Generalized uncertainty principle in three-dimensional gravity and the BTZ black hole
Alfredo Iorio, Gaetano Lambiase, Pablo Pais, Fabio Scardigli
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 concerning 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 $M\ensuremath{\ne}0$ Ba\~nados-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, $\ensuremath{\ell}=1/\sqrt{\ensuremath{-}\mathrm{\ensuremath{\Lambda}}}$, as a possible alternative to ${R}_{g}$, and present a condensed matter analog realization of this scenario.