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REVIEW 3 major objections 5 minor 114 references

Effects of matter with anisotropic pressure on the Fan-Wang regular black hole shadows

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Dressing a Fan-Wang regular black hole in a Kiselev-like anisotropic fluid with negative pressures enlarges its shadow and can create three horizons.

desk verdict Routine but careful shadow catalogue for Fan-Wang plus an anisotropic fluid; the headline EHT constraints are invalid because the cosmological horizon sits at ~10M, far inside Earth's distance. read the letter →

arxiv 2507.18787 v1 pith:S42HYCBN submitted 2025-07-24 gr-qc

Yergali Kurmanov , Orlando Luongo , Daulet Berkimbayev , Kuantay Boshkayev , Talgar Konysbayev , Marco Muccino , Guldana Rabigulova This is my paper · ORCID
classification gr-qc
keywords regularblackholesFan-WangspacetimeanisotropicfluidholeshadowphotonspheresphericalaccretionKiselevsolutionEventHorizonTelescope
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 argues that the Fan-Wang regular black hole, when surrounded by a Kiselev-like anisotropic fluid whose radial and tangential pressures are constant and negative (state parameter $\omega \in (-1,-1/3)$, so the Zel'dovich limit is violated), is described by the metric function $f(r) = 1 - \frac{2M r^2}{(r+l)^3} - \frac{a}{r^{3\omega+1}}$. In this combined spacetime the black hole can develop three horizons, and its photon sphere and shadow impact parameter move outward: for $\omega=-0.7$, $l=4/27$, $a=0.05$, $M=1$, the event horizon grows from $1.51$ to $1.71\,M$, the photon sphere from $2.34$ to $2.56\,M$, and the impact parameter from $4.35$ to $5.73\,M$. The same metric changes the images produced by static and infalling spherical accretion, and the EHT angular diameters of Sgr A* and M87* translate into a narrow allowed range for the normalization constant $a$ when $\omega=-2/3$. If the construction is right, black-hole shadow sizes become a direct probe of a fluid that emulates dark energy without being quintessence.

What carries the argument

The engine is the modified metric function $f(r) = 1 - \frac{2M r^2}{(r+l)^3} - \frac{a}{r^{3\omega+1}}$, built by superposing the Fan-Wang regular black hole (magnetic-charge parameter $l$) and the Kiselev anisotropic fluid (normalization $a$, constant equation-of-state parameter $\omega$). Null geodesics are governed by the effective potential $V_{\mathrm{eff}}(r) = (L^2/r^2) f(r)$. The photon sphere is located by $r f'(r) - 2 f(r) = 0$, the impact parameter is $b_{ph} = r_{ph}/\sqrt{f(r_{ph})}$, and the shadow angular diameter is $\Omega = 2b_{ph}/D$. These three relations carry the argument from the metric to the horizons, photosphere, accretion images, and EHT constraints.

What would settle it

Compute the Einstein tensor of the metric $f(r) = 1 - \frac{2M r^2}{(r+l)^3} - \frac{a}{r^{3\omega+1}}$ and compare it with the sum of the Fan-Wang and Kiselev energy-momentum tensors; any mismatch at any radius would show that the three-horizon and shadow numbers are not a general-relativistic prediction for this fluid. A simpler check is whether the combined fluid's radial and tangential pressures remain constant and equal to $\omega$ times the energy density everywhere.

Watch

Extended reading notes

Core claim

The central claim is that adding the Kiselev anisotropic-fluid term to the Fan-Wang regular black hole yields a valid spacetime whose observable null-geodesic features are calculable and measurably different. Concretely, with $M=1$, $a=0.05$, and $l=4/27$, the state parameter $\omega=-0.7$ produces three horizons and enlarges the event horizon by about 13%, the photon sphere by about 9%, and the shadow impact parameter by about 32% relative to the fluid-free Fan-Wang black hole. The shadow's angular diameter is then $\Omega = 2 b_{ph}/D$, and matching it to the EHT measurements for Sgr A* and M87* restricts $a$ to the intervals $0.090$-$0.103$ and $0.102$-$0.118$ at $\omega=-2/3$ for $l=8/27$. The paper also computes the specific intensity of the shadow for static and infalling spherical accretion and finds that the fluid raises the luminosity of the photon ring while leaving the overall shadow size only moderately changed.

