Bending of Light from Reissner-Nordstr\"om-de Sitter-Monopole Black Hole

M. Haluk Se\c{c}uk; \"Ozg\"ur Delice

REVIEW 3 major objections 4 minor 34 references

Bending of Light from Reissner-Nordstr\"om-de Sitter-Monopole Black Hole

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives a complete weak-field formula for light deflection around a charged black hole with a global monopole and a cosmological constant, separating mass/charge/monopole and Λ contributions.

desk verdict Plausible second-order deflection formulas for RN-dS-monopole, but the key orbit solution is asserted rather than shown; worth a referee with a request for the missing algebra. read the letter →

arxiv 1908.00509 v2 pith:BQ2ZJ2PC submitted 2019-08-01 gr-qc

M. Haluk Seçuk , Özgür Delice This is my paper · ORCID

classification gr-qc MSC 83C1083C57

keywords gravitationallensinglightdeflectionglobalmonopoleReissner-Nordström-deSitterblackholecosmologicalconstantweak-fieldapproximationnullgeodesicssolidangledeficit

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 tries to establish exactly how much a passing light ray is bent by a black hole that combines three ingredients: electric charge, a global monopole, and a cosmological constant. The central claim is that in the weak-field limit the total deflection angle is the sum of a mass/charge/monopole piece and a separate cosmological-constant piece, with the latter given by a polynomial in the impact parameter. The result includes new cross-terms, in particular couplings between the monopole parameter and the cosmological constant, which earlier work did not contain. If the formulas are right, they give an analytic handle for comparing lensing observations against this class of spacetimes and for constraining the monopole strength. The paper also identifies the range of impact parameters and radial positions in which the deflection formula is physically meaningful.

What carries the argument

The load-bearing object is the metric function $\Delta_r$ together with the second-order perturbative orbit $u(\varphi)=1/r(\varphi)$ given in Eq. (16). The orbit is the solution of the modified Binet equation that also satisfies the first-integral orbit equation; the paper discards antisymmetric $\sin\varphi$ terms on symmetry grounds and fixes the remaining arbitrary functions through Eq. (11). The deflection is then computed from the small-angle expression $\psi\approx [\sqrt{\Delta_r/r^2}\,r|dr/d\varphi|^{-1}]_{\varphi=\pi/2}$, which is the mechanism that converts the non-flat metric into a finite bending angle. Expanding this expression along the orbit separates the cosmological-constant contribution from the asymptotically-flat part and produces the polynomial (30).

What would settle it

Integrate the null geodesic equations (4)–(6) numerically for a few parameter sets with $0<b^2<1$ and small $\Lambda$, measure the deflection angle by tracking the asymptote of the orbit, and compare with Eqs. (23) and (30); any disagreement beyond the stated truncation order would show that the perturbative orbit (16) is not the true solution.

Watch

Extended reading notes

Core claim

Working in the equatorial plane of the line element $ds^2 = -\frac{\Delta_r}{r^2}dt^2 + \frac{r^2}{\Delta_r}dr^2 + r^2(d\theta^2+\sin^2\theta\,d\varphi^2)$ with $\Delta_r = b^2 r^2 - 2Mr - \frac{\Lambda}{3}r^4 + Q^2$ and $b^2 = 1 - 8\pi\eta^2$, the paper solves the null-geodesic orbit equation to second order in $M$, $Q^2$, and $\eta'=8\pi\eta^2$. Applying the method of [5] for non-asymptotically flat geometries, it obtains the total deflection angle $\delta\varphi = \delta\varphi_M + \delta\varphi_\Lambda$, with $\delta\varphi_M$ given by Eq. (23) in terms of impact parameter $R$ and equivalently by Eq. (26) in terms of minimum distance $r_{\min}$, and with $\delta\varphi_\Lambda = \Lambda(L_0 + L_1 R + L_2 R^2 + L_3 R^3 + L_4 R^4)$ and explicit coefficients. The monopole contributes both a leading angle-deficit deflection $4\pi^2\eta^2$ and second-order couplings to mass, charge, and the cosmological constant; the effect of the monopole is to increase the bending, while the charge opposes it.

Load-bearing premise

The calculation stands on the unshown premise that the second-order perturbative orbit written in Eq. (16), with all antisymmetric $\sin\varphi$ terms discarded and with arbitrary functions fixed through the first-integral equation, is the exact orbit to that order; if a term is missing or the fixing is inconsistent, Eqs. (23), (26), and (30) all inherit the error.

