Recent advances in classical electromagnetic theory

Abstract

The early Sections of the present Thesis utilise a metric-free and connection-free approach so as derive the foundations of classical electrodynamics. More specifically, following a tradition established by Kottler [65], Cartan [14] and van Dantzig [137], Maxwell's theory is introduced without making reference to a notion of distance or parallel transport. With very few exceptions, the relevant concepts are derived from first principles. Indeed, Maxwell's theory is constructed starting from three experimentally justified axioms: (i) electric charge is conserved, (ii) the force acting on a test charge due to the electromagnetic field is the standard Lorentz one, (iii) magnetic flux is conserved. To be precise, a strictly deductive approach requires that three further postulates are introduced, as explained in the manual [41] by Hehl and Obukhov. Nevertheless, a shortened formalism is observed to be adequate for the purpose of this work. In nearly all cases, the electromagnetic medium is demanded to be local and linear. Moreover, the propagation of light is studied in the approximate geometrical optics regime. Lindell's astute derivation of the dispersion equation [80] is reformulated in the widespread mathematical language of tensor indices. The method devised in Ref. [80] is integrated with the analysis due to Dahl [16] of the space encompassing the physically viable polarisations. As a result, the geometry associated with the dispersion equation is investigated with considerable rigour. From the literature it is known that, to a great extent, the notion of distance can be viewed as a by-product of Maxwell's theory. In fact, imposing that the constitutive law is electric-magnetic reciprocal and skewon-free determines, albeit non-uniquely, a Lorentzian metric. A novel proof of this statement is examined. In addition, the unimodular forerunner of electric-magnetic reciprocity, defined in earlier works by Lindell [79] and Perlick [112], is shown to preserve the energy-momentum tensor.