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Observing the eye of the storm I: testing regular black holes with LVK and EHT observations

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read EHT shadow sizes and LVK inspiral phases tie the eye-of-the-storm black hole to near-Schwarzschild geometry.

desk verdict EHT half is a clean standard analysis and the shadow bounds are likely right; the GW half has a factor-of-two stationary-phase error and a frequency-convention muddle, so the quoted LVK constraints need repair before use. read the letter →

arxiv 2411.13897 v1 pith:AWEJMS43 submitted 2024-11-21 gr-qc

Carlos A. Benavides-Gallego , Swarnim Shashank , Haiguang Xu This is my paper · ORCID
classification gr-qc MSC 83C5783C1083C35
keywords regularblackholeseyeofthestormmetricGhosh-Simpson-Visserspacetimeno-hairtheoremgravitationalwaveconstraintsholeshadowEventHorizonTelescopeparameterizedpost-Einsteinianframework
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 how far a singularity-free black hole can deviate from the Schwarzschild solution before current observations notice. The model tested is the 'eye of the storm' metric, a rotating regular black hole whose only extra freedom is a dimensionless parameter $\ell$ that softens the mass near the center. Using shadow angular diameters from the Event Horizon Telescope, the paper finds $0\leq \ell \leq 0.148$ for Sgr A* and $0\leq \ell \leq 0.212$ for M87*. Using inspiral gravitational waves from the LVK catalogs, the tightest bounds are $\ell=0.041^{+0.106}_{-0.041}$ and $\ell=0.050^{+0.165}_{-0.050}$ for two events under the SEOB waveform model. The takeaway is that the exterior of this regular black hole must look nearly Schwarzschild, so a singularity-free core is not excluded but leaves almost no observable trace in current data.

What carries the argument

The central object is the eye-of-the-storm line element, obtained from the Kerr metric by the mass replacement $M \to M e^{-\ell M/r}$, which removes the central singularity while preserving separability of the Hamilton-Jacobi equations. The identity that carries the gravitational-wave argument is $\ell = -\frac{9}{80}\varphi_2\delta\varphi_2$, obtained by matching the leading $\ell$ correction in the Fourier phase to the PPE phase parameter $\beta u^{-1}$; for shadows, the operative relations are the photon-sphere condition $\partial V_{\mathrm{eff}}/\partial r = 0$, the shadow radius $b_{\mathrm{ph}} = r_{\mathrm{ph}}/\sqrt{1 - 2M(r_{\mathrm{ph}})/r_{\mathrm{ph}}}$, and the angular-diameter formula $\Theta = 2b_{\mathrm{ph}}/D$.

What would settle it

Recompute Kepler's law for circular equatorial orbits in the EOS metric from the coordinate angular velocity $\Omega = d\varphi/dt$ via the metric components $g^{\mu\nu}$; if the $\ell$ term appears with a different coefficient or at a different order than in Eq. (28), the identification $\ell = -\frac{9}{80}\varphi_2\delta\varphi_2$ is wrong and the quoted gravitational-wave constraints do not apply to this model.

Watch

Extended reading notes

Core claim

The paper establishes that the Ghosh-Simpson-Visser 'eye of the storm' regular black hole is observationally pinned close to Schwarzschild. The EHT angular-diameter measurements give $0\leq \ell \leq 0.148$ for Sgr A* and $0\leq \ell \leq 0.212$ for M87*, while the most stringent gravitational-wave constraints from the inspiral phase are $\ell=0.041^{+0.106}_{-0.041}$ (GW191204-171526) and $\ell=0.050^{+0.165}_{-0.050}$ (GW190924-021846) for the SEOB model. The argument maps the deformed Schwarzschild effective-one-body Hamiltonian into the parameterized post-Einsteinian phase correction, yielding the identification $\ell = -\frac{9}{80}\varphi_2\delta\varphi_2$, and then reads the $\delta\varphi_2$ posteriors from the LVK catalogs. For the shadow part, the photon-sphere radius and impact parameter are computed from the deformed metric, giving an angular diameter $\Theta = 2b_{\mathrm{ph}}/D$ that shrinks as $\ell$ grows.