Editorial extensions

If this is right

Monopole strength $\eta$ adds a deflection $4\pi^2\eta^2$ plus higher-order terms, so a global monopole acts as a magnifying lens, increasing the bending relative to a purely Reissner-Nordström-de Sitter black hole.
The cosmological constant makes a concrete, impact-parameter-dependent contribution to lensing even in the weak-field regime; the polynomial structure in $R$ could in principle be distinguished from mass and charge terms observationally.
The monopole enlarges the photon-sphere radius and shrinks the cosmological horizon, so the allowed parameter window for observing deflection narrows as $\eta$ grows.
The second-order charge terms, though negligible for typical astrophysical charges, can become sizable in scenarios with large effective tidal charge, such as brane-world black-hole analogues.

Reading between the lines

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

A direct numerical ray-tracing test of Eqs. (23) and (30) on the exact geodesic equations would be a cheap way to check whether the discarded antisymmetric terms in the perturbative orbit are truly harmless; the paper does not perform such a check.
Because the solid-angle-deficit structure is shared by radial string-hedgehog configurations, the same deflection polynomial may carry over to those spacetimes, with the monopole parameter reinterpreted—this is an application the paper mentions but does not develop.
The explicit coupling terms between $\Lambda$ and $\eta'$ suggest that measurements of weak lensing by clusters could, in principle, place a bound on the monopole parameter, if systematic uncertainties in mass and charge can be controlled; turning this into an observational strategy would require a separate statistical analysis.

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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

Colleague,

Quick read of Seçuk and Delice. The paper's new content is the combined second-order weak-field deflection angle for RN-dS with a global monopole, including monopole–cosmological-constant couplings (Eq. 30). That is a real extension of the Rindler–Ishak line of work, and if the formulas are right they give a useful reference for lensing in these spacetimes.

What the paper does well: it recovers the standard limits—Einstein deflection, the monopole solid-angle term, the RN charge terms, and the Rindler–Ishak SdS Λ-term. The domain-of-validity discussion (photon sphere, cosmological horizon, impact parameter bounds) is a plus. The comparison with [27] and coordinate rescaling is careful.

The soft spot is exactly what the stress-test note flags. Equation (16) is the foundation for everything, and it is presented as the result of 'some lengthy calculation process' with arbitrary functions fixed using Eq. (11). No verification is shown. The same applies to Eq. (30), which appears after 'some long calculations.' This is not necessarily a fatal flaw—the limits match and the structure is standard—but it means the novel cross-terms (η′ε, η′ν, εν and the L0–L4 coefficients) have no independent check in the manuscript. The claim that [32] has a missing 37 sin φ term inherits the same risk.

One more thing: the paper states that solution (16) satisfies both the Binet equation and the first-integral equation. Given that Eq. (14) is obtained by differentiating Eq. (11), this is not automatic—the integration constants matter. If the authors can provide a clean derivation, or even a short symbolic computation, the result would be much stronger.

Net: this is a serious analytic calculation in a specialized area. It deserves a referee, but the referee should ask for the missing algebra or a reproducible check before accepting. I would not cite it in my own work yet, mainly because the derivation isn't auditable. If it checks out, it becomes a standard reference.

Referee Report

3 major / 4 minor

Summary. The paper studies null geodesics in the Reissner-Nordström–de Sitter–global-monopole spacetime (Eqs. (1)-(2)). The authors solve the modified Binet equation (14) perturbatively to second order in ε=M/R, ν=Q²/R², and η'=8πη², obtaining the orbit solution (16). For Λ=0 they use the Rindler-Ishak method to derive the deflection angle δφ_M in terms of the impact parameter R (Eq. (23)) and of the minimum distance r_min (Eq. (26)). For Λ≠0 they absorb Λ/3 into the definition of R, so the orbit equation takes the Λ=0 form, and they obtain the total deflection 2ψ=δφ_M+δφ_Λ, with δφ_Λ=Λ(L0+L1R+L2R²+L3R³+L4R⁴) given by Eq. (30), to first order in Λ and first order in M, Q², η' where multiplied by Λ. The paper also discusses the domain of validity of the Rindler-Ishak construction in Section 4.1.

