  In this article we carefully distinguish the notion of bi-refringence (a
polarization-dependent doubling in photon propagation speeds) from that of
bi-metricity (where the two photon polarizations ``see'' two distinct metrics).
We emphasise that these notions are logically distinct, though there are
special symmetries in ordinary (3+1)-dimensional nonlinear electrodynamics
which imply the stronger condition of bi-metricity.
  To illustrate this phenomenon we investigate a generalized version of
(3+1)-dimensional nonlinear electrodynamics, which permits the inclusion of
arbitrary inhomogeneities and background fields. [For example dielectrics (a la
Gordon), conductors (a la Casimir), and gravitational fields (a la
Landau--Lifshitz).] It is easy to demonstrate that the generalized theory is
bi-refringent: In (3+1) dimensions the Fresnel equation, the relationship
between frequency and wavenumber, is always quartic. It is somewhat harder to
show that in some cases (eg, ordinary nonlinear electrodynamics) the quartic
factorizes into two quadratics thus providing a bi-metric theory. Sometimes the
quartic is a perfect square, implying a single unique effective metric. We
investigate the generality of this factorization process.
