Hyperbolic Metamaterials and Massive Klein-Gordon Equation in 2+1 Dimensional de Sitter Spacetime

TL;DR

This paper demonstrates that the wave equation in hyperbolic metamaterials can be interpreted as a massive Klein-Gordon equation in 2+1 dimensional de Sitter spacetime, linking metamaterial physics with cosmological models.

Contribution

It introduces a novel interpretation of the wave equation in hyperbolic metamaterials as a massive Klein-Gordon field in de Sitter spacetime, connecting metamaterials with cosmological theories.

Findings

Wave equation in hyperbolic metamaterials maps to a massive Klein-Gordon equation in dS spacetime.

The scalar field mass relates to the cosmological constant, Planck constant, and speed of light.

Identifies a gapless mode with linear dispersion in de Sitter spacetime.

Abstract

The wave equation obeyed by the extraordinary component of the electric field in a hyperbolic metamaterial was shown to be a massless Klein-Gordon field living in a flat spacetime with two timelike and two spacelike dimensions. Such a wave equation, unexpectedly, allows dispersionless propagation albeit having two spatial dimensions. Here we show that the same equation can be naturally interpreted as a particular massive Klein-Gordon equation with the usual one timelike and two spacelike dimensions in a de Sitter (dS) background spacetime. The mass parameter of the scalar field is given in terms of the cosmological constant, Planck constant and the speed of light as $m =\sqrt{\Lambda}\frac{\hbar}{c}$ which corresponds to the point for which the left and right conformal weights of the boundary conformal field theory (CFT) (via the de Sitter/CFT correspondence) are equal. This particular mass corresponds to the gapless mode in the dS spacetime for which the dispersion relation is linear in the wave number.