  While the postulate of covariance of Maxwell's equations for all inertial
observers led Einstein to special relativity, it was the further demand of
general covariance -- form invariance under general coordinate transformations,
including between accelerating frames -- that led to general relativity.
Several lines of inquiry over the past two decades, notably the development of
metamaterial-based transformation optics, has spurred a greater interest in the
role of geometry and space-time covariance for electrodynamics in ponderable
media. I develop a generally covariant, coordinate-free framework for
electrodynamics in general dielectric media residing in curved background
space-times. In particular, I derive a relation for the spatial medium
parameters measured by an arbitrary time-like observer. In terms of those
medium parameters I derive an explicit expression for the Finsler-like optical
metric of birefringent media and show how it reduces to a Riemann-like optical
metric for nonbirefringent media. This formulation provides a basis for a
unified approach to ray and congruence tracing through media in curved
space-times that may smoothly vary among positively refracting, negatively
refracting, and vacuum.
