On the instability of the fundamental mode of the Regge-Wheeler effective potential

TL;DR

This work reassesses the reported instability of the fundamental Regge-Wheeler mode under metric perturbations by constructing a physically motivated, analytic approximation to the potential and perturbation. Using an analytic, plateau-based approximation to $V_{RW}$ and closed-form region solutions, the authors derive a semi-analytic perturbation framework in which the QNM shifts exhibit an exponential sensitivity to the perturbation location $x_c$, tracing a spiral in the complex frequency plane. They show that while the fundamental mode remains stable (${\rm Im}\,\omega < 0$) under realistic perturbations, the magnitude of its frequency shift can be substantial (up to ~50% in some estimates), and the spiral period is governed mainly by ${\rm Re}\,\omega$, with asymptotic infinity properties and energy norms playing a key role in the shift magnitude. The analytic results align reasonably with numerical calculations and suggest that black hole spectroscopy in the era of gravitational-wave astronomy must carefully account for spacetime infinity effects and perturbation energy content, though time-domain waveforms remain largely insensitive to these perturbations.

Abstract

It was recently pointed out that the fundamental mode of the Regge-Wheeler effective potential is unstable against an insignificant Gaussian metric perturbation, which, in turn, might substantially challenge the black hole spectroscopy. This intriguing result has been interpreted by some authors as arising from essentially replacing the black hole's effective potential and its perturbation with two disjoint potential barriers. We argue that such an analysis may have oversimplified the real physical scenario. To be more precise, a metric perturbation planted farther away from the black hole horizon might not always be appropriately approximated by a disjoint minor barrier. Particularly, for the perturbed Pöschl-Teller potential, joint and disjoint metric perturbations might lead to drastically different stability properties for the low-lying modes. Following this line of thought, this study conducts a refined analysis of the stability of the fundamental mode of the Regge-Wheeler effective potential by closely examining a few physically relevant ingredients. While our analysis qualitatively confirms the main findings of previous studies, as the stability of the fundamental mode is primarily determined by the imaginary part of the quasinormal frequency, we show that specific features of both the effective potential at spatial infinity and the metric perturbation can have a sizable impact on the instability. In contrast, the spiral period, governed by the real part of the quasinormal frequency, appears largely insensitive to the details of the black hole metric or its perturbations. The analytic estimates are in reasonable agreement with the numerical results.

Figures (2)

• Figure 1: The resulting QNMs of the approximate Regge-Wheeler effective potential compared against those of the original Regge-Wheeler potential. Left: The calculated QNMs of the approximate effective potential Eq. \ref{['V_RW_app0']} (shown in filled red squares) compared with those of the Regge-Wheeler potential Eq. \ref{['V_RW']} (indicated by green triangles). Right: The calculated QNMs of the approximate effective potential (shown in filled red squares) compared with those obtained by separated regions of the piecewise function defined in the first and second lines of Eq. \ref{['V_RW_app0']} (indicated by filled blue circles and orange triangles). The numerical calculations are carried out using the wavefunctions given by Eqs. \ref{['RWApxf']}-\ref{['RWApxg']}, where one assumes the parameters $\ell=2$, $s=0$, $x_0=-0.9575$, and $x_1=2.4637$. One notes that the fundamental mode of the approximate potential is very close to that of the original black hole.
• Figure 2: The resulting fundamental mode as a function of the location of the metric perturbation $x_c$ in the approximate Regge-Wheeler effective potential. As the coordinate $x_c$ varies from 15 to 30 with an increment of 0.05, the resulting fundamental mode spirals away from the original fundamental mode in the counterclockwise direction. The fundamental modes of the perturbed metric are presented by empty blue circles, and the location of the corresponding unperturbed mode $\omega_0 = 1.0717 - i 0.14909$ is indicated by the filled red circle (left panel) and solid red lines (right panel). The numerical calculations are carried out using the analytic wavefunctions given by Eqs. \ref{['RWAsolf']}-\ref{['RWAsoluH']}, and \ref{['RWAsolg2']}, where one adopts the metric parameters $\ell=2$, $s=0$, $x_0=-0.9575$, $x_1=2.4637$, and $\epsilon=0.01$.