Load-bearing premise

The paper assumes that simply adding the Kiselev anisotropic-fluid term $-a/r^{3\omega+1}$ to the Fan-Wang metric produces an exact solution of Einstein's equations; if that superposition is not a real solution, none of the derived horizons, photon spheres, or shadow sizes follow.

Editorial extensions

If this is right

  • For $M=1$, $a=0.05$, and $\omega=-0.7$, the Fan-Wang black hole with $l=4/27$ acquires a cosmological horizon at $r_c\approx 13.19\,M$ in addition to the inner and event horizons, and $r_h$, $r_{ph}$, and $b_{ph}$ all exceed their fluid-free values.
  • At $\omega=-2/3$ and $l=8/27$, the EHT angular diameters bound the fluid strength to $0.090 < a < 0.103$ for Sgr A* and $0.102 < a < 0.118$ for M87*.
  • For fixed $\omega$, larger $l$ shrinks $r_h$, $r_{ph}$, and $b_{ph}$ and raises the peak intensity; for fixed $l$, more negative $\omega$ enlarges $r_h$ and $b_{ph}$ and pulls the cosmological horizon inward.
  • In both static and infalling spherical accretion, the observed specific intensity peaks at the critical impact parameter $b=b_{ph}$; the infalling model gives lower peak intensities but preserves the ordering of shadow sizes and ring luminosities.
  • The fluid's effect on the shadow is moderate in the representative case, so detecting it requires precision measurements rather than a qualitative change in the image.

Reading between the lines

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

  • Beyond the paper, the same intensity formulas could be used to predict photon-ring brightness asymmetries for a non-spherical or tilted accretion flow, where the $\omega$ dependence of the peak at $b_{ph}$ would show up more sharply than in angular diameter alone.
  • A natural extension is to map the allowed $(a,\omega)$ region for both EHT sources, since the paper reports bounds only at $\omega=-2/3$ but provides the machinery for any $\omega$ in $(-1,-1/3)$.
  • If a future derivation shows the superposition is not an exact Einstein solution, the shadow formulas would still apply to any metric of that functional form, but the physical interpretation of $a$ and $\omega$ as a fluid's equation of state would need to be revised.
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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 / 5 minor

Summary. The paper studies the Fan-Wang regular black hole metric supplemented by a Kiselev-like anisotropic fluid term, writing f(r)=1-2Mr^2/(r+l)^3 - a/r^{3ω+1} with ω in (-1,-1/3). It computes the inner, event, and cosmological horizons, the photon sphere and impact parameter, and then models the shadow and photon ring under static and infalling spherical accretion, comparing angular diameters to EHT results for Sgr A* and M87*. The central claimed results are that the anisotropic fluid with negative pressure modifies the shadow size and brightness, and that EHT data constrain the parameter a (e.g., 0.090<a<0.103 for Sgr A* and 0.102<a<0.118 for M87* at ω=-2/3 and l=8/27).

Significance. If the model and its observer assumptions were valid, the paper would provide a useful extension of regular-black-hole shadow phenomenology to anisotropic equation-of-state matter, a topic of current interest. The geodesic and horizon calculations are standard, and the numerical tables appear internally consistent; the accretion-intensity machinery follows the common framework of Ref. [76]. However, the paper's main observational constraints are undermined by the presence of a cosmological horizon, and the stress-energy interpretation is not derived. The useful part of the paper is the metric-level dictionary between parameters (l,a,ω) and shadow features, which is independent of the EHT comparison.

