Covariant electrodynamics in linear media: Optical metric

TL;DR

This paper develops a covariant, coordinate-free framework for electrodynamics in dielectric media within curved space-times, deriving optical metrics for birefringent and nonbirefringent media, facilitating unified ray tracing approaches.

Contribution

It introduces a generally covariant, coordinate-free formulation of electrodynamics in media in curved space-times, including explicit optical metrics for birefringent and nonbirefringent materials.

Findings

Derived a relation for spatial medium parameters for arbitrary observers.

Explicitly formulated the Finsler-like optical metric for birefringent media.

Reduced the optical metric to a Riemann-like form for nonbirefringent media.

Abstract

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.