Load-bearing premise

The gravitational-wave constraints hinge on identifying the angular frequency in the phase integral with $L = r^2\Omega$ using the proper-time angular velocity; if that substitution is not the coordinate frequency a distant observer assigns to the orbit, the derived mapping $\ell = -\frac{9}{80}\varphi_2\delta\varphi_2$ and the resulting bounds change.

Editorial extensions

If this is right

  • The EOS metric can deviate from Schwarzschild by at most about $\ell = 0.2$ in the region probed by shadows, so any singularity-free core must be hidden deep inside an essentially Schwarzschild exterior.
  • Gravitational-wave inspiral data from the LVK catalogs are consistent with $\ell = 0$; the most constraining events place $\ell$ below 0.05 at the median, so current merger observations do not demand a regular-core modification.
  • For the allowed range of $\ell$, the predicted shadow angular diameters stay within the EHT error bars, meaning higher-resolution images or additional sources are needed to see the deformation.
  • The widening of the lensed- and photon-ring impact-parameter intervals with $\ell$ implies that, if the deformation is near its upper bound, future very-long-baseline observations of the photon ring could detect it.

Reading between the lines

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

  • The paper's $a=0$ assumption means $\ell$ could trade off against spin in a rotating fit; X-ray reflection spectroscopy, which the authors say is in progress, is the natural way to break that degeneracy.
  • A hierarchical analysis treating $\ell$ as a population hyper-parameter would convert per-event upper limits into a statement about whether all regular black holes share one mass-distribution parameter; the paper stops short of that.
  • The proper-time angular-velocity issue in the Kepler-law derivation could rescale the gravitational-wave bounds without affecting the shadow bounds; redoing the derivation in coordinate time would show whether the reported $\ell$ values shift.
  • If future shadow measurements shrink the uncertainty below about 2 $\mu$as, the upper limits on $\ell$ improve roughly linearly with the angular-diameter error, so the method's power scales with interferometric resolution.
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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 / 6 minor

Summary. The paper tests the regular black hole metric of Ghosh and Simpson-Visser (the 'EOS' metric), which adds a free parameter ℓ to the mass and spin, against two independent observational probes. For gravitational waves, the authors restrict to a=0, derive a leading-order correction to the binding energy and orbital phase in an effective-one-body setup, map this correction to the PPE parameter δφ2, and then convert LVK posterior samples from GWTC-1/2/3 into constraints on ℓ. For EHT, they compute the nonrotating shadow radius and angular diameter, compare the predicted values for SgrA* and M87* with the reported EHT measurements, and generate intensity profiles and images for static, infalling, and thin-disk accretion models. The headline results are EHT bounds 0≤ℓ≤0.148 (SgrA*) and 0≤ℓ≤0.212 (M87*), and GW bounds for which the most stringent events are GW191204-171526 and GW190924-021846 with ℓ≈0.04-0.05 for the SEOBNRv4 model.

Significance. If the GW derivation is repaired, the paper offers a useful two-channel test of a popular regular black hole family: EHT shadow-size data and LVK inspiral phase data constrain the same deformation parameter, and the authors correctly use public posterior samples rather than fitting a new waveform model. The EHT shadow part is clean and internally consistent, and the photon-ring classification into direct, lensed, and photon-ring emission is thorough and well illustrated. The GW part is the main quantitative asset, but its headline numbers are currently affected by internal algebraic errors in the stationary-phase mapping and by a frequency-convention issue in the modified Kepler law; these need to be corrected before the quoted constraints can be used.

