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REVIEW 4 major objections 5 minor 2 cited by

Influence of Perfect Fluid Dark Matter on Shadow Observables of Yang-Mills modified charged black holes

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A rotating Yang-Mills-charged black hole immersed in perfect-fluid dark matter casts a shadow whose size and shape depend on the dark-matter parameter and the Yang-Mills charge, and current shadow observations already bound these parameters

desk verdict A standard shadow calculation on a metric that is never shown to solve the stated field equations, with stress tensors that contradict the action. read the letter →

arxiv 2509.03507 v1 pith:X75KAPOW submitted 2025-09-03 gr-qc

Md Sabir Ali , Abhishek Negi , Sanjay Pant This is my paper
classification gr-qc MSC 83C5783C1083C15 PACS 04.70.-s95.35.+d
keywords blackholeshadowperfectfluiddarkmatterYang-Millschargerotatingnullgeodesicsenergyemissionratecircularityparameterconstraints
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

The paper extends a Yang-Mills-modified charged black hole by embedding it in a perfect-fluid dark matter (PFDM) environment, producing a rotating spacetime whose metric contains a logarithmic dark-matter term. It then follows null geodesics to the unstable photon sphere and derives the shadow boundary, together with observables such as shadow area, oblateness, circularity deviation, shadow diameter, and energy emission rate. The central claim is that the PFDM parameter α and the Yang-Mills charge q_YM leave opposite, measurable imprints: larger q_YM enlarges the shadow, while increasing α shrinks and distorts it. Using the circularity and diameter bounds from the M87 and SgrA* shadow observations, the paper derives parameter ranges—for instance q_YM ≤ 0.7 and α ≤ 0.35 under the M87 circularity limit—and argues these observables can constrain dark-matter models and test non-Kerr geometries. A sympathetic reader would care because the calculation turns an invisible dark-matter background into a concrete, imageable shadow feature.

What carries the argument

The load-bearing object is the metric function (2.4) with its logarithmic PFDM term (α/r)ln(r/|α|), and its rotating version Δ=r^2-2Mr+Q^2+Q_YM r^{4-4p}+a^2+α r ln(r/|α|). That term modifies the photon effective potential, shifting the unstable photon-sphere radius. The argument is carried by the critical impact parameters η_crit and ξ_crit from the simultaneous conditions V_eff=V'_eff=0: they define the shadow boundary through R_s^2=η_crit+ξ_crit^2, and every subsequent observable—area, oblateness, circularity deviation, diameter, and energy emission rate—is computed from them. The PFDM term is what makes the shadow respond to dark matter rather than to charge or spin alone.

What would settle it

Substitute the metric (2.4) into the field equations (2.2) and verify that the (r,r) and (θ,θ) components vanish identically for the stated Maxwell, Yang-Mills, and PFDM stress tensors; if the logarithmic PFDM term fails any component, the shadow and emission observables refer to a spacetime that is not a solution of the action (2.1).

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the PFDM-modified Yang-Mills black hole metric f(r)=1-2M/r+Q^2/r^2+Q_YM/r^{4p-2}+(α/r)ln(r/|α|), when made rotating, has a photon effective potential whose unstable circular orbit depends explicitly on α and q_YM. The critical impact parameters η_crit and ξ_crit, obtained from V_eff=V'_eff=0, combine into the shadow radius R_s and the celestial coordinates of the shadow boundary. Numerical evaluation shows the shadow enlarges with q_YM and shrinks with α, with the characteristic dent appearing for small q_YM or larger spin. From the shadow boundary the paper computes the area A, oblateness D, circularity deviation ΔC, shadow diameter d_sh, and

Load-bearing premise

The paper assumes, without demonstrating it, that the metric (2.4) actually solves the claimed Einstein field equations with the Maxwell, Yang-Mills, and PFDM stress tensors; the PFDM tensor is written as a single diagonal component rather than a full anisotropic perfect-fluid stress tensor, so this is not automatic.

Editorial extensions

If this is right

  • PFDM imprints on the shadow: larger q_YM enlarges the shadow while increasing α shrinks and distorts it, giving an image-based handle on dark-matter density around a black hole.
  • The M87 circularity bound already restricts the model, roughly to q_YM ≤ 0.7 and α ≤ 0.35.
  • Shadow-diameter measurements are tighter than circularity: for M87 at inclination 17°, the 1σ bounds give a ∈ [0.010, 0.559] and q_YM ∈ [0.010, 0.767].
  • Energy emission rates grow and peak at higher photon frequencies as q_YM or α increases, so the Hawking spectrum carries the same environmental imprint as the shadow.

