# The Reissner-Nordstr\"om black hole with the fastest relaxation rate

## TL;DR

This paper analytically investigates the relaxation rates of Reissner-Nordstr"om black holes, revealing that the fastest relaxation occurs at a specific charge parameter value around 0.73, confirming previous numerical findings.

## Contribution

It provides an analytical proof that the Reissner-Nordstr"om black hole with a charge parameter approximately 0.73 has the fastest relaxation rate in the eikonal regime.

## Key findings

- The fastest relaxation rate occurs at a charge parameter of about 0.73.
- Analytical methods confirm previous numerical results.
- The relaxation time is minimized at this specific charge value.

## Abstract

Numerous {\it numerical} investigations of the quasinormal resonant spectra of Kerr-Newman black holes have revealed the interesting fact that the characteristic relaxation times $\tau({\bar a},{\bar Q})$ of these canonical black-hole spacetimes can be described by a two-dimensional function ${\bar \tau}\equiv \tau/M$ which increases monotonically with increasing values of the dimensionless angular-momentum parameter ${\bar a}\equiv J/M^2$ and, in addition, is characterized by a non-trivial ({\it non}-monotonic) functional dependence on the dimensionless charge parameter ${\bar Q}\equiv Q/M$. In particular, previous numerical investigations have indicated that, within the family of spherically symmetric charged Reissner-Nordstr\"om spacetimes, the black hole with ${\bar Q}\simeq 0.7$ has the {\it fastest} relaxation rate. In the present paper we use {\it analytical} techniques in order to investigate this intriguing non-monotonic functional dependence of the Reissner-Nordstr\"om black-hole relaxation rates on the dimensionless physical parameter ${\bar Q}$. In particular, it is proved that, in the eikonal (geometric-optics) regime, the black hole with ${\bar Q}={{\sqrt{51-3\sqrt{33}}}\over{8}}\simeq 0.73$ is characterized by the {\it fastest} relaxation rate (the smallest dimensionless relaxation time ${\bar \tau}$) within the family of charged Reissner-Nordstr\"om black-hole spacetimes.