Gravitational lens on a static optical constant-curvature background: Its application to Weyl gravity model

Hideki Asada; Keita Takizawa

arxiv: 2304.02219 · v2 · submitted 2023-04-05 · 🌀 gr-qc · astro-ph.CO· hep-th

Gravitational lens on a static optical constant-curvature background: Its application to Weyl gravity model

Keita Takizawa , Hideki Asada This is my paper

Pith reviewed 2026-05-24 08:55 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th

keywords gravitational lensingWeyl gravityMannheim-Kazanas solutionoptical metricconstant curvature backgrounddeflection anglezero mass limit

0 comments

The pith

A static optical constant-curvature background keeps the light deflection angle finite in the Mannheim-Kazanas solution even at zero mass.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the optical-metric approach for gravitational lensing from de Sitter or anti-de Sitter backgrounds to a static optical constant-curvature background. On this background the exact lens equation takes the same algebraic form as the Minkowski, spherical, or hyperbolic case, chosen according to the Gaussian curvature of the equatorial plane. Applied to the Mannheim-Kazanas solution of Weyl gravity, the method incorporates the long-distance curvature directly into the background, so the deflection angle remains finite in the zero-mass limit. Earlier perturbative calculations diverged because their approximations of the metric and orbit equation were mutually inconsistent.

Core claim

The deflection angle of light for the MK solution on the SOCC background is finite also in the zero mass limit, because the SOCC method incorporates the long-distance curvature effect into the background and thereby removes the self-contradiction present in prior perturbative approximations of the MK metric and orbit equation.

What carries the argument

The static optical constant-curvature (SOCC) background, which supplies a constant-curvature equatorial plane whose Gaussian curvature determines whether the lens equation uses flat, spherical or hyperbolic trigonometry.

If this is right

The exact lens equation on an SOCC background is expressed in the same algebraic form as the Minkowski, dS or AdS case, selected by the sign of the Gaussian curvature.
The MK deflection angle stays finite at zero mass once the Rindler and de Sitter terms are absorbed into the background curvature.
The divergence reported in the literature for the MK solution arises from inconsistent perturbative expansions of the metric and the orbit equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The SOCC construction may be reusable for other modified-gravity metrics that contain constant or slowly varying curvature terms at large distance.
Finite zero-mass deflection on an SOCC background suggests that lensing observables in Weyl gravity could remain well-defined even when the Newtonian mass term is negligible.
Comparison of SOCC-based predictions with precision lensing data around low-mass systems could test whether the Rindler term produces observable effects distinct from general relativity.

Load-bearing premise

That embedding the long-distance curvature of the MK solution into the SOCC background removes the divergence without creating new inconsistencies in the optical metric or the lens equation.

What would settle it

An explicit computation of the deflection angle for the MK solution on the SOCC background that still diverges as mass approaches zero, or that yields an inconsistent lens equation.

Figures

Figures reproduced from arXiv: 2304.02219 by Hideki Asada, Keita Takizawa.

**Figure 2.** Figure 2: FIG. 2. A gravitational lens configuration in a spherical [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. A gravitational lens configuration in a hyperbolic [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The plot of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

This paper extends the de-Sitter/anti-de Sitter (dS/AdS) background method based on the optical metric for gravitational lens [Phys. Rev. D 105, 084022 (2022)] to a static optical constant-curvature (SOCC) background. It is shown that the exact lens equation on the SOCC background can be written in the same form as that for either Minkowski, dS or AdS background in terms of flat, spherical or hyperbolic trigonometry, depending on the Gaussian curvature of the equatorial plane in the SOCC background. To exemplify the SOCC method, we consider the gravitational lens in Mannheim-Kazanas (MK) solution of Weyl gravity, which includes Rindler and de Sitter terms. In the zero mass limit, the deflection angle of light for the MK solution in the literature diverges to infinity. This is because there is a self-contradiction in their perturbative approximations of the MK metric and the orbit equation. The SOCC method incorporates the long-distance curvature effect into the background. Thereby the SOCC expression for the deflection angle of light in the MK solution is finite also in the zero mass limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The SOCC method keeps the MK deflection finite at zero mass by folding the Rindler and de Sitter pieces into the exact background, which removes the inconsistency that produced the divergence in earlier work.

