From Entropy to Echoes: Counting the quasi-normal modes and the quantum limit of silence

TL;DR

This paper estimates black hole entropy by analyzing quasi-normal modes and their modifications due to quantum effects, revealing a fundamental timescale for echoes related to the black hole's entropy and temperature.

Contribution

It introduces a novel approach to estimate black hole entropy through quasi-normal mode analysis, incorporating quantum dissipation effects and identifying a universal echo timescale.

Findings

Small classical entropy from mode counting without dissipation

Dissipation alters quasi-normal modes, leading to a log(Entropy)/Temperature timescale

Reproduces Bekenstein-Hawking entropy with Planck-scale dissipation

Abstract

We estimate the canonical entropy of a quantum black hole by counting its quasi-normal modes. We first show that the partition function of a classical black hole, evaluated by counting the quasi-normal modes with a thermodyanmic Boltzmann weight, leads to a small entropy of order unity due to the small contribution from higher angular modes. We then discuss how this will be modified when taking into account dissipation effects near the horizon due to interaction with the quantum black hole microstates. The structure of quasi-normal modes drastically changes, yielding a fundamental frequency of the inverse of $t_{\rm echo} \sim$ log(Entropy)/Temperature. This is the time-scale for reflection from the microstates (or the quantum time limit of silence, followed by echoes), $\textit{independent of the strength of dissipation}$, and is comparable to the scrambling time proposed by Sekino & Susskind. Setting the dissipation constant to Planck time, we reproduce the Bekenstein-Hawking entropy of $\sim$ (Horizon area)/(Planck area). This result suggests the possibility of simulating black hole entropy in analog horizons realized in condensed matter systems.