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REVIEW 3 major objections 4 minor 56 references

Probing black holes via quasinormal modes in a dark energy-induced dark matter

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that dark matter and gravitational shielding measurably lower black hole ringdown frequencies and slow their decay.

desk verdict First QNM computation for this dark-matter metric, but the central tables use parameters where no horizon exists, so the numbers are not black hole quasinormal modes. read the letter →

arxiv 2412.07172 v2 pith:A6OTRZY7 submitted 2024-12-10 gr-qc

Jie Liang , Dong Liu , Hao-Jie Lin , Zheng-Wen Long This is my paper · ORCID
classification gr-qc PACS 04.70.-s04.30.-w95.35.+d
keywords quasinormalmodesdarkmatterenergygravitationalshieldingblackholeringdownWKBapproximationtime-domainmethodM87*
topics Dark Matter
open problems Dark Matter
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 argues that the ringdown gravitational waves from a supermassive black hole like M87* carry a measurable imprint of dark matter and dark energy. It works with a metric in which dark energy gives gravitons an effective mass, producing a Yukawa-like suppression of gravity at a length scale λG and a dark-matter parameter η. By computing quasinormal mode frequencies for scalar, electromagnetic, and axial gravitational perturbations, it finds that increasing η or λG lowers the real oscillation frequency and shrinks the magnitude of the negative imaginary frequency, meaning weaker, longer-lived oscillations. The implication, if correct, is that observed ringdown waveforms could distinguish this dark-matter environment from a bare Schwarzschild black hole and constrain the parameters.

What carries the argument

The load-bearing object is the metric function $f(r)=e^{-2\eta/r}e^{-r/\lambda_G}-2M/r$, with $\lambda_G=1/\sqrt{2\Lambda}$ acting as a gravitational shielding length and $\eta$ parametrizing the dark matter distribution. From this $f$, the paper derives three effective potentials — scalar ($s=0$), electromagnetic ($s=1$), and axial gravitational ($s=2$) — and reduces each perturbation to a Schrödinger-like equation in the tortoise coordinate. It then solves those equations with the sixth-order WKB scheme and with a time-domain finite-difference scheme plus Prony extraction, using the $\eta=0$, $M=1/2$ Schwarzschild mode to calibrate the numerics.

What would settle it

Evaluate $f(r)=e^{-2\eta/r}e^{-r/2}-1/r$ for $M=1/2$ and $\lambda_G=2$ with any $\eta>0$: the function stays negative for all $r>0$, so a horizon search finds no root and the standard quasinormal-mode boundary conditions cannot be imposed; this direct calculation would settle whether the quoted frequencies are legitimate black hole modes.

Watch

Extended reading notes

Core claim

The central discovery is a parameter-dependent shift in the quasinormal-mode spectrum of the metric $f(r)=e^{-2\eta/r}e^{-r/\lambda_G}-2M/r$. For fixed multipole $l=2$, as the dark matter parameter $\eta$ rises (with $\lambda_G=2$) or as the gravitational shielding length $\lambda_G$ rises (with $\eta=0.4$), the real part of the fundamental mode frequency decreases monotonically and the imaginary part, though negative, decreases in absolute value across all three perturbation types. The $\eta=0$ limit is taken to reproduce the Schwarzschild quasinormal frequencies, and the same trend appears both in sixth-order WKB approximations and in time-domain Prony extraction, so the paper treats the trend as a robust signature of the model.

Load-bearing premise

The load-bearing premise is that metric (2) with $M=1/2$ and $\lambda_G=2$ describes a black hole, but in fact for those values the metric function is negative at every radius, so the spacetime has no event horizon; if it is not a black hole, the imposed boundary conditions and the quoted 'black hole' quasinormal modes are not physically applicable.

Editorial extensions

If this is right

  • If the model is right, ringdown observations of supermassive black holes should show lower frequencies and longer damping times than Schwarzschild predictions for the same mass.
  • The M87* 3σ shadow constraints ($0.09978 \leq \lambda_G \leq 6.113$ at $\eta=0.4$; $0 \leq \eta \leq 0.94$ at $\lambda_G=2$) define a concrete window in which QNM deviations of order 10–20% in the real part are expected.
  • Because the imaginary part stays negative for all parameter choices studied, the spacetime is linearly stable against these perturbations within the considered range.
  • The two independent numerical methods agree closely, so the monotonic frequency shift should be reproducible by other quasinormal-mode codes.

