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REVIEW 4 major objections 4 minor 51 references

Complementarity of Gravitation Collapse (II) XOB and the Damping Gravitational Waveform as Evidences

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that binary black-hole merger waveforms damp only if black holes have extended inner structure that deforms like a banana during merger.

desk verdict The new waveform calculation is real but the central exclusivity claim is contradicted by the standard black-hole perturbation theory cited in the paper's own references. read the letter →

arxiv 2505.14749 v2 pith:27ZAQPR5 submitted 2025-05-20 gr-qc

Ding-fang Zeng This is my paper
classification gr-qc
keywords gravitationalcollapsecomplementaritybinaryblackdynamicsrelativityseries
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

XOB models a binary system as two point particles, each moving in a Schwarzschild-like geometry built around the other body. The two geometries rotate together, so each particle behaves like a test mass in a static black-hole background. From this Lagrangian the paper derives a one-body Hamiltonian and expands it in powers of 1/c. It claims this expansion matches standard post-Newtonian theory to all orders, but the displayed H0, H2, H4 terms are described in the same text as not coinciding with the standard PN Hamiltonian.

The waveform calculation starts from the usual quadrupole formula and multiplies it by a factor zeta = sin(4GMz/a)/(4GMz/a). The variable z is supposed to measure a banana-shape deformation of the merging black holes, and its time evolution is set by a chosen exponent nbnn and a scale zf, with optional extra coefficients c1 and so on. For the equal-mass case the author sets zf=1.2 and nbnn=2 and reports a 0.7062 match against the EOBNRv2 waveform. The abstract instead promises 99% agreement with numerical relativity, which does not match the comparison reported inside the paper.

On this basis the paper argues that numerical relativity actually describes black holes with extended internal mass and no true event horizon, because a real horizon would prevent finite-time merger and would kill the damping tail. It also derives upper and lower bounds on the lowest ringdown frequency from the Hamiltonian and compares them with GWTC catalog data. The logic is circular in an important way: the inner-structure assumption enters through the deformation factor, and the resulting waveform agreement is then offered as evidence for that inner structure.

Extended reading notes

Core claim

The load-bearing assertion, stated in the abstract and conclusion, is that the damping tail with quasi-normal-mode features in binary black-hole merger waveforms is possible only when the black holes possess extended inner mass distributions and experience banana-shape deformation, so observed damping is evidence for the inner structure proposed in the complementarity of gravitational collapse. If correct, numerical relativity's apparent-horizon boundary conditions would be interpreted as describing horizonless, extended objects, and ringdown frequencies would be traceable to this interior structure.

Load-bearing premise

The argument depends on the claim that a true event horizon would make the observed merger and damping impossible, because two point-like singularities could not become a circular-line singularity in finite Schwarzschild time. This coordinate-time argument appears in the Figure 1 caption and Introduction. If the reasoning is wrong, the inference from a damping tail to 'no event horizon' loses its foundation, independent of any waveform fitting.

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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

4 major / 4 minor

Summary. This paper proposes an 'exact one-body' (XOB) description of binary black-hole merger dynamics, in which each body is treated as a test particle in a Schwarzschild-like geometry generated by the other, supplemented by a dissipative quadrupole radiation force. It claims that the weak-field-low-speed expansion of XOB matches the post-Newtonian series to all orders for the conservative dynamics, uses a 'banana-shape deformation' reduction factor zeta in the quadrupole formula to generate merger waveforms, reports a match of 0.7062 to EOBNRv2 for an equal-mass binary, and derives upper and lower bounds on the real part of the fundamental quasi-normal-mode frequency. The paper concludes that damped QNM waveforms are possible only if black holes have no true event horizon, possess extended internal mass distributions, and undergo banana-shape deformation, interpreting numerical relativity's apparent-horizon treatment as evidence for such an interior structure.

