Asymptotical quasinormal mode spectrum for piecewise approximate effective potential

TL;DR

This paper demonstrates that even a minor discontinuity in the effective potential drastically alters the asymptotic quasinormal mode spectrum of black holes, shifting it from the imaginary axis to the real axis, with implications for astrophysics.

Contribution

It provides analytical and numerical evidence that a single discontinuity in the potential significantly changes the asymptotic quasinormal mode spectrum, regardless of its size or position.

Findings

Discontinuity causes spectrum to shift along the real axis.

Small or distant steps still significantly affect the spectrum.

Analytical derivations support the numerical results.

Abstract

It was pointed out that the black hole quasinormal modes resulting from a piecewise approximate potential are drastically distinct from those pertaining to the original black hole metric. In particular, instead of lining up parallel to the imaginary axis, the spectrum is found to stretch out along the real axis. In this work, we prove that if there is a single discontinuity in the effective potential, no matter how insignificant it is, the asymptotical behavior of the quasinormal modes will be appreciably modified. Besides showing numerical evidence, we give analytical derivations to support the above assertion even when the discontinuity is located significantly further away from the maximum of the potential and/or the size of the step is arbitrarily small. Moreover, we discuss the astrophysical significance of the potential implications in terms of the present findings.