d-Dimensional Black Hole Entropy Spectrum from Quasi-Normal Modes

TL;DR

The paper addresses how to obtain a discrete Bekenstein-Hawking entropy spectrum for $d$-dimensional black holes by leveraging the large-damping quasi-normal mode frequency as a fundamental vibrational mode. By formulating an adiabatic invariant $I=\int dE/\omega_{QNM}(E)$ and applying Bohr-Sommerfeld quantization, it derives an equally spaced entropy spectrum $S_{BH}=\frac{4\pi\alpha^{(d)}}{d-3}\,n$ with $\alpha^{(d)}=\frac{d-3}{4\pi}\ln(m_0)$, leading to $\Omega=\exp(S_{BH})=(m_0)^n$. In 4D, this yields $S_{BH}=n\ln(3)$, aligning with loop quantum gravity predictions for an appropriate Immirzi parameter, and generalizes to $d$ dimensions. The work provides a semiclassical bridge between QNM dynamics and microscopic state counting, offering a testable constraint on quantum gravity theories and suggesting possible connections to horizon conformal symmetry.

Abstract

Starting from recent observations\cite{hod,dreyer1} about quasi-normal modes, we use semi-classical arguments to derive the Bekenstein-Hawking entropy spectrum for $d$-dimensional spherically symmetric black holes. We find that the entropy spectrum is equally spaced: $S_{BH}=k \ln(m_0)n$, where $m_0$ is a fixed integer that must be derived from the microscopic theory. As shown in \cite{dreyer1},4-$d$ loop quantum gravity yields precisely such a spectrum with $m_0=3$ providing the Immirzi parameter is chosen appropriately. For $d$-dimensional black holes of radius $R_H(M)$, our analysis requires the existence of a unique quasinormal mode frequency in the large damping limit $ω^{(d)}(M) = α^{(d)}c/ R_H(M)$ with coefficient $α^{(d)} = {(d-3)/over 4π} \ln(m_0)$, where $m_0$ is an integer and $Γ^{(d-2)}$ is the volume of the unit $d-2$ sphere.