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The Image of Scalar Hairy Black Holes with Asymmetric Potential

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Pith's one-line read Scalar hairy black hole shadows match EHT measurements only for narrow parameter windows, and the phi1 = 0.5 branch is excluded.

desk verdict Competent imaging pipeline for a hairy black hole, but the EHT constraints are computed outside the solution's own validity domain, so the central claim does not stand as written. read the letter →

arxiv 2411.13049 v1 pith:IJ7CA3XL submitted 2024-11-20 gr-qc

Carlos A. Benavides-Gallego , Eduard Larrañaga This is my paper · ORCID
classification gr-qc
keywords blackholeshadowscalarhairyasymmetricpotentialphotonsphereaccretiondiskimagingEHTobservationsSgrA*M87*
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 sets out to establish whether a class of black holes that evade the no-hair theorem—static, spherically symmetric solutions of Einstein gravity minimally coupled to a scalar field with an asymmetric potential—can produce the same shadow observations as the supermassive black holes at the centers of M87 and the Milky Way. It computes the photon sphere, the ISCO, the shadow radius, and the thin-disk images for these scalar hairy black holes, and compares the predicted angular diameter of the shadow with the EHT measurements for Sgr A* and M87*. The central claim is that the shadow angular diameter grows with the potential parameter $\phi_0$, so the observed sizes select narrow windows for $\phi_0$ for each value of $\phi_1$, and the case $\phi_1 = 0.5$ is excluded because its predicted shadows are too small. If this claim is right, shadow imaging becomes a direct observational test of no-hair violations driven by non-positive scalar potentials.

What carries the argument

The central object is the numerical scalar hairy black hole metric $ds^2 = -N(r)e^{-2\sigma(r)}dt^2 + N^{-1}(r)dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\varphi^2)$, with $N(r) = 1 - 2m(r)/r$ and metric functions $m$, $\sigma$, and scalar field $\phi$ obtained by solving the Einstein-Klein-Gordon equations for the asymmetric potential $V(\phi)$. The argument is carried by the photon-sphere impact parameter $b = r e^{\sigma(r)}/\sqrt{N(r)}$ evaluated at the photon sphere radius $r_{\rm ps}$, where $rN'(r)-2N(r)=0$; this value is the shadow radius $r_{\rm shadow} = b(r_{\rm ps})$, and it converts directly into the angular diameter $\Theta = 2r_{\rm shadow}/D$ used in the EHT comparison. The same metric, in the Hamiltonian formalism, produces the isoradial curves, redshift factor, and Page-Thorne energy flux that make up the synthetic images.

What would settle it

Recompute the shadow angular diameter for a rotating version of the same asymmetric-potential scalar hairy black hole and compare with the EHT ring diameters; if spin moves the predicted values for $\phi_1 = 0.5$ into the observed range, or shifts the $\phi_0$ windows by more than the measurement uncertainties, the paper's parameter constraints would be overturned.

Watch

Extended reading notes

Core claim

On the paper's own terms, a scalar hairy black hole is not just a Schwarzschild black hole in disguise: the asymmetric potential parameters $\phi_0$ and $\phi_1$ change the metric functions $N(r)$ and $\sigma(r)$ near the horizon, which in turn moves the photon sphere and the shadow boundary. The paper's key quantitative result is that for fixed $\phi_1$ the shadow angular diameter $\Theta = 2 r_{\rm shadow}/D$ increases with $\phi_0$, so matching the EHT values $51.8 \pm 2.3\,\mu$as for Sgr A* and $42 \pm 3\,\mu$as for M87* yields finite allowed intervals. For $\phi_1 = 1.0$ the allowed range is approximately $0.89 \leq \phi_0 \leq 0.92$; for $\phi_1 = 2.0$, $3.0$, and $5.0$ the windows are wider, while for $\phi_1 = 0.5$ the angular diameter never reaches the observed lower bounds. The paper also shows that the intensity of the photon ring weakens and the ring radius grows as $\phi_0$ increases, and that the Doppler-boosted side of the disk dominates the bolometric flux distribution.

