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REVIEW 4 major objections 5 minor 5 cited by

EHT shadow measurements place a bound on the hair parameter of a rotating Horndeski black hole, a step toward testing whether black holes in this modified-gravity theory look different from Kerr black holes.

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For a rotating Horndeski black hole, the hair parameter h shrinks and deforms the shadow, and EHT shadow sizes allow but do not uniquely confirm a non-zero h.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Useful shadow image catalog for a Kerr-like Horndeski metric, but the EHT 'confirmation' is unsupported—metric validity is unverified and the constraint section has unit and overclaim problems. the 4 major comments →

arxiv 2509.05803 v1 pith:ZZ4T6PQO submitted 2025-09-06 gr-qc

Probing Horndeski Gravity via Kerr Black Hole: Insights from Thin Accretion Disks and Shadows with EHT Observations

Xiao-Xiong Zeng , Chen-Yu Yang , M. Israr Aslam , Rabia Saleem This is my paper
classification gr-qc MSC 83C5783C10 PACS 04.70.-s
keywords black hole shadowHorndeski gravityhair parameterthin accretion diskray tracingphoton ringEHT constraintsKerr-like metric
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.

The reading

This paper asks whether shadow and accretion-disk images of a rotating black hole can distinguish Horndeski gravity from general relativity. It works with a Kerr-like Horndeski black hole whose only extra feature over Kerr is a hair parameter h in the metric, and simulates its appearance under two kinds of illumination: a distant celestial sphere and a thin accretion disk. The simulations show that h shrinks and distorts the shadow and the photon ring, while the spin parameter a controls the frame-dragging asymmetry. The authors then compare the predicted shadow angular diameter with the EHT measurements of M87* and Sgr A* and find that the observations keep h within the 1–2 sigma bands, which they read as support for Horndeski gravity.

Core claim

The central claim is that the hair parameter h of the rotating Horndeski black hole leaves a measurable imprint on strong-field images, and that existing EHT observations already constrain it. For fixed spin, increasing h moves the shadow from a near-circle toward a 'D' shape, shrinks the photon ring, and shifts the shadow contour to the right; decreasing h does the opposite. Under thin-disk illumination, h changes the size of the inner shadow, the lensing bands, and the redshift pattern, with the effects reversed between prograde and retrograde flows. Using the observed shadow angular diameters of M87* and Sgr A*, the paper states that h stays inside the 1σ–2σ confidence intervals, thereby

What carries the argument

The engine of the analysis is the Boyer-Lindquist-style metric of Eq. (1), with Δ = r^2 + a^2 − 2Mr + h r ln(r/2M), which adds a logarithmic hair term to the Kerr spacetime. On that geometry the paper runs null geodesic ray tracing through a fisheye camera projection, and for disk images combines a Keplerian-plus-plunging thin-disk emission model with redshift factors and a horizon-to-disk intensity integral. The observables that lead to EHT constraints are the shadow radius R_d and circularity deviation δ_d, converted to angular diameter via Eq. (22).

Load-bearing premise

The metric in Eqs. (1)–(2) is an exact rotating black hole solution of Horndeski gravity, something the paper takes from Refs. [26, 73] rather than verifying; if it is only a Kerr-like construction, the shadow images and EHT bounds do not describe Horndeski gravity.

What would settle it

Substitute the metric (1)–(2) into the Horndeski field equations and compute the residual; a nonzero residual means the metric is not a solution and the constraints are void. Observationally, measure the shadow angular diameter and asymmetry of Sgr A* at percent-level precision: the predicted h-dependence separates from spin because h changes diameter while spin mostly changes asymmetry.

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

If this is right

  • If h is nonzero and inside the EHT band, the M87* and Sgr A* shadows are compatible with a one-parameter Horndeski deformation of Kerr, and shadow angular diameter becomes the cleanest probe of that deformation.
  • Larger h implies smaller, more D-shaped shadows with a smaller photon-ring radius, so future higher-resolution images could separate h from spin by morphology.
  • The redshift and lensing-band maps differ measurably between prograde and retrograde accretion flows; matching those patterns to accretion models can reduce parameter degeneracies.
  • The same angular-diameter conversion can be applied to any future black hole shadow measurement, making the constraint scheme reusable.

