Quanta of Geometry and Rotating Black Holes

TL;DR

This paper proposes a method to fix the Immirzi parameter in loop quantum gravity by comparing classical bounds on black hole angular momentum with quantum predictions, leading to a consistent quantum description of extremal rotating black holes.

Contribution

The paper introduces an argument that determines the Immirzi parameter value by matching classical and quantum bounds on black hole angular momentum.

Findings

The Immirzi parameter is fixed to unity.

Quantum extremal rotating black holes have all spin concentrated in a single puncture.

Classical and quantum bounds on angular momentum are consistent for extremal 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.