Asymptotic quasinormal modes, echoes, and black hole spectral instability: a brief review

TL;DR

This review surveys analytical progress on black hole spectral instability, focusing on how small ultraviolet metric perturbations can markedly deform the full QNM spectrum, including both high overtones and low-lying modes, with significant implications for gravitational-wave spectroscopy. It connects asymptotic QNMs to echo modes, Regge poles, and greybody factors, highlighting methods such as pseudospectra, Wronskian analysis, and geometric-series formulations of Green's functions to understand causality and observability. Key findings include the sensitivity of high overtones to perturbations, the potential emergence of echo modes as a separate spectral branch, and the relative robustness of greybody factors via Regge-pole decompositions. The work underscores open questions about the physical relevance of perturbed spectra, the time-domain imprint on waveforms, and the viability of spectroscopy under spectral instability, advocating further analytic and numerical investigations.

Abstract

We present a short review of the analytical aspects of recent progress in the study of black hole spectral instability and its potential observational consequences. This topic, inspired by earlier foundational works, has attracted considerable attention in the recent literature. It has been demonstrated that both the low-lying modes and high overtones of black hole quasinormal spectra can be substantially influenced by ultraviolet metric perturbations. The temporal evolution of gravitational wave signals is primarily governed by the first few low-lying quasinormal modes. In contrast, the asymptotic behavior of high overtones is closely associated with the phenomenon of black hole echoes. We review relevant studies on spectral instability in both regimes, highlighting their potential to produce substantial observational signatures in gravitational wave data. Additionally, recent proposals of Regge poles and reflectionless modes as alternative stable observables for probing black hole spacetimes are summarized.

Figures (8)

• Figure 1: (Color Online) Spectral instability in high overtone QNMs. Top row: the staircase (left) and piecewise linear (right) approximations for the Regge-Wheeler effective potential. Bottom row: the QNM spectra obtained from the approximated effective potentials, using, on the left, $N=4, 16, 64, 256$, and $1024$ in the staircase approximation, and, on the right, using $N=3, 5, 6$, and $7$ line segments in the piecewise linear appproximation. The plots are excerpted from Refs. agr-qnm-35agr-qnm-instability-11.
• Figure 2: (Color Online) Pseudospectra of the Pöschl-Teller (left) and Regge-Wheeler (right) effective potentials under deterministic sinusoidal perturbations $\delta \tilde{V}_{\mathrm{r}}$ with given wave vector $k$ of increasing magnitudes. In the top row, the ratio between the condition numbers $\kappa$ measures the degree to which the orthogonality between eigenvectors is broken in the non-Hermitian system. The obtained pseudospectra are bounded by the (solid white) contour lines, and the pseudospectra not attained by the perturbation remain essentially unchanged. The plots are excerpted from Ref. agr-qnm-instability-07.
• Figure 3: (Color Online) Asymptotic QNM in the perturbed Pöschl-Teller effective potentials under ultraviolet metric perturbations. Left: The pseudospectrum when a metric perturbation is implemented by a minor discontinuity placed at the peak of the potential. Asymptotic QNMs are evaluated using different approaches, which correspond to the analytic results given in agr-qnm-Poschl-Teller-03agr-qnm-Poschl-Teller-04 (empty blue squares), the improved matrix method (filled red circles), and the refined semi-analytic results (empty green triangles) derived in agr-qnm-lq-matrix-12. The inset of the figure shows a zoomed-in section of a few of the lowest-lying modes, indicated by the dashed square box. It is observed that all the results are in reasonable agreement, where those obtained by the matrix method are closer to the refined semi-analytic ones. Right: The bifurcation and purely imaginary modes for the metric perturbation implemented by a minor discontinuity moving away from the black hole. The discontinuity is placed at the tortoise coordinate $x_c = 0.35$. The plots are excerpted from Ref. agr-qnm-lq-matrix-12.
• Figure 4: (Color Online) Stability of the low-lying modes in the perturbed Pöschl-Teller effective potential. The filled green stars correspond to the modes of the original Pöschl-Teller effective potential. Left: The spiral motion of the fundamental mode as the metric perturbation planted at $x_\mathrm{step}$, indicated by the numerical values, moves outward. The fundamental mode is manifestly stable as it spirals inward as the perturbation moves away from the potential. Right: The evolution of the first overtone. Unlike the fundamental mode, the first overtone is unstable as it spirals outward as the perturbation moves away from the potential. The plots are excerpted from Ref. agr-qnm-instability-55.
• Figure 5: (Color Online) An illustration of two types of echoes, and their interplay with the late-time tail. Top-left: The effective potential of a relatively dense star that possesses two maxima. The potential $V(x)$ is given as a function of the tortoise coordinate $x$, and the dotted black line indicates the point of discontinuity due to the star's surface. Top-right: The temporal evolution of smaller perturbations, where echoes are observed. The echo period is found to be roughly twice the distance between the two maxima, irrelevant to the position of the discontinuity. Middle-left: The effective potential of a less-dense star that does not have any local maximum but features a minor discontinuity at the star's surface. Middle-right: The corresponding temporal evoluton of smaller perturbations that is also characterized by echoes. The echo period is found to be approximately twice the distance between the maximum and the discontinuity of the effective potential. Bottom-left: The effective potential of a Damour-Solodukhin type wormhole utilized for the numerical simulations. Bottom-right: The corresponding temporal evolution of smaller perturbations that demonstates an inerplay between echoes and late-time tail. The waveform is shown in log-log and semilog (inset) scales. Both the echo period and the slope of the late-time tail are in good agreement with analytical estimations. The plots are excerpted from Refs. agr-qnm-echoes-35agr-qnm-echoes-45.