Where Pith is reading between the lines

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

  • The phrase 'confirming the validity of Horndeski gravity' is stronger than what the calculation establishes: the EHT comparison bounds one parameter in one metric family, not the full theory.
  • If the metric were independently verified to solve the Horndeski field equations, the same ray-tracing pipeline could predict polarized images and jet emission, offering independent tests of h.
  • The D-shape deformation is the most distinctive geometric signature; a targeted measurement of shadow asymmetry at a few percent precision could either detect h or push its upper bound below current values.
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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

4 major / 5 minor

Summary. The manuscript studies a rotating, Kerr-like metric labeled as a Horndeski-gravity black hole, with line element (1) and Δ(r) = r² + a² − 2Mr + h r ln(r/2M) in Eq. (2). Using Hamilton–Jacobi separability and a fisheye-camera/backward ray-tracing pipeline, it computes shadow contours, photon-region impact parameters, and the Hioki–Maeda observables R_d and δ_d. It then produces images for two illumination models: a celestial sphere and a geometrically thin accretion disk, including prograde/retrograde flows, redshift maps, and lensing bands. Section V compares predicted shadow angular diameters with EHT measurements of M87* and Sgr A* and claims constraints on the hair parameter h, with the abstract and conclusion stating that the results confirm the validity of Horndeski gravity.

Significance. If the metric in Eqs. (1)–(2) were proven to be an exact rotating solution of Horndeski gravity, the paper would provide a broad phenomenological survey of shadow and accretion-disk images for a non-Kerr hair parameter, plus a first EHT-based constraint on that parameter. The imaging pipeline follows standard references and the qualitative trends (dependence of shadow size, photon ring, and lensing bands on h and a) are coherent. However, the physical interpretation as a test of Horndeski theory is entirely contingent on the exact-solution status of the metric, which is not established in the manuscript. In addition, the EHT constraint section contains a unit/conversion error, an incorrect M87* mass/distance, and an undefined relationship between R_d as plotted and R_d as used in Eq. (22). These issues currently block the advertised conclusion.

major comments (4)
  1. [Sec. II, Eqs. (1)–(2)] The metric is presented as a rotating Horndeski black hole by citing Refs. [26,73], but the paper never verifies that (1)–(2) solve the Horndeski field equations, and it does not specify the Horndeski action or scalar-field profile. A Kerr-like metric with a deformed Δ is not automatically a solution of a scalar-tensor theory; the logarithmic term is characteristic of Newman–Janis-generated Kerr-like seeds. Since every subsequent result — geodesics, photon region, shadow, disk images, and EHT constraints — depends on this metric, the interpretation as a Horndeski test is unsupported. Please provide an explicit field-equation check (or a reference that contains the derivation, including the scalar field configuration) for nonzero a and h. If the metric is only a Kerr-like ansatz, state this and revise the claims accordingly.
  2. [Sec. V, Eq. (22)] As printed, Eq. (22) is dimensionally incorrect: the conversion factor from (M/M_sun)(1 kpc/D_obs) to microarcseconds is 9.87098×10⁻⁶, not 9.87098, so the predicted angular diameters are too large by a factor of 10⁶. The section also quotes the M87* mass as (6.5±0.7)×10⁶ M_sun and the distance as 16.8 kpc; the commonly adopted values are (6.5±0.7)×10⁹ M_sun and 16.8 Mpc. These errors make Fig. 13 and the quoted allowed h intervals unreproducible. Use the correct conversion factor, correct mass/distance values, and recompute Fig. 13.
  3. [Sec. II / Fig. 3 and Sec. V] The shadow radius R_d plotted in Fig. 3 is ≈0.05, while the shadow contours in Fig. 2 have radii ≈5M. The manuscript does not define the units of R_d. If Fig. 3 reports an angular radius at r_obs = 100 (≈5/100 rad), then it cannot be substituted into Eq. (22), which requires a shadow radius in units of M. This ambiguity is load-bearing because Eq. (22) and Fig. 13 depend on R_d. Please define R_d consistently and either recompute the EHT comparison with the correct R_d (in units of M) or use the appropriate angular formula.
  4. [Abstract and Sec. V] The statement that EHT observations 'confirm the validity of Horndeski gravity' is not supported by the analysis. Consistency of the predicted shadow diameter with the 1σ/2σ EHT band for some h interval is a necessary but very weak test; the Kerr metric itself also fits these diameter measurements. The paper performs no model comparison, no marginalization over spin, inclination, mass, and distance systematics, and no full image/shape comparison. The conclusion should be weakened to: under the assumption that (1)–(2) is an exact Horndeski solution, EHT shadows place constraints on h.
minor comments (5)
  1. [Throughout] Typos and grammatical errors: 'Schwarszchild' (Sec. II), 'space-dragging' for frame-dragging, 'The red-shift factors' for redshift, and repeated 'the the'. A careful language edit is needed.
  2. [Sec. IV, Fig. 5] The text says Fig. 5 is computed with θ_obs = 80°, but the caption states θ_obs = 70°. Please correct the inconsistency.
  3. [Sec. IV, Table I] The text states that g_max decreases with increasing a and increases with increasing h, but the table shows the opposite trend: for fixed h, g_max grows with a, and for fixed a, g_max decreases with h. Either the table or the text is mislabeled.
  4. [Sec. V] The distance to M87* is quoted as 16.8 kpc; the accepted value is 16.8 Mpc. This is not just a typo in the EHT analysis and must be corrected along with the mass quoted.
  5. [References] Several references are arXiv preprints or self-citations; some may be appropriate, but Refs. [26,73] should be checked carefully to confirm whether they actually prove that the metric (1)–(2) is an exact Horndeski solution.

