Strong Gravitational Lensing for Photon Coupled to Weyl Tensor in Kiselev Black Hole

Asif Mahmood; G. Abbas; M. Zubair

REVIEW 3 major objections 6 minor 59 references

Strong Gravitational Lensing for Photon Coupled to Weyl Tensor in Kiselev Black Hole

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a Kiselev black hole, photons coupled to the Weyl tensor follow different lensing paths depending on their polarization, so the photon sphere, deflection angle, and observable images all depend on the coupling α and the parameter σ.

desk verdict Solid standard extension undermined by swapped critical-coupling labels and a shadow section disconnected from the coupling. read the letter →

arxiv 1909.06433 v2 pith:LMLSQVTH submitted 2019-08-06 gr-qc

G. Abbas , Asif Mahmood , M. Zubair This is my paper · ORCID

classification gr-qc PACS 95.30.Sf04.20.Dw04.70.Bw98.62.S

keywords gravitationallensingstrongWeyltensorcouplingphotonpolarizationKiselevblackholesphereshadowdeflectionangle

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 claims that when photons couple to the Weyl tensor, strong gravitational lensing by a Kiselev black hole is not a single universal phenomenon: the photon-sphere radius, the strong-deflection coefficients, the deflection angle, the relativistic image positions, the angular separation of images, and the relative magnifications all split according to photon polarization and shift with the coupling $\alpha$ and the Kiselev parameter $\sigma$. The authors derive two light-cone conditions from the Weyl-corrected Maxwell equation in the geometric-optics limit, encode them in an effective metric, and run the standard strong-deflection program on that metric. The result matters because it turns lensing observables and the black hole shadow into potential probes of a curvature–photon coupling and of the background parameter $\sigma$, rather than assuming they only measure mass. It also extends the Schwarzschild case treated in Ref. [1] to the more general Kiselev spacetime, recovering those results when $\sigma \to 0$.

What carries the argument

The central object is an effective metric of the form $ds^2 = -A(r)\,dt^2 + B(r)\,dr^2 + r^2 W(r)^{-1}(d\theta^2 + \sin^2\theta\, d\varphi^2)$, with $A(r)=1-2M/r-\sigma r$, $B(r)=1/A(r)$, and $W(r)$ equal to $(r^3-8\alpha M)/(r^3+16\alpha M)$ for one polarization and its reciprocal for the other. The mechanism is geometric optics: the Weyl-corrected Maxwell equation gives two light-cone conditions that this metric packages as a single null-geodesic problem, and the photon-sphere equation $W[A'C-AC']+ACW'=0$ together with the strong-deflection expansion of Ref. [27] converts the coupling $\alpha$ into a logarithmic deflection formula with coefficients $\bar{a}$ and $\bar{b}$.

What would settle it

Recompute the matrix equation (26) from the definitions of the polarization vectors $l_\mu$ and $m_\mu$, trace which effective $W(r)$ follows from each light-cone condition, and independently integrate the deflection integral (57) for both $W$ functions; if the branch with increasing $r_{ps}(\alpha)$ and larger deflection is the one the paper labels PPL rather than PPM, the polarization-tagged plots are reversed.

Watch

Extended reading notes

Core claim

For the Kiselev metric $f(r)=1-2M/r-\sigma r$, the paper shows that a photon coupled to the Weyl tensor propagates along null geodesics of an effective metric whose angular part is rescaled by $W(r)^{-1}$; the two photon polarizations correspond to two reciprocal forms of $W$, so one branch's photon sphere shrinks as $\alpha$ grows while the other's grows. This yields a critical window $\alpha_{c2}<\alpha<\alpha_{c1}$, with $\alpha_{c1}=-2\alpha_{c2}$ given by Eq. (55), outside which no photon sphere lies outside the horizon and the strong-deflection formula (67) ceases to apply. In the limit $\sigma \to 0$ the window reduces to $-M^2/2 < \alpha < M^2$. Numerical evaluation then gives polarization-dependent values for the strong-deflection coefficients, the deflection angle, and the observables $\theta_\infty$, $s$, and $r_m$, and the shadow size increases with both $M$ and $\sigma$.

Load-bearing premise

The polarization-specific conclusions rest on the paper's assignment of the label PPL to one of the two light-cone conditions and PPM to the other; if those two branches are interchanged, every polarization-dependent trend reported in the paper would be reversed.

