Graphene and the Zermelo Optical Metric of the BTZ Black Hole

TL;DR

This paper explores how the geometry of curved graphene sheets can be modeled using optical metrics related to BTZ black holes, revealing geometric limitations in extending these models to the horizon.

Contribution

It establishes a connection between graphene's electronic properties, Zermelo optical metrics, and BTZ black hole geometry, highlighting geometric constraints.

Findings

Modeling electric fields with Zermelo metrics conformal to BTZ black holes

Using Randers metrics to represent magnetic fields in graphene

Identifying geometric obstacles near the black hole horizon

Abstract

It is well known that the low energy electron excitations of the curved graphene sheet $\Sigma$ are solutions of the massless Dirac equation on a 2+1 dimensional ultra-static metric on ${\Bbb R} \times \Sigma$. 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.