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arxiv: 2602.16237 · v2 · submitted 2026-02-18 · 🌀 gr-qc · astro-ph.CO· hep-th

Recognition: 2 theorem links

· Lean Theorem

Rotating Black Holes with Primary Scalar Hair: Shadow Signatures in Beyond Horndeski Gravity

Kourosh Nozari , Milad Hajebrahimi , Sara Saghafi , G. Mustafa , Emmanuel N. Saridakis
Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:43 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th
keywords rotating black holesscalar hairbeyond Horndeski gravityblack hole shadowEvent Horizon TelescopeM87*
0
0 comments X

The pith

Rotating black holes with primary scalar hair produce shadows whose size and distortion depend on the sign of the hair parameter Q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper constructs rotating black hole solutions that carry primary scalar hair in beyond Horndeski gravity. It computes the photon regions for these metrics and derives the resulting shadow boundaries. The scalar hair parameter Q changes the shadow in a sign-dependent way: negative Q produces a larger and rounder shadow, while positive Q produces a smaller and more distorted shadow. Applying the EHT measurements of shadow diameter and circularity for M87* maps out the combinations of spin and Q that remain consistent with the data.

Core claim

The constructed rotating metric with scalar hair parameter Q leads to photon regions whose boundaries determine shadows with characteristic modifications. Negative Q enlarges the shadow and reduces its oblateness, while positive Q shrinks the shadow and enhances its distortion. Imposing EHT bounds on the shadow diameter and circularity deviation for M87* restricts the viable (a, Q) parameter space, with stronger constraints for positive Q, yet does not rule out such black holes.

What carries the argument

The scalar hair parameter Q embedded in the rotating black hole metric derived from beyond Horndeski gravity, which alters the effective potential for photon geodesics and thus the shadow boundary.

If this is right

  • Current EHT observations permit a range of negative Q values alongside various spins.
  • Positive Q values are more tightly constrained by the circularity and size measurements.
  • Scalar-hair effects produce shadow deviations on the order of microarcseconds.
  • Next-generation horizon-scale imaging could detect or further limit these deviations.

Where Pith is reading between the lines

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

  • Similar shadow signatures might appear in other strong-field tests such as gravitational wave ringdown or accretion disk images.
  • Higher-precision measurements of Sgr A* could provide independent constraints on the same (a, Q) space.
  • If the metric construction holds, it suggests a pathway to test no-hair violations without invoking secondary hair from matter fields.

Load-bearing premise

That the constructed rotating metric is a consistent solution of the beyond-Horndeski field equations and that the photon-region analysis accurately predicts the observable shadow boundary without significant higher-order corrections or frame-dragging effects omitted in the model.

What would settle it

A measurement of the M87* shadow diameter or circularity deviation that lies outside all predicted values for any combination of spin a and hair parameter Q in the model.

Figures

Figures reproduced from arXiv: 2602.16237 by Emmanuel N. Saridakis, G. Mustafa, Kourosh Nozari, Milad Hajebrahimi, Sara Saghafi.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

The Event Horizon Telescope (EHT) image of M87* provides a direct test of strong-field gravity, measuring an angular shadow diameter $\theta_{d}=42\pm 3~\mu\mathrm{as}$ and a circularity deviation $\Delta C\leq 0.1$. Such observations allow quantitative tests of the Kerr paradigm and of possible deviations from the no-hair theorem. In scalar-tensor extensions of gravity, black holes may possess primary scalar hair, introducing an additional independent parameter beyond mass and spin. In this work, we construct a rotating configuration inspired by black hole solutions with primary scalar hair in beyond Horndeski gravity and analyze their photon regions and shadow formation. We show that the scalar hair parameter $Q$ induces characteristic modifications of the shadow, and in particular negative $Q$ enlarges the shadow and reduces its oblateness, while positive $Q$ shrinks and enhances its distortion. Adopting M87* as a representative case within this framework and imposing the EHT bounds on $\theta_{d}$ and $\Delta C$, we identify the viable $(a,Q)$ parameter space. We find that current observations do not exclude rotating black holes with primary scalar hair, although the allowed region is significantly restricted for $Q>0$. Finally, the scalar-hair-induced deviations are of order $\mathcal{O}(\mu\mathrm{as})$, placing them near the sensitivity threshold of present instruments and within reach of next-generation horizon-scale imaging.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs rotating black hole spacetimes with primary scalar hair in beyond Horndeski gravity by generalizing known static solutions. It computes the photon regions and resulting shadows, demonstrating that the hair parameter Q modifies the shadow diameter and circularity deviation (negative Q enlarges the shadow and reduces oblateness; positive Q shrinks it and increases distortion). Adopting M87* EHT bounds on angular diameter θ_d = 42 ± 3 μas and ΔC ≤ 0.1, the work maps the viable (a, Q) parameter space and concludes that rotating black holes with primary scalar hair remain allowed, though the region is restricted for Q > 0, with deviations of order O(μas).

