Backwards on Minkowski's road. From 4D to 3D Maxwellian electromagnetism

Abstract

Abstract. Minkowski's concept of a four-dimensional physical space is a central paradigm of modern physics. The three-dimensional Maxwellian electrodynamics is uniquely generalized to the covariant four-dimensional form. Is the (1+3) decomposition of the covariant four-dimensional form unique? How do the different sign assumptions of electrodynamics emerge from this decomposition? Which of these assumptions are fundamental and which of them may be modified? How does the Minkowski space-time metric emerge from this preliminary metricfree construction? In this paper we are looking for answers to the problems mentioned. Our main result is the derivation of four different possible sets of electrodynamic equations which may occur in different types of isotropic electromagnetic media. The wave propagation in each of these media is described by the Minkowskian optical metrics. Moreover, the electric and magnetic energies are nonnegative in all cases. We also show that the correct directions of the Lorentz force (as a consequence of the Dufay and the Lenz rules) hold true for all these cases. However, the differences between these four types of media must have a physical meaning. In particular, the signs of the three electromagnetic invariants are different.

Maxwell's equations in Minkowski's world: their premetric generalization and the electromagnetic energy-momentum tensor

“…The signs in (5) are not conventional. They are rather dictated by the Lenz rule, see Itin et al [25,26]. If we put the latter two equations into matrix form in order to be able to compare them later with the results of Minkowski, we find…”

“…compare (15) 2 . Note that (25) belongs to the principal part (1) χ ijkl . In order to compare (24) with experiments, we have to split it into time and space parts.…”

“…The laws (36) are also premetric. The Lenz rule, as the reason for the relative sign difference between the time derivatives in (32) 2 and (36) 1 , is discussed in [25,26]. We have then the Maxwellian set 10 dH = J and dF = 0 .…”

On Kottler's path: Origin and evolution of the premetric program in gravity and in electrodynamics

“…Perhaps surprisingly, we derived also the Lorentz signature, see [58,77,76]. Because of the importance of this result, we will discuss it separately in Sec.…”

Generally covariant Maxwell theory for media with a local response: Progress since 2000

“…5) Signature of the metric can be derived from electrodynamics: In [18] it has been pointed out that the signature of the metric can be derived from the Lenz rule and the positivity of the energy of the electromagnetic field. The formalism has been straightened out, improved, and enriched by Itin et al [27], [33]. 6) Four incarnations of the Tellegen gyrator: The Tellegen gyrator [43], [72], [78], [79] is an element in linear and 1 (iv) The hypothetical axion particle of elementary particle physics, see Wilczek [82] and Weinberg [81, pp.…”