Circularity Check

0 steps flagged

No significant circularity: EHT constraints are an independent parameter-estimation exercise; heavy self-citation is methodological, not load-bearing.

full rationale

The paper's central advertised result is a constraint on the hair parameter h of the rotating Horndeski black hole from EHT angular-diameter measurements of M87* and Sgr A*. This is not circular: the shadow radius R_d is computed from the assumed metric (Eqs. 1-2) via null geodesics, the angular diameter is obtained from Eq. (22), and the result is compared with externally measured EHT values. No EHT data are used as input to determine h before the comparison; the paper scans h and reads off the allowed range from the 1-sigma and 2-sigma bands. Thus the 'prediction' is not equivalent to a fitted input by construction. The metric itself is imported from Refs. [26,73] rather than re-derived from Horndeski field equations; that is a real correctness/verification risk, but it is not a circularity because those references are external to the present authors and the paper does not define Horndeski gravity in terms of the shadow quantities it later predicts. Numerous self-citations (e.g., Refs. [44,45,55,57,69,72]) appear, but they support the imaging methodology and prior similar studies, not the specific conclusion that EHT data confirm Horndeski gravity. The final sentence 'confirming the validity of Horndeski gravity' is an overinterpretation of a parameter constraint, but overinterpretation is not circular reasoning. No step reduces to its own input by definition or by a self-citation chain, so the circularity score is low.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The paper introduces no new particle, field, or conserved quantity; the hair parameter h is carried over from the cited Horndeski solution. The main ledger burdens are the assumed exactness of the rotating metric, the thin-disk model, and the unverified unit conversion in the EHT angular-diameter formula.

free parameters (4)
  • hair parameter h = not fitted; scanned from 0.001 to 3.1 and loosely bounded by EHT at fixed a=0.5
    The central model parameter whose effect on shadows, disks, and EHT diameter is the paper's subject.
  • spin parameter a = 0.5 fixed for EHT constraints; 0.001 to 0.9 in images
    The EHT constraint depends on choosing a=0.5; no marginalization over spin or inclination is performed.
  • observer inclination theta_obs = 70 or 80 degrees depending on figure
    Shadow shape and the EHT diameter comparison depend on the assumed observer angle, which is fixed by hand.
  • emissivity coefficients rho_1, rho_2 = -1/2, -2
    Adopted from Ref. [42] for the disk emissivity profile; affects intensity maps and lensed-band appearance.
axioms (5)
  • domain assumption The metric in Eqs. (1)-(2) is an exact rotating solution of Horndeski gravity.
    Stated in Sec. II as 'given as follows [26,73]'; no field equations or solution-generating procedure are shown.
  • domain assumption The modified metric admits the same Hamilton-Jacobi separability and Carter constant as Kerr.
    Sec. II Eqs. (3)-(6) use Carter's separability prescription without demonstrating it for the new Delta(r).
  • domain assumption The accretion disk is geometrically and optically thin, equatorial, geodesic, and split into ISCO and plunging regions.
    Sec. IV Eqs. (13)-(17) are adopted from Ref. [52]; no magnetohydrodynamics or radiation transport is included.
  • domain assumption The EHT-reported masses, distances, and shadow diameters are correct inputs.
    Sec. V uses values from Refs. [80,81], but the M87* mass and distance are misprinted in the text.
  • ad hoc to paper The angular diameter formula Eq. (22) correctly converts R_d to microarcseconds.
    As printed the formula lacks the 10^-6 unit conversion, and the relation between the fisheye R_d of Fig. 3 and the dimensionless R_d in Eq. (22) is not stated.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Probing Horndeski Gravity via Kerr Black Hole: Insights from Thin Accretion Disks and Shadows with EHT Observations." pith.science (2026). https://pith.science/paper/ZZ4T6PQO

