Quasinormal resonances of rapidly-spinning Kerr black holes and the universal relaxation bound

TL;DR

This paper analytically demonstrates that all rapidly-spinning Kerr black holes adhere to the universal relaxation bound, resolving previous ambiguities in the near-extremal regime by analyzing perturbation modes across different parameter ranges.

Contribution

The paper provides a rigorous analytical proof that the universal relaxation bound holds for all rapidly-spinning Kerr black holes, including challenging parameter regimes.

Findings

All black-hole perturbation modes satisfy the relaxation bound.

Modes in the regime $m^{-1}\\ll MT_{\text{BH}}/\hbar\ll1$ have relaxation times with $\\pi\tau\times T_{\text{BH}}/\hbar\geq1$.

The relaxation bound is universally valid for Kerr black holes regardless of spin.

Abstract

The universal relaxation bound suggests that the relaxation times of perturbed thermodynamical systems is bounded from below by the simple time-times-temperature (TTT) quantum relation $\tau\times T\geq {{\hbar}\over{\pi}}$. It is known that some perturbation modes of near-extremal Kerr black holes in the regime $MT_{\text{BH}}/\hbar\ll m^{-2}$ are characterized by normalized relaxation times $\pi\tau\times T_{\text{BH}}/\hbar$ which, in the approach to the limit $MT_{\text{BH}}/\hbar\to0$, make infinitely many oscillations with a tiny constant amplitude around $1$ and therefore cannot be used directly to verify the validity of the TTT bound in the entire parameter space of the black-hole spacetime (Here $\{T_{\text{BH}},M\}$ are respectively the Bekenstein-Hawking temperature and the mass of the black hole, and $m$ is the azimuthal harmonic index of the linearized perturbation mode). In the present compact paper we explicitly prove that all rapidly-spinning Kerr black holes respect the TTT relaxation bound. In particular, using analytical techniques, it is proved that all black-hole perturbation modes in the complementary regime $m^{-1}\ll MT_{\text{BH}}/\hbar\ll1$ are characterized by relaxation times with the simple dimensionless property $\pi\tau\times T_{\text{BH}}/\hbar\geq1$.