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REVIEW 4 major objections 9 minor 50 references

Constraining Dirty Black Holes and pseudo-complex General Relativity with the Gravitational Waves Transient Catalog 3.0

T0 review · 4 major / 9 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Using nine merging black-hole signals from the third gravitational-wave transient catalog, this paper reports that horizonless pseudo-complex general relativity and dirty-black-hole solutions are excluded at 1PN and, for the first time…

desk verdict A genuinely new 1.5PN exclusion attempt with a marginal statistical foot; worth refereeing, but the headline claim needs a proper hierarchical treatment. read the letter →

arxiv 2505.10199 v2 pith:DH62HLJI submitted 2025-05-15 gr-qc

Yehu I. Maimon , Alex B. Nielsen , Ofek Birnholtz This is my paper · ORCID
classification gr-qc
keywords pseudo-complexgeneralrelativitydirtyblackholeshorizonlesssolutionspost-Newtonianphasegravitational-waveastronomyparametrizedtestsofholehorizons
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

This paper asks whether a family of modified black-hole metrics that can remove the horizon entirely—pseudo-complex general relativity (pcGR) and dirty-black-hole models with a running mass function—is still compatible with the gravitational waves observed from merging black holes. Using nine events from the third gravitational-wave transient catalog, it computes, for each event, the smallest deviation strength $b_c$ that would make the remnant horizonless, then evaluates the post-Newtonian phase shift such a deviation would leave in the inspiral waveform and compares it with the catalog's measured bounds on deviations from general relativity. The central result is that at $b_c$ the predicted 1PN phase ratio $\delta\phi$ lies outside both the general and restricted bounds, and the 1.5PN ratio, tested here for the first time, lies outside the general bounds. At 2PN, 3PN, and 3.5PN the predicted ratios remain inside the bounds, so horizonless solutions at those orders are not excluded. If the result holds, any horizonless pcGR or dirty-black-hole description of these binaries must place its leading deviation at 2PN or higher.

What carries the argument

The load-bearing object is the family of modified Kerr metrics with a mass function $m(r)=M[1-b(M/r)^n]$, representing a dirty black hole in general relativity and a vacuum solution in pcGR. The horizonless threshold is the critical value $b_c$ at which $\Delta = r^2 + a^2 - 2m(r)r$ has no positive real roots, so no Killing horizon exists; the paper evaluates $b_c$ from the remnant spin $\chi$ via Eq. (5). The observable imprint is the pcGR post-Newtonian phase coefficient $PN^{pcGR}_n = \frac{20}{(n-4)(2n-5)}\left(\frac{(n+2)(n+1)}{3}Q+\rho\right)b$, derived through the stationary-phase approximation, and the argument proceeds by comparing the ratio $\delta\phi = PN^{pcGR}_n / PN^{GR}_n$ with the catalog's general and restricted bounds at each order. This ratio converts an abstract no-horizon condition into a measurable waveform-phase prediction.

What would settle it

Recompute $\delta\phi$ for all nine events using the mass-weighted effective inspiral spin (or the component spins) in place of the remnant spin in Eq. (5), and check whether the 1PN and 1.5PN ratios still fall outside the catalog's 90% bounds; if they do not, the exclusion of horizonless pcGR solutions at these orders does not follow from the current data.

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Extended reading notes

Core claim

The paper's central claim is that horizonless pcGR/dirty-black-hole solutions are observationally excluded at two post-Newtonian orders. For each of the nine events, setting the deviation parameter $b$ to the critical horizonless value $b_c$ (Eq. 5, evaluated with the remnant spin $\chi$) produces a phase ratio $\delta\phi = PN^{pcGR}_n / PN^{GR}_n$ that exceeds the measured bounds at 1PN and 1.5PN: at 1PN the ratio lies outside both the General bounds (which allow deviation coefficients to vary across events) and the Restricted bounds (which require a common deviation), while at 1.5PN—tested here for the first time—it lies outside the General bounds. GW200115, the low-spin outlier, provides the most stringent 1PN and 1.5PN exclusions and comes closest to excluding 2PN. The paper concludes that the earlier 1PN exclusion is confirmed and that 1.5PN horizon-removing deviations are newly ruled out, while 2PN, 3PN, and 3.5PN horizonless solutions remain within the observational bounds.

Load-bearing premise

The exclusion rests on assuming that the deviation strength needed to erase the horizon, computed from the merged remnant's final spin, also sets the size of the phase deviation during the early inspiral; if the relevant spins are instead the inspiral component spins or a mass-dependent $b_c$, the predicted phase shifts change and the claimed exclusion could fail.

