Hierarchy of Angular Instabilities in Scalarized Black Holes

Fech Scen Khoo; Jose Luis Bl\'azquez-Salcedo; Jutta Kunz; Luis Manuel Gonz\'alez-Romero; Pablo Navarro Moreno

REVIEW 2 major objections 5 minor 40 references

Hierarchy of Angular Instabilities in Scalarized Black Holes

T0 review · 2 major / 5 minor · reviewed 2026-07-09 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Black hole instabilities cascade from high to low angular modes

desk verdict Solid numerical work unifying quadrupole and eikonal instabilities under one empirical scaling law; the law itself is fitted, not derived. read the letter →

arxiv 2607.07208 v1 pith:ALEZIRM5 submitted 2026-07-08 gr-qc

Jose Luis Blázquez-Salcedo , Luis Manuel González-Romero , Fech Scen Khoo , Jutta Kunz , Pablo Navarro Moreno This is my paper · ORCID

classification gr-qc

keywords hierarchyinstabilitiesstableangularblackholesinstabilityscalarized

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies scalarized black holes — objects that acquire a surrounding scalar field through a spontaneous symmetry-breaking mechanism in a modified theory of gravity combining Einstein gravity with scalar and curvature terms (Einstein-scalar-Gauss-Bonnet-Ricci theory). These black holes come in continuous families parameterized by mass and coupling constants. The authors compute the full spectrum of nonspherical (polar) perturbations across all angular multipoles, from the quadrupole (l=2) to the eikonal limit (l→∞). They discover that instability does not appear independently in each angular sector. Instead, as one moves along a branch of solutions, instability first appears at very high angular multipole number (the eikonal regime), then systematically cascades downward to lower multipoles — l=6, then l=4, eventually reaching the quadrupole l=2 — while the dipole sector (l=1) remains stable throughout. The critical threshold values at which each multipole becomes unstable lie on smooth curves described by a common power-law scaling, and these curves converge to a finite limit as l→∞ that matches an independently derived angular-Laplacian instability threshold. This means two previously known but seemingly disconnected instabilities — the quadrupole instability at low l and the angular-Laplacian instability at high l — are actually the two ends of a single continuous hierarchy. Radial (spherical) stability, by contrast, changes only at discrete turning points of the branch and is structurally separate from this angular cascade.

What carries the argument

The mechanism is a spectral analysis of polar perturbations of static, spherically symmetric scalarized black hole solutions. The authors decompose metric and scalar-field perturbations into spherical harmonics labeled by angular multipole l, reparametrize for purely ingoing/outgoing boundary conditions at the horizon and infinity, and discretize using Chebyshev spectral methods on a compactified radial coordinate. This yields a quadratic eigenvalue problem whose eigenvalues ω determine stability (Im(ω)>0 means instability). By solving this system for each l individually and tracking the critical horizon scalar-field value φ_H at which the zero mode (ω=0) appears, they reconstruct the full l

What would settle it

Compute the instability threshold for an angular multipole l significantly beyond the range used in the fit (e.g., l=30 or l=50) for one of the β values in Table I and check whether it lies on the extrapolated scaling curve. A systematic deviation from φ_H^cr(l) = φ_H^{l=∞} + A·l^{-p} at large l would break the claimed continuous hierarchy.

Watch

Extended reading notes

Core claim

The central discovery is that nonspherical deformation instabilities in scalarized black holes form an ordered hierarchy across angular multipole number l: instability always appears first in the eikonal (large-l) regime and then extends to progressively lower l values as one moves along a solution branch, with each threshold obeying the scaling law φ_H^cr(l) = φ_H^{l=∞} + A·l^{-p}. This hierarchy connects the previously known quadrupole instability (l=2) and the angular-Laplacian eikonal instability (l→∞) as endpoints of a single continuous structure, while the dipole sector (l=1) is dynamically excluded from the hierarchy entirely.

Load-bearing premise

The scaling law connecting instability thresholds across all angular multipoles is an empirical fit to numerically computed points for a finite range of l, not derived from the underlying equations of motion. The claimed unification of the quadrupole and eikonal instabilities through a common scaling law depends on this power-law form holding all the way to l→∞, though the numerical coincidence with an independently derived eikonal threshold provides partial corroboration.

