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REVIEW 3 major objections 4 minor 25 references

For the massless scalar on Schwarzschild–anti-de Sitter black holes, any nonzero deformation of the Dirichlet boundary condition adds a mode that grows exponentially in time.

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2026-08-01 22:27 UTC pith:B2I562H2

load-bearing objection Excellent Dirichlet benchmarks and an honest spectral study; the claimed instability is a real eigenvalue but rests on an unproven admissibility assumption, so the physics remains open. the 3 major comments →

arxiv 2607.15731 v1 pith:B2I562H2 submitted 2026-07-17 gr-qc

Scalar quasinormal modes of Schwarzschild--anti-de Sitter black holes: spectral analysis and generalized boundary conditions

Davide Batic , Alan S. Cornell , Denys Dutykh This is my paper
classification gr-qc PACS 04.70.-s
keywords quasinormal modesSchwarzschild-anti-de Sitter black holesmassless scalar fieldboundary-condition instabilityspectral methodquadratic matrix pencilDirichlet boundary conditionAdS boundary conditions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies quasinormal modes of a massless scalar field around Schwarzschild–anti-de Sitter black holes, first benchmarking a high-precision spectral method that reproduces the known stable Dirichlet spectrum across small, intermediate, and large black holes. It then deforms the boundary condition at the AdS boundary from the standard vanishing-field condition to a one-parameter relation A cosθ + B sinθ = 0 between the two asymptotic coefficients of the scalar. The central result is that every nonzero deformation probed—down to θ = π/64—adds a mode with positive imaginary part, meaning exponential growth in time, for all three black-hole sizes studied. The Dirichlet endpoint alone is stable. The paper argues that the standard spectrum is therefore not generic: stability is tied to the specific boundary condition, not to the scalar field or the background itself.

Core claim

For the minimally coupled massless scalar on four-dimensional Schwarzschild–anti-de Sitter, keeping both asymptotic falloffs at the conformal boundary and imposing A cosθ + B sinθ = 0—with A and B the coefficients of the slow and fast branches—defines a one-parameter family of generalized coefficient boundary conditions. The paper finds that the Dirichlet endpoint θ = 0 reproduces the standard, stable scalar spectrum, but for every nonzero θ examined an additional quasinormal mode appears with Im Ω > 0 under the e^{-iωt} convention, hence growing exponentially in time. This unstable branch is seen for small (x₊ = 0.01), intermediate (x₊ = 1), and large (x₊ = 50) black holes, with growth rate

What carries the argument

The argument rests on rewriting the radial scalar equation as a quadratic matrix pencil in the dimensionless frequency Ω = ωR after compactifying the exterior domain and factoring out the ingoing horizon behavior. For the Dirichlet case the AdS falloff is built into the ansatz; for the generalized condition both asymptotic branches are kept and A cosθ + B sinθ = 0 is imposed as a spectral boundary row on the expansion coefficients, using A = χ(1) and B = −(4/3)χ⁽³⁾(1). Because the boundary row is frequency-independent, the leading matrix becomes singular, and the problem is solved by reducing onto the null-space of the constraint. The same machinery lets Dirichlet, coefficient-Neumann, Robin

Load-bearing premise

The load-bearing premise is that A cosθ + B sinθ = 0 is an admissible boundary condition for the quasinormal-mode problem; if it does not correspond to a well-posed evolution problem at the timelike AdS boundary, the unstable eigenvalue computed from it is not a genuine physical instability.

What would settle it

Carry out a time-domain evolution of the massless scalar on Schwarzschild–anti-de Sitter with the same generalized boundary condition from compactly supported smooth initial data: if the field amplitude does not grow at the exponential rate Im Ω computed from the spectral problem, or if no global-in-time solution exists for that boundary condition, the central instability claim fails. Equivalently, check whether the spatial operator with this boundary condition admits a nonnegative self-adjoint extension in the standard admissibility framework; if not, the positive-imaginary eigenvalue is a sp