Reading between the lines

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

  • Beyond the paper's claims: a natural next check is to repeat the calculation for parameter sets where $f(r)$ actually has an event horizon; if the monotonic trends persist there, the signal is physical, and if not, the reported numbers are artifacts of boundary conditions applied to a horizonless spacetime.
  • The same metric could be tested against echo or shadow signatures; a naked-singularity interpretation would change the expected late-time ringdown entirely.
  • The connection between $\lambda_G$ and a cosmological constant suggests the QNM shift could be translated into a bound on $\Lambda$ using a single ringdown observation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript studies quasinormal modes (QNMs) of scalar, electromagnetic, and axial gravitational perturbations of a spherically symmetric dark-energy-induced dark matter spacetime given by Eqs. (1)–(2). It fixes M=1/2 and λG=2, uses EHT M87* shadow constraints to delimit the dark matter parameter η, and computes fundamental QNM frequencies with sixth-order WKB and time-domain/Prony methods (Tables II–VII). The central reported result is that increasing η or λG decreases both the real part of the QNM frequency and the absolute value of its negative imaginary part. However, for the central parameter choice M=1/2, λG=2 the metric function has no zero for any η≥0, so the spacetime has no event horizon and the QNM boundary conditions used in the paper are not defined.

Significance. The intended application is timely: ringdown QNMs are a promising probe of dark-matter environments around supermassive black holes, and the paper applies two standard numerical methods with WKB/time-domain cross-checks. The verification of the Prony extraction against the known Schwarzschild fundamental frequency is a good idea. However, the significance is undermined by a load-bearing geometric error: the spacetime used for Tables II, IV, VI and Figures 2, 4 has no event horizon for λG=2, so the computed frequencies are not black-hole quasinormal modes in the sense required by the stated boundary conditions. In addition, the paper's claim that η=0 degenerates to Schwarzschild is false for λG=2, making the Schwarzschild comparison baseline inconsistent. The physical interpretation and the central claim therefore do not follow from the calculations as presented.

major comments (3)
  1. [Section II, Eq. (2); Tables II–VII] For M=1/2 and λG=2, the metric function is f(r)=e^{-2η/r}e^{-r/2}-1/r. For every η≥0 and r>0, e^{-2η/r}≤1 while the maximum of r e^{-r/2} is 2/e<1, so e^{-r/2}<1/r and hence f(r)<0 for all r>0. Thus f has no zero, no event horizon exists, and the spacetime is a naked singularity. Consequently the tortoise coordinate (7) does not map a horizon to r*→−∞, and the QNM boundary conditions (25)–(26) are not defined. Since all central numerical tables fix λG=2, the computed frequencies are not black-hole quasinormal modes.
  2. [Below Eq. (2); Section IV; Tables II, IV, VI] The claim that η=0 degenerates to the Schwarzschild black hole is incorrect for the parameter values used. With λG=2, the η=0 metric is f(r)=e^{-r/2}-1/r, which has no horizon and is not Schwarzschild; the Schwarzschild metric would require λG→∞. The η=0 rows of Tables II, IV, and VI report the exact Schwarzschild QNM frequencies (e.g., 0.967284−0.193532i for scalar perturbations), but these values cannot be produced by the metric actually employed. The reported percentage shifts relative to Schwarzschild are therefore not meaningful.
  3. [Section II, Eqs. (9), (18), and (23)] The effective potentials are not reliably derived as printed. In Eq. (9), the bracket contains terms of different length dimensions: l(l+1)/r^2 has dimension 1/L^2, while terms such as 2M/r^2 have dimension 1/L; the expression also has unbalanced parentheses. For η=0 it does not reduce to the standard scalar/electromagnetic potential f[l(l+1)/r^2+(1−s)2M/r^3]. Similarly, Eq. (18) mixes dimensionless, length, and inverse-length terms, so the axial gravitational potential (23) is not trustworthy. These issues are secondary to the absence of a horizon, but they would need to be corrected in any revision.
minor comments (4)
  1. [Throughout] There are numerous typographical and language issues, including 'the spherical harmonic function of Hamilton' (should be 'of the spherical harmonics'), inconsistent spacing in section headings, and incomplete sentences in the abstract and introduction.
  2. [Figures 2 and 3] The horizontal axis labels 'r*/2M' are confusing because the potentials are plotted versus r*, and the scale extends to negative r*; the axis label and tick values should be made consistent with the variable actually used.
  3. [Section III.B, Eq. (29)] The discretization formula contains signs and factors that are not fully derived; a reference to the standard null-cone scheme or a brief derivation would improve reproducibility.
  4. [Section IV, Tables III, V, VII] The text states that these tables show the effect of λG, but the chosen range includes values for which the metric has no horizon as well as values where a horizon may exist; the physical interpretation of the trend in λG depends on the horizon issue raised above.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QNMs are computed from an externally sourced metric and EHT-constrained parameters, not fitted to the target frequencies.