Significance. If the central claim were established, the paper would offer a radically different interpretation of merger ringdown, connecting observed damping to interior structure rather than to horizon boundary conditions, and it would reinterpret numerical relativity as describing horizonless extended objects. The manuscript is commendably transparent about its numerical limitations: it explicitly reports the modest 0.7062 match score and acknowledges that additional parameters in Eq. (17) can tune the waveform arbitrarily. However, the central exclusivity claim is not supported. Standard black-hole perturbation theory, cited in the paper's own references [17]-[20], already produces damped QNM ringdown from perturbed Schwarzschild/Kerr geometries with genuine event horizons, and the paper's coordinate-time argument against horizons does not survive scrutiny. The abstract's all-orders PN and 99% agreement claims are also contradicted by the body of the paper. As it stands, the paper does not provide a valid test of its proposed inner structure.

major comments (4)
  1. [Abstract; Section 3, Eqs. (9)-(11); Fig. 5] The abstract claims that XOB's weak-field-low-speed expansion matches the post-Newtonian series to all orders for the conservative dynamics and that waveforms agree with numerical relativity to 99%. Both claims are contradicted later in the text: after Eqs. (9)-(11) the paper states 'This does not coincide with the PN expansion of ref. [4] & [5]', and Fig. 5 reports a match score of 0.7062, described as 'not a high score'. The all-orders claim is an internal inconsistency, not merely a wording issue, because the paper's clarified notion of exactness is that the equations of motion differ from standard PN, and the 99% statement is the headline quantitative validation in the abstract. These claims must be corrected or reconciled before the paper's main assertions can be assessed.
  2. [Introduction, Fig. 1 caption, Conclusion] The load-bearing premise that QNM damping is impossible with true event horizons rests on the assertion that two point-like initial singularities cannot become a circular-line singularity in finite Schwarzschild time. This is not a physical obstruction. In a dynamical binary spacetime there is no global Schwarzschild time; the infinite infall time in the static exterior is a coordinate property of Schwarzschild coordinates, and the final ring singularity in the maximal extension is not an evolved worldline of the initial singularities. Moreover, the paper's own references [17]-[20] show that damped quasi-normal ringdown is a standard solution of black-hole perturbation theory on Schwarzschild/Kerr backgrounds with genuine event horizons, where the damping follows from boundary conditions at the horizon and at infinity, not from an extended inner mass distribution. Since the 'only when' statement in the Conclusion depends on this premise, the central inference from observed damping to banana-shaped interiors loses its foundation.
  3. [Eqs. (14)-(17), Fig. 5] The waveform comparison is not an independent validation. The damping tail is produced by the reduction factor zeta = sin(4GMz/a)/(4GMz/a) in Eq. (16), with z modeled by Eq. (17); the match in Fig. 5 uses fitted parameters zf = 1.2, nbnn = 2, and the text explicitly states that 'by switching on the additional banana-shape deformation parameters in (17), such as c1, ..., our waveforms can be tuned to match those of EOBNRv2 as closely as desired.' Thus the QNM feature is inserted through the free deformation parameters and then cited as evidence for the inner structure that the parameters represent. This is a fitting exercise rather than a test of the proposed interior structure, and it cannot support the exclusivity claim.
  4. [Fig. 5 upper panel and adjacent text] The claimed lower and upper bounds on omega_re^022 are asserted without derivation. The text states that setting H = MA + MB or H to the minimal allowed value in Eq. (13) yields bounds on the real part of the lowest quasi-normal frequency, but no mapping from the Hamiltonian value to the ringdown frequency is given, and no justification is provided for taking the minimal-H configuration as the lower bound or H = MA + MB as the upper bound. Because these bounds are presented as the quantitative observational support for the model, the missing derivation is a load-bearing gap. The agreement with GWTC1-3 data cannot be assessed without this derivation.
minor comments (4)
  1. [Eq. (9)] The displayed expression for H0 contains garbled notation, reading 'n ≡ q/q' or similar; the intended dot product n·p should be clarified.
  2. [Fig. 1 caption] The caption states 'The waveform of EOB has no QNM feature', but the text and Fig. 5 describe EOBNRv2 waveforms with QNM features obtained by EOB+NR+BHPT; this apparent discrepancy should be reconciled.
  3. [Throughout] The manuscript contains numerous typographical and grammatical errors, such as 'ultilizes' in the Conclusion and inconsistent article use, and several references are incomplete; a careful language and reference edit is needed.
  4. [Fig. 5 lower panel] The match-score comparison omits important details, including the number of cycles used, the alignment procedure, and the mass-ratio and spin parameters of the EOBNRv2 template; without these details the 0.7062 score is under-specified.
Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central evidence chain relies on the author's earlier XOB and inner-structure papers, and on a deformation model whose fitted parameters produce the waveform agreement. The QNM frequency bounds are the only genuinely predictive element, and their derivation is not shown.

