Extra-dimensional metamaterials: simple models of inflation and metric signature transitions

Lattices of topological defects, such as Abrikosov lattices and domain wall lattices often arise as metastable ground states in higher-dimensional field theoretical models. We demonstrate that such lattice states may be described as extra-dimensional metamaterials via higher-dimensional effective medium theory. A 4+1 dimensional extension of Maxwell electrodynamics with a compactified time-like dimension has been considered as an example. It is demonstrated that from the point of view of macroscopic electrodynamics an Abrikosov lattice state in such a 4+1 dimensional spacetime may be described as a uniaxial hyperbolic metamaterial. Extraordinary photons perceive this medium as a 3+1 dimensional Minkowski spacetime in which one of the original spatial dimensions (the optical axis of the metamaterial) plays the role of a new time-like coordinate. Since metric signature of this effective space-time depends on the Abrikosov lattice periodicity, the described model may be useful in studying metric signature transitions. A particular kind of metric signature transition understood as a macroscopic medium effect may emulate cosmological inflation.