full rationale

The derivation chain is not circular. The metric (2) is adopted from the external reference [47] (R. C. Pantig, Phys. Dark Univ. 45, 101550 (2024)), which is not by the present authors. The parameters η and λG are constrained from the M87* EHT shadow-radius range (Table I), again using [47], with no quasinormal-mode data involved in the fit. The QNM frequencies in Tables II–VII are then computed by solving the perturbed master equations with the sixth-order WKB method and the time-domain/Prony method; no computed QNM is fed back into the model or used to determine any parameter. The monotonic decrease of Re(ω) and |Im(ω)| with η and λG follows from the explicit effective potentials (9) and (23), so it is a derived consequence rather than a renamed fit. Author self-citations ([15]–[19], [28]) appear only as background, methodology, or related prior applications, and none is load-bearing for the central result. There is no uniqueness theorem imported from the authors' own prior work, and no ansatz is smuggled in via such a citation. Separately, there is a serious physical correctness issue: for the central choice M=1/2, λG=2, the metric function f(r)=e^{-2η/r}e^{-r/2}-1/r is negative for all r>0, so the spacetime has no event horizon and the η=0 row is not Schwarzschild; this invalidates the QNM boundary conditions and the Schwarzschild baseline comparison, but that is an internal-consistency/correctness error, not circularity. Because the circularity rules require exhibiting a reduction of the target result to its inputs, and no such reduction exists here, the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central numerical results depend on two parameters, η and λG, imported from previous model-building and constrained by EHT shadow observations. The QNM calculation itself adds no new free parameters.

free parameters (2)
  • η (dark matter distribution parameter) = 0 to 0.94 (3σ range for λG=2)
    Constrained from M87* EHT shadow radius; varied from 0 to 0.8 in QNM tables. It is an input from prior literature, not fitted to QNMs.
  • λG (gravitational shielding length) = 0.09978 to 6.113 (3σ range for η=0.4)
    Defined via λG=1/sqrt(2Λ); constrained from M87* shadow; varied 1 to 5 in QNM tables.
assumptions (4)
  • domain assumption The metric (2) from Pantig [47] describes a black hole surrounded by dark matter.
    Taken from prior work; the paper does not derive it and uses it as background.
  • domain assumption The parameter ranges from M87* shadow constraints are valid.
    The EHT shadow radius range 2.546M ≤ Rsh ≤ 7.846M is taken from literature [47] and used to constrain η and λG.
  • ad hoc to paper Standard QNM boundary conditions (incoming at horizon, outgoing at infinity) apply to this spacetime.
    The spacetime is not asymptotically flat (f→0 as r→∞) and may lack a horizon for the chosen λG; boundary conditions are not justified.
  • ad hoc to paper The effective potentials (9) and (23) are correct.
    Eq. (9) as printed is dimensionally inconsistent, so the computation relies on a corrected but unstated version.

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Cite this review

Pith. "Pith review of Probing black holes via quasinormal modes in a dark energy-induced dark matter." pith.science (2026). https://pith.science/paper/A6OTRZY7

@misc{pith2026241207172,
  author       = {Pith},
  title        = {Pith review of: Probing black holes via quasinormal modes in a dark energy-induced dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A6OTRZY7}},
  note         = {Machine review of arXiv:2412.07172}
}
read the original abstract

This study delves into the existence of dark matter around supermassive black holes in galactic cores using a novel gravitational model. By analyzing gravitational waves emitted during the ringdown phase of black holes under different field perturbations, we explore the potential for detecting dark matter. The model hypothesizes that the dark matter distribution around black hole is driven by a mechanism where dark energy endows gravitons with mass, thereby forming a new spacetime structure. Results reveal that as relevant parameters increase, the quasinormal modes (QNMs) exhibit a gradual reduction in real parts, with negative imaginary parts whose absolute values also decrease. Moreover, compared to gravitational wave signals from Schwarzschild black hole without dark matter, this system demonstrates significant differences in oscillation modes and frequencies. This achievement not only validates the self-consistency of the new gravitational model but also lays a theoretical foundation for subsequent gravitational wave detection within dark matter. Simultaneously, it provides new theoretical support for understanding the mechanism of dark energy in large-scale cosmic structures and broadens the research perspective on the relationships between black hole physics, dark matter, and dark energy.

Figures

Figures reproduced from arXiv: 2412.07172 by the authors.

Figure 1
Figure 1. FIG. 1: Constraints on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: It shows how the effective potential energies of the scalar field, electromagnetic field and gravitational field [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: It shows how the effective potential energies of the scalar field, electromagnetic field and gravitational field [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: It shows the comparison of the perturbation time-domain profiles (for the massless scalar field, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: It shows the comparison diagrams of the time-domain profiles (for the massless scalar field, electromagnetic [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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