Reading between the lines

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

  • Beyond the paper, the same logarithmic dark-matter term should also shift photon-ring radii, lensing time delays, and quasinormal-mode frequencies, giving independent observational channels.
  • A natural next step is to tie α to a concrete dark-matter density profile; if that can be done, shadow size becomes a direct probe of halo concentration around black holes.
  • Combining shadow area and oblateness at two observer inclinations could break the degeneracies the paper identifies among (a,q_YM), (a,α), and (a,p).
  • If the solution is genuine, the Yang-Mills exponent p becomes measurable from shadow morphology, turning an internal gauge-action parameter into an observational target.
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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 / 5 minor

Summary. The paper constructs a rotating Yang-Mills-inspired charged black hole surrounded by perfect fluid dark matter (PFDM), starting from a static seed metric (2.4) and applying the Azreg-Aïnou algorithm to obtain the rotating metric (2.5)-(2.6). It then computes null geodesics, shadow shapes and observables, constrains parameters using M87* and SgrA* EHT data, and evaluates the energy emission rate. The central physical output is that the Yang-Mills charge q_YM, PFDM parameter α, and power p produce observable deviations in shadow size, shape, circularity, and emission spectrum. All of these conclusions rest on the claim that the metric (2.4) solves the field equations (2.2) with the stated matter sources.

Significance. If the metric were a genuine solution, the work would be a useful addition to the growing literature on non-Kerr shadows, with a systematic parameter-estimation pipeline and explicit EHT constraints. The Hamilton-Jacobi and shadow-observable formalism is standard, and the contour analysis over (a, q_YM), (a, α), and (a, p) is thorough. However, the foundational step is not established: the Yang-Mills stress tensor written in Eq. (2.2) does not follow from the power-law action (2.1), the PFDM energy-momentum tensor is incompatible with the spherical ansatz, and the metric is obtained by solving only the (t,t) component. Because Sections 3-7 inherit this unsupported metric, the reported signatures cannot be attributed to the claimed physical model.

major comments (4)
  1. [§2, Eq. (2.2)] The Yang-Mills stress tensor is written in a p-independent Maxwell-like form. For L_YM = (Tr F^2)^p, the metric variation produces a factor p (Tr F^2)^{p-1} multiplying the canonical tensor plus a term -g_μν L_YM. Eq. (2.2) omits both the factor and the trace term. Since the metric function (2.4) depends on p through Q_YM/r^{4p-2}, the claimed solution does not follow from the stated field equations unless those additional terms vanish identically, which is not shown.
  2. [§2, Eqs. (2.3)-(2.4)] The text states that after solving the (t,t) component of Eq. (2.2), the metric function (2.4) is obtained. For the ansatz (2.3), the mixed Einstein tensor satisfies G^t_t = G^r_r, so any solution also requires T^t_t = T^r_r. The PFDM tensor T^{(PFDM)}_ν^μ = diag(-ρ,0,0,0) contributes zero to T^r_r, and no verification is given that the Maxwell and Yang-Mills parts satisfy the radial, polar, and azimuthal equations. Thus Eq. (2.4) is reverse-engineered from a single field-equation component, and all subsequent shadow and emission results describe a spacetime not shown to be a solution of the stated theory.
  3. [§7, Eq. (7.4)] The Hawking temperature is asserted without derivation. For a rotating spacetime, the temperature follows from the surface gravity at the event horizon; the expression T_h = T_Kerr + (α r_h - Q^2 - Q_YM)/(4π a^2 r_h + 4π r_h^3) is not derived from metric (2.5)-(2.6). The energy emission rate in Eq. (7.3) depends directly on this temperature, so the reported emission spectra and their parameter dependence are not supported.
  4. [§2, PFDM stress tensor] The tensor T^{(PFDM)}_ν^μ = diag(-ρ,0,0,0) has vanishing radial and tangential pressures, so it is not a perfect-fluid energy-momentum tensor. This conflicts with the paper's terminology and with the cited PFDM model (e.g., Li and Yang, Ref. [13]), which is anisotropic. This is not merely a naming issue: the missing radial pressure is the reason the compatibility condition G^t_t = G^r_r cannot be checked, as noted above.
minor comments (5)
  1. [§6] The sentence beginning 'there is a surge of interest to test other rotating black holes in of the EHT observations become unprecedentedly useful' is garbled and should be rewritten.
  2. [§6] A stray 'pl' appears at the end of the paragraph after 'constraints on the various parameters of the rotating black holes.'
  3. [§7, Eq. (7.2)] The coordinates X_r, X_l, Y_t, Y_b and X_t used in the approximate shadow radius are not defined in the text; define them for clarity.
  4. [Figures 7-8] Figure 8 is captioned for SgrA* but the surrounding text discusses both M87 and SgrA*; check that captions and text agree.
  5. [General] Several typographical errors occur throughout, including 'Schwazrschild', 'espression', and 'osculating'; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; metric is assumed as a model input and EHT data are used only to constrain parameters, not to fabricate predictions.

full rationale

The paper's derivation chain is a model-building exercise: the metric (2.4) is introduced as a solution of the field equations, and shadow/emission observables are then computed from that metric using standard geodesic and black-hole formulas. The EHT measurements of M87* and SgrA* are used only to place bounds on the free parameters (a, q_YM, alpha), not to fit the predictions themselves. The Yang-Mills and PFDM terms are imported from external references, not from self-citations by the present authors, and no uniqueness theorem or prior work by the same authors is used to force the chosen form. The main weakness identified—that only the (t,t) component of the field equations is solved and the full stress-energy content is not verified—is a correctness/derivation gap, not a circular reduction. Because the paper does not define its outputs in terms of its inputs or rename a fitted parameter as a prediction, no circular step is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper's central computation rests on a metric that is asserted rather than derived. The free parameters (q_YM, p, α, a) are scanned and constrained, but not obtained from a deeper theory. No new physical entity is introduced.