Significance. If the rotating metrics satisfy the field equations, the results supply concrete, falsifiable predictions for how primary scalar hair alters strong-field observables, extending no-hair tests to beyond Horndeski theories at the sensitivity of current and next-generation horizon-scale imaging. The explicit mapping from Q to shadow observables and the derived (a, Q) constraints constitute a useful benchmark for EHT analyses.

major comments (2)
  1. [§2] §2 (metric construction): The rotating metric is introduced as 'inspired by' the static hairy solutions. No explicit verification is provided that the ansatz satisfies the beyond-Horndeski field equations for simultaneous nonzero a and Q. Because the photon-region and shadow calculations rest on this metric being a solution of the theory, this step is load-bearing and requires either direct substitution into the equations or a reference to a prior derivation that confirms the equations hold.
  2. [§4] §4, photon-orbit conditions: The effective potential for null geodesics is derived from the given metric components, but the text does not address whether frame-dragging or scalar-field stress-energy contributions introduce additional terms that could shift the unstable photon orbit radii beyond the reported O(μas) level. Clarification is needed on the completeness of the geodesic equation used.
minor comments (3)
  1. [Abstract] Abstract and §1: The phrase 'beyond Horndeski gravity' is used inconsistently with and without hyphen; adopt a single convention throughout.
  2. [Figure 4] Figure 4 (shadow contours): Axis labels and the color scale for ΔC are legible but the boundary curves corresponding to the θ_d = 42 ± 3 μas constraint could be highlighted with a distinct linestyle for immediate readability.
  3. [§3.2] §3.2: The definition of the circularity deviation ΔC follows the standard EHT prescription but omits an explicit formula; insert the expression used for the numerical evaluation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will incorporate clarifications and revisions as needed to strengthen the presentation.

read point-by-point responses

  1. Referee: [§2] §2 (metric construction): The rotating metric is introduced as 'inspired by' the static hairy solutions. No explicit verification is provided that the ansatz satisfies the beyond-Horndeski field equations for simultaneous nonzero a and Q. Because the photon-region and shadow calculations rest on this metric being a solution of the theory, this step is load-bearing and requires either direct substitution into the equations or a reference to a prior derivation that confirms the equations hold.

    Authors: We agree that explicit confirmation is essential for the validity of the subsequent calculations. The rotating metric was obtained by generalizing the static hairy solutions via a Boyer-Lindquist-type ansatz that reduces correctly when a=0. In the revised manuscript we will add either a direct (albeit abbreviated) substitution of the ansatz into the beyond-Horndeski field equations for the key components or a reference to a companion derivation that establishes the solution for nonzero a and Q. revision: yes

  2. Referee: [§4] §4, photon-orbit conditions: The effective potential for null geodesics is derived from the given metric components, but the text does not address whether frame-dragging or scalar-field stress-energy contributions introduce additional terms that could shift the unstable photon orbit radii beyond the reported O(μas) level. Clarification is needed on the completeness of the geodesic equation used.

    Authors: Beyond Horndeski gravity is a metric theory; therefore the geodesic equation for photons is determined solely by the spacetime metric. Frame-dragging is already encoded in the g_tφ components, and the scalar-field stress-energy affects only the background metric coefficients, not the form of the geodesic equation itself. We will insert a short clarifying paragraph in the revised §4 stating this explicitly and confirming that no additional terms arise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; shadow modifications computed directly from independent metric parameters

full rationale

The paper introduces the scalar hair parameter Q as an independent input in a rotating metric constructed by ansatz from known static solutions. The photon-region analysis and resulting shadow observables (diameter enlargement for Q<0, distortion enhancement for Q>0) are obtained by direct integration of the metric's null geodesics. These computed signatures are then compared against external EHT data on M87* to delineate the allowed (a,Q) region. No step reduces a reported prediction to a quantity fitted from the same data, nor does any self-citation or uniqueness theorem close a loop that makes the central claim equivalent to its inputs by construction. The derivation remains self-contained against the external observational benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of rotating solutions with primary scalar hair in beyond Horndeski gravity and on the validity of the photon-region calculation for the resulting metric.

free parameters (1)
  • Q
    Scalar hair parameter introduced by the beyond-Horndeski theory; its value is constrained rather than derived from first principles.
axioms (1)
  • domain assumption Beyond Horndeski gravity admits black-hole solutions with primary scalar hair that can be extended to rotating cases
    Invoked to justify the construction of the rotating configuration.

pith-pipeline@v0.9.0 · 5588 in / 1355 out tokens · 46514 ms · 2026-05-15T21:43:42.298492+00:00 · methodology

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Reference graph

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