# Graphene and the Zermelo optical metric of the BTZ black hole

## 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 R Multiplication-Sign {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 a fundamental geometric obstacle to obtaining a model that extends all the way to the black hole horizon. - Highlights: Black-Right-Pointing-Pointer Electrons on graphene sheets obey the Dirac equation in ultra-static metrics. Black-Right-Pointing-Pointer External magnetic fields on give ultra-stationary metrics. Black-Right-Pointing-Pointer These metrics are of optical-Zermelo form. Black-Right-Pointing-Pointer Embedded surfaces corresponding to BTZ black holes are given.

- Authors:

- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396 (United States)
- (Slovenia)
- D.A.M.T.P., University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22157109

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 327; Journal Issue: 11; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; BLACK HOLES; COSMOLOGICAL CONSTANT; COSMOLOGY; DIRAC EQUATION; ELECTRIC FIELDS; ELECTRONS; EXCITATION; GRAPHENE; MAGNETIC FIELDS; MATHEMATICAL SOLUTIONS; METRICS; ONE-DIMENSIONAL CALCULATIONS; POTENTIALS; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Cvetic, M., Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, and Gibbons, G.W, E-mail: gwg1@damtp.cam.ac.uk.
```*Graphene and the Zermelo optical metric of the BTZ black hole*. United States: N. p., 2012.
Web. doi:10.1016/J.AOP.2012.05.013.

```
Cvetic, M., Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, & Gibbons, G.W, E-mail: gwg1@damtp.cam.ac.uk.
```*Graphene and the Zermelo optical metric of the BTZ black hole*. United States. doi:10.1016/J.AOP.2012.05.013.

```
Cvetic, M., Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, and Gibbons, G.W, E-mail: gwg1@damtp.cam.ac.uk. Thu .
"Graphene and the Zermelo optical metric of the BTZ black hole". United States. doi:10.1016/J.AOP.2012.05.013.
```

```
@article{osti_22157109,
```

title = {Graphene and the Zermelo optical metric of the BTZ black hole},

author = {Cvetic, M. and Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor and Gibbons, G.W, E-mail: gwg1@damtp.cam.ac.uk},

abstractNote = {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 R Multiplication-Sign {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 a fundamental geometric obstacle to obtaining a model that extends all the way to the black hole horizon. - Highlights: Black-Right-Pointing-Pointer Electrons on graphene sheets obey the Dirac equation in ultra-static metrics. Black-Right-Pointing-Pointer External magnetic fields on give ultra-stationary metrics. Black-Right-Pointing-Pointer These metrics are of optical-Zermelo form. Black-Right-Pointing-Pointer Embedded surfaces corresponding to BTZ black holes are given.},

doi = {10.1016/J.AOP.2012.05.013},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 11,

volume = 327,

place = {United States},

year = {2012},

month = {11}

}