free parameters (4)
  • q_YM (Yang-Mills charge) = 0.495-0.654 (1σ) or ≤0.7
    Scanned as a model parameter; constrained by EHT contour plots but not derived from any deeper principle.
  • p (Yang-Mills power) = 0 < p < 3/4
    Chosen by hand to keep Q_YM positive; enters the metric as Q_YM/r^(4p-2).
  • α (PFDM parameter) = α = -0.1 in most plots; bounds like [-0.155,0.081] or ≤0.35
    Controls the dark matter term; varied and constrained via EHT observables.
  • a (spin parameter) = 0.010 to 1.000 in scans
    Free rotation parameter, constrained but not fitted to a theory.
assumptions (6)
  • ad hoc to paper The metric (2.4) solves the field equations (2.2) with the given stress tensors.
    No derivation is shown; the stress tensors are introduced by hand and the Yang-Mills stress tensor is p-independent despite the power-law action.
  • ad hoc to paper The Yang-Mills stress tensor takes the standard p=1 form for the power-law Lagrangian L_YM = Tr(F^2)^p.
    Eq. (2.2) writes T^YM without any dependence on p, which is not justified for p≠1.
  • domain assumption The PFDM stress tensor is diag(-ρ,0,0,0) with ρ = -α/(8πr^3).
    This is a single-component tensor, not the anisotropic perfect fluid normally used for PFDM; sign and consistency with the metric are not checked.
  • standard math The Azreg-Aïnou algorithm gives a valid rotating metric for this seed.
    Standard technique, but application relies on the seed being a solution.
  • standard math The shadow boundary is given by V_eff = V'_eff = 0 and the observables A, D, ΔC, δ, d_sh follow from the standard definitions.
    Standard methods in the shadow literature.
  • ad hoc to paper The Hawking temperature is T_h = T_Kerr + (α r_h - Q^2 - Q_YM)/(4π a^2 r_h + 4π r_h^3).
    Eq. (7.4) is stated without derivation; no surface gravity computation is given.

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Cite this review

Pith. "Pith review of Influence of Perfect Fluid Dark Matter on Shadow Observables of Yang-Mills modified charged black holes." pith.science (2026). https://pith.science/paper/X75KAPOW

@misc{pith2026250903507,
  author       = {Pith},
  title        = {Pith review of: Influence of Perfect Fluid Dark Matter on Shadow Observables of Yang-Mills modified charged black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X75KAPOW}},
  note         = {Machine review of arXiv:2509.03507}
}
read the original abstract

We investigate the influence of perfect fluid dark matter (PFDM) on Yang--Mills--inspired charged black holes, with a particular focus on the resulting modifications to key black hole observables. By embedding a PFDM term into the spacetime geometry, we examine the alterations in shadow morphology, photon geodesics, and the associated energy emission spectra. Our analysis reveals that PFDM induces notable deviations in the shadow size, shape, and circularity, and significantly impacts the stability of circular orbits. Furthermore, the energy emission rate exhibits a strong dependence on both the Yang--Mills charge and the dark matter distribution. These results indicate that environmental effects arising from dark matter can imprint observable signatures on black hole shadows and radiation processes, offering a potential pathway to constrain dark matter models and probe non-Kerr geometries with forthcoming high-precision observations such as those from the Event Horizon Telescope and next-generation interferometers.

Figures

Figures reproduced from arXiv: 2509.03507 by the authors.

Figure 1
Figure 1. The behavior of the g rr = ∆ vs the radial coordinate (r) for a set of values of qYM (Left) and a set of values of α (Right). Each of the plot admits two distinct horizons, namely, the inner horizon (Cauchy horizon) and the outer horizon (black hole event horizon). The relation QY M = − 2 2p−1 4p−3 q 2p Y M tells us that the parametric values of the Yang-Mills parameter can be both positive and negative depending on… view at source ↗
Figure 2
Figure 2. Parametric three-dimensional plot for the parameters (i) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Contour plots showing the behavior of: (i) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: (i) Plot for potential for different value of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Parametric plots of the charged rotating black holes shadow in Yang-Mills theory endowed with a [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Contour plots of shadow area observables [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: , the contour plots of the circularity deviation of M87 black holes in the parameter spaces (a, qYM) and (a, α) [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: (i) The Circularity deviation ∆C for SgrA∗ supermassive black hole in the parameter space (a, qYM) for p = 0.6 and α = −0.1 (left) and (ii) in the parameter space (a, α) for p = 0.6 qYM = 0.5 (right) In [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: On the upper panel, we depict the contour plots of the shadow deviation parameter [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: On the upper panel, we depict the contour plots of the shadow deviation parameter [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: On the upper panel, we depict the contour plots of the shadow deviation parameter [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: On the upper panel, we depict the contour plots of the shadow deviation parameter [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Evolution of the energy emission rate with the photon frequency [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

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Reviewed August 5, 2026 · model on record in the stance chip above.