Quasi-normal modes and microscopic description of 2D black holes

TL;DR

This paper explores using quasi-normal modes to understand the microscopic structure of 2D AdS black holes, proposing a model of black holes as ensembles of quantum oscillators and linking QNMs to entropy and conformal quantum mechanics.

Contribution

It extends QNM spectrum analysis to massive scalar perturbations and introduces a microscopic model of 2D black holes as coherent states and oscillator ensembles, connecting to holography.

Findings

QNMs are purely imaginary and scale linearly with overtone number.

Black hole entropy is recovered as the Gibbs entropy of oscillators, with sub-leading corrections.

A holographic link between QNMs and conformal quantum mechanics is established.

Abstract

We investigate the possibility of using quasi-normal modes (QNMs) to probe the microscopic structure of two-dimensional (2D) anti-de Sitter (AdS$_2$) dilatonic black holes. We first extend previous results on the QNMs spectrum, found for external massless scalar perturbations, to the case of massive scalar perturbations. We find that the quasi-normal frequencies are purely imaginary and scale linearly with the overtone number. Motivated by this and extending previous results regarding Schwarzschild black holes, we propose a microscopic description of the 2D black hole in terms of a coherent state of $N$ massless particles quantized on a circle, with occupation numbers sharply peaked on the characteristic QNMs frequency $\hat \omega$. We further model the black hole as a statistical ensemble of $N$ decoupled quantum oscillators of frequency $\hat\omega$. This allows us to recover the Bekenstein-Hawking (BH) entropy $S$ of the hole as the leading contribution to the Gibbs entropy for the set of oscillators, in the high-temperature regime, and to show that $S=N$. Additionally, we find sub-leading logarithmic corrections to the BH entropy. We further corroborate this microscopic description by outlining a holographic correspondence between QNMs in the AdS$_2$ bulk and the de Alfaro-Fubini-Furlan conformally invariant quantum mechanics. Our results strongly suggest that modelling a black hole as a coherent state of particles and as a statistical ensemble of decoupled harmonic oscillators is always a good approximation in the large black-hole mass, large overtone number limit.