Generalized Uncertainty Principle in three-dimensional gravity and the BTZ black hole

TL;DR

This paper explores the Generalized Uncertainty Principle in three-dimensional gravity, focusing on the BTZ black hole, and examines its effects on black hole thermodynamics and potential analogs in condensed matter systems.

Contribution

It proposes a specific gravitational radius for the GUP in 3D, based on the BTZ black hole horizon, and analyzes the resulting modifications to Hawking temperature and entropy.

Findings

The event horizon of the BTZ black hole provides a consistent minimal length scale.

The GUP induces measurable corrections to black hole thermodynamics.

An alternative length scale from the cosmological constant is also considered.

Abstract

We investigate the structure of the gravity-induced Generalized Uncertainty Principle in three dimensions. The subtleties of lower dimensional gravity, and its important differences with respect to four and higher dimensions, are duly taken into account, by considering different possible candidates for the gravitational radius, $R_g$, that is the minimal length/maximal resolution of the quantum mechanical localization process. We find that the event horizon of the $M \neq 0$ Ba\~{n}ados-Teitelboim-Zanelli micro black hole furnishes the most consistent $R_g$. This allows us to obtain a suitable formula for the Generalized Uncertainty Principle in three dimensions, and also to estimate the corrections induced by the latter on the Hawking temperature and Bekenstein entropy. We also point to the extremal $M=0$ case, and its natural unit of length introduced by the cosmological constant, $\ell = 1 / \sqrt{-\Lambda}$, as a possible alternative to $R_g$, and present a condensed matter analog realization of this scenario.