major comments (3)
  1. [Sec. IV B, Eqs. (37)-(44)] The stationary-phase factor is off by a factor of two. Since ΨGW(f)=2ϕ(t0) and the stationary point satisfies ν(t0)=f/2, the ℓ-correction in Eq. (35), which scales as (2πν)^{-1}, must be evaluated at ν=f/2 and then multiplied by 2. This yields δΨ=-(5/12)η^{-2/5}ℓu^{-1}, not the coefficient -(5/24) shown in Eq. (38). Equating this corrected coefficient with the LVK PPE form β=(3/128)φ2δφ2η^{-2/5} gives ℓ=-(9/160)φ2δφ2, not Eq. (44)'s -(9/80)φ2δφ2. Consequently every entry in Table I is a factor of two too large in magnitude; for example GW191204-171526 would shift from ℓ≈0.041 to ℓ≈0.020. This is an internal algebraic error, independent of the LVK data, and must be fixed before the GW constraints are quoted.
  2. [Sec. IV A, Eq. (28)] The modified Kepler law is derived using L=r²Ω, but L in Eqs. (19)-(24) is the proper-time specific angular momentum, L=r²dφ/dτ, while Ω=dφ/dt is the coordinate angular velocity. For a static, spherically symmetric metric the correct relation is Ω=L(1-2M(r)/r)/(E r²), so replacing L by r²Ω is not valid at the order retained. Indeed, setting ℓ=0 in Eq. (28) gives Ω²=(m/r³)(1+3m/r+9m²/r²+...), which disagrees with the exact Schwarzschild coordinate-frequency result Ω²=m/r³. Since the ℓ-dependent term in Eq. (28) enters at the same 1PN order at which the proper-time/coordinate-time distinction first appears, the mapping from ℓ to the orbital phase in Eqs. (31)-(35) is built on an incorrect intermediate quantity. This needs to be rederived from dφ/dt obtained directly from the geodesic equations.
  3. [Abstract, Sec. V, and Sec. VII] The EHT constraints are computed only for the nonrotating subfamily (a=0), but the abstract and conclusion present them as constraints on ℓ of the EOS spacetime without this qualification. Since the underlying model includes spin and the observed sources are not known to be nonrotating, the bounds should be explicitly labeled as applying to the a=0 slice of the parameter space, or accompanied by a quantitative estimate of how the shadow diameter depends on spin.
minor comments (6)
  1. [Throughout] The collaboration name appears as 'L VK' with a space in the title, abstract, and several section headings; it should be 'LVK'.
  2. [Sec. IV B and Table I] The waveform model is written as 'IMPRPhenomPv2' in the text and table; this should be 'IMRPhenomPv2'.
  3. [Table I caption and Sec. VII] The table caption reports 90% confidence while the discussion section reports 95% confidence; the two should be made consistent.
  4. [Eq. (33)] The orbital phase integral as written has no explicit lower limit and the change of variables leading to ∫(1/˙E)(dE/dΩ)ΩdΩ is not shown; adding the frequency limits would make the SPA step easier to verify.
  5. [Sec. V A, Fig. 5 caption] The caption says 'Central and left panels' but the figure contains central and right panels; the wording should be corrected.
  6. [Throughout] There are numerous typographical errors, including 'panle', 'colunms', 'usign', 'sourroundings', 'ETH observations' in the Sec. V heading, and an incomplete bibliographic entry in reference [19]; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EOS metric is mapped to PPE and EHT observables through explicit geodesic/PN derivations, then compared with external LVK and EHT data.

full rationale

The paper's derivation chain is self-contained with respect to external data. The GW constraints come from deriving the leading-order correction to the binding energy and orbital phase from the EOS metric (Eqs. (24)-(38)), then mapping to the PPE phase via Eq. (44) and sampling external LVK posterior samples for φ2 and δφ2 (Sec. IV B). The EHT constraints are obtained by computing the shadow impact parameter bph from the photon effective potential (Eqs. (45)-(47)), converting to angular diameter (Eqs. (48)-(49)), and comparing with the published EHT angular diameters for SgrA* and M87* (Sec. V A). ℓ is not defined in terms of the measured quantities; it is a metric parameter, and the mapping from ℓ to observables is derived, not fitted. The only self-citations (e.g., Refs. [67], [81]) are used as methodological precedents for assuming GR radiation reaction, and the central claim does not reduce to them. A possible factor-of-two issue in the stationary-phase mapping (ν(t0)=f/2 vs the coefficient used in Eq. (38)) is an internal algebraic/correctness concern, not a circularity, and does not change this verdict.