Significance. If the derivation is correct, the paper provides a unified weak-field lensing formula for a charged black hole embedded in a global monopole with a cosmological constant, including new cross-terms between the monopole parameter and M, Q², and Λ. The paper's main strengths are the recovery of known limits: the Einstein 4M/R deflection, the standard Reissner-Nordström charge terms, the pure global-monopole solid-angle-deficit term, and the Rindler-Ishak Schwarzschild–de Sitter Λ term. The formulas are falsifiable predictions, and no constants are fitted to data, so there is no circularity. The principal weakness is that the central perturbative orbit (16) and the coefficients (30) are presented as results of unshown algebra, making the new cross-terms difficult to verify independently. The paper does not include a reproducibility artifact such as a computer-algebra notebook.

major comments (3)

[Section 2.3, Eq. (16)] The perturbative orbit solution (16) is the foundation of every subsequent result, but the manuscript does not show how it satisfies both the Binet equation (14) and the first-integral equation (11). Because (16) contains secular terms (φ sinφ, φ² cosφ) and all mixed second-order terms η'ε, η'ν, εν, an error in any of these terms would propagate directly into Eqs. (17), (23), (26), and (30). Please include an explicit substitution (or a supplementary notebook) verifying (16) to the claimed order, and identify the arbitrary functions fixed by Eq. (11). This verification is also needed to support the claim in Section 4 that Eq. (10) of Ref. [32] has a missing 37 sinφ term.
[Section 4, Eq. (30)] The coefficients L0–L4 are the main new Λ-dependent result, but they are introduced after 'some long calculations' and no derivation is provided. Since the expressions contain terms such as Q^6/M^4 and η'^2 Q^4/M^4 that diverge as M→0, it is not possible to check the truncation and the cancellation structure without a reproducible calculation. Please provide a step-by-step derivation of Eq. (28)→Eq. (30), or a computer-algebra file, and state explicitly which terms are kept at each order in M, Q², η', and Λ.
[Section 3, Eq. (26)] The coefficient of the η²Q² term appears to contain a typo. Combining Eq. (23)'s term -12π²η²Q²/R² = -(3π/2)η'Q²/R² with the η' correction that arises when -3πQ²/(4R²) is converted via Eq. (25) gives a net coefficient -3π/4 η'Q²/r_min² = -6π²η²Q²/r_min², whereas Eq. (26) displays -6πη²Q²/r_min². The printed expression appears to be missing one factor of π. Please check and correct this coefficient.

minor comments (4)

[Section 2.3, Eq. (16)] Equation (16) is hard to parse because the placement of denominators is ambiguous in the typeset expression. Please use explicit parentheses, e.g. -ε(cos2φ-3)/(2R) and η'(cosφ+φ sinφ)/(2R), so that each term is unambiguous.
[Section 3, Eqs. (19)-(20)] The expressions (19) and (20) contain terms such as 15πη'ν/(32ε), where a ratio of perturbation parameters appears. It would aid the reader to display these results as ordered expansions in ε, ν, and η' rather than as ratios involving 1/ε.
[Section 4, Eqs. (27)-(30)] The paper redefines the impact parameter via 1/R²=E²/L²+Λ/3 and then uses Eq. (23) inside Eq. (28). Please state explicitly in the final formulas that R denotes the redefined impact parameter, and give the relation to the physical impact parameter b=L/E to first order in Λ, so that the comparison with Refs. [30]-[32] is unambiguous.
[Abstract and Introduction] The phrase 'up to second-order' is used without specifying the perturbation parameters. Please state more explicitly that the Λ=0 deflection is second order in M, Q², and η', while the Λ contribution is first order in Λ and first order in M, Q², and η' in the terms multiplied by Λ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the deflection-angle results follow from the explicitly stated metric, orbit solution, and Rindler-Ishak formula without fitted parameters or self-citation used as evidence.

full rationale

The paper's central results, Eqs. (23), (26), and (30), are obtained by direct substitution: the metric function Δr from Eq. (2), the conserved-energy geodesic equations (4)-(6), the perturbative orbit solution (16), and the Rindler-Ishak bending-angle formula (18) are combined algebraically. No parameter is fitted to data or to a target deflection angle, and no 'prediction' is constructed from a previously fitted quantity. The orbit solution (16) is introduced as a perturbative ansatz whose consistency with Eq. (11) is asserted rather than demonstrated, but that is an unverified derivation premise or correctness risk, not a circular reduction: the final deflection formulas are not assumed as inputs. The redefinition of the impact parameter in Sec. 4, 1/R^2 = E^2/L^2 + Λ/3, is a legitimate change of notation that leaves the Binet equation form unchanged, while the cosmological-constant contribution still enters through the explicit -Λr^2/3 term in the metric function inside Eq. (18), so it is not inserted by hand. Comparisons with prior works such as Refs. [5, 18, 30, 31, 32] are consistency checks rather than calibration steps. The only self-citation, Ref. [13] by one of the authors, is used for the incidental interpretation of the spacetime as a string-hedgehog configuration and plays no role in the deflection-angle derivation. Accordingly, no significant circularity is present.