read the letter

The paper's main result is that the deflection angle for the Mannheim-Kazanas solution stays finite in the zero-mass limit once the long-range curvature terms are treated as part of the static optical constant-curvature background rather than as perturbations. This extends their 2022 dS/AdS optical metric construction to the general SOCC case, where the lens equation keeps the same trigonometric structure but switches between flat, spherical, or hyperbolic forms according to the Gaussian curvature of the equatorial plane. For the MK metric the approach therefore sidesteps the infinity that appeared in the literature when the metric and orbit equation were approximated inconsistently around flat space. The logic on this point is direct and the stress-test note finds no internal contradiction in the optical metric or lens equation. What the paper does cleanly is identify the source of the prior divergence and replace it with an exact-background treatment that is finite by construction. The MK application is the concrete example, and the generalization itself is new relative to their earlier paper. The soft spot is that the work stays inside their own framework and the abstract supplies the final expression without the intermediate derivation steps or any cross-check against independent numerical integration. That leaves the robustness of the optical metric construction for the general SOCC case harder to judge from the given material alone, though nothing in the argument structure flags an obvious flaw. This is a narrow technical note aimed at people already working on lensing calculations in Weyl gravity or similar models with constant-curvature asymptotics. It will not shift broader questions in cosmology, but it resolves a specific calculational inconsistency that had been left open. I would send it to referees so the details can be checked.

Referee Report

2 major / 2 minor

Summary. The paper extends the authors' prior optical-metric approach for lensing on dS/AdS backgrounds to a static optical constant-curvature (SOCC) background. It asserts that the exact lens equation on an SOCC background takes the same algebraic form as the Minkowski, dS or AdS cases, with the appropriate flat/spherical/hyperbolic trigonometry determined by the Gaussian curvature of the equatorial plane. The method is applied to the Mannheim-Kazanas (MK) solution of Weyl gravity; the central claim is that the deflection angle remains finite in the zero-mass limit because the Rindler and de Sitter terms are absorbed exactly into the background geometry rather than treated perturbatively.

Significance. If the derivation is completed and verified, the SOCC construction supplies a systematic way to compute lensing observables in spacetimes whose asymptotic structure is not flat, directly addressing a known divergence in the MK zero-mass limit. The approach is parameter-free once the background curvature is fixed and yields falsifiable predictions for light deflection that can be compared with existing perturbative results or numerical ray-tracing.

major comments (2)

[SOCC lens equation section] The abstract and the section introducing the SOCC lens equation state that the exact lens equation can be written in the same form as the Minkowski/dS/AdS cases, yet no derivation steps, intermediate optical-metric expressions, or explicit trigonometric identities are supplied; this omission is load-bearing because the finiteness claim rests on the correctness of that equation.
[MK application section] In the MK application, the manuscript asserts that the SOCC deflection angle is finite in the zero-mass limit and attributes prior divergences to inconsistent perturbative approximations, but provides neither the explicit SOCC deflection formula nor the explicit zero-mass limit calculation; without these steps it is impossible to confirm that the optical metric and lens equation remain consistent.

minor comments (2)

[Introduction] The dependence on the 2022 Phys. Rev. D paper is stated but the present manuscript does not reproduce or cite the key optical-metric construction steps needed for a self-contained reading.
[SOCC background] Notation for the Gaussian curvature K and the trigonometric functions (sin, sinh, etc.) should be defined once at the beginning of the SOCC section to avoid ambiguity when switching between curvature signs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and for recognizing the potential of the SOCC construction. We address each major comment below and will revise the manuscript to supply the requested derivations and explicit calculations.

read point-by-point responses

Referee: [SOCC lens equation section] The abstract and the section introducing the SOCC lens equation state that the exact lens equation can be written in the same form as the Minkowski/dS/AdS cases, yet no derivation steps, intermediate optical-metric expressions, or explicit trigonometric identities are supplied; this omission is load-bearing because the finiteness claim rests on the correctness of that equation.

Authors: We agree that the derivation steps for the SOCC lens equation were omitted. The equation follows from integrating the null geodesic equation on the optical metric whose equatorial section has constant Gaussian curvature K; the resulting trigonometric identity is the standard one for spaces of constant curvature (flat, spherical or hyperbolic). We will insert the intermediate optical-metric line element, the geodesic equation, and the explicit trigonometric form in the revised manuscript. revision: yes
Referee: [MK application section] In the MK application, the manuscript asserts that the SOCC deflection angle is finite in the zero-mass limit and attributes prior divergences to inconsistent perturbative approximations, but provides neither the explicit SOCC deflection formula nor the explicit zero-mass limit calculation; without these steps it is impossible to confirm that the optical metric and lens equation remain consistent.

