arxiv: 2601.06864 · v2 · submitted 2026-01-11 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Boundary-only weak deflection angles from isothermal optical geometry

Ali \"Ovg\"un , Reggie C. Pantig

Authors on Pith no claims yet

Pith reviewed 2026-05-16 15:38 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords weak deflection angleGauss-Bonnet theoremoptical geometryisothermal coordinatesfinite distancegravitational lensingblack hole

0 comments

The pith

Weak gravitational deflection angles reduce to boundary integrals evaluated on a flat reference ray.

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

The paper develops a method to compute weak deflection angles of light in gravitational fields at finite distances using only boundary data. By introducing isothermal coordinates on the two-dimensional optical manifold, the Gaussian curvature term in the Gauss-Bonnet theorem is converted into a boundary contribution through integral identities. This allows the leading deflection to be expressed as one-dimensional integrals along a straight reference path, with finite distance effects captured solely by the positions of source and receiver. The approach is shown to be independent of the choice of normalization for the isothermal radius. Validation is performed on standard black hole spacetimes including Schwarzschild, Reissner-Nordström, and Schwarzschild-de Sitter.

Core claim

The leading weak deflection angle is obtained by evaluating elementary one-dimensional integrals on a flat reference ray in the conformal plane, where finite distance dependence enters only through the endpoint coordinates, and the method is invariant under residual normalization freedom of the isothermal radius.

What carries the argument

Isothermal coordinates on the equatorial optical manifold, in which the optical metric is conformal to flat and the Gaussian curvature is the Laplacian of the conformal factor, turning the area integral of Gauss-Bonnet into pure boundary terms via Green identities.

If this is right

Finite-distance weak deflection angles can be computed without performing area integrals over the optical manifold.
The same boundary-only formula applies to both asymptotically flat and non-asymptotically flat spacetimes.
Physical observables remain unchanged under changes in the normalization of the isothermal radius.
Leading charge corrections and cosmological constant contributions are recovered directly from endpoint data.

Where Pith is reading between the lines

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

Similar coordinate choices might simplify strong deflection calculations or higher-order post-Newtonian expansions.
The method could be extended to axisymmetric spacetimes or include plasma effects by modifying the optical metric accordingly.
Direct comparison with ray-tracing simulations in numerical spacetimes would test the accuracy beyond the weak-field limit.

Load-bearing premise

The equatorial optical manifold must remain two-dimensional and admit globally isothermal coordinates in which the Gaussian curvature reduces exactly to the Laplacian of the conformal factor, with all closure terms controlled.

What would settle it

A direct comparison of the derived deflection angle for the Schwarzschild metric at finite distance with the known exact weak deflection formula would fail if the boundary-only reduction misses any contribution.

read the original abstract

We develop a boundary only method for computing weak gravitational deflection angles at finite source and receiver distances within the Gauss-Bonnet theorem formulation of optical geometry. Exploiting the fact that the relevant equatorial optical manifold is two dimensional, we introduce isothermal (conformal) coordinates in which the optical metric is locally conformal to a flat reference metric and the Gaussian curvature reduces to a Laplacian of the conformal factor. Such an identity converts the curvature area term in the Gauss-Bonnet theorem into a pure boundary contribution via Green/Stokes-type relations, yielding a deflection formula that depends only on boundary data and controlled closure terms. The residual normalization freedom of the isothermal radius is isolated as an additive freedom in the conformal factor and is shown to leave physical observables invariant, eliminating the need for orbit dependent calibration prescriptions. We explicitly implement the boundary only formalism in weak deflection, where the leading bending reduces to elementary one-dimensional integrals evaluated on a flat reference ray in the conformal plane, with finite distance dependence entering solely through endpoint data. We validate the construction by reproducing finite distance weak deflection for Schwarzschild, deriving the leading finite distance charge correction for Reissner-Nordstr\"om, and applying the same boundary only framework to the Kottler (Schwarzschild-de Sitter) geometry as a representative non-asymptotically flat test case, recovering the standard finite distance expansion including the explicit $\mathcal{O}(\Lambda)$ and mixed $\mathcal{O}(\Lambda M)$ contributions to the total deflection angle.

