Time travel in transformation optics: Metamaterials with closed null geodesics

We apply the methods of transformation optics to theoretical descriptions of spacetimes that support closed null geodesic curves. The metric used is based on frame dragging spacetimes, such as the van Stockum dust or the Kerr black hole. Through transformation optics, this metric is analogous to a material that in theory should allow for communication between past and future. Presented herein is a derivation and description of the spacetime and the resulting permeability, permittivity, and magnetoelectric couplings that a material would need in order for light in the material to follow closed null geodesics. We also address the paradoxical implications of such a material and demonstrate why such a material would not actually result in a violation of causality. A full derivation of the Plebanski equations is also included.