Maxwell’s equations as mechanical law

Abstract

Electrodynamics and special relativity are so deeply interconnected that it is difficult decide which of them is the logical consequence of the other, without running the risk of circular reasoning. Following the historical development by introducing electromagnetism phenomenologically first, this issue is not resolved but at best minimized, because even the Lorentz force law itself is fundamentally relativistic. Here we propose a derivation of Maxwell’s equations at a beginning graduate level with the goal of circumventing this difficulty. Based on prior work that establishes special relativity independently of any electrodynamic postulates, it is indeed possible to derive Maxwell’s equations from relativity by adding suitable assumptions. The contribution of the present work is to choose these assumptions in such a way that they can be regarded as extensions of the same postulates on which special-relativistic point-particle dynamics itself is based in the first place. The biggest leap is to extend the law of energy- and momentum conservation to the electromagnetic field itself. This hinges on the energy–momentum tensor, and the fact that the latter must account not only for the Lorentz force on a charge, but also for the forces exerted by that same charge on its environment.