Quasi-normal frequencies: Semi-analytic results for highly damped modes

TL;DR

This paper investigates the conditions under which highly-damped quasi-normal frequencies of Schwarzschild--deSitter black holes exhibit a predictable offset plus imaginary multiple pattern, highlighting the role of surface gravity ratios.

Contribution

It provides a semi-analytic analysis showing the non-universality of the frequency pattern and links the rationality of surface gravity ratios to QNF periodicity.

Findings

Offset + i n pattern is common but not universal.

Rational surface gravity ratios relate to QNF periodicity.

The results are consistent across different black hole spacetimes.

Abstract

Black hole highly-damped quasi-normal frequencies (QNFs) are very often of the form (offset)} + i n (gap). We have investigated the genericity of this phenomenon for the Schwarzschild--deSitter (SdS) black hole by considering a model potential that is piecewise Eckart (piecewise Poeschl-Teller), and developing an analytic ``quantization condition'' for the highly-damped quasi-normal frequencies. We find that the (offset) + i n(gap) behaviour is common but not universal, with the controlling feature being whether or not the ratio of the surface gravities is a rational number. We furthermore observed that the relation between rational ratios of surface gravities and periodicity of QNFs is very generic, and also occurs within different analytic approaches applied to various types of black hole spacetimes. These observations are of direct relevance to any physical situation where highly-damped quasi-normal modes are important.