Quasinormal Modes, the Area Spectrum, and Black Hole Entropy

TL;DR

The paper connects classical black hole quasinormal mode frequencies to quantum geometry by applying Bohr correspondence to relate area quanta to QNM energies. This yields a fixed area change $\Delta A=4\ln 3\; l_P^2$ and, when combined with the area spectrum, determines the Immirzi parameter as $\gamma=\frac{\ln 3}{2\pi\sqrt{2}}$ and the minimal spin $j_{\min}=1$. Consequently, the black hole entropy matches the Bekenstein–Hawking result, $S=A/(4 l_P^2)$, and suggests the gauge group of quantum gravity is SO(3). The work elegantly links a classical oscillation spectrum to quantum geometric data and motivates reconsideration of the underlying gauge structure, while acknowledging the need for an analytic proof of the QNM result and extensions to rotating black holes.

Abstract

The results of canonical quantum gravity concerning geometric operators and black hole entropy are beset by an ambiguity labelled by the Immirzi parameter. We use a result from classical gravity concerning the quasinormal mode spectrum of a black hole to fix this parameter in a new way. As a result we arrive at the Bekenstein - Hawking expression of $A/4 l_P^2$ for the entropy of a black hole and in addition see an indication that the appropriate gauge group of quantum gravity is SO(3) and not its covering group SU(2).

Figures (2)

• Figure 1: In canonical quantum gravity the area of a surface is quantized. If a surface intersects a spin network edge with label $j$ it acquires an area of $8\pi\gamma l_P^2\sqrt{j(j+1)}$. The parameter $\gamma$ is called the Immirzi parameter.
• Figure 2: This figure shows the first 124 quasinormal mode frequencies of a Schwarzschild black hole.