major comments (3)
  1. [Section III, Eq. (19), Fig. 8] The EHT constraints are invalid as stated. For the parameters used (ω=-2/3, l=8/27, a≈0.09-0.118), the metric has a cosmological horizon at r_c≈1/a≈8-11 M. The Earth distances D=8.127 kpc and D=16.8 Mpc correspond to D/M≈4×10^10 and ≈6×10^10, so Earth is far outside r_c, where f(r)<0 and no static observer exists. Thus Eq. (19), which assumes an asymptotic static observer, cannot be applied. The paper itself says the observer is 'proximate to the cosmological horizon,' which is inconsistent with substituting terrestrial distances into Eq. (18). These ranges and Fig. 8 must be removed or recomputed with an observer at finite radius r_obs<r_c and the correct angular formula, e.g., sin α = b√f(r_obs)/r_obs.
  2. [Section II, Eq. (4)] The superposition ansatz f = f_Fan-Wang + f_Kiselev is not justified. The paper states that the metric function is derived by adding the Kiselev term, but it does not compute the Einstein tensor or the stress-energy tensor. Because the Einstein tensor is nonlinear in f, the total matter content is not the linear sum of the Fan-Wang and Kiselev sources; the claim that the surrounding fluid has constant radial and tangential equations of state is therefore unverified. For the combined metric one expects cross-terms, so the pressure anisotropy may not be of the asserted form. The authors should either derive T_μν for Eq. (4) and check p_r/ρ, p_t/ρ and the energy conditions, or explicitly frame the metric as a phenomenological ansatz and soften the physical interpretation accordingly.
  3. [Section IV, Eqs. (20)-(28), Figs. 9-12] The intensity and shadow images are computed without specifying the observer's radius. In a spacetime with a cosmological horizon at finite r_c, there is no asymptotic region, so 'distant observer' and the unbounded integrals in Eqs. (23) and (28) are not well-defined. The integration limits should depend on r_obs, and as r_obs approaches r_c the observed angular size and intensity profile change because f(r_obs)→0 suppresses the angular radius. Please state r_obs and the integration limits for each impact-parameter class (b<b_ph, b=b_ph, b>b_ph); otherwise the comparison of luminosities in Figs. 10 and 12 is ambiguous.
minor comments (5)
  1. [Abstract and Section I] The statement that negative pressures 'de facto violate the Zel'dovich limit' is incorrect: the Zel'dovich limit p ≤ ρ is satisfied by negative pressures. The relevant statement would be that the fluid violates the strong energy condition (ρ+3p<0 for ω<-1/3). This phrasing appears in the abstract and in Sections I and V.
  2. [Eq. (10a) and Section IV] The sign in ˙t = -E/f differs from the usual ˙t = E/f with E=-p_t; this is harmless where only E² enters, but the statement in Section IV that Eq. (10a) gives k^t=1/b is unclear and should be justified.
  3. [Throughout] The term 'photosphere' is used for the photon sphere; in astrophysics 'photosphere' denotes the visible surface of a star or accretion flow. Consider replacing it with 'photon sphere' or defining the intended meaning explicitly.
  4. [Section IV, Eqs. (23) and (28)] The emissivity normalization is dropped in Eqs. (23) and (28); the plotted intensities are thus in arbitrary units. Please state this explicitly and specify the proportionality constant.
  5. [Fig. 8] The left panel is reproduced from Ref. [76], but the caption does not state the source; the caption should include the reproduction credit.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the shadow, horizon, and intensity results follow directly from the explicit metric ansatz, and the cited prior work is not doing load-bearing circular work.

full rationale

The paper's central results are derived from an explicit metric ansatz, Eq. (4), f(r) = 1 - 2Mr^2/(r+l)^3 - a/r^(3w+1). The horizon radii follow from f(r)=0, the photon sphere from rf'(r)-2f(r)=0, the impact parameter from b_ph = r_ph/sqrt(f(r_ph)), and the shadow intensities from the standard geodesic redshift and proper-length integrals of Eqs. (20)-(28). These are direct consequences of the stated spacetime, not outputs of a fit that is later renamed a prediction. The EHT comparison in Fig. 8 is likewise a constraint procedure: with M, D, w, and l fixed, the angular-diameter relation Eq. (19) is inverted to bracket a, and the paper presents the resulting ranges as restrictions on a rather than as independent predictions. The paper does contain self-citations, notably [98] for the condition that Fan-Wang event horizons exist only for l <= 8/27, and [57, 61, 99] for related regular-black-hole and accretion contexts. However, the l <= 8/27 condition is a simple analytic property of the Fan-Wang lapse function and is not a uniqueness theorem or an unverified imported result; it is parameter-free and externally checkable, so it does not constitute load-bearing circularity. The weakest assumption in the paper, that the Kiselev anisotropic-fluid term can simply be added to the Fan-Wang lapse function to form a valid solution, is a physical-validity concern rather than a circularity concern: the paper does not derive the combined stress-energy tensor from a Lagrangian or verify the Einstein equations, but this is a correctness risk, not a reduction of the output to the input. For the same reason, the skeptical note about the observer distance and the cosmological horizon is an applicability concern for Eq. (19), not a circular step. Overall, the derivation chain is self-contained once the metric ansatz is accepted, and no prediction is equivalent to an input by construction.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim depends on three hand-chosen parameters (a, omega, l) and on two physical assumptions: the Fan-Wang metric's validity and the unproven superposition of the Kiselev fluid. No new particle, field, or conserved quantity is introduced; the 'quintessence-like' fluid is the known Kiselev anisotropic matter.