Load-bearing premise

The constraint analysis assumes that the ring angular diameter measured for Sgr A* and M87* can be equated with the shadow angular diameter $\Theta = 2r_{\rm shadow}/D$ of a static, spherically symmetric, non-rotating scalar hairy black hole.

Editorial extensions

If this is right

  • Accepting the EHT-shadow identification, the observed angular diameters of Sgr A* and M87* confine the asymmetric-potential parameter $\phi_0$ to finite windows for $\phi_1 = 1.0$, $2.0$, $3.0$, and $5.0$.
  • The branch with $\phi_1 = 0.5$ is excluded by both EHT measurements, because its shadow angular diameter stays below the observed lower bounds.
  • Increasing $\phi_0$ pushes the photon sphere and shadow outward and weakens the bright photon ring, so the same scalar hair that enlarges the shadow also makes it harder to see.
  • In the thin-disk images, the approaching side of the disk remains blueshifted and carries the bulk of the bolometric flux, with the overall flux magnitude set by $\phi_0$, $\phi_1$, and the inclination angle.

Reading between the lines

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

  • Beyond the paper: the constraints are computed for a static, non-rotating metric, whereas Sgr A* and M87* are expected to spin; a rotating version of the same hairy solution could shift the shadow size enough to alter the $\phi_0$ windows or even rescue the $\phi_1 = 0.5$ branch.
  • Beyond the paper: the quoted windows depend on the adopted mass and distance of each black hole, so improved astrometric distance or mass measurements for Sgr A* and M87* would directly rescale the allowed parameter ranges.
  • Beyond the paper: the predicted dimming of the photon ring as $\phi_0$ grows is a testable trend, since future higher-resolution interferometric images that measure the ring-to-shadow brightness contrast would discriminate among the allowed parameter windows.
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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 studies the optical appearance of static, spherically symmetric scalar hairy black holes with an asymmetric potential, originally constructed by Corichi et al. (2006). Using a Hamiltonian ray-tracing formalism, the authors compute the effective potential, photon sphere, ISCO, shadow radius, specific intensity for spherical accretion, and isoradial curves, redshift maps, and bolometric flux for a thin equatorial accretion disk in the Page-Thorne framework. The predicted angular shadow diameters are compared with EHT measurements of Sgr A* and M87*, leading to claimed constraints on the potential parameter φ0 for several values of φ1 and the claimed exclusion of φ1=0.5.

Significance. The paper treats a nontrivial numerical black hole solution and produces a broad set of standard observables, which is useful if the underlying solution is correctly parameterized. Its most promoted result, however—the EHT-based constraints on φ0—is undermined by the paper's own domain condition, since almost all quoted allowed intervals lie in the region 2φ0>φ1 where the asymmetric potential is not the WEC-violating one described in Eq. (2). If the results are recomputed on the valid parameter domain, the central constraint claim may disappear entirely. The paper's other sections, based on standard formulas, are more robust but still need revisions.

