Spectral instability in modified Pöschl-Teller effective potential triggered by deterministic and random perturbations

TL;DR

The paper addresses spectral instability of black hole quasinormal modes under metric perturbations using a Pöschl-Teller toy model, with $V_ ext{eff}=V_ ext{PT}+V_ ext{pert}$ and $V_ ext{PT}=V_0\text{sech}^2\left(\frac{r_*}{b}\right)$, computing QNMs by zeros of the Wronskian via a matrix method in hyperboloidal coordinates. For deterministic perturbations $V_ ext{pert}^ ext{det}=\epsilon\sin(k\pi r_*)$ (here $V_0=b=5$), the deformation arises first at high overtones and, as $k$ increases, bifurcates into two branches, with possible purely imaginary modes; perturbations are exponentially suppressed at $|r_*|\to\infty$. For random perturbations $V_ ext{pert}^ ext{ran}=x\varepsilon$, even tiny perturbations produce significant high-overtone deformation and bifurcation, with spectrum uncertainty across realizations; combining random and deterministic perturbations reveals competition between effects and stronger instability as $\varepsilon$ grows. Localization studies show moving perturbations away from the horizon can reduce spectral instability under physically motivated scaling $\varepsilon=\varepsilon_0\left(\frac{r_0}{r_c}\right)^2$, underscoring the importance of perturbation placement for black hole spectroscopy. Overall, the work clarifies when and how QNM spectra destabilize under metric perturbations and highlights potential suppression mechanisms with implications for observational inferences of black hole spacetimes.

Abstract

Owing to its substantial implications for black hole spectroscopy, spectral instability has attracted considerable attention in the literature. While the emergence of such instability is attributed to the non-Hermitian nature of the gravitational system, it remains sensitive to various factors. About the spatial scale of the metric deformation, spectral instability is particularly susceptible to ``ultraviolet'' metric perturbations. In this work, we conduct a focused analysis of black hole spectral instability using the Pöschl-Teller potential as a toy model. We investigate the dependence of the resulting spectral instability on the magnitude, spatial scale, and localization of deterministic and random perturbations in the effective potential of the wave equation, and discuss the underlying physical interpretations. It is observed that small perturbations in the potential initially have a limited impact on the less damped black hole quasinormal modes with deviations typically around their unperturbed values, a phenomenon first derived by Skakala and Visser in a more restrictive context. In the higher overtone region, the deviation propagates, amplifies, and eventually gives rise to spectral instability and, inclusively, bifurcation in the quasinormal mode spectrum. While deterministic perturbations give rise to a deformed but well-defined quasinormal spectrum, random perturbations lead to uncertainties in the resulting spectrum. Nonetheless, the primary trend of the spectral instability remains consistent, being sensitive to both the strength and location of the perturbation. However, we demonstrate that the observed spectral instability might be suppressed for perturbations that are physically appropriate.

Figures (8)

• Figure 1: The spectral instability in the perturbed Pöschl-Teller effective potential Eq. \ref{['Veff_MPT']} triggered by deterministic metric perturbation Eq. \ref{['V_per_der']}, where one assumes $V_0=b=5$ and $\epsilon=0.001$. From left to right and top to bottom, we compare in pairs the resulting QNM spectra for different metric perturbations using the wave numbers $k=0$ (unperturbed metric, empty red circles), $1.0$ (empty blue squares), $2.0$ (empty orange diamonds), $3.0$ (empty gray triangles), $4.0$ (empty magenta flipped triangles), $5.0$ (filled cyan circles), and $6.0$ (filled pink squares). The numerical calculations are carried out using the matrix method, and the results have been verified using different grid sizes.
• Figure 2: The spectral instability in the perturbed Pöschl-Teller effective potential Eq. \ref{['Veff_MPT']} triggered by deterministic metric perturbation Eq. \ref{['V_per_der']}, where one assumes $V_0=b=5$ and $k=3.0$. From left to right and top to bottom, we compare in pairs the resulting QNM spectra for different metric perturbations using the magnitudes $\epsilon=1\times 10^{-8}$ (empty red circles), $1\times 10^{-7}$ (empty blue squares), $1\times 10^{-6}$ (empty orange diamonds), $1\times 10^{-5}$ (empty gray triangles), $1\times 10^{-4}$ (empty magenta flipped triangles), $1\times 10^{-3}$ (filled cyan circles), and $1\times 10^{-2}$ (filled pink squares). The numerical calculations are carried out using the matrix method, and the results have been verified using different grid sizes.
• Figure 3: A summary of the deformed QNM spectra shown in Fig. \ref{['fig_PT_det_frequency']} (left) and Fig. \ref{['fig_PT_det_magnitude']} (right).
• Figure 4: The spectral instability in the perturbed Pöschl-Teller effective potential Eq. \ref{['Veff_MPT']} triggered by random metric perturbation Eq. \ref{['V_per_ran1']}, where one assumes $V_0=b=5$. For random metric perturbations of given strength, two sets of results are presented in each panel. From left to right and top to bottom, we show the resulting QNM spectra for different metric perturbations using the magnitudes $\varepsilon=1\times 10^{-3}$ (empty red circles and empty blue squares), $1\times 10^{-5}$ (empty blue squares and empty orange diamonds), $1\times 10^{-10}$ (empty orange diamonds and empty gray triangles), $1\times 10^{-15}$ (empty gray triangles and empty magneta flipped triangles), $1\times 10^{-20}$ (empty magenta flipped triangles and filled cyan cirlcles), $1\times 10^{-30}$ (filled cyan circles and filled dark-yellow squares). Due to the nature of random metric perturbations, the perturbed QNM spectra, particularly the high overtones, do not coincide precisely, even though a consistent trend is observed. The numerical calculations are carried out using the matrix method.
• Figure 5: The spectral instability in the perturbed Pöschl-Teller effective potential Eq. \ref{['Veff_MPT']} triggered by random metric perturbation Eq. \ref{['V_per_ran2']}, where one assumes $V_0=b=5$ and for the deterministic perturbation $k=1$ with strength $\epsilon=1\times 10^{-3}$. Similar to Fig. \ref{['fig_PT_ran_white']}, for random metric perturbations of given strength, two sets of results are presented in each panel. From left to right and top to bottom, we show the resulting QNM spectra for different metric perturbations using the magnitudes $\varepsilon=1\times 10^{-3}$ (empty red circles and empty blue squares), $1\times 10^{-5}$ (empty blue squares and empty orange diamonds), $1\times 10^{-10}$ (empty orange diamonds and empty gray triangles), $1\times 10^{-30}$ (filled cyan circles and filled pink squares), a magnified view of the slightly perturbed low-lying modes of the last panel with identical $\varepsilon=1\times 10^{-30}$ (filled cyan circles and filled pink squares), $1\times 10^{-50}$ (filled dark-yellow diamonds and filled pink squares). The numerical calculations are carried out using the matrix method.