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

Abstract

Using electromagnetism to study analogue space-times is tantamount to considering consistency conditions for when a given (meta-)material would provide an analogue space-time model or --- vice versa --- characterizing which given metric could be modelled with a (meta-)material. While the consistency conditions themselves are by now well known and studied, the form the metric takes once they are satisfied is not. This question is mostly easily answered by keeping the formalisms of the two research fields here in contact as close to each other as possible. While fully covariant formulations of the electrodynamics of media have been around for a long while, they are usually abandoned for (3+1)- or 6-dimensional formalisms. Here we shall use the fully unified and fully covariant approach. This enables us even to generalize the consistency conditions for the existence of an effective metric to arbitrary background metrics beyond flat space-time electrodynamics. We also show how the familiar matrices for permittivity $\epsilon$, permeability ${\mu }^{-1}$, and magneto-electric effects $\zeta$ can be seen as the three independent pieces of the Bel decomposition for the constitutive tensor $Z^{abcd}$, i.e., the components of an orthogonal decomposition with respect to a given observer with four-velocity $V^a$. Finally, we shall use the Moore--Penrose pseudo-inverse and the closely related pseudo-determinant to then gain the desired reconstruction of the effective metric in terms of the permittivity tensor ${\epsilon }^{ab}$, the permeability tensor $[{\mu }^{-1}{]}^{ab}$, and the magneto-electric tensor ${\zeta }^{ab}$, as an explicit function $g_{\text{eff}}(\epsilon ,{\mu }^{-1},\zeta )$.

arXiv commentV1: 24 pages; V2: 14 pages (reformatted, minor typos fixed, no physics changes)

References in corpus (10)

• The Dynamics of General Relativity
• Analogue Gravity
• Dynamical laws of superenergy in General Relativity
• Riemannian light cone from vanishing birefringence in premetric vacuum electrodynamics
• Gravito-electromagnetic analogies
• Linear pre-metric electrodynamics and deduction of the light cone
• Non-minimal coupling of photons and axions
• Causal structure and algebraic classification of area metric spacetimes in four dimensions
• The non-birefringent limit of all linear, skewonless media and its unique light-cone structure
• Optical metrics and birefringence of anisotropic media

Cited by in corpus (10)

• Covariant electrodynamics in linear media: Optical metric
• Vorticity in analogue spacetimes
• Atomic Quantum Technologies for Quantum Matter and Fundamental Physics Applications
• Towards a Gordon form of the Kerr spacetime
• Bespoke analogue space-times: Meta-material mimics
• Boyer-Lindquist space-times and beyond: Meta-material analogues
• Optical Aharonov-Bohm effect due to toroidal moment inspired by general relativity
• Optical solitons in curved spacetime
• Electromagnetic analogue space-times, analytically and algebraically
• The geometry of induced electromagnetic fields in moving media