free parameters (3)
  • a = 0.05 (main runs); EHT bounds 0.090-0.103 (Sgr A*) and 0.102-0.118 (M87*) for omega=-2/3
    Normalization constant of the Kiselev-type term in Eq. (4). Chosen by hand to produce three horizons; constrained by EHT angular diameters in Sect. IV.
  • omega = -0.4, -0.5, -0.6, -0.7, -0.8 (scanned)
    Constant equation-of-state parameter for the anisotropic fluid. Scanned to show sensitivity; main figures use -0.5 and -0.7.
  • l = 0, 2/27, 4/27, 6/27, 8/27 (scanned)
    Magnetic-charge-type regularization parameter of the Fan-Wang metric. Values below the extremal bound l<=8/27 are chosen for comparability.
assumptions (3)
  • domain assumption The Fan-Wang metric f(r) = 1 - 2M r^2/(r+l)^3 is a valid regular black hole solution.
    Adopted from Refs [95-99]; not re-derived in this paper.
  • ad hoc to paper The total stress-energy tensor is the linear sum of the Fan-Wang source and the Kiselev anisotropic fluid, permitting f(r) = 1 - 2M r^2/(r+l)^3 - a/r^(3 omega + 1).
    Eq. (4) is stated as a metric ansatz without a derivation from field equations; the fluid's stress tensor is inferred from the metric.
  • domain assumption EHT angular-diameter measurements for Sgr A* and M87* can be interpreted through the static spherical accretion shadow model.
    Used in Sect. IV to constrain a; ignores disk geometry, spin, and more realistic accretion models.

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Pith. "Pith review of Effects of matter with anisotropic pressure on the Fan-Wang regular black hole shadows." pith.science (2026). https://pith.science/paper/S42HYCBN

@misc{pith2026250718787,
  author       = {Pith},
  title        = {Pith review of: Effects of matter with anisotropic pressure on the Fan-Wang regular black hole shadows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S42HYCBN}},
  note         = {Machine review of arXiv:2507.18787}
}
read the original abstract

We here investigate the consequences of an exotic fluid, exhibiting negative radial and tangential pressures, \emph{de facto} violating the Zel'dovich limit, on a regular solution that easily generalizes the Schwarzschild black hole. More precisely, we focus on the regular Fan-Wang spacetime, computing how the black hole shadow images, surrounded by the quoted fluid, is modified through the presence of \emph{negative} equations of state for the two pressure components. Even though quite different from quintessence, we consider constant radial and tangential equations of state with the aim of emulating, but not reproducing, dark energy effects. Moreover, we explore the main properties of infalling spherical accretion flows and, accordingly, the influence of the equations of state on the horizons, photosphere, and impact parameter of the Fan-Wang black hole. Afterwards, we examine the luminosities of the shadow and the photon ring in two distinct spherically accretion flows, as well as the observed specific intensity of the shadow itself. Last but not least, we physically interpret the impact of negative pressures on our findings and discuss possible extensions to the isotropic case.

Figures

Figures reproduced from arXiv: 2507.18787 by the authors.

Figure 1
Figure 1. FIG. 1: The mass of the BH as a function of the radial coor [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. depicts the behavior of the function f(r) with M = 1, a = 0.05, and ω = −0.7. The f(r) function has three real roots, which correspond to r−, rh, and rc. Tab. II shows that increasing the absolute value of the ω parameter reduces the difference between the event and cosmological horizons. ω=-0.7 0 5 10 15 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r M r-=0.4610 rh=0.8558 rc=13.2648 Mmax=2.28145 Mmin=0.97109 M=1 FIG. 1: The ma… view at source ↗
Figure 3
Figure 3. FIG. 3: The variation of the effective potential with respect [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The effective potential [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The dependence of the photon radius on the param [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Photon sphere radius as a function of parameter [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The polar plots depict the paths of light rays surroun [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The relationship between the parameter [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The observed intensity [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Shadows and photon rings for static spherical accre [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The observed intensity [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Shadows and photon rings for infalling spherical ac [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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