Hawking radiation from black holes in 2+1 dimensions

TL;DR

This paper models 2+1 dimensional black holes' quantum geometry, deriving entropy close to Bekenstein-Hawking and a modified Hawking spectrum using a length spectrum and local observer effects.

Contribution

It introduces a quantized length spectrum for black hole horizons in 2+1 dimensions and derives entropy and Hawking radiation spectra from this model.

Findings

Entropy is close to the Bekenstein-Hawking value.

Hawking spectrum is obtained with a temperature modified by the Tolman factor.

Horizon quantization involves elementary quanta of length $8\\pi \\ell_{P} n$.

Abstract

The paper develops a model to understand the effective quantum geometry of a black hole horizon and the emission of Hawking spectrum in $2+1$ dimensions. Using the algebra of Hamiltonian charges on the horizon, we establish that one should view the black hole horizon as formed out of quantised lengths of elementary quanta of value $8\pi \ell_{P}\, n$, where $n\in \mathbb{N}$, and $\ell_{P}$ is the Planck length. We determine the black hole entropy using this equidistant length spectrum in the microcanonical ensemble and show that its value is close to the Bekenstein-Hawking entropy. To evaluate the Hawking spectrum, we note that, to an observer near the black hole horizon, the entropy (or length of horizon cross-section) is related to the black hole energy. Hence, one may develop a formulation of length ensemble (similar to the area canonical ensemble of Krasnov) from which the black body spectrum may be obtained directly. This local observer perceives a Hawking spectrum whose temperature is modified by the Tolman factor.