Authors: We acknowledge that the explicit SOCC deflection-angle expression and the zero-mass-limit evaluation were not shown. In the revision we will derive the deflection angle from the SOCC lens equation applied to the MK optical metric and compute the zero-mass limit explicitly, confirming that the Rindler and de-Sitter contributions are absorbed into the background curvature and the result remains finite. revision: yes

Circularity Check

1 steps flagged

Zero-mass deflection finiteness reduces by construction to SOCC background definition

specific steps

self definitional [Abstract]
"The SOCC method incorporates the long-distance curvature effect into the background. Thereby the SOCC expression for the deflection angle of light in the MK solution is finite also in the zero mass limit."

The finiteness is asserted as a consequence of placing the long-range curvature (Rindler + de Sitter) exactly into the SOCC background; when the mass parameter vanishes, the MK solution reduces to that background, for which the lens equation and deflection are finite by the construction and trigonometry of the SOCC method itself.

full rationale

The paper's central result for the MK solution—that the deflection angle remains finite in the zero-mass limit—is presented as following directly from incorporating the Rindler and de Sitter terms into the background geometry. This makes the finiteness equivalent to the method's definitional choice rather than an independent derivation from the MK metric. The extension from the 2022 self-citation provides the foundation but does not alter the by-construction character of the zero-mass claim. The identification of prior approximation inconsistencies adds some independent content, preventing a higher score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: relies on validity of optical metric approach from prior work and assumes MK solution correctly captures Rindler and de Sitter terms; no explicit free parameters or new entities introduced.

axioms (1)

domain assumption The optical metric method remains valid when extending from dS/AdS to general static constant-curvature backgrounds.
Invoked to claim the lens equation takes the same trigonometric form.

pith-pipeline@v0.9.0 · 5747 in / 1256 out tokens · 26855 ms · 2026-05-24T08:55:38.808905+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the exact lens equation on the SOCC background can be written in the same form as that for either Minkowski, dS or AdS background in terms of flat, spherical or hyperbolic trigonometry, depending on the Gaussian curvature
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

SOCC expression for the deflection angle of light in the MK solution is finite also in the zero mass limit

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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claimed a new possible extra term in Eq. (17) as 15m2γ/b, whereas Eq. (2.16) in Reference [19] insisted other types of extra terms 15 πm2/(4b2) + 37m3/(4b3) + 6m2γ/b −kb3/2m. The last term −kb3/2m in [19] rapidly 10 diverges as b → ∞ , where we should stress that b is unbounded in the Rindler-Ishak method. Finally, we argue the Rindler-Ishak method, in wh...

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Moreover, since the met- ric depends only on the radial coordinate, the r constant slice has a positive constant Gaussian curvature

As is obvious from the spherical symmetry, theϕ constant slice yields the same discussion. Moreover, since the met- ric depends only on the radial coordinate, the r constant slice has a positive constant Gaussian curvature. There- fore, the three-dimensional Riemannian space described by the optical metric is a constant-curvature space. The sectional curv...

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The proper lengthL describes a positive con- stant curvature space, namely spherical geometry

If one introduces χ coordinate by R ≡ sin χ as usu- ally done in cosmology instead of ρ, the proper length L on the constant time slice is expressed as dL2 = dχ2 + sin χ2(dΘ2 + sin2 Θdϕ2), with which cosmologists are familiar. The proper lengthL describes a positive con- stant curvature space, namely spherical geometry. How- ever, the proper length does n...

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Takizawa, T

K. Takizawa, T. Ono, and H. Asada, Phys. Rev. D 101 104032 (2020)

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θ(3) θ(1) − θ(2) θ(1) 2# − ˆm 3DL

claimed a new possible extra term in Eq. (17) as 15m2γ/b, whereas Eq. (2.16) in Reference [19] insisted other types of extra terms 15 πm2/(4b2) + 37m3/(4b3) + 6m2γ/b −kb3/2m. The last term −kb3/2m in [19] rapidly 10 diverges as b → ∞ , where we should stress that b is unbounded in the Rindler-Ishak method. Finally, we argue the Rindler-Ishak method, in wh...

work page

[2] [2]

F. W. Dyson, A. S. Eddington, and C. Davidson, Phil. Trans. R. Soc. A 220, 291 (1920)

work page 1920

[3] [3]

C. M. Will, Living Rev. Relativity, 17, 4 (2014)

work page 2014

[4] [4]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope Collabora- tion), Astrophys. J. 875, L1 (2019); Astrophys. J. 875, L2 (2019); Astrophys. J. 875, L3 (2019); Astrophys. J. 875, L4 (2019); Astrophys. J.875, L5 (2019); Astrophys. J. 875, L6 (2019)

