Title: "Premetric Classical Electrodynamics: Achievements and Prospects"

by

Friedrich W. Hehl and Jonathan Lux

Abstract:
Classical electrodynamics can be based on the conservation laws of
electric chargeQ and magnetic flux &Phi . Both laws
are 'topological', that is, independent of the metric and the linear
connection of spacetime. This framework is called premetric
electrodynamics and, in order to establish a predictive theory, it
requires additionally constitutive laws for the matter involved and
for vacuum, respectively. The elimination of the metric in the
fundamental charge and flux laws allows to formulate them only in
terms of electromagnetic notions without intervention of the
gravitational potential (the metric). This clear separation of concepts
pays off in a more transparent structure of the fundamentals of
electrodynamics and in the formulation of some applications:
(i) With a local and linear constitutive law and a ban on
birefringence (Itin, Lämmerzahl, Obukhov, Rubilar, fwh), one can
derive the light cone of spacetime, including its signature. The
Lenz rule, the positivity of the energy of the electromagnetic field,
and the Lorentz signature are irresolvably interrelated.
(ii) The
linear constitutive magnetoelectric tensor can be classified à la
Segrè (Schuller et al.). It allows a better classification of
materials and provides insight into metamaterials (Favaro and
Bergamin).
(iii) The axionic piece of the magnetoelectric tensor of
Cr_{2}O_{3} is non-vanishing and can be measured (Obukhov, Rivera,
Schmid, fwh). Thus, the so-called Post constraint (Lakhtakia) is
invalid.
(iv) The Cauchy problem can be streamlined in a premetric
formulation (Perlick).
(v) The energy-momentum current can be
formulated effectively and the Abraham-Minkowski controversy clarified
(Obukhov).
(vi) Metamaterials can be better understood from a
4-dimensional point of view (Itin and Friedman).
(vii) The properties
of the premetric Tamm-Rubilar tensor can be understood in a
geometrical context as being related to generalized Fresnel-Kummer
surfaces (Rubilar). These and other application will be
discussed.