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, the standard vanishing-field scalar spectrum is not representative: a small-but-nonzero deformation of the boundary condition changes the stability of the spacetime.
  • The instability is independent of black-hole size across the probed range, so it is a property of the boundary condition rather than of horizon-scale physics.
  • No finite critical angle was found down to θ = π/64, so either a critical angle exists below that value, or the Dirichlet point is an isolated stable endpoint.
  • The spectral-boundary-row formulation extends directly to coefficient-Neumann, Robin, and general linear conditions, and the same machinery can be applied to electromagnetic and gravitational perturbations.
  • The unstable branch behaves as a separate mode, not as a damped quasinormal mode crossing the real axis, which gives a target for analytic follow-up.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A conservative reading is that the generalized condition has not been shown to define a well-posed evolution problem; if it is inadmissible at the conformal boundary, the growing eigenvalue would be an artifact of an ill-posed spectral problem rather than a physical instability. Nothing in the paper rules this out.
  • If the instability is physical, a natural testable extension is to evolve the field in time with the same boundary condition and check that generic initial data grow at the rate Im Ω; this would also reveal whether the growth is a finite-norm mode or a spectral artifact.
  • One could connect this to the electromagnetic sector, where two reflective branches are known; the same matrix-pencil method could determine whether the second branch also yields an unstable mode, which would suggest boundary-condition-induced instability is generic in AdS.
  • An analytic continuation in θ near zero might show the unstable frequency diverging or moving off the physical sheet as θ → 0, which would explain why no critical angle is visible numerically; the paper leaves this open.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a high-precision Chebyshev spectral method for scalar quasinormal modes on four-dimensional Schwarzschild–anti-de Sitter spacetimes. The standard Dirichlet problem is reformulated as a quadratic matrix pencil after compactification, and the resulting spectra are benchmarked extensively against Horowitz–Hubeny, Konoplya, Cardoso–Konoplya–Lemos, AIM, Lin–Qian, and continued-fraction results, with excellent agreement. The central new claim concerns a one-parameter generalized coefficient boundary condition A cosθ + B sinθ = 0 at the conformal AdS boundary. For every nonzero θ examined, the authors report an additional spectral mode with Im Ω > 0, which they interpret as an exponential-in-time instability induced by the boundary-condition deformation. The Dirichlet endpoint θ=0 is stable and reproduces the standard spectrum; no critical angle is found down to θ=π/64.

Significance. If the instability claim is correct, the paper would demonstrate a striking sensitivity of scalar SAdS QNM stability to the choice of AdS boundary condition, and the spectral method itself is a useful high-precision tool. The Dirichlet-sector validation is a genuine strength: agreement with six independent methods, including long overtone sequences, small- and large-black-hole limits, and the approach to pure-AdS normal modes, gives confidence in the numerical machinery. The code is made available. However, the new result is conditional on a load-bearing premise that the paper does not establish: that Eq. (60) defines an admissible boundary condition in the Ishibashi–Wald / Warnick sense. Without this, the positive-imaginary eigenvalue is a property of a formally defined spectral problem rather than a demonstrated dynamical instability of SAdS. The paper's own caveats at the end of §IVB and in §V explicitly leave the analytic origin and physical interpretation open.

major comments (3)
  1. [§II.2, Eq. (60); §IVB, Table XII] The central new claim rests on treating A cosθ + B sinθ = 0 as a legitimate QNM boundary condition. The paper distinguishes this from the usual alternative-quantization Robin family, but it never proves admissibility. For the massless scalar, the two branches in Eq. (24) are ψ ∼ A z + B/z²; A is the non-normalizable branch. For θ ≠ 0, Eq. (60) generically forces A ≠ 0, so the solutions lie outside the standard finite-energy sector. No evolution semigroup, self-adjoint-extension argument, or energy-flux check is supplied. The interpretation of Im Ω > 0 as 'exponential growth in time' therefore requires a well-posed initial-boundary-value problem that has not been shown to exist. The manuscript itself acknowledges in §IVB that 'a definitive characterization of its analytic origin... remains open' and in §V that the result is 'a statement about this deformed massless scalar spectral problem
  2. [§IVB, Figs. 2–3; Table XII; §III, Eq. (107)] The unstable branch is produced solely by the authors' spectral implementation; unlike the Dirichlet sector, there is no independent cross-check. The triplet-matching filter at N=190, 195, 200 with τ=10⁻⁴ is a practical stability diagnostic, but it does not exclude spurious eigenvalues introduced by the null-space reduction of the boundary row in Eq. (107), especially because the generalized problem has a singular leading matrix and a boundary row independent of Ω. A residual check, a second independent discretization, or a time-domain simulation is needed to confirm that the positive-imaginary mode is an actual mode of a well-posed problem rather than a spectral artifact. In addition, the near-Dirichlet scan stops at θ=π/64 and the mode vanishes discontinuously at θ=0; the data therefore cannot support the claim that the instability appears 'as soon as the Dirichlet condition is deforme
  3. [§IVB, Table XII] For x₊=0.01, the imaginary part of the unstable mode is reported as 3.1934×10⁻⁵ for θ=π/8, π/4, and 3π/8 to the displayed precision, with only the real part changing. This near-constancy is not explained; it may have a simple analytic reason, but it could also indicate that the mode is dominated by a numerical or boundary-row artifact in the near-global-AdS regime. Please provide either an explanation or a more detailed convergence/residual analysis for this case.
minor comments (4)
  1. [Table XII caption] The caption says the frequencies were checked against 'N∈{90,195}'; this should read 'N∈{190,195}'.
  2. [Eq. (99)] The derivation of the spectral boundary row is correct, but it would be clearer to show explicitly the substitution of T_j^{(3)}(1) from Eq. (98), including the factor 1/15, before writing the final compact form with 4/45.
  3. [§IVB, Fig. 2] In the left panel, the growth-rate axis is logarithmic and the data for x₊=0.01 are very close to 10⁻⁵; a short comment clarifying that the unrounded values are strictly positive and stable under the resolution triplet would help the reader distinguish genuine positivity from rounding.
  4. [Sec. V / Abstract] The abstract's phrase 'signaling a boundary-condition-induced instability' is stronger than the more careful formulation in Sec. V, where the result is called a statement about the deformed spectral problem. Aligning the abstract with the qualified conclusion would be appropriate.

