Causal structure and algebraic classification of non-dissipative linear optical media

Abstract

In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.

A Restatement of the Algebraic Classification of Area Metrics on 4-Manifolds

“…The motivation for studying area metrics is that they appear as a natural generalization of Lorentz metrics in physics [17]. For example, in relativistic electromagnetics, a Lorentz metric always describes an isotropic medium, but using an area metric one can also model anisotropic medium, where differently polarized waves can propagate with different wave-speeds.…”

Electromagnetic potential in pre-metric electrodynamics: Causal structure, propagators and quantization

“…With the introduction of M in (14) by combining M with an ε tensor density we can rephrase the real principal type condition (B1) for pre-metric electrodynamics as…”

“…Sometimes it is useful to consider the completion A cpl of A in its natural 14 topology. Consider the continuous extension of A ⊗n to the map (denoted by the same symbol)…”

“…It follows from the properties of the quantum field A, that an admissible n-point distribution ω n must be a weak solution of the field equation in each argument 13 We denote the complex conjugate of z byz. 14 The "natural" topology of A is that induced (via the direct sum, quotient and subspace topology) by the test function topology on Ω 3 c ðMÞ. This uses the fact that the field algebra is the quotient of the tensor algebra ⨁ n S ⊗n sc by the commutation relations.…”

“…The Fresnel tensor density and the Fresnel polynomial play a central role in the analysis of the partial differential Eq. (4)-essentially, the Fresnel tensor defines the underlying causal structure of pre-metric electrodynamics [14], while requiring hyperbolicity of the Fresnel polynomial guarantees that the initial value problem of the field equations is well posed [15], independent of the existence of a Lorentzian metric. We discuss the connection between causality and the properties of the Fresnel polynomial in more detail in Sec.…”