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REVIEW 3 major objections 5 minor 73 references

This paper claims the dilaton-Euler-Heisenberg black hole is stable against polar metric-dilaton perturbations: every fundamental quasinormal frequency computed for dilaton modes l=0–3 and gravitational modes l=2–3 has a negative imaginary

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2026-08-03 09:31 UTC pith:WKPH2SDD

load-bearing objection First QNM dataset for dEH black holes, but a wrong printed metric and an under-documented master-equation reduction mean the numbers should not be trusted until corrected. the 3 major comments →

arxiv 2601.13521 v2 pith:WKPH2SDD submitted 2026-01-20 gr-qc

Polar perturbations of dilaton-Euler-Heisenberg black holes

Sheng-Yuan Li , Yun Soo Myung , Ming Zhang , Xufen Zhang , De-Cheng Zou This is my paper
classification gr-qc MSC 83C5783C35 PACS 04.70.Bw04.30.-w
keywords quasinormal modespolar perturbationsdilaton-Euler-Heisenberg black holesstabilitydilaton hairEuler-Heisenberg electrodynamicsdirect integration methodmatrix-valued continued fraction
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 tries to establish that the dilaton-Euler-Heisenberg black hole — a magnetically charged, spherically symmetric solution of Einstein–Maxwell–dilaton theory with a dilaton-coupled Euler–Heisenberg nonlinear term — is dynamically stable against polar (even-parity) metric and dilaton perturbations. The authors compute the fundamental quasinormal-mode frequencies for dilaton modes with angular momentum l=0,1,2,3 and gravitational modes with l=2,3, using two independent numerical methods whose results agree to better than a few percent. Every computed frequency has a negative imaginary part, meaning every mode decays in time rather than growing. They also find that the damping rates behave differently depending on the sign of the coupling parameter ε=α−β, especially near extremality.

Core claim

The paper studies the linearized polar (even-parity) perturbations of the spherically symmetric dilaton-Euler-Heisenberg black hole with dilaton hair, a magnetically charged solution of Einstein–Maxwell–dilaton theory with a dilaton-coupled Euler–Heisenberg term. In the Regge–Wheeler gauge the polar sector forms four coupled equations for the metric variables and the dilaton; using an algebraic identity and a reduction procedure, the authors obtain two coupled second-order master equations, one for the gravitational channel and one for the dilaton channel. They compute the fundamental (n=0) quasinormal frequencies by direct integration and by a matrix-valued continued fraction method, report

What carries the argument

The load-bearing object is the algebraic identity (19), obtained from the (r,r) component of the linearized Einstein equations, which eliminates one of the polar metric variables and allows the four coupled first-order equations to be reduced to two coupled second-order master equations (20) and (23) in the tortoise coordinate. These master equations, with potentials (21)–(26), turn the stability question into a complex eigenvalue problem whose eigenvalues are the quasinormal frequencies: a positive imaginary part would signal growth, a negative imaginary part means damping. Two independent numerical schemes — direct integration and matrix-valued continued fraction — are used to solve that e

Load-bearing premise

The stability conclusion rests on the reduction of the four linearized polar equations to the two master equations being complete and not over-constrained — a step the paper itself flags as risky.

What would settle it

Solve the original first-order system (15)-(18) with the same horizon and infinity boundary conditions but without using the algebraic identity (19); if any eigenvalue with positive imaginary part appears, the stability claim fails. Separately, verify that the background metric printed in Eq. (11) is asymptotically flat — as r grows it reads A(r)≈2Mr/Q_m^2, which cannot reproduce the Schwarzschild frequencies listed for Q_m=0.01.

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

If this is right

  • The dEH black hole remains stable against monopole and dipole dilaton perturbations (l=0,1) and against quadrupole and octupole gravitational and dilaton perturbations (l=2,3), at least at the fundamental-mode level and for the parameter range studied.
  • The real part of the dilaton-mode frequency grows with magnetic charge Q_m, so stronger magnetic hair makes the black hole ring at a higher frequency; the gravitational l=2 mode behaves differently for ε=1 (peaking and declining) than for ε=−1 (monotonic rise).
  • Near extremality, the damping rate of the ε=−1 gravitational l=2,3 modes reverses and grows, while the ε=1 modes keep declining — the sign of the dilaton coupling leaves an observable fingerprint in the ringdown.
  • Because all computed imaginary parts are negative, the spacetimes are dynamically stable in this sector, and deviations of their quasinormal frequencies from Schwarzschild values could be used to constrain Q_m and ε with future gravitational-wave ringdown observations.