free parameters (3)
  • zf = 1.2 (equal-mass case)
    Banana-deformation amplitude in Eq. (17); chosen by hand in Fig. 5 to raise match with EOBNRv2 to 0.7062.
  • nbnn = 2 (equal-mass case)
    Exponent controlling when banana deformation begins, Eq. (17); chosen by hand in Fig. 5.
  • c1, ... = unspecified (tunable)
    Additional deformation coefficients in Eq. (17); paper states they can tune waveform to match EOBNRv2 as closely as desired.
assumptions (6)
  • ad hoc to paper The two-patch synchronous Schwarzschild geometry in Eq. (1) describes the full conservative dynamics of the binary merger.
    Introduced as the key idea of XOB; no derivation from Einstein equations is given in this paper.
  • domain assumption Center-of-mass fixing plus synchronous inspiral gives rA=MB r/M and EB=MB^3/M^2 in Eqs. (5)-(6).
    Needed to reduce the two-body Lagrangian to one-body form; the text notes eccentric orbits are more complicated and neglects O[dot-r^2/(omega^2 r^2)] corrections.
  • ad hoc to paper The dissipative force is given by the quadrupole formula with banana-shape reduction factor zeta=sin(4GMz/a)/(4GMz/a).
    Borrowed from ref. [27]; this paper states no derivation of zeta or of Eq. (17).
  • domain assumption A real event horizon would prevent the observed finite-time merger and damping because two point singularities cannot become a circular singularity in finite Schwarzschild time.
    Used in the Introduction and Figure 1 to conclude NR's apparent horizon implies no event horizon.
  • ad hoc to paper Setting H=MA+MB or H=H_min in Eq. (13) yields valid upper and lower bounds on the real part of the lowest QNM frequency.
    Asserted in the Fig. 5 discussion, but no derivation of the bound mapping is shown.
  • standard math The Legendre transform and double series expansion in 1/q and p^2 are valid.
    Used to produce H0, H2, H4 in Eqs. (8)-(11).
invented entities (1)
  • Black holes with extended internal mass distribution, no true event horizon, and banana-shape deformation during merger. independent evidence
    purpose: Explains the damping tail of binary merger waveforms and connects ringdown features to black-hole interior structure.
    The paper predicts upper and lower bounds on the lowest QNM frequency and compares them with GWTC1-3 data, a falsifiable handle outside the paper. However, the deformation parameters themselves are fitted, and the bounds are not formally derived.

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Pith. "Pith review of Complementarity of Gravitation Collapse (II) XOB and the Damping Gravitational Waveform as Evidences." pith.science (2026). https://pith.science/paper/27ZAQPR5

@misc{pith2026250514749,
  author       = {Pith},
  title        = {Pith review of: Complementarity of Gravitation Collapse (II) XOB and the Damping Gravitational Waveform as Evidences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27ZAQPR5}},
  note         = {Machine review of arXiv:2505.14749}
}
read the original abstract

This is the second paper of our working series on the complementarity of gravitational collapse. In this paper we prove that dynamics of XOB (an eXact One-Body Method) under the weak-field-low-speed expansion matches with the post-newtonian series to all orders for the conservative part of binary dynamics in general relativity. Using XOB and an inner-structure modulated quadrupole formula, we generate gravitational waveforms for the black hole binary merger process agreeing with numeric relativity to 99\% degree. Basing on this agreement, we argue that black holes in numerical relativities have inner structures we propose in the complementarity of gravitational collapse.

Figures

Figures reproduced from arXiv: 2505.14749 by the authors.

Figure 1
Figure 1. FIG. 1: Schwarzschild BHs in NR contain no singularity, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The effective potential [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The variation of relative motion orbit parameters. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The banana-shape deformation and GWforms of an [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.