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Using electromagnetism to study analogue space-times is tantamount toconsidering consistency conditions for when a given (meta-)material wouldprovide an analogue space-time model or --- vice versa --- characterizing whichgiven metric could be modelled with a (meta-)material. While the consistencyconditions themselves are by now well known and studied, the form the metrictakes once they are satisfied is not. This question is mostly easily answeredby keeping the formalisms of the two research fields here in contact as closeto each other as possible. While fully covariant formulations of theelectrodynamics of media have been around for a long while, they are usuallyabandoned for (3+1)- or 6-dimensional formalisms. Here we shall use the fullyunified and fully covariant approach. This enables us even to generalize theconsistency conditions for the existence of an effective metric to arbitrarybackground metrics beyond flat space-time electrodynamics. We also show how thefamiliar matrices for permittivity $\epsilon$, permeability $\mu^{-1}$, andmagneto-electric effects $\zeta$ can be seen as the three independent pieces ofthe Bel decomposition for the constitutive tensor $Z^{abcd}$, i.e., thecomponents of an orthogonal decomposition with respect to a given observer withfour-velocity $V^a$. Finally, we shall use the Moore--Penrose pseudo-inverseand the closely related pseudo-determinant to then gain the desiredreconstruction of the effective metric in terms of the permittivity tensor$\epsilon^{ab}$, the permeability tensor $[\mu^{-1}]^{ab}$, and themagneto-electric tensor $\zeta^{ab}$, as an explicit function$g_{\text{eff}}(\epsilon,\mu^{-1},\zeta)$.

Effective metrics and a fully covariant description of constitutive tensors in electrodynamics