proof of invariance.