Editorial extensions

If this is right

  • Horizonless pcGR and dirty-black-hole solutions with 1PN or 1.5PN leading deviations are incompatible with the nine events considered, so any surviving horizonless model must have $b < b_c$ at those orders.
  • The 2PN, 3PN, and 3.5PN parameter regions remain observationally open, so horizonless solutions whose deviations first appear at those orders are not ruled out by this dataset.
  • The low-spin event GW200115 yields the most stringent 1PN and 1.5PN bounds and comes closest to excluding 2PN, indicating that low-spin mergers carry the most exclusion power per event.
  • Compared with earlier single-event bounds, the updated constraints are narrower at 1PN and have shifted the allowed 2PN and 3PN intervals in a direction more accommodating of pcGR horizonless objects.
  • Applying the same ratio test to the next observing run's catalog should refine these exclusions and may decide whether 2PN-order horizonless deviations become excluded as sensitivity improves.

Reading between the lines

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

  • Because $b_c$ grows as spin decreases, the exclusion power of this method is spin-dependent; a catalog with more low-spin mergers, or a different choice of which spin enters Eq. (5), could either sharpen or erode the stated bounds, a direction the paper does not quantify.
  • If horizonless deviations are forced to 2PN or higher, their inspiral-phase signatures shrink, so the cleanest future tests would move to merger/ringdown observables such as the photon ring and quasinormal-mode spectra, which the paper flags as its planned next step.
  • The same 'horizonless threshold to inspiral phase ratio' mapping should transfer to any Kerr-like metric with a similar mass-function form, letting the method constrain other parametrized deviations without building new waveform models.
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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 / 9 minor

Summary. This paper updates constraints on the pseudo-complex General Relativity (pcGR) and dirty-black-hole metric parameter b, for post-Newtonian orders n = 1, 1.5, 2, 3, 3.5, using the inspiral phases of the nine GWTC-3 O3b events that passed the LVK parameterized-test selection. The authors evaluate the horizonless threshold b_c of a single spinning object (Eq. 5) at the averaged final-spin posterior of each event, compute the pcGR-to-GR phase-coefficient ratio delta_phi = PN_pcGR/PN_GR (Eq. 10, with PN_pcGR from Eq. 9 at b = b_c), and compare delta_phi with the 90% 'General' and 'Restricted' bounds of the LVK parameterized tests. They conclude that 1PN horizonless pcGR is excluded (extending earlier work), that 1.5PN horizonless pcGR is excluded for the first time, and that 2PN-3.5PN remain allowed. The analysis code is publicly available.

Significance. A robust 1.5PN exclusion would be a genuinely new constraint on an exotic-metric family, and the paper's mechanics are partly commendable: the analytic b_c construction is transparent; the quoted GR phase coefficients at 1PN (6.44) and 1.5PN (-16*pi) match the standard TaylorF2 values, confirming the mapping; the code is public; and the use of GWTC-3 rather than O1 events is a clear step forward. The 1PN confirmation, whose median predictions lie far outside both quoted bounds, is credible. However, the headline 1.5PN claim is not established by the analysis as presented, for the reasons detailed below; the result becomes publishable only after a proper statistical treatment of the comparison.

