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.