Quanta of Geometry and Rotating Black Holes

Abstract

In the loop approach to quantum gravity the spectra of operators corresponding to such geometrical quantities as length, area and volume become quantized. However, the size of arising quanta of geometry in Planck units is not fixed by the theory itself: a free parameter, sometimes referred to as Immirzi parameter, is known to affect the spectrum of all geometrical operators. In this paper I propose an argument that fixes the value of this parameter. I consider rotating black holes, in particular the extremal ones. For such black holes the ``no naked singularity condition'' bounds the total angular momentum J by A_H/8 pi G, where A_H is the horizon area and G Newton's constant. A similar bound on J comes from the quantum theory. The requirement that these two bounds are the same fixes the value of Immirzi parameter to be unity. A byproduct of this argument is the picture of the quantum extremal rotating black hole in which all the spin entering the extremal hole is concentrated in a single puncture.