Editorial extensions

If this is right

For either polarization branch, every strong-lensing quantity—photon-sphere radius, impact parameter, deflection coefficients, and observable image positions—depends on $\alpha$ and $\sigma$, so lensing is no longer purely mass-driven.
The two photon polarizations produce different photon-sphere radii, with opposite monotonicity in $\alpha$, so a suitably placed source behind the black hole would show two sets of relativistic images separated by polarization.
The critical window $\alpha_{c2}<\alpha<\alpha_{c1}$ sets a bound: for couplings outside it the photon sphere merges with or disappears inside the horizon, and the strong-deflection logarithmic formula no longer applies.
When $\sigma \to 0$ the results reduce to the Schwarzschild coupled-photon case, with critical couplings $M^2$ and $-M^2/2$, giving a consistency check.
The shadow cast by the Kiselev black hole grows with both mass $M$ and $\sigma$, so a shadow measurement carries information about the background parameter.

Reading between the lines

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

Because the two branches are distinguished only by polarization, polarimetric imaging of the photon ring or of relativistic images could in principle separate the two predicted image sets, making vacuum birefringence a directly observable signature.
The same effective-metric construction could be applied to rotating black holes or other static backgrounds; if the split persists there, the shadow shape or image separation would distinguish a curvature–photon coupling from ordinary geometric effects.
A cheap consistency check before using any of the numerical results is to verify which $W(r)$ belongs to which polarization label, since the paper's plotted trends reverse if the two labels are exchanged.
The existence of a critical window suggests that a strong enough Weyl coupling could suppress the photon sphere entirely; translating $\alpha$ into a physical length scale would tell whether realistic astrophysical black holes sit inside the window.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: this is a routine but mostly careful extension of Chen and Jing's photon-Weyl coupling lensing from Schwarzschild to Kiselev. The derivation of the effective metric and the Bozza strong-deflection machinery is reproduced correctly. The new content is the sigma-dependence of the photon sphere and the lensing coefficients, and those numerical results probably follow from the equations. But the paper has a load-bearing internal inconsistency in the critical couplings, and the shadow section is disconnected from the paper's own coupling setup.

I checked the branch labels. The stress-test is right: the W functions in (47) and (48) do match the light-cone conditions (40) and (41), so there's no swap there. The real problem is in the critical values. At sigma=0, W(2M)=0 for the W in (47), which the paper calls PPL, gives alpha=M^2; the W in (48), called PPM, gives alpha=-M^2/2. But Eq. (55) and the conclusion assign alpha_c1=M^2 to PPM and alpha_c2=-M^2/2 to PPL. So the branch labels on the critical values are exactly reversed. Worse, the PPL branch still has photon-sphere roots for alpha well below -M^2/2 (e.g., alpha=-M^2), so alpha_c2 cannot be an existence threshold for that branch. Every polarization-specific statement built on those critical values is therefore unreliable as written.

The shadow section is a separate weakness. It uses the uncoupled Kiselev metric, not the effective metric with W(r). Eq. (89) for the shadow radius is stated without derivation and looks numerically wrong for moderate sigma; the abstract promises shadows of the coupled photon, and this section doesn't deliver that.

Credit where due: the light-cone derivation, the import of Bozza's formalism, and the numerical work on a-bar, b-bar, and the observables are competent. The paper cites Chen and Jing properly, and it acknowledges Visser's critique of the Kiselev solution. No code or data, so the numerics can't be reproduced, which is a minor but real frustration.

This paper deserves a serious referee—the core machinery is correct and the flaws are fixable—but the current version should not be accepted as is. I'd ask the authors to re-derive the critical couplings with consistent branch labels, check whether alpha_c2 is any kind of threshold on the PPL branch, and either connect the shadow calculation to the coupled effective metric or drop the shadow claim.

For a reading group, maybe—it's a useful example of how a small extension can carry a labeling error through to every conclusion.

Referee Report

3 major / 6 minor

Summary. The paper extends the Chen–Jing treatment of photons coupled to the Weyl tensor from Schwarzschild to Kiselev spacetime. It derives modified light-cone conditions, an effective metric with polarization-dependent function W(r), the null geodesic equation, the photon-sphere equation, critical coupling values for photon-sphere existence, strong-deflection coefficients a-bar and b-bar, deflection angles, and lensing observables. A shadow of the Kiselev black hole is also presented. The central claim is that the photon–Weyl coupling parameter alpha and the Kiselev parameter sigma, together with the photon polarization state, modify the photon sphere, the strong-deflection coefficients, and the observable lensing quantities.