@misc{pith2026250905803,
  author       = {Pith},
  title        = {Pith review of: Probing Horndeski Gravity via Kerr Black Hole: Insights from Thin Accretion Disks and Shadows with EHT Observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZ4T6PQO}},
  note         = {Machine review of arXiv:2509.05803}
}
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abstract

In this study, we have considered the Kerr-like black hole (BH) model in Horndeski gravity and analyse the visual characteristics of shadow images under two illumination models, such as a celestial light source and a thin accretion disk. To capture the BH shadow images, we utilise a recent fisheye camera model and ray-tracing procedures. In this view, we carefully addressed the influence of the spin parameter $a$ and the hair parameter $h$ on the BH shadow images. The results indicate that for smaller values of $h$, the BH shadow contours shift noticeably towards the right side of the screen, while for larger values of $h$, the nearly circular shadow gradually deforms into a ``D'' shape profile. For a celestial light source, the larger values of $h$ lead to a reduction in the corresponding radius of the photon ring, while the space-dragging effect becomes more pronounced with increasing $a$. We further discuss the distinctive characteristics of images observed in both prograde and retrograde accretion disk scenarios. The results reveal that variations in $h$ significantly affect both the inner shadow and the resulting Einstein ring. Subsequently, we also discussed the distinct features of red-shift configurations on the disk for both direct and lensed images, which are closely related to the accretion flow and the relevant parameters. We also attempt to use the recent observational data from M$87^{\ast}$ and Sgr $A^{\ast}$ and constrain the hair parameter $h$, confirming the validity of Horndeski gravity.

Figures

Figures reproduced from arXiv: 2509.05803 by Chen-Yu Yang, M. Israr Aslam, Rabia Saleem, Xiao-Xiong Zeng.

Figure 1
Figure 1. Figure 1: FIG. 1: Physical behaviour of ∆( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The shadow contours are plotted for fixed values of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The shadow observable [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Optical images of rotating BHs in Horndeski gravity with different values of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Optical images of rotating BHs in Horndeski gravity with different values of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The red-shift factors of direct images of rotating BHs in Horndeski gravity with different values of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The red-shift factors of lensed images of rotating BHs in Horndeski gravity with different values of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The lensing bands of rotating BHs in Horndeski gravity with different values of [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Optical images of rotating BHs in Horndeski gravity for different values of [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The red-shift factors of direct images of rotating BHs in Horndeski gravity for different values of [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The red-shift factors of lensed images of rotating BHs in Horndeski gravity for different values of [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The lensing bands of rotating BHs in Horndeski gravity for different values of [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Plots showing the approximated shadow angular diameter [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗

discussion (0)

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Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Reshaping the inner shadow of a Kerr black hole by a torn accretion disk

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    Torn accretion disks around Kerr black holes erode the inner shadow and create bifurcated, crescent, and multi-ring shadow features driven by sub-disk discontinuities and outer tilt angle.

  2. Reshaping the inner shadow of a Kerr black hole by a torn accretion disk

    gr-qc 2026-04 unverdicted novelty 6.0

    Torn accretion disks around Kerr black holes erode the inner shadow and produce bifurcated, crescent, and multi-order ring morphologies hard to obtain with standard equatorial disks.

  3. Probing Lorentz-violating effects via precession and accretion disk images of a rotating bumblebee black hole

    gr-qc 2026-04 conditional novelty 5.0

    Lorentz violation in a rotating bumblebee black hole suppresses Lense-Thirring precession, increases periastron precession, shrinks the inner shadow, and enhances the lensed ring while leaving the critical curve nearl...

  4. Imaging and Polarimetric Signatures of Konoplya-Zhidenko Black Holes with Various Thick Disk

    astro-ph.HE 2025-10 unverdicted novelty 5.0

    Horizon-scale synchrotron images and polarization maps of Konoplya-Zhidenko black holes with thick disks show expanded photon rings, darker centers, and viewing-angle-dependent asymmetries that vary with the deformati...

  5. Optical images of Kerr-Sen black hole illuminated by thick accretion disks

    astro-ph.HE 2026-04 unverdicted novelty 4.0

    Increasing charge Q shrinks photon rings and central shadows in Kerr-Sen black hole images while spin creates brightness asymmetry; polarization patterns follow lensing and frame dragging.

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.