Where Pith is reading between the lines

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

  • If the reduction is exact, the same two-channel scheme should reproduce the ε=0 limit of this family smoothly; the paper's tables approach Schwarzschild at small Q_m but do not tabulate ε=0, so an independent check at ε=0 would test continuity.
  • The paper computes only n=0 modes, but the l=0,1 effective potentials are positive definite; a spectral argument would likely extend dilaton-mode stability to higher overtones, a step the paper does not take.
  • The sign asymmetry near extremality tracks the different horizon structures of ε=+1 (single horizon) and ε=−1 (two horizons up to an extremal limit); connecting the damping reversal near Q_m≈0.826 to the disappearance of the second horizon is a natural next calculation.
  • A direct waveform prediction is within reach: the two coupled master equations give not only frequencies but also channel mixing, so ringdown templates could be generated to assess how much ε and Q_m are actually measurable in a single event.

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 / 5 minor

Summary. The manuscript studies linear polar (even-parity) metric-dilaton perturbations of the magnetically charged dilaton–Euler–Heisenberg (dEH) black hole with dilaton hair. Starting from four radial equations (15)–(18) and an algebraic identity (19), it reduces the polar sector to two coupled second-order master equations for K and S with potentials (21)–(26). The authors compute fundamental quasinormal frequencies using direct integration and matrix-valued continued-fraction/AIM methods for l=0,...,3 and ϵ=±1, compare against Schwarzschild QNMs for Q_m=0.01, and report that all computed modes have negative imaginary parts, from which they conclude stability. They also identify different near-extremal damping behavior for ϵ=+1 and ϵ=−1.

Significance. If correct, the paper would be one of the first stability/ringdown studies of the dEH family and would strengthen the case for using ringdown measurements to constrain dilaton-EH couplings. The numerical work has clear strengths: no output parameters are fitted; the Q_m=0.01 limit reproduces the standard Schwarzschild values (Tables III–IV; e.g. 0.373681−0.0889637i vs 0.3737−0.08896i for l=2 gravitational); and two independent solvers agree to sub-percent in most cases. The l=0,1 potential plots are positive definite. However, the central reduction is asserted rather than demonstrated, the background metric as printed is not asymptotically flat, and the stability conclusion exceeds what a finite set of n=0 modes can show.

major comments (3)
  1. [§III and Appendix A, Eqs. (19)–(26)] The load-bearing step is the reduction of the four-equation first-order system to the two master equations (20) and (23). The paper states identity (19) without proof and Appendix A only lists the ansatz coefficients, citing the Mathematica notebook of Ref. [68]. The numerical agreement between DI and CFM does not test this reduction because both methods solve the same reduced equations. Please provide the full derivation or the notebook, including how H0 and R1 are eliminated and how over-constraining of the first-order system is avoided. Without this, the frequencies—and hence the stability conclusion—cannot be independently checked.
  2. [§II, Eq. (11)] As printed, the background metric function A(r) is not asymptotically flat and does not reduce to Schwarzschild: for r→∞, A(r) ≈ 1 − 4M^2/Q_m^2 + 2Mr + O(r^{-5}), and the Q_m→0 limit is singular. The original metric in Eqs. (8)–(9) is asymptotically flat, so this appears to be a coordinate/transcription error, likely a missing denominator involving Q_m^2 + sqrt(Q_m^4+4M^2r^2). Since the potentials in Eqs. (21)–(26) are constructed from these A and B, the tables cannot be reproduced from the printed equations. Please correct Eq. (11) and state the coordinate relation to Eqs. (8)–(9).
  3. [§V and Conclusion] The inference that 'all negative imaginary quasinormal frequencies imply stability' is too strong. The work computes only n=0 fundamental frequencies for l≤3; for l=0,1 potential positivity is argued, but for the coupled l=2,3 system no such argument is given. A finite sample of damped modes is evidence, not proof, of linear stability. Please either restrict the claim to 'no unstable modes found in the computed set' or add a stability proof (e.g., S-deformation, potential-positivity, or absence of unstable eigenvalues) for the coupled l≥2 system.
minor comments (5)
  1. [Tables I–II] The table captions use 'bEH' instead of 'dEH'; also the discrepancy is called ΔDA for l=0,1 and ΔDC for l≥2. Standardize the notation.
  2. [Sec. IV and Appendix B] Tables I–II use the asymptotic iteration method (AIM), but Section IV only describes direct integration and matrix-valued continued fraction. Clarify that the l=0,1 results use AIM (Appendix B) while l≥2 uses DI/CFM.
  3. [Figs. 5–6] The legends and line styles for gravitational versus dilaton modes are hard to follow, especially in Figs. 5(b) and 6(b). Please use explicit labels such as 'Im ω' and distinguish modes more clearly.
  4. [Abstract and Sec. I] The phrase 'All negative imaginary quasinormal frequencies' should be qualified as 'all computed fundamental quasinormal frequencies' to avoid overstating the result, consistent with the major comment above.
  5. [Appendix A, Ref. [68]] Since the reduction is central, please provide a stable reference or link to the exact version of the Mathematica notebook used, and state which simplifications were applied to eliminate exponential factors and derivatives of A and B.

