QNMs of Schwarzschild-like Black Holes

• Quasinormal modes are the damped oscillations triggered by perturbations in Schwarzschild-like spacetimes, encoding key geometric and physical information.
• Hyperboloidal compactification and Chebyshev-Lobatto spectral methods enable precise mapping of spectral instability induced by realistic metric deformations.
• Physically motivated perturbations, modeled via the RZ parametrization, cause overtone migration and degeneracy that challenge unambiguous black hole spectroscopy.

Quasinormal modes (QNMs) of Schwarzschild-like black holes refer to the characteristic damped oscillations that arise in response to field perturbations on static, spherically symmetric, asymptotically flat or non-flat spacetimes with black hole horizons. In the context of "Schwarzschild-like" backgrounds—spacetimes which generalize or deform the Schwarzschild solution—such as those arising via physically-motivated metric parametrizations (e.g., the Rezzolla-Zhidenko or RZ parametrization), or with environmental/quantum corrections, the QNM spectrum encodes essential information on the geometry and the physical environment. The spectral (in)stability and pseudospectral properties of these QNMs have direct importance for the robustness of black hole spectroscopy, gravitational wave signal interpretation, and the prospects for distinguishing modified gravity or exotic physics effects from general relativity with observational data.

1. Mathematical Structure: Schwarzschild-like Metrics and RZ Parametrization

The generic Schwarzschild-like black hole line element is

$$ds^2 = -a(r)\, dt^{2} + \frac{1}{b(r)} dr^{2} + r^{2}(d\theta^{2} + \sin^{2}\theta\, d\phi^{2}),$$

where $a(r), b(r)$ are functions encoding deviations from Schwarzschild geometry. The RZ parametrization expresses $a(r)$ and $b(r)$ in terms of $x = 1 - r_0/r$ (with $r_0$ the horizon radius) and deformation parameters $\epsilon, a_i, b_i$: $a(r) = x A(x), \quad b(r) = \frac{x A(x)}{B(x)^2},$

$$A(x) = 1 - \epsilon(1-x) + (a_0-\epsilon)(1-x)^2 + \tilde{A}(x)(1-x)^3,$$

$B(x) = 1 + b_0(1-x) + \tilde{B}(x)(1-x)^2,$

where $\tilde{A}(x), \tilde{B}(x)$ are continued fractions in the higher-order $a_i, b_i$. Schwarzschild is recovered for all deformation parameters vanishing.

Axial perturbations with frequency $\omega$ satisfy a wave equation

$$\frac{d^2\psi}{dr_*^2} + [\omega^2 - V(r)]\,\psi = 0,$$

where the tortoise coordinate is $dr_*/dr = 1/\sqrt{a(r)b(r)}$, and the general effective potential is

$$V(r) = a(r) \left[ \frac{\ell(\ell + 1)}{r^2} - \frac{6 m(r)}{r^3} + \frac{m'(r)}{r^2} \right],$$

$m(r) = \frac{r[1 - b(r)]}{2}.$

The boundary conditions for QNMs are outgoing at infinity and ingoing at the event horizon: $$\psi(r) \sim \exp(\pm i\omega r_*), \quad r_* \to \pm \infty.$$

2. Hyperboloidal Formulation and Pseudospectrum Analysis

A key computational and conceptual advance is the recasting of the QNM problem into a hyperboloidal slicing framework, where the standard Cauchy foliation is replaced with compactified hyperboloidal coordinates $(\tau, \sigma)$; $\sigma \in [0,1]$ maps spatial infinity ($\sigma=0$) and the event horizon ($\sigma=1$) to finite coordinate values. The master equation becomes a matrix eigenvalue problem: $L\,\mathbf{u} = s\,\mathbf{u},$ with $s = -i r_0 \omega$ and $L$ a differential operator acting on the function vector $\mathbf{u}$ associated with field perturbation variables.

The $\varepsilon$-pseudospectrum,

$$\varsigma^{\varepsilon}(L) = \{ s \in \mathbb{C} : \left|\left| (s\,\mathbb{I} - L)^{-1} \right|\right|^{-1} < \varepsilon \},$$

quantifies the spectral sensitivity: it is the set of $s$ where under perturbations of size up to $\varepsilon$, the eigenvalues of $L$ may "migrate".

An energy inner product is defined,

$$\|\mathbf{u}\|_E^2 = \frac{1}{2}\int_0^1 \left[ w(\sigma) |\bar\phi|^2 + p(\sigma)|\partial_\sigma\bar\psi|^2 + q(\sigma) |\bar\psi|^2 \right] d\sigma,$$

where $w,p,q$ are weight functions determined by the background geometry and perturbation variables, setting the norm in which pseudospectra are computed.