major comments (4)
  1. [Sec. IV / Table I / Fig. 3] The claim that 1.5PN horizonless pcGR is 'rule[d] out' rests on comparing the predicted delta_phi with the 'General' 90% bound (-3%, 3%) only. The same table quotes the 'Restricted' (common-deviation) 90% bound (-5%, -1%) for 1.5PN, and the predicted values in Table I overlap that interval: -3.7% to -6.5% for the chi=0.7 group and -4.9% to -9.1% for GW200115. Because b is a universal parameter of pcGR (the per-event variation enters only through the q-dependent factor Q in Eq. 8), the common-deviation comparison is the pertinent one, and a defensible exclusion requires a joint/hierarchical likelihood over b across the nine events, or at minimum a comparison against the Restricted bound. As it stands, the 1.5PN exclusion is not established against the appropriate bound.
  2. [Sec. III / Table I] The quantity delta_phi is evaluated at point estimates (averaged posteriors) of q and chi, and the reported ranges are obtained by substituting the 90% bounds of the inputs one parameter at a time, ignoring correlations in the joint posterior. For the n=1.5, chi=0.7 case, the upper end of the quoted range (-3.7%) lies only 0.7 percentage points inside the General bound (-3%, 3%); under a full posterior predictive computation using the joint q, chi_1, chi_2, chi_f posteriors, a non-negligible fraction of the predictive mass of delta_phi will fall inside the bound, so the 90%-confidence exclusion language is not supported. The authors should compute the full predictive distribution of delta_phi (or of b directly), for example by reweighting the public LVK posterior samples of the parameterized-test parameters, and report the posterior mass outside the relevant bound.
  3. [Sec. III / Eq. (5)] The horizonless threshold b_c is evaluated with the remnant (final) spin and then applied to a two-body inspiral phase coefficient. The justification given, namely that the final spin exceeds every component spin for all events considered and thus provides a more restrictive bar, is asserted but not demonstrated for the nine events, and no derivation is provided for mapping a single-object horizon condition to the inspiral PN coefficient. The authors should either derive the appropriate effective spin entering the inspiral horizonless condition or demonstrate robustness with a sensitivity analysis (e.g., using effective inspiral spin or the component spins in Eq. 5) and quantify the resulting change in delta_phi. I note that if the final-spin ordering failed, b_c and |delta_phi| would increase and the exclusion would be strengthened; nevertheless, the physical basis of the chosen threshold must be part of the argument.
  4. [Sec. II / Eq. (9) vs Table I] Evaluating Eq. (9) at the tabulated b_c for the chi=0, q=1 rows gives PN_pcGR values larger than those tabulated by a systematic factor of roughly 2^(n-1) (n=1.5: 6.14 vs 4.33; n=2: 23.7 vs 11.8; n=3: -112 vs -28.1; n=3.5: -172 vs -30.4), while the n=1 row agrees. Because delta_phi is the central quantity of the paper, the authors must reconcile Eq. (9) with the numbers actually used in Table I, give the exact convention connecting B(f) in Eq. (7) to the LVK delta_phi-hat_n parameters (including any powers of 2 arising from the (M 2*pi*f)^(2n/3) factor), and confirm that the GitHub code implements the convention printed in the paper.
minor comments (9)
  1. [Sec. II / Eq. (6)] Equation (6) prints the SPA prefactor as (pi*G*M*f)^(5/3); the standard stationary-phase expression has (pi*G*M*f)^(-5/3). The exponent should be corrected; the ratio delta_phi is unaffected, but the printed formula is wrong as it stands.
  2. [Sec. II / first paragraph] The sentence 'pcGR converges with GR predictions in the weak-field regime for larger (or more precisely (M/r)^n)' is garbled; it should say that (M/r)^n tends to zero at large r.
  3. [Table I / quoted bounds] As quoted in Table I, the Restricted 1.5PN interval (-5%, -1%) excludes delta_phi=0 at 90%, a statement that would itself be notable; the authors should verify this number against the source (arXiv:2112.06861) and comment on its meaning, and in general should cite the specific table or figure from which each General and Restricted bound is taken.
  4. [References] Reference [7] spells the author name 'Birnholz' whereas the current paper and reference [6] use 'Birnholtz'; the typo should be fixed.
  5. [Acknowledgments] The acknowledgment thanks the 'annonymous referee'; this should read 'anonymous referee'.
  6. [Table I / presentation] Table I mixes input parameters (q, chi with their 90% bounds) with derived quantities (b_c, PN_pcGR, PN_GR, delta_phi); splitting the table or adding explicit column labels for medians and upper/lower 90% values would greatly aid reproducibility, and the caption should state that the chi=0, q=1 rows are not GWTC-3 events but a comparison with [6].
  7. [Sec. IV / 1PN claim] The statement that 'The 1PN deviation lies outside both the General and Restricted bounds' is not strictly true for the full quoted range: for the chi=0.7 group the lower end of the quoted delta_phi range is 13%, inside the General upper bound of 14%. The claim should either be restricted to medians or be supported by the full comparison.
  8. [Sec. III / event grouping] Grouping the eight non-GW200115 events at a fixed 'averaged' chi=0.7 discards the per-event posterior information shown in Fig. 2; using each event's own posterior would be more principled and would connect directly to the hierarchical comparison requested in the major comments.
  9. [Abstract vs Sec. V] Section V's 'effectively ruled out' is weaker than the abstract's 'for the first time rule out 1.5PN'; the abstract, main text, and summary should make the same, appropriately qualified claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the pcGR phase deviations are derived from the analytic horizonless threshold and compared against independent LVK bounds, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim is that horizonless pcGR solutions are excluded at 1PN and (for the first time) 1.5PN by comparing the predicted phase-deviation ratio delta-phi = PN_pcGR/PN_GR to the GWTC-3 bounds. The prediction is not equivalent to its inputs by construction. The horizonless threshold b_c is obtained analytically from Eq. (5), which solves the horizon condition r^2 + a^2 - 2M[1 - b(M/r)^n]r = 0; it uses only the measured final spin and mass ratio, not the LVK PN-deviation bounds. Equation (10) then evaluates delta-phi at b = b_c, and Table I compares these values to the independent General and Restricted bounds from GWTC-3. No parameter is fitted to the target result, and no quantity that is being 'ruled out' is re-inserted as the constraint. The self-citations to the authors' prior works [6,7] supply the pcGR waveform-phase model, but that model is restated in the text (Eqs. 2-9) and traces to the pcGR framework (refs [2-5]) rather than to the GWTC-3 bounds being tested. The cited prior results are parameter-free with stated assumptions that do not include the target exclusion, and they are externally falsifiable against the LVK data. Concerns about point-estimate versus full posterior comparison, or about whether the General or Restricted bound is the appropriate one for a universal parameter b, are statistical and modeling matters, not circularity. The derivation is self-contained against an external benchmark, so the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central exclusion rests on two imported ingredients: the pcGR phase formula from the authors' prior work and the LVK bounds. The only ad hoc modeling choice specific to this paper is using the final spin to set b_c. No parameters are fitted to the gravitational wave data.