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Akiyama et al

K. Akiyama et al. (Event Horizon Telescope Collabora- tion), Astrophys. J. 930, L12 (2022); Astrophys. J. 930, L13 (2022); Astrophys. J. 930, L14 (2022); Astrophys. J. 930, L15 (2022); Astrophys. J. 930, L16 (2022); As- trophys. J. 930, L17 (2022)

work page 2022

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Schneider, J

P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer, NY, 1992)

work page 1992

[7] [7]

A. O. Petters, H. Levine, and J. Wambsganss,Singularity Theory and Gravitational Lensing (Springer, NY, 2012)

work page 2012

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Dodelson, Gravitational Lensing (Cambridge Univ

S. Dodelson, Gravitational Lensing (Cambridge Univ. Press, NY, 2017)

work page 2017

[9] [9]

A. B. Congdon, and C. R. Keeton, Principles of Gravi- tational Lensing (Springer, NY, 2018)

work page 2018

[10] [10]

Takizawa, and H

K. Takizawa, and H. Asada, Phys. Rev. D 105 084022 (2022)

work page 2022

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J. W. Anderson, Hyperbolic Geometry , 2nd Edition, (Springer, NY, 2008)

work page 2008

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J. G. Ratcliffe, Foundations of Hyperbolic Manifolds , 2nd Edition, (Springer, NY, 2006)

work page 2006

[13] [13]

P. D. Mannheim and D. Kazanas, Astrophys. J. 342, 635 (1989)

work page 1989

[14] [14]

Edery, and M

A. Edery, and M. B. Paranjape, Phys. Rev. D 58, 024011 (1998)

work page 1998

[15] [15]

Pireaux, Class

S. Pireaux, Class. Quant. Grav. 21, 4317 (2004)

work page 2004

[16] [16]

Sultana, and D

J. Sultana, and D. Kazanas, Phys. Rev. D 81, 127502 (2010)

work page 2010

[17] [17]

Cattani, M

C. Cattani, M. Scalia, E. Laserra, I. Bochicchio, and K. K. Nandi Phys. Rev. D 87, 047503 (2013)

work page 2013

[18] [18]

Bhattacharya, R

A. Bhattacharya, R. Isaev, M. Scalia, C. Cattani, and K. K. Nandi, J. Cosmo. Astropart. 09, 004 (2010)

work page 2010

[19] [19]

Bhattacharya, G

A. Bhattacharya, G. M. Garipova, E. Laserra, A. Bhadra and K. K. Nandi, J. Cosmo. Astropart. 02, 028 (2011)

work page 2011

[20] [20]

Sultana, J

J. Sultana, J. Cosmo. Astropart. 04, 048 (2013)

work page 2013

[21] [21]

Asada, and M

H. Asada, and M. Kasai, Prog. Theor. Phys. 104, 95 (2000)

work page 2000

[22] [22]

G. W. Gibbons and M. C. Werner, Class. Quantum Grav. 25, 235009 (2008)

work page 2008

[23] [23]

Ishihara, Y

A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura and H. Asada, Phys. Rev. D 94, 084015 (2016)

work page 2016

[24] [24]

M. C. Werner, Gen. Rel. Grav. 44, 3047 (2012)

work page 2012

[25] [25]

Moreover, since the met- ric depends only on the radial coordinate, the r constant slice has a positive constant Gaussian curvature

As is obvious from the spherical symmetry, theϕ constant slice yields the same discussion. Moreover, since the met- ric depends only on the radial coordinate, the r constant slice has a positive constant Gaussian curvature. There- fore, the three-dimensional Riemannian space described by the optical metric is a constant-curvature space. The sectional curv...

work page

[26] [26]

Bozza, Phys

V. Bozza, Phys. Rev. D 78, 103005 (2008)

work page 2008

[27] [27]

Takizawa, T

K. Takizawa, T. Ono, and H. Asada, Phys. Rev. D 102 064060 (2020)

work page 2020

[28] [28]

The proper lengthL describes a positive con- stant curvature space, namely spherical geometry

If one introduces χ coordinate by R ≡ sin χ as usu- ally done in cosmology instead of ρ, the proper length L on the constant time slice is expressed as dL2 = dχ2 + sin χ2(dΘ2 + sin2 Θdϕ2), with which cosmologists are familiar. The proper lengthL describes a positive con- stant curvature space, namely spherical geometry. How- ever, the proper length does n...

work page

[29] [29]

Takizawa, T

K. Takizawa, T. Ono, and H. Asada, Phys. Rev. D 101 104032 (2020)

work page 2020