major comments (4)
  1. [§II and §IV.A/Table I] The parameter constraints in Table I violate the domain condition 0<2φ0<φ1 stated in §II immediately after Eq. (2). For φ1=1.0 the valid range is φ0<0.5, yet Table I lists allowed intervals 0.8967≤φ0≤0.9131 (M87*) and 0.89≤φ0≤0.92 (Sgr A*). For φ1=2.0 the valid range is φ0<1.0, yet the table gives 1.39826≤φ0≤1.4775 and 1.375≤φ0≤1.498. For φ1=3.0 the upper bounds 1.532 and 1.57 exceed the valid value 1.5. For these parameters the potential in Eq. (2) is non-negative and does not violate the weak energy condition, so the numerical hairy solution from Ref. [16] is not the object being studied. The associated claims—that φ0 is constrained for φ1=1,2,3 and that φ1=0.5 is excluded—are therefore unsupported. This is a load-bearing inconsistency in the paper's central comparison.
  2. [§IV.A, Eqs. (36)–(37)] The EHT comparison identifies the measured ring angular diameter (51.8±2.3 μas for Sgr A* and 42±3 μas for M87*) with the shadow diameter Θ=2r_shadow/D computed from a static, spherically symmetric, non-rotating metric. Realistic EHT targets are expected to be rotating, and the observed ring is an emission feature rather than the geometric shadow. The paper does not propagate the mass and distance uncertainties into the bounds in Table I, nor does it discuss how spin would affect the shadow size. This assumption should be stated explicitly and its impact on the constraints assessed, at least via a consistency check with the Kerr prediction for the same mass/distance inputs.
  3. [§II and Data Availability] The metric functions m(r), σ(r), φ(r) are numerical solutions taken from Ref. [17]. The manuscript does not provide the numerical data nor describe the interpolation procedure used to evaluate the metric at arbitrary φ0 between the plotted sample points. The Data Availability Statement says 'This manuscript has no associated data,' which conflicts with the paper's reliance on shared numerical data mentioned in the acknowledgments. Without access to the metric data or the interpolation details, the quantitative results in Table I and Figure 10 cannot be reproduced or checked.
  4. [Captions of Fig. 12 and Table II] Several displayed parameter combinations lie outside the allowed domain 0<2φ0<φ1. The caption of Fig. 12 uses (φ1,φ0)=(0.5,0.3), (1.0,0.6), and (3.0,1.6); Table II uses (3.0,2.0). These do not correspond to valid scalar-hairy-black-hole solutions with the asymmetric potential defined in Eq. (2). The physical interpretation of the corresponding curves, images, and fluxes is therefore not well defined.
minor comments (4)
  1. [§IV.A] The sentence 'for ϕ1=1.0, the possible values for ϕ0 belong to the interval 0.89 ≤ ϕ0 ≤ 0.92 and ≤ ϕ0 ≤ for M87* and Sgr A*' is incomplete; the second interval is missing its entries.
  2. [Throughout] Several typos appear throughout: 'introducation' (§IV.B), 'infintiy' (§IV.B.1), 'Hece' (§IV.B.1), 'Schawarzchild' (§V), and 'Hereido' (§II).
  3. [Conclusion] In the conclusion, the reference 'Fig. reffig:accretionDiskProfiles' is broken and should be replaced with the actual figure number.
  4. [Data Availability Statement] The Data Availability Statement says 'no observational data related to this article' although EHT data are used; this is at least misleading and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shadow, photon-ring, and accretion-disk image predictions are computed from the imported hairy-black-hole metric, and the EHT angular sizes are used as an external post-hoc benchmark rather than as fitted inputs.

full rationale

The derivations in Sections III and IV are self-contained given the numerical metric from Refs. [16,17]: the photon-sphere condition (Eq. 24), shadow radius (Eq. 35), angular diameter (Eqs. 36-37), specific intensity (Eq. 42), and Page-Thorne flux (Eq. 53) all follow algebraically from the line element, without any parameter fitted to the EHT ring sizes. The EHT comparison in Sec. IV.A and Table I is applied after the shadow radii have been computed; it constrains (phi0, phi1) but does not enter the construction of the metric, so it is an external benchmark rather than a circular input. The imported solution is prior work by other authors, and while the paper relies on numerically shared data from Ref. [17], that is ordinary dependence on an external computation, not a self-citation chain or a definitional equivalence. A genuine concern, noted here for completeness, is that the stated potential-domain condition '0 < 2phi0 < phi1' immediately after Eq. (2) is violated by most Table I intervals (e.g., phi1=1.0 with phi0 in [0.8967,0.9131]), and the EHT ring diameter is approximated by the static spherical shadow diameter; however, these are physical and correctness risks about the validity of the parameter space and the benchmark identification, not reductions of a predicted quantity to an input by construction. No equation in the paper is equivalent to its own input, and no load-bearing claim is justified solely by a self-citation. Hence the circularity score is zero.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles or fields are introduced. The paper uses an existing numerical spacetime, standard geodesic and disk models, and external EHT data. The main free parameters are the potential parameters ϕ0 and ϕ1, which are model inputs rather than fitting parameters in a derivation, although the EHT comparison restricts them.

