Transformation Optics for the Modelling of Waves in a Universe with Nontrivial Topology

Abstract

We consider how transformation optics and invisibility cloaking can be used to construct models in subsets ${\mathbb{R}}^3$ with a varying metric, where the time-harmonic waves for a given angular wavenumber $k$, are equivalent to the waves in some closed orientable manifold. The obtained models could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by $({\mathbb{R}}_{+}\times M,-d{t}^2+g)$, where $(M,g)$ is a closed, orientable, $C^{\infty }$-smooth, 3-dimensional Riemannian manifold. Thus the obtained construction could be used to simulate cosmological models using metamaterial devices.

42 pages. Added 3 figures. Layout of chapter 3 changed for clarity. More motivation and a new subsection on cloaking and black holes added to the introduction. Additional references added. Typos fixed

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