Graphene and the Zermelo Optical Metric of the BTZ Black Hole
arXiv:1202.2938 · doi:10.1016/j.aop.2012.05.013
Abstract
It is well known that the low energy electron excitations of the curved graphene sheet $Σ$ are solutions of the massless Dirac equation on a 2+1 dimensional ultra-static metric on ${\Bbb R} \times Σ$. An externally applied electric field on the graphene sheet induces a gauge potential which could be mimicked by considering a stationary optical metric of the Zermelo form, which is conformal to the BTZ black hole when the sheet has a constant negative curvature. The Randers form of the metric can model a magnetic field, which is related by a boost to an electric one in the Zermelo frame. We also show that there is fundamental geometric obstacle to obtaining a model that extends all the way to the black hole horizon.
10 pages Latex, no figures, substantial revisions, relation between magnetic and electric fields and Randers and Zermelo forms clarified