free parameters (2)
  • ϕ0 = scanned over 0.001 to 2.1; constrained intervals e.g. 0.89-0.92 for ϕ1=1.0
    Parameter of the asymmetric scalar potential; controls the shape of the metric and is constrained by EHT angular diameter.
  • ϕ1 = discrete values 0.5, 1.0, 2.0, 3.0, 5.0
    Parameter of the asymmetric scalar potential; different values produce different metric behaviors; ϕ1=0.5 is excluded by the shadow comparison.
assumptions (4)
  • domain assumption The numerical SHBH metric functions m(r), σ(r), ϕ(r) from Refs. [16,17] are correct and can be interpolated over continuous ranges of ϕ0.
    The entire analysis uses these functions without re-solving the field equations.
  • domain assumption The spacetime is static, spherically symmetric, asymptotically flat, and regular on and outside the horizon, with a weak energy condition violation near the horizon.
    This is the established property of the Corichi et al. solution, used to justify the line element and horizon structure.
  • domain assumption The thin accretion disk is geometrically thin, optically thick, and composed of gas on equatorial circular geodesics, following Luminet and Page-Thorne.
    These assumptions underlie the isoradial curves, redshift, and flux calculations in Sec. IV.
  • ad hoc to paper The EHT angular diameter of the ring equals the shadow angular diameter of a static spherical metric.
    This identification is used to derive the ϕ0 constraints in Table I; it is not independently justified for rotating astrophysical black holes.

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Pith. "Pith review of The Image of Scalar Hairy Black Holes with Asymmetric Potential." pith.science (2026). https://pith.science/paper/IJ7CA3XL

@misc{pith2026241113049,
  author       = {Pith},
  title        = {Pith review of: The Image of Scalar Hairy Black Holes with Asymmetric Potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IJ7CA3XL}},
  note         = {Machine review of arXiv:2411.13049}
}
read the original abstract

Black hole accretion disks are a fascinating topic in astrophysics, as they play a crucial role in several high-energy situations. This paper investigates the optical appearance of scalar hairy black holes (SHBHs) with asymmetric potential, a numerical solution obtained in Phys. Rev. D 73, 084002 (2006) and discussed in Phys.Rev.D 108 (2023) 4, 044020. Since the solution is spherically symmetric and surrounded by a thin accretion disk, we base our analysis on the work of J.~P. Lumininet (1979). We discuss the behavior of the effective potential for massive and massless particles, the innermost stable circular orbits (ISCO), and the photon sphere radius for different SHBHs. The study includes the plots of isoradial curves and spectral shifts arising from gravitational and Doppler shifts by considering direct and secondary images. Based on the work of Page and Thorne (1974), we also investigate the intrinsic intensity of radiation emitted by the disk at a given radius, which allows the calculation of the distribution of observed bolometric flux. We use the angular size of the shadow reported by the EHT for Sagittarius A* and M87* to constrain the SHBHs parameters.

Figures

Figures reproduced from arXiv: 2411.13049 by the authors.

Figure 1
Figure 1. FIG. 1. Behavior of the functions [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Metric components [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Effective potential ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Radius of the photon sphere [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Photon trajectories with different values of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effective potential as a function of the radial coordinate [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Radius of the ISCO depending on the parame [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Effective potential for a test particle moving with the angular momentum required at the ISCO as a function of the [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Size of the black hole shadows perceived by an asymp [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Observed specific intensity [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Shadows and photon rings of the SHBH in the case of static spherical accretion. To illustrate the behavior, we chose [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Coordinate system at the image plane of the observer [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Direct images of a thin accretion disk surrounding the SHBH under different values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Redshift [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Observed energy flux of the direct images cast by the SHBH for different values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

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