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 paper reduces weak deflection to boundary integrals alone by switching the optical metric to isothermal coordinates and applying Green's theorem to the curvature term.

read the letter

The main thing to know is that this work converts the area integral of Gaussian curvature in the Gauss-Bonnet optical approach into pure boundary contributions. By placing the equatorial optical metric in isothermal form, the curvature becomes the Laplacian of the conformal factor, so Green's theorem moves everything to the endpoints along a flat reference ray. Finite-distance dependence then enters only through source and receiver positions, and the additive normalization freedom in the conformal factor cancels in the final angle. That step looks new relative to earlier Gauss-Bonnet lensing papers and removes the need for orbit-dependent calibration. They then work out the leading weak-deflection integrals explicitly and test the formula on Schwarzschild, Reissner-Nordström, and Kottler. The Schwarzschild case recovers the standard finite-distance expansion, the RN case adds the expected charge correction at leading order, and the Kottler case produces both the pure Lambda term and the mixed Lambda-M piece. Those checks are direct and reassuring. The soft spots are limited. The construction stays inside static spherical symmetry and the weak-deflection regime, so the reference-ray approximation and global existence of isothermal coordinates are not tested outside those conditions. No higher-order terms or error bounds appear, which is reasonable for a methods paper but leaves the practical range of accuracy for later work. This is aimed at people who compute analytic lensing expressions in black-hole spacetimes and want a shortcut that avoids full geodesic integration, especially when finite distances or a cosmological constant matter. A reader already comfortable with optical geometry will pick up the technical reduction quickly. The algebra is consistent with the checks, so the paper deserves a serious referee. I would send it for review; the boundary reduction is a useful technical step that could be cited in follow-up calculations.

Referee Report

1 major / 2 minor

Summary. The paper develops a boundary-only method for computing weak gravitational deflection angles at finite source and receiver distances by applying the Gauss-Bonnet theorem to the equatorial optical manifold after introducing globally isothermal coordinates. In these coordinates the Gaussian curvature reduces to a Laplacian of the conformal factor, allowing the curvature area integral to be converted exactly into boundary contributions via Green's theorem. The residual normalization freedom in the isothermal radius is shown to cancel in physical observables. The leading weak-deflection bending is reduced to elementary one-dimensional integrals evaluated along a flat reference ray, with finite-distance effects entering only through endpoint data. The construction is validated by recovering the known finite-distance weak-deflection expansions for Schwarzschild, Reissner-Nordström (including the leading charge correction), and Kottler (Schwarzschild-de Sitter) geometries, including explicit O(Λ) and O(ΛM) terms.

Significance. If the central reduction holds, the method supplies a streamlined, parameter-free route to finite-distance deflection angles that depends only on boundary data and controlled closure terms, removing the need for orbit-dependent calibration. The explicit recovery of standard results across asymptotically flat and non-asymptotically flat spacetimes, together with the use of standard 2D Riemannian identities (Gauss-Bonnet plus Laplacian identity), provides a reproducible check on the formalism and could simplify calculations in gravitational lensing and light-propagation studies. The invariance proof under constant shifts of the conformal factor is a clear technical strength.

major comments (1)

Applications sections: the manuscript states that the leading bending reduces to explicit one-dimensional integrals on the flat reference ray and that these recover the known O(Λ) and O(ΛM) terms for Kottler, yet the explicit integral expressions and any accompanying error estimates are not displayed. Including these expressions (or an appendix tabulating the integrands and their evaluated forms) is necessary to allow independent verification of the finite-distance corrections.

minor comments (2)

The invariance of observables under additive shifts of the conformal factor is asserted but would be strengthened by a short explicit calculation showing the cancellation inside the boundary integrals.
Notation for the isothermal radius and the reference flat metric could be introduced in a dedicated short subsection to improve readability for readers unfamiliar with optical geometry.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the constructive comment on the applications section. We address the point below and have revised the manuscript to incorporate the requested additions.

read point-by-point responses

Referee: Applications sections: the manuscript states that the leading bending reduces to explicit one-dimensional integrals on the flat reference ray and that these recover the known O(Λ) and O(ΛM) terms for Kottler, yet the explicit integral expressions and any accompanying error estimates are not displayed. Including these expressions (or an appendix tabulating the integrands and their evaluated forms) is necessary to allow independent verification of the finite-distance corrections.