free parameters (1)
  • ρ (emission model parameter) = 0
    Set to zero per footnote 2; this choice favors GW emission and makes the exclusion harder, so it is conservative for the paper's claim.
assumptions (5)
  • domain assumption The pcGR metric ansatz with m(r) = M(1 - b(M/r)^n) describes the binary spacetime.
    From [2-5], used in Eqs. (2)-(3) as the starting point for the waveform deviation.
  • standard math The horizonless condition is obtained from the roots of Eq. (4), with b_c given by Eq. (5) and assumed independent of M.
    Algebraic root condition for the Kerr-like metric; the M-independence of b_c is stated in Section II.
  • domain assumption The stationary phase approximation waveform (Eq. 6) from [19] and the pcGR phase correction (Eq. 7) from [6,7] correctly map the metric parameter b to a PN phase coefficient.
    The mapping between metric deformation and gravitational wave phase is imported from the authors' prior work rather than re-derived here.
  • domain assumption The LVK parameterized-test bounds on PN coefficient deviations from [14] apply directly to the pcGR δφ ratio.
    The comparison in Table I treats the GWTC-3 General/Restricted 90% intervals as exclusion boundaries for the pcGR model.
  • ad hoc to paper The final remnant spin determines the effective horizonless condition for the inspiral signal.
    Section III uses the final spin χ to compute b_c for each event; this is the paper's key modeling choice and is not derived from the binary inspiral formalism.

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Pith. "Pith review of Constraining Dirty Black Holes and pseudo-complex General Relativity with the Gravitational Waves Transient Catalog 3.0." pith.science (2026). https://pith.science/paper/DH62HLJI

@misc{pith2026250510199,
  author       = {Pith},
  title        = {Pith review of: Constraining Dirty Black Holes and pseudo-complex General Relativity with the Gravitational Waves Transient Catalog 3.0},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DH62HLJI}},
  note         = {Machine review of arXiv:2505.10199}
}
read the original abstract

We use data from the Gravitational Wave Transient Catalog 3.0 to update constraints on parameterized deviations from General Relativity, as encountered in pseudo-complex general relativity (pcGR) theory and models of dirty black holes. The pcGR framework extends Einstein's theory of general relativity by introducing additional parameters that diverge from standard predictions in the strong-field regime, potentially excluding black hole horizons for specific parameter choices. We analyze gravitational wave signals from coalescing compact objects to obtain new bounds on these parameters. Our results modify existing constraints and identify previously unexplored regions of parameter space, exploring the observational viability of dirty black holes and horizonless solutions in pcGR. We confirm the exclusion of 1PN deviations sufficient to avoid a horizon, and for the first time rule out 1.5PN as well. We also discuss implications for current and future gravitational wave observations in refining these constraints

Figures

Figures reproduced from arXiv: 2505.10199 by the authors.

Figure 1
Figure 1. FIG. 1: Plot of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Histograms of the posterior densities of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Post-Newtonian (PN) Parameter Plot for GW Events, showing both the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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