Semi-analytic results for quasi-normal frequencies

TL;DR

This paper introduces a semi-analytic model using a piecewise Eckart potential to analyze black hole quasi-normal frequencies, revealing conditions under which the typical frequency pattern holds or fails.

Contribution

It provides a semi-analytic framework that captures key features of black hole QNFs and identifies when the common frequency pattern occurs based on potential asymptotics.

Findings

The (offset)+ i n (gap) pattern is common but not universal.

The pattern holds when the ratio of exponential falloff rates is rational.

The results likely extend to black holes with cosmological horizons.

Abstract

The last decade has seen considerable interest in the quasi-normal frequencies [QNFs] of black holes (and even wormholes), both asymptotically flat and with cosmological horizons. There is wide agreement that the QNFs are often of the form omega_n = (offset) + i n (gap), though some authors have encountered situations where this behaviour seems to fail. To get a better understanding of the general situation we consider a semi-analytic model based on a piecewise Eckart (Poeschl-Teller) potential, allowing for different heights and different rates of exponential falloff in the two asymptotic directions. This model is sufficiently general to capture and display key features of the black hole QNFs while simultaneously being analytically tractable, at least for asymptotically large imaginary parts of the QNFs. We shall derive an appropriate "quantization condition" for the asymptotic QNFs, and extract as much analytic information as possible. In particular, we shall explicitly verify that the (offset)+ i n (gap) behaviour is common but not universal, with this behaviour failing unless the ratio of rates of exponential falloff on the two sides of the potential is a rational number. (This is "common but not universal" in the sense that the rational numbers are dense in the reals.) We argue that this behaviour is likely to persist for black holes with cosmological horizons.