Editorial extensions

If this is right

Stability assessments of scalarized black holes that examine only one or two angular multipoles are incomplete — the full angular spectrum must be surveyed to determine whether a solution is genuinely stable.
The finite eikonal limit of the hierarchy defines a sharp, calculable boundary for the angularly stable region of parameter space, which can be used to constrain which scalarized black hole solutions are astrophysically viable.
The common scaling law across different coupling constants β suggests the hierarchy is a structural property of the theory rather than a numerical artifact, and may extend to other higher-curvature gravity theories with scalarization.
The exclusion of the dipole sector from the hierarchy indicates a dynamical distinction between l=1 and l≥2 perturbations that may reflect an underlying selection rule or symmetry not yet identified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the hierarchy generalizes to rotating scalarized black holes — which the authors explicitly flag as a future direction — the angular instability cascade may interact with the richer phase structure of spinning solutions, potentially producing a three-dimensional instability surface spanning mass, spin, and angular multipole.
The power-law exponent p varying weakly with β (from ~0.77 to ~0.83) hints that the scaling may itself carry information about the underlying curvature couplings, suggesting a possible analytic derivation from the perturbation equations rather than a purely empirical fit.
The clean separation between radial (l=0) and angular (l≥2) instability channels raises the question of whether intermediate mixed radial-angular modes or time-dependent deformations could bridge the two sectors in dynamical scenarios such as gravitational collapse or merger.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know: this paper shows that the quadrupole instability (Ref. [28]) and the angular-Laplacian eikonal instability (Ref. [29]) in EsGBR scalarized black holes are endpoints of a single continuous hierarchy. As you move along a branch, instability first appears at large l, then marches down to l=2. The thresholds follow a power-law fit that approaches a finite eikonal limit, and that limit matches the independently derived angular-Laplacian threshold from a different group. That cross-check is the strongest piece of evidence in the paper and gives real credibility to the unification claim. The dipole sector stays stable throughout, which is reported but not explained — a minor loose end, not a flaw in the argument. The numerical machinery (spectral Chebyshev decomposition, quadratic eigenvalue problem) is well-established from the authors' prior work, and the fit residuals are small (RMSE ~10^{-3} to 10^{-4}). The eikonal frequency scaling (Fig. 3) provides independent support for the large-l identification. This is a genuine structural insight — the ordering of instability onset across angular sectors is new and not obvious from the equations. The stress-test concern about the scaling law being empirical rather than derived is legitimate but somewhat overstated. The paper never claims Eq. 11 is a first-principles result; it's presented as an empirical parametrization. The real load-bearing evidence is the numerical coincidence of the l→∞ limit with Ref. [29], not the specific power-law form. If the true asymptotics have logarithmic corrections, the hierarchy picture survives; only the specific functional form would need revision. The fitted exponents (p ≈ 0.77–0.83 across three β values) are comparable but not identical, and the paper doesn't claim universality of p — it says 'common scaling law,' which is a softer statement. One genuine gap: no systematic convergence study of the spectral method is provided. The small RMSE values could partly reflect overfitting rather than genuine numerical precision, and the reader can't fully assess this without seeing how the thresholds converge with grid size Np. This is the kind of thing a referee should ask for. Overall: the central result holds up. The hierarchy is real, the cross-validation is convincing, and the numerical method is sound. The paper is for people working on black hole stability in higher-curvature gravity — it organizes previously disconnected phenomena into a coherent picture. It deserves a serious referee who should push for convergence tests and a clearer statement of what the scaling law does and doesn't claim.