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

The central claim rests on four imported or assumed ingredients: the effective metric (1)-(2) for the monopole-RN-dS system, the smallness of 8 pi eta^2, the Rindler-Ishak definition of the bending angle, and the correctness of the perturbative solution (16). No free parameters are fitted to data, and no new entities are introduced. The least documented item is the perturbative solution, which the paper asserts without showing the verification.

assumptions (4)

domain assumption The metric (1)-(2) with Delta_r = b^2 r^2 - 2Mr - (Lambda/3) r^4 + Q^2 and b^2 = 1 - 8 pi eta^2 correctly describes the far field of a global monopole swallowed by an RN-dS black hole.
The line element is the starting point of every calculation; if the metric is not the correct far-field description, all formulas inherit the error.
domain assumption The monopole strength is small, 8 pi eta^2 << 1, so the parameter eta' = 8 pi eta^2 can be used as a perturbative expansion parameter around flat spacetime.
The perturbation scheme and all second-order formulas assume eta' is small; the paper states this is the physically significant choice.
domain assumption The Rindler-Ishak formula (18), borrowed from [5], gives the physically meaningful bending angle in non-asymptotically flat spacetimes when evaluated at phi = pi/2.
The entire computation of delta_phi_M and delta_phi_Lambda rests on this interpretation; the paper notes the topic is controversial but does not derive the formula.
ad hoc to paper The particular solution (16), with sin phi terms discarded and arbitrary functions fixed via Eq. (11), is the correct perturbative solution to the orbit equations to second order.
The paper asserts this but does not exhibit the verification. If this solution is incomplete, the deflection angles in Eqs. (23), (26), and (30) are wrong.

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Pith. "Pith review of Bending of Light from Reissner-Nordstr\"om-de Sitter-Monopole Black Hole." pith.science (2026). https://pith.science/paper/BQ2ZJ2PC

@misc{pith2026190800509,
  author       = {Pith},
  title        = {Pith review of: Bending of Light from Reissner-Nordstr\"om-de Sitter-Monopole Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQ2ZJ2PC}},
  note         = {Machine review of arXiv:1908.00509}
}

read the original abstract

We study light deflection from a Reissner-Nordstr\"om-de Sitter black hole in the gravitational monopole background. We first calculate the orbit equation and the contribution of the monopole and black hole parameters to the deflection angle up to second-order for the vanishing cosmological constant case using the Rindler-Ishak method. We also obtain the contribution of the cosmological constant to light deflection in this geometry in the weak field limit using the same method.

Figures

Figures reproduced from arXiv: 1908.00509 by the authors.

**Figure 2.** Graph showing the change of the observational domain (shaded area) of bending of light from a RN-dS [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [27]

Light deﬂection by a rotating global monopole spacetime,

K. Jusuﬁ, M. C. Werner, A. Banerjee, and A. ¨Ovg¨ un, “Light deﬂection by a rotating global monopole spacetime,” Phys. Rev. D , vol. 95, no. 10, p. 104012, 2017

work page 2017

[2] [32]

Contribution of the cosmological constant to the bending of light in kerr–de sitter spacetime,

J. Sultana, “Contribution of the cosmological constant to the bending of light in kerr–de sitter spacetime,” Phys. Rev. D , vol. 88, p. 042003, 2013

work page 2013

[3] [1]

First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole,

E. H. T. Collaboration et al., “First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole,” Astrophys. J. Lett., vol. 875, no. 1, p. L1, 2019

work page 2019

[4] [2]

A determination of the deﬂection of light by the Sun’s gravitational ﬁeld, from observations made at the total eclipse of May 29, 1919,

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deﬂection of light by the Sun’s gravitational ﬁeld, from observations made at the total eclipse of May 29, 1919,”Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character , vol. 220, pp. 291–333, 1920

work page 1919

[5] [3]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects . Cambridge University Press, 2000

work page 2000

[6] [4]

Gravitational ﬁeld of a global monopole,

M. Barriola and A. Vilenkin, “Gravitational ﬁeld of a global monopole,” Phys. Rev. Lett. , vol. 63, no. 4, p. 341, 1989

work page 1989

[7] [5]

Contribution of the cosmological constant to the relativistic bending of light revisited,

W. Rindler and M. Ishak, “Contribution of the cosmological constant to the relativistic bending of light revisited,” Phys. Rev. D , vol. 76, no. 4, p. 043006, 2007

work page 2007

[8] [6]

The relevance of the cosmological constant for lensing,

M. Ishak and W. Rindler, “The relevance of the cosmological constant for lensing,” General Relativity and Gravitation, vol. 42, no. 9, pp. 2247–2268, 2010

work page 2010