Significance. If the results hold, the paper provides a nontrivial extension of an established formalism (Chen and Jing) to a two-parameter black-hole family and yields falsifiable predictions for strong-field lensing: polarization-dependent photon-sphere radii, deflection angles, and relativistic-image observables. The derivation is analytic with numerical root-finding only for solving existing equations; there is no fitting to data, so the predictions are in principle testable. The paper also correctly cites Visser and Boonserm et al. regarding the status of the Kiselev spacetime. However, the shadow section is disconnected from the coupled-photon effective metric, and the critical-coupling formula is garbled and underived, so the present version does not yet establish all of its central claims.

major comments (3)

[Sec. 4.2 (Shadow of Black Hole), Eqs. (85)-(89)] The shadow is computed from the uncoupled Kiselev metric (11) and the standard Carter-separability geodesic equations (77)-(83). No quantity involving the effective metric W(r) of Eqs. (47)-(48) or the coupling parameter alpha appears in the celestial coordinates or in Eq. (89). Since Sec. 3 establishes that coupled photons propagate on the effective metric (43), the shadow for Weyl-coupled photons should follow from the impact parameter u(r)=sqrt(C/(A W)) and the coupled photon-sphere equation (54). As written, the shadow result is unrelated to the photon-Weyl coupling that is the paper's subject, and the abstract's inclusion of shadows as part of the coupled-photon study is not supported.
[Sec. 3, Eq. (55) and Conclusions] Eq. (55) is presented without derivation and the printed expression is garbled. Read literally, the displayed formula gives (1/8)[1+3(1-8 sigma M)^{1/2}+3(1-8 sigma M)+(1-8 sigma M)^{3/2}]/(8 sigma^3 M)=(1+sqrt(1-8 sigma M))^3/(64 sigma^3 M), which diverges as sigma -> 0, contradicting the Conclusions' statement that alpha_c1 -> M^2. The consistent limit requires a factor (1-sqrt(1-8 sigma M))^3, corresponding to the black-hole horizon r_{h-} of Eq. (14) rather than r_{h+} of Eq. (13). The authors must correct the formula and show how it follows from Eq. (54).
[Sec. 3, Eq. (54), Fig. 1, and Sec. 5] The assignment of critical values to the two polarizations is not justified by the text. At sigma=0, a naive evaluation of Eq. (54) at the horizon r=2M gives the condition W(2M)=0 when W is finite, which returns alpha=M^2 for the W of Eq. (47) (the branch the paper calls PPL) and alpha=-M^2/2 for the W of Eq. (48) (the branch called PPM), opposite to the paper's assignment alpha_c1=M^2 to PPM and alpha_c2=-M^2/2 to PPL. The actual existence boundary is the pole of W at the horizon, not the zero, but this is not stated. Without an explicit derivation showing why the pole condition selects alpha_c1 for PPM and alpha_c2 for PPL, the polarization attribution of every subsequent PPL/PPM statement (Figs. 2-6, Sec. 5) remains ambiguous.

minor comments (6)

[Abstract and throughout] There are numerous grammatical and typographical errors (e.g., "phenomena" in the abstract, "the observables theta_infinity and r_m is the increasing function" in Sec. 4.3); the paper would benefit from a thorough language edit.
[Eq. (55)] The typesetting of Eq. (55) contains OCR artifacts; the exponents "1 2" and "3 2" should be superscripts, and the signs in the numerator should be checked against the derivation.
[Sec. 4.2, Eqs. (85)-(86)] The modified celestial coordinates are introduced without a clear statement of whether the factor (1-sigma) multiplies the whole square root or the impact parameter; please clarify the notation and give the derivation leading to Eq. (89).
[Sec. 3, Fig. 1 caption and text] The text says the photon sphere occurs when "alpha_c1 > alpha > alpha_c2 both for PPM and PPL cases", while the figure caption says "alpha < alpha_c1 for PPM and alpha > alpha_c2 for PPL"; these should be unified to avoid confusion.
[Sec. 4.1, Eq. (67)] The range of alpha over which the effective metric retains a Lorentzian signature outside the horizon is not discussed; for some alpha values below -M^2/2 the function W changes sign in the exterior region, and the physical admissibility of such solutions should be stated.
[Sec. 4.2, Eq. (89)] The domain of validity of Eq. (89), in particular the condition sigma M < 1/8 for the existence of horizons and the choice of photon-sphere branch in the derivation, should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the polarization-dependent lensing results are derived from the action and metric via standard strong-field formalism, not fitted or renamed inputs.