Referee Report

2 major / 5 minor

Summary. This Letter investigates the stability of scalarized black holes in Einstein-scalar-Gauss-Bonnet-Ricci (EsGBR) theory, focusing on nonspherical (polar) perturbations along fundamental branches. The authors compute instability thresholds for individual angular multipoles l and find that instability first appears in the eikonal (large-l) regime, then extends to progressively lower multipoles as one moves along the branch, down to the quadrupole (l=2), while the dipole (l=1) sector remains stable. The thresholds are fitted to an empirical scaling law (Eq. 11), and the large-l limit is shown to coincide with an independently derived angular-Laplacian instability threshold from Ref. [29]. The central claim is that the previously known quadrupole instability [28] and the eikonal angular-Laplacian instability [29] are connected by a continuous hierarchy of instability thresholds. The numerical method (spectral decomposition with Chebyshev polynomials, compactified coordinate, quadratic eigenvalue problem) follows the authors' prior work [30, 31].

Significance. The main result—a continuous hierarchy of angular instabilities connecting the quadrupole and eikonal sectors—is a novel structural observation that goes beyond individual-sector stability analyses. The cross-validation of the large-l fit limit against the independently derived angular-Laplacian threshold of Ref. [29] (different authors, different method) is a meaningful consistency check and strengthens the claim. The phase diagram (Fig. 2) and the identification of a finite eikonal boundary for the angularly stable region provide a falsifiable, quantitative prediction. The work is well-motivated and addresses a genuine gap in understanding how instabilities organize across angular sectors in higher-curvature gravity.

major comments (2)

No systematic convergence study of the spectral method is presented. Eq. (10) introduces a Chebyshev decomposition with grid size Np, and Table I reports RMSE values as small as 2.71×10^{-4}, but the reader cannot assess whether the computed zero-mode thresholds φ_H^{cr}(l) are numerically converged to the precision implied by these residuals. A brief convergence test (e.g., showing stability of the critical φ_H values as Np is increased, for at least one representative case) would substantiate the precision of the thresholds and rule out the possibility that the small RMSE reflects overfitting rather than genuine numerical accuracy. This is load-bearing because the scaling-law fit (Eq. 11) and the claimed coincidence with Ref. [29]'s threshold both depend on the accuracy of the individual φ_H^{cr}(l) values.
The range of l values used to fit Eq. (11) is not stated explicitly. Fig. 2 shows data points up to approximately l~20–100, but the text does not specify the minimum l included in the fit or whether the fit quality changes when the fitting window is restricted to larger l. Since the claim of a 'common scaling law' across β=2, 5, 12.5 rests on the fitted exponent p being 'comparable' (0.77, 0.80, 0.83), it is important to document the fitting range and to show that the fit is robust to its choice. A plot of residuals versus l, or a statement of the l-range used, would address whether the power-law form is genuinely supported across the full range or only in a subset.

minor comments (5)

The exponent p varies from 0.77 to 0.83 across the three β values. The text describes these as 'comparable,' but no quantitative criterion for this comparison is given. A brief justification for why this spread is acceptable, or a note that the variation is within expected numerical uncertainty, would strengthen the claim of a common scaling law.
In the paragraph following Eq. (11), the text states that the residual errors 'decrease with increasing β' and that the exponents are 'comparable,' but does not discuss whether the variation of p with β is physically meaningful or merely a fitting artifact. A sentence addressing this would clarify the intended scope of the universality claim.
Figure 1: the l=∞ curve is labeled with a dotted style, but in the figure it can be difficult to distinguish from other curves. Consider adjusting line styles or adding a legend entry that makes the eikonal boundary more visible.
The statement 'no unstable dipole mode was found in the parameter range investigated' could be made more precise by specifying the range of l=1 modes tested (e.g., which β values and φ_H ranges were scanned), or by noting whether the absence is based on the full spectrum search or only on a subset.
Reference [31] is dated 2026 in the bibliography. If this is a forward reference or preprint, please confirm the proper publication date and citation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected

