Invisibility Cloaking and Transformation Optics for Three Dimensional Manifolds and Applications in Cosmology

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. Our model can be used in two ways. First, the model makes it possible to do mathematical analysis on a compact 3-manifold $M$ by smoothly embedding the set $M\setminus L$, where $L$ is a finite union of closed curves, to an open subset of $\mathbb R^3$. Second, the model 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 $(\mathbb{R}_+\times M, -dt^2 +g)$, where $(M,g)$ is a closed, orientable, $C^\infty$-smooth, 3-dimensional Riemannian manifold. Thus the obtained device could be used to simulate cosmological models using metamaterial devices.