Circularity Check

0 steps flagged

No significant circularity: QNM frequencies are direct eigenvalues of a derived ODE system with external Schwarzschild checks; self-citations are ancillary.

full rationale

The derivation chain is: action (1) -> background (10)-(11) from the external Ref. [9] -> first-order polar system (15)-(18) -> master equations (20)-(26) -> QNM eigenvalues -> stability conclusion. No step inverts this chain. The quasinormal frequencies are computed as eigenvalues of an ODE system with dissipative boundary conditions; no parameter is fitted to the target frequencies, and the Schwarzschild limits (46)-(49) are external benchmarks from Refs. [63-66], not outputs of this paper. The epsilon=+/-1 scan is internal parameter variation, not fitting. The self-citations are not load-bearing: Ref. [62] is used only as a prior 'suggests' statement ('This suggests that the dilaton propagating around the dEH black holes is stable against the l=0 polar perturbation'), and Ref. [50] is a general reference for expanding the field equations; the actual polar equations (15)-(18) are stated in the paper. The paper's own warning 'to prevent the system from being over-constrained' (Sec. IV.A.1) and the condensed Appendix A ('we adopt the procedure of Ref. [68] and acknowledge the authors for using their Mathematica notebook on GitHub') flag an omitted derivation, and the typeset background (11) is not asymptotically flat as printed; these are correctness/reproducibility risks, not circularity, because the reduced equations are not used to define their own inputs and the agreement between DI and CFM validates numerics rather than the reduction itself. No 'prediction' is equivalent by construction to an input, so there is no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No parameters are fitted to data: M=1 sets the scale and Q_m, l, and ε=±1 (with ε=α−β) are theory/scan parameters of the model, not knobs tuned to obtain the results. The axioms are the solution's validity, the QNM boundary-condition selection, the gauge reduction, the correctness of the long master potentials, the l=0,1 dilaton-only reduction, and the equivalence of the l=0 potential to Ref [9]. No new particles, fields, or entities are invented; the dilaton and Euler-Heisenberg term come from the pre-existing theory.

axioms (6)
  • domain assumption The dEH metric (8)-(11) is an exact solution of the field equations (4)-(6).
    Taken from Ref [9]; the paper builds all perturbation equations on this background without re-validating the solution.
  • domain assumption QNM boundary conditions — purely ingoing at the horizon, purely outgoing at infinity — select the physical spectrum.
    §IV.A.1, Eqs (28)-(32); standard for QNM computations but an assumption about the mode content.
  • standard math The Regge-Wheeler gauge is sufficient: H0, H1, H2, K with H2=H0 and the algebraic identity (19) fully determine the polar sector.
    §III; the paper itself notes the system must be arranged to avoid over-constraining (§IV.A.1).
  • ad hoc to paper The elimination to the two master equations (20),(23) with potentials (21)-(26) is complete and correct.
    Appendix A supplies the coefficients α,β,γ,λ but the enormous potentials are asserted; any transcription error changes all frequencies.
  • domain assumption For l=0,1 the metric perturbation variables are gauge-redundant, so the dilaton equation alone (Eqs 42, 44) governs the mode.
    §V.A-B; asserted from the redundancy of the Zerilli variables for low l.
  • domain assumption The l=0 effective potential (43) reduces to Eq (6.5) of Ref [9].
    Asserted without demonstration in §V.A: 'they reduce to the same result when the background solution is applied.'

pith-pipeline@v1.3.0-alltime-deepseek · 19716 in / 25417 out tokens · 235299 ms · 2026-08-03T09:31:42.235341+00:00 · methodology

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read the original abstract

We investigate the quasinormal modes of polar metric-dilaton perturbations around the dilaton-Euler-Heisenberg (dEH) black holes with dilaton hair. The dEH black holes are obtained from the Einstein-Maxwell-dilaton theory with two dilaton coupling parameters ($\alpha,\beta$) to the nonlinear Euler-Heisenberg term. We compute the quasinormal mode spectra by making use of two numerical techniques: direct integration and matrix values continued fraction methods. An excellent agreement is found between two approaches, confirming the robustness of our computation. We present the fundamental quasinormal frequencies for both gravitational and dilaton modes and analyze their dependence on the magnetic charge ($Q_m$), angular momentum quantum number ($l$), and coupling parameter ($\epsilon=\alpha-\beta$). All negative imaginary quasinormal frequencies for polar metric-dilaton perturbations imply that the dEH black hole with dilaton hair is stable against dilaton with $l=0,1,2,3$ and gravitational modes with $l=2,3$. Also, our results reveal distinct qualitative behaviors between $\epsilon=1$ and $\epsilon=-1$, particularly in the damping rates near the extremality.

Figures

Figures reproduced from arXiv: 2601.13521 by De-Cheng Zou, Ming Zhang, Sheng-Yuan Li, Xufen Zhang, Yun Soo Myung.

Figure 1
Figure 1. Figure 1: FIG. 1: Graphs for the potential of the [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Variation of fundamental ( [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Graphs for the potential of the [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of fundamental QNM frequencies (real and imaginary parts) with [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of real and imaginary parts for fundamental QNM frequencies with [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of real and imaginary parts for fundamental QNM frequencies with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

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Reference graph

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