Universal Bound on Dynamical Relaxation Times and Black-Hole Quasinormal Ringing

TL;DR

It is shown that black holes comply with the bound on the relaxation time of a perturbed system, and may actually saturate it, so that when judged by their relaxation properties, black holes are the most extreme objects in nature.

Abstract

From information theory and thermodynamic considerations a universal bound on the relaxation time $τ$ of a perturbed system is inferred, $τ\geq \hbar/πT$, where $T$ is the system's temperature. We prove that black holes comply with the bound; in fact they actually {\it saturate} it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory.

Figures (2)

• Figure 1: Imaginary part of Kerr black-hole QNM frequencies as a function of the black-hole rotation parameter $a$. The numerical results are for the fundamental (least damped) gravitational resonances with $l=m=2$. The resonant frequencies conform to the bound $\omega_{I} \leq \pi T_{BH}/\hbar$, and saturate it in the extremal limit $a/M \to 1$.
• Figure 2: Imaginary part of Kerr-Newman black-hole QNM frequencies as a function of the black-hole rotation parameter $a$. We display results for $Q/M=0.2$ (lower curve) and $Q/M=0.8$ (upper curve, for which $a/M \leq 0.6$). The numerical results are for the fundamental resonances with $l=m=2$. The resonant frequencies conform to the bound $\omega_{I} \leq \pi T_{BH}/\hbar$, and saturate it in the extremal limit $a^2+Q^2 \to M^2$.