Circularity Check

0 steps flagged

No significant circularity: the instability is a computed eigenvalue of the stated spectral problem; the only caveat is an unproven admissibility/well-posedness step, which is a correctness issue, not a circular reduction.

full rationale

The paper's central derivation is not circular. The Dirichlet-sector computation is validated against independent external benchmarks (Horowitz-Hubeny [7], Konoplya [8], Cardoso-Konoplya-Lemos [9], Lin-Qian [11], Daghigh-Green-Morey [12]) and only incidentally uses the asymptotic iteration method [10], which shares a co-author (Cornell); that self-citation is used to correct a single known mismatch in a published overtone and is not load-bearing for the paper's new claim. The generalized boundary condition (60) is not an output of the calculation: it is imposed as a spectral row (99), derived exactly from the asymptotic expansion (65) and the endpoint relations (95). The 'additional mode with Im Ω > 0' is then obtained as an eigenvalue of the explicitly assembled quadratic matrix pencil, with triplet-matching across resolutions; it is not fitted, renamed, or defined in terms of the conclusion. No equation used to derive the instability presupposes that instability. The manuscript's own caveats—'A definitive characterization of its analytic origin, and in particular of its limiting behaviour as θ→0+, remains an open question' (Sec. IV B) and 'A complementary time-domain study would also be valuable' (Sec. V)—point to an unproven step: whether boundary condition (60) is admissible in the Ishibashi-Wald/Warnick framework, so that the positive-imaginary eigenvalue genuinely corresponds to exponential growth in a well-posed evolution problem. That is a correctness/well-posedness gap, not a circularity, because the spectral computation stands on its own terms. Score 2 reflects the lone minor, non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted to data: the code computes eigenvalues of a fixed quadratic pencil. The deformation parameter θ is scanned, not fitted. The central new claim rests on the admissibility of the boundary condition and on the spectral filtering procedure.

axioms (4)
  • domain assumption The SAdS line element (1) with f(r) as in (3) describes the gravitational background; the scalar field is minimally coupled and massless.
    Standard background for the QNM problem; defines the spectral problem.
  • ad hoc to paper The generalized coefficient boundary condition (60) A cosθ+B sinθ=0 defines a legitimate QNM spectral problem.
    The paper imposes this condition without proving admissibility/well-posedness; this is the load-bearing premise for the instability claim.
  • domain assumption The factorization ansatz (61)-(62) with χ regular captures the complete QNM spectrum of the deformed problem.
    The paper assumes no mode is lost by this ansatz; the O(1/z^4) behavior of H and the fixed z^-3 coefficient in (65) restrict the solution space.
  • ad hoc to paper The triplet-matching criterion at N=190,195,200 with τ=10^-4 identifies physical modes and excludes spurious spectral modes.
    Used for all reported modes; not a rigorous convergence proof.

pith-pipeline@v1.3.0-alltime-deepseek · 25970 in / 8955 out tokens · 72725 ms · 2026-08-01T22:27:27.651773+00:00 · methodology

0 comments
read the original abstract

We study quasinormal modes (QNMs) of a minimally coupled massless scalar field on four-dimensional Schwarzschild--anti-de Sitter black holes using a Chebyshev spectral method. After compactifying the exterior domain, the radial problem is formulated as a quadratic matrix pencil in the dimensionless frequency. For the standard Dirichlet, or vanishing-field, boundary condition at the conformal AdS boundary, we reproduce the known scalar spectra across small, intermediate, and large black holes, including long overtone sequences and the expected approach to pure-AdS normal modes in the small black hole limit. We then deform the AdS boundary condition by imposing a generalized relation between the two independent asymptotic coefficients of the massless scalar. This deformation is treated as a generalized coefficient boundary condition for the massless scalar, and not as the usual alternative quantization for scalars in the Breitenlohner-Freedman window. The Dirichlet endpoint recovers the stable standard spectrum. For every non-Dirichlet value examined, and for representative small, intermediate, and large black holes, we find an additional mode with positive imaginary part, signaling a boundary-condition-induced instability. A near-Dirichlet refinement finds no finite critical angle down to the smallest deformation probed.

Figures

Figures reproduced from arXiv: 2607.15731 by Alan S. Cornell, Davide Batic, Denys Dutykh.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

discussion (0)

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Reference graph

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