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

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.