full rationale

The paper's central claim—that quadrupole and eikonal instabilities are connected by a continuous hierarchy of angular instability thresholds—is supported by independently computed numerical data: for each multipole l, the zero-mode threshold φ_H^cr(l) is obtained by solving the polar perturbation equations (Eqs. 9) via a spectral Chebyshev method. The scaling law (Eq. 11) is an empirical fit to these computed thresholds, and the paper is transparent about this, using language like 'accurately described by' and presenting it as a parametrization rather than a first-principles derivation. The large-l limit of the fit is then cross-checked against the angular-Laplacian instability threshold from Ref. [29] (Minamitsuji, Mukohyama, Tsujikawa—entirely different authors), providing genuine external validation of the endpoint. The quadrupole instability from Ref. [28] (which shares author Kunz) is cited as a prior finding being connected, not as a load-bearing premise that defines the present result. The spectral method of Refs. [30, 31] (overlapping authors) is a standard numerical technique, not an ansatz that constrains the outcome. No step in the derivation chain reduces to its own inputs by construction. The skeptic's concerns about the empirical nature of the scaling law and the absence of a convergence study are legitimate correctness/robustness issues, but they do not constitute circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or fields. The free parameters (φ_H^{l=∞}, A, p, a, b) are fitting constants in empirical scaling relations, not new physical degrees of freedom. The axioms are standard mathematical decompositions plus two domain-specific assumptions (spectral convergence, functional form of the scaling law) that are typical of numerical relativity papers but not independently proven here.

free parameters (4)

φ_H^{l=∞} = 0.51417 (β=2), 0.42229 (β=5), 0.29840 (β=12.5)
Fitted parameter in the scaling law Eq. (11), representing the finite eikonal limit of the instability threshold.
A = 0.57630 (β=2), 0.33336 (β=5), 0.19148 (β=12.5)
Fitted amplitude parameter in the scaling law Eq. (11).
p = 0.76756 (β=2), 0.80343 (β=5), 0.82533 (β=12.5)
Fitted exponent parameter in the scaling law Eq. (11).
a, b = Not reported numerically
Empirical parameters in the eikonal frequency scaling Eq. (13), used to parametrize the asymptotic approach of ω_I/√(l(l+1)) to a constant.

assumptions (4)

standard math The polar perturbation equations can be consistently decomposed into spherical harmonics Y_lm with m=0 by spherical symmetry.
Standard decomposition for spherically symmetric backgrounds; invoked in Eqs. (7)-(8).
domain assumption The spectral decomposition in Chebyshev polynomials (Eq. 10) converges to the true eigenvalues of the ODE system.
Implicit in the numerical method; convergence is assumed but not explicitly demonstrated with resolution tests in this paper.
ad hoc to paper The scaling law φ_H^cr(l) = φ_H^{l=∞} + A·l^{-p} captures the true functional form of the instability threshold hierarchy.
This functional form is chosen empirically to fit the data (Table I); no derivation from the underlying equations is provided.
domain assumption The absence of unstable l=1 modes in the parameter range studied implies the dipole sector is dynamically distinct from the deformation instability hierarchy.
Stated in the Angular Instabilities section; this is an inference from the numerical domain explored, not a proven theorem.

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Cite this review

Pith. "Pith review of Hierarchy of Angular Instabilities in Scalarized Black Holes." pith.science (2026). https://pith.science/paper/ALEZIRM5

@misc{pith2026260707208,
  author       = {Pith},
  title        = {Pith review of: Hierarchy of Angular Instabilities in Scalarized Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALEZIRM5}},
  note         = {Machine review of arXiv:2607.07208}
}

read the original abstract

We investigate the stability of scalarized black holes in Einstein-scalar-Gauss-Bonnet-Ricci theory along their fundamental branches. We show that initially stable solutions first lose nonspherical stability in the eikonal regime, while lower multipoles remain stable. As the branch is continued, instability extends systematically toward lower multipoles, forming an ordered hierarchy of deformation instabilities extending down to the quadrupole mode, while the dipole sector remains stable. The instability thresholds obey a common scaling law and approach finite eikonal limits, defining the boundary of the angularly stable region. We demonstrate that the previously identified quadrupole and angular-Laplacian instabilities are connected by a continuous hierarchy of instability thresholds spanning the angular sectors of the theory. This hierarchy is distinct from radial stability, which changes only at branch turning points, and reveals a previously unexplored angular organization of instabilities in scalarized black holes.

Figures

Figures reproduced from arXiv: 2607.07208 by the authors.

**Figure 2.** FIG. 2. Angular-instability phase diagram. The dots are the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Eikonal scaling of the unstable modes for two so [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Scalarized black-hole branches in the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.

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