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

Black hole highly-damped quasi-normal frequencies (QNFs) are very often of the form ω_n = (offset) + in (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 Poschl–Teller), and developing an analytic "quantization condition" for the highly-damped quasi-normal frequencies. We find that the ω_n = (offset) + in (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.