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

The central claim rests on the choice of metric family, the EOB mapping for GW, the GR radiation reaction, and a simplified shadow model. The free parameter ℓ is the target of the fit. No new particles or fields are introduced by this paper.

free parameters (1)
  • ℓ (EOS regularity parameter) = ℓ ≤ 0.148 (EHT Sgr A*), ℓ ≤ 0.212 (EHT M87*); ℓ = 0.041^{+0.106}_{-0.041} (GW191204-171526, SEOB)
    The central claim is that these intervals constrain the deviation from Schwarzschild; ℓ is the target parameter fitted to external data.
assumptions (5)
  • domain assumption The EOS metric (Eq. 1) is assumed to describe astrophysical black holes.
    The entire analysis tests this specific metric family; there is no independent evidence that astrophysical BHs follow this geometry.
  • domain assumption A binary of non-spinning EOS BHs can be modeled as a test particle moving in the effective non-rotating EOS metric with total mass m and reduced mass ratio η (EOB mapping, Sec. IV A).
    This mapping is standard in EOB theory but here it is applied to a deformed metric without the full EOB potential calibration; the paper assumes the geodesic dynamics in the effective metric directly controls the two-body inspiral.
  • domain assumption The dissipative sector (gravitational wave emission) is described exactly by the Einstein quadrupole formula, and only the conservative dynamics are modified by ℓ (Sec. IV B).
    This assumption is stated explicitly after Eq. (33); if the EOS metric modifies the radiation reaction, the mapping to PPE would change.
  • domain assumption The expansion is truncated to first order in ℓ (ℓ << 1), and all results are valid only in this regime.
    The paper acknowledges this for the starred events in Table I, and uses it in Eqs. (21), (26)-(27), (31), and (38).
  • domain assumption The EHT shadow angular diameter can be computed from the non-rotating EOS metric using fixed values of mass M and distance D for Sgr A* and M87*.
    The paper uses M = 4.14e6 M_sun, D = 8.127 kpc for Sgr A* and M = 6.2e9 M_sun, D = 16.8 Mpc for M87* without propagating uncertainties; spin and accretion disk effects are not included in the shadow-size constraint.

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Cite this review

Pith. "Pith review of Observing the eye of the storm I: testing regular black holes with LVK and EHT observations." pith.science (2026). https://pith.science/paper/AWEJMS43

@misc{pith2026241113897,
  author       = {Pith},
  title        = {Pith review of: Observing the eye of the storm I: testing regular black holes with LVK and EHT observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWEJMS43}},
  note         = {Machine review of arXiv:2411.13897}
}
abstract

According to the celebrated singularity theorems, space-time singularities in general relativity are inevitable. However, it is generally believed that singularities do not exist in nature, and their existence suggests the necessity of a new theory of gravity. In this paper, we investigated a regular astrophysically viable space-time (regular in the sense that it is singularity-free) from the observational point of view using observations from the LIGO, Virgo, and KAGRA (LVK), and the event horizon telescope (EHT) collaborations. This black hole solution depends on a free parameter $\ell$ in addition to the mass, $M$, and the spin, $a$, violating, in this way, the non-hair theorem/conjecture. In the case of gravitational wave observations, we use the catalogs GWTC-1, 2, and 3 to constrain the free parameter. In the case of the EHT, we use the values of the angular diameter reported for SgrA* and M87*. We also investigated the photon ring structure by considering scenarios such as static spherical accretion, infalling spherical accretion, and thin accretion disk. Our results show that the EHT observations constrain the free parameter $\ell$ to the intervals $0\leq \ell \leq 0.148$ and $0\leq \ell \leq 0.212$ obtained for SgrA* and M87*, respectively. On the other hand, GW observations constrain the free parameter with values that satisfy the theoretical limit, particularly those events for which $\ell<<1$. Our results show that the most stringent constraints on $\ell$ correspond to the events GW191204-171526 ($\ell=0.041^{+0.106}_{-0.041}$) and GW190924-021846 ($\ell=0.050^{+0.165}_{-0.050}$) for the SEOB model.

Figures

Figures reproduced from arXiv: 2411.13897 by the authors.

Figure 1
Figure 1. FIG. 1. Plot of ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Violin plots for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Left panel: the effective potential for photons for different values of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Left panel: the apparent boundary (shadow) vs. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. First row: Examples of photon trajectories for different values of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Dependence of the observer specific intensity [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. First row: The observed intensity, [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. First row: the total number of photons orbits, [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Left and central panels: Observational appearance of a thin, optically thin disk of emission near the EOS BH (with [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Left column: total emission intensities [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Left column: total emission intensities [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The Cartesian coordinates ( [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Image of the EOS BH for different values of [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

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Reference graph

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