full rationale

The derivation is self-contained: starting from the Weyl-photon action (1) and the Kiselev metric (11)-(12), the paper derives the modified light-cone conditions (40)-(41), the effective metric (43)-(48), the null geodesic equation (51), the photon-sphere condition (54), and then applies Bozza's standard strong-deflection expansion to obtain the deflection coefficients and observables (56)-(68). The critical couplings in Eq. (55) are obtained by imposing the photon-sphere condition at the horizon, i.e. by solving the derived equation (54), not by fitting parameters to data; the numerical lensing plots are evaluations of these derived formulas. The reliance on Chen and Jing [1] and Bozza [27] is attribution to external prior work with stated provenance, and no load-bearing step reduces to an unverified self-citation. Although the paper appears to swap the PPL/PPM assignments of the critical values at the Schwarzschild limit relative to the W-functions in Eqs. (47)-(48), that issue is an internal consistency or correctness concern, not circularity of the kind where a prediction is equivalent to its input by construction. Accordingly, no circular step is identified and the score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. It relies on four input parameters and the four listed assumptions. The photon-Weyl coupling alpha and the Kiselev parameter sigma are model inputs, not fitted quantities. The geometric optics and Bozza formalisms are standard. The Haroon celestial coordinate modification is a domain assumption taken from the literature.

free parameters (3)

alpha (photon-Weyl coupling)
Dimensionful coupling constant in the action (1), treated as an input parameter and varied in plots; not fitted to data.
sigma (Kiselev parameter)
Parameter of the Kiselev metric (11)-(12), treated as an input and varied in plots; not fitted.
M (black hole mass) = 0.5 in numerical plots
Mass scale set to 0.5 in the numerical examples; a dimensionful input, not fitted.

assumptions (4)

domain assumption Geometric optics approximation: photon wavelength is much smaller than the curvature scale but larger than the electron Compton wavelength.
Used to derive the photon equation of motion (10) from the Maxwell equation with Weyl coupling; cited from Drummond and Hathrell [2].
domain assumption Kiselev spacetime (11)-(12) is a valid static spherically symmetric black hole solution.
The paper uses this metric throughout; it acknowledges Visser's correction [54] that the Kiselev solution is not a quintessence perfect-fluid solution, but still adopts it as the background.
standard math Bozza's strong-deflection limit formalism is applicable.
Used to derive the logarithmic deflection formula (67) and the coefficients a-bar and b-bar; this is a standard technique from [27].
domain assumption The modified celestial coordinates of Haroon et al. [59] apply to asymptotically non-flat spacetimes.
Used in Eqs. (85)-(86) for the shadow; no derivation is given in this paper.

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Cite this review

Pith. "Pith review of Strong Gravitational Lensing for Photon Coupled to Weyl Tensor in Kiselev Black Hole." pith.science (2026). https://pith.science/paper/LMLSQVTH

@misc{pith2026190906433,
  author       = {Pith},
  title        = {Pith review of: Strong Gravitational Lensing for Photon Coupled to Weyl Tensor in Kiselev Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMLSQVTH}},
  note         = {Machine review of arXiv:1909.06433}
}

abstract

The ambition of the present work is to highlight the phenomena of strong gravitational lensing and deflection angle for the photons coupling with Weyl tensor in a Kiselev black hole. Here, we have extended the prior work of Chen and Jing \cite{1} for Schwarzschild black hole to Kiselev black hole. For this purpose, the equation of motion for the photons coupled to Weyl tensor, null geodesic and equation of photon sphere in a Kiselev black hole spacetime have been formulated. It is found that the equation of motion of the photons depends not only on the coupling between photon and Weyl tensor, but also on the polarization direction of the photons. There is a critical value of the coupling parameter $\alpha$ for existence of the marginally circular photon orbit outside the event horizon, which depends on the parameters of black hole and the polarization direction of photons. Further, the polarization directions of coupled photon and the coupling parameter $\alpha$, both modify the features of the photon sphere, the angle of deflection and the functions $(\bar{a}$ and $\bar{b})$ for the strong gravitational lensing in Kiselev black hole spacetime. In addition to this, the observable gravitational lensing quantities and the shadows of the Kiselev black hole spacetime are presented in detail.

Figures

Figures reproduced from arXiv: 1909.06433 by the authors.

**Figure 2.** Variation of photon sphere radius for PPM with coupling par [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Variation of photon sphere radius for PPL with coupling par [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Variation of strong deflection limit function ¯a [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Variation of strong deflection limit function [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Variation of deflection angle α(θ) with coupling parameter α for different Kiselev parameter σ for PPM and PPL cases, where M = 0.5. Pr = f(r) −1 r,˙ (71) Pθ = r 2 ˙θ, (72) Pφ = r 2 sin2 θφ˙ = L, (73) where E known as energy and L defines the angular momentum per unit…

**Figure 7.** Figure 7: Shadow cast by Kiselev black hole spacetime at [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Variation of innermost relativistic image [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Variation of angular separation s with coupling parameter [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Variation of relative magnitude rm with coupling parameter α for different Kiselev parameter σ for the cases of PPL and PPM, where M = 0.5. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

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