Waveform stability for the piecewise step approximation of Regge-Wheeler potential

TL;DR

This work addresses the robustness of Schwarzschild ringdown against environmental perturbations by approximating the Regge-Wheeler potential with a piecewise step model and analyzing the resulting QNM spectrum and time-domain waveforms via Green's function methods. A transfer-matrix approach is used to compute QNM frequencies from the piecewise potential and to obtain excitation factors $E_n$, while the piecewise construction employs a Chebyshev-Lobatto grid to efficiently approximate the potential. The authors find that the magnitudes of the reflection and transmission amplitudes, $|H(\omega)|$ and $|F(\omega)|$, converge toward the Schwarzschild values as the number of steps $N_{\text{st}}$ increases, though phase differences $\delta(\omega)$ and $\eta(\omega)$ persist at certain frequencies; time-domain waveforms remain stable with both delta-function and Gauss-bump sources, with broader initial bumps (larger $\sigma$) being more sensitive probes of outer potential differences. The results reinforce the view that ringdown observables are robust to small environmental deformations and suggest strategies for probing a black hole's exterior by tailoring the initial perturbation shape, contributing to black-hole spectroscopy and tests of strong-field gravity.

Abstract

By interpreting the difference between the original Regge-Wheeler potential and its piecewise step approximation as perturbative effects induced by the external environment of the black hole, we investigate the stability of Schwarzschild black hole time-domain waveforms. In this work, we employ the Green's function method under the assumption of an observer at spatial infinity to obtain the waveform. For two cases in which the initial Gauss bump near the event horizon or at spatial infinity, we derive analytic expressions for the corresponding waveforms. Our results demonstrate that the waveform is indeed insensitive to tiny modifications of the effective potential, thereby confirming its stability. More importantly, we find that broader initial bumps imprint the influence of small environmental modifications more clearly on the waveform, which may provide theoretical guidance for probing the exterior environment of black holes.

Figures (9)

• Figure 1: Comparisons between the R-W potential (red curves) and the effective potential obtained via piecewise step potential (blue curves) for $l=2$ and $s=2$ case with $N_{\text{st}}=5$, $10$, $15$, $20$, $25$, and $30$, respectively.
• Figure 2: For $l=2$ and $s=2$, QNM spectra with various $N_{\text{st}}$ are displayed. Magenta $\bullet$ stands for $N_{\text{st}}=5$, cyan $\blacksquare$ stands for $N_{\text{st}}=10$, green $\blacklozenge$ stands for $N_{\text{st}}=15$, red $\star$ stands for $N_{\text{st}}=20$, blue $\blacktriangle$ stands for $N_{\text{st}}=25$, and black $\blacktriangledown$ stands for $N_{\text{st}}=30$. The displayed range for the QNM spectra is $[0,3]\times[-0.3,0]$. In addition, the discrimination criterion of our Muller method is $|A_{\text{in}}|<10^{-100}$.
• Figure 3: The absolutes of the QNM EFs with various $N_{\text{st}}$ are displayed, where the horizontal axis is the overtone number. Overtones are sorted by the absolutes of the imaginary parts of spectra. For each panels, magenta $\bullet$ stands for $N_{\text{st}}=5$, cyan $\blacksquare$ stands for $N_{\text{st}}=10$, green $\blacklozenge$ stands for $N_{\text{st}}=15$, red $\star$ stands for $N_{\text{st}}=20$, blue $\blacktriangle$ stands for $N_{\text{st}}=25$, and pink $\blacktriangledown$ stands for $N_{\text{st}}=30$.
• Figure 4: The magnitude of $H(\omega)$ and the phase $\delta(\omega)$ are shown in the left panels. The differences between $|H_P(\omega)|$ and $|H_S(\omega)|$, the differences between $\delta_P(\omega)$ and $\delta_S(\omega)$ are shown in the right panels. Different colors represent the piecewise step potential case with different $N_{\text{st}}$, where black color represents the Schwarzschild case. In the left bottom panel, the black dashed line corresponds to the result of subtracting $2\pi$ from the black solid line and is used to obtain the phase difference shown in the right bottom panel. Note that the phase unwrapping is imposed on the principal argument function $\operatorname{Arg}$ in the process of getting a continuous phase.
• Figure 5: The left panel is the waveforms of $|\Psi_{\text{H},P}(u)|$ and $|\Psi_{\text{H},S}(u)|$. The right panel is the mismatch between $\Psi_{\text{H},P}(u)$ and $\Psi_{\text{H},S}(u)$ with the end time $u_{\text{max}}$ varying, where the dashed vertical line is $u_{\text{max}}=5$. The black color represents the result of Schwarzschild R-W potential, while other colors represent the results of piecewise step potential with different $N_{\text{st}}$.