Spectral analysis is implemented with Chebyshev-Lobatto collocation methods, with mesh-refinement to resolve the potential's gradients, and hyperboloidal compactification ensuring regularity at the physical boundaries.

3. Spectral Instability of Schwarzschild-like QNMs

Physically-motivated deformations parametrized by nonzero $\epsilon, a_i, b_i$ in the RZ formalism induce strong instability in the QNM spectrum, even for small parameter values. High-overtone QNMs rapidly deviate from the Schwarzschild case as $a_3$ increases, with overtone branches opening and aligning horizontally in the complex frequency plane; both $a_3$ (branch structure) and $\epsilon$ (instability threshold) are critical tuneable parameters. Notably, the QNM instability is not smoothed by the presence of realistic, structured metric perturbations.

The $\varepsilon$-pseudospectra of such backgrounds do not manifest closed or localized contours around QNM eigenvalues—unlike the artificially stabilized spectrum observed for Schwarzschild black holes subjected to random perturbations [cf. PhysRevX.11.031003]. This signals strong spectral instability: small, realistic perturbations to the metric persistently induce large changes in the QNM spectrum for overtones, and subsequent, even smaller perturbations can substantially relocate these modes.

Introducing small "ad-hoc" (sinusoidal) perturbations, $\delta q = \eta \sin(2\pi k \sigma)$ with moderate to large $k$, produces dramatic displacements in the QNM spectrum, overlapping with the spectra of other deformations. This degeneracy complicates attributing the origins of observed instabilities in QNM data, especially when multiple perturbative sources (e.g., environmental effects, quantum corrections, deviations from GR) are present.

4. Physical and Observational Implications

• Random vs. physically-motivated perturbations: Artificial random spectrum displacement imposes a false sense of stability in the QNM spectrum. However, realistic metric deformations leave the spectrum acutely sensitive—no shielding of high overtones is present.
• Degenerate Spectral Signatures: When multiple deformations coexist (metric plus effective potential), distinct physical origins can produce nearly overlapping QNM spectra. Analysis of ringdown or overtone data may then be ambiguous.
• Implications for Black Hole Spectroscopy: High-overtone QNMs, often invoked for precision tests of gravity, cannot be reliably used as spectroscopic probes unless the spacetime geometry and all sources of perturbation are independently constrained.
• Robustness at Fundamental Mode: While overtones are sensitive, the fundamental QNM remains relatively robust under small physically motivated deformations; however, its use alone may limit potential to discriminate alternative theories or environmental effects.

Comparison Table: Spectral Stability Under Perturbations

Scenario	Overtone QNM Spectral Stability	Notes
Random potential perturbation	Appears artificially stable	Pseudospectrum contours closed, modes localized
RZ (physically-motivated) perturbation	Spectrally unstable, high sensitivity	Contours not closed, QNMs migrate substantially
Multiple perturbations	Compounded instability, degeneracy	QNM spectra of different sources can overlap

5. Methodological Advances and Theoretical Significance

The methodology—combining hyperboloidal compactification, high-accuracy spectral methods, and rigorous pseudospectrum analysis—provides a transparent and robust means of quantifying QNM instability under both random and physically motivated metric deformations. Controlled numerical experiments with the RZ parametrization, mesh refinement, and tailored norm selection enable the elucidation of instability mechanisms distinct from prior approaches relying solely on random or "bump" perturbations.

This framework enables a detailed mapping between the space of metric deformations and QNM spectral behavior, clarifying which deviations are spectrally benign and which force overtone localization to break down. The analysis underscores the necessity of moving beyond classical mode analysis for interpreting gravitational wave data in realistic astrophysical or beyond-GR scenarios, emphasizing the fundamental limits imposed by spectral instability on black hole spectroscopy.

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In summary, the QNM spectrum of Schwarzschild-like black holes deformed via realistic metric perturbations (e.g., RZ parameterization) is generically unstable for overtones, as revealed by detailed pseudospectrum analysis (Siqueira et al., 23 Jan 2025). The spectrum remains highly sensitive to further perturbations, and observed spectral features can reflect overlapping influences from environmental, quantum, or alternative gravity effects, complicating their unambiguous identification and the use of high-overtone data for black hole spectroscopy. This sets critical practical constraints for the fidelity and interpretation of gravitational wave ringdown analyses in the presence of realistic spacetime deformations.