Authors: We agree that displaying the explicit integrands would enhance independent verification. In the revised manuscript we add a dedicated appendix that tabulates the one-dimensional integrands for the Kottler weak-deflection contributions (including the separate O(Λ) and mixed O(ΛM) pieces), shows their evaluation along the flat reference ray, and supplies the resulting closed-form expressions together with the leading error estimates inherent to the weak-field truncation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard theorems

full rationale

The paper's central construction applies the Gauss-Bonnet theorem to the equatorial optical manifold in globally isothermal coordinates, converting the Gaussian curvature integral into boundary contributions via the Laplacian identity and Green's theorem. This is a direct consequence of any 2D Riemannian metric on a simply-connected domain and does not depend on quantities defined by the authors themselves. The invariance of observables under additive shifts in the conformal factor follows immediately from the scaling behavior of curvature (inverse square) and area (square) under constant rescaling of the optical metric. Finite-distance dependence enters only through explicit endpoint data on the reference ray, with no fitted parameters or self-referential predictions. Validation recovers known weak-deflection expansions for Schwarzschild, Reissner-Nordström, and Kottler without introducing new calibrations. No load-bearing step reduces by construction to an input defined within the paper; the result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the mathematical identity that the Gaussian curvature of a 2D conformal metric equals the Laplacian of the conformal factor, together with the standard application of the Gauss-Bonnet theorem to the optical manifold; no new free parameters or postulated entities are introduced beyond the background spacetime metrics.

axioms (2)

standard math Gauss-Bonnet theorem holds for the optical manifold with the usual topological contribution from the geodesic curvature at the boundary
Invoked to equate total deflection to the integrated Gaussian curvature plus boundary terms
standard math Any 2D Riemannian metric admits local isothermal coordinates in which the metric is conformal to the flat metric
Used to reduce the curvature integral to a Laplacian that becomes a boundary integral via Green/Stokes identities

pith-pipeline@v0.9.0 · 5561 in / 1472 out tokens · 39322 ms · 2026-05-16T15:38:52.828703+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 unclear

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

In isothermal coordinates the optical metric is conformal to a flat reference metric, and this special structure collapses the intrinsic curvature to a Laplacian acting on the conformal factor... K = −e^{-2φ}(∂_u²φ + ∂_v²φ)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

the curvature area term in the Gauss-Bonnet theorem into a pure boundary contribution via Green/Stokes-type relations

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

70 extracted references · 70 canonical work pages · 25 internal anchors

[1]

the spacetime lapse function (or metric functions) definingg tt and the induced equatorial optical metricg opt ab

work page
[2]

(III.16) (including the choice of integration constants and the asymptotic/normalization convention)

the isothermal radial map ρ(r)determined by Eq. (III.16) (including the choice of integration constants and the asymptotic/normalization convention)

work page
[3]

the conformal factorφ(ρ)enteringdℓ 2 =e 2φ(dρ2 +ρ 2dϕ2)

work page
[4]

the Gaussian curvatureK(or equivalently∆φ) and the resulting boundary representation of RR D K dS

work page
[5]

the closing curve choice CΓ and the explicit evaluation of its contribution (including the orientation and the cancellation withΦ RS when applicable)

work page
[6]

p 1−b 2u2 R uR + p 1−b 2u2 S uS # + ΛM b 6

the weak-field/weak-deflection approximation used (e.g. straight-line zeroth-order ray) together with the order counting inM/b,Q 2/b2,Λb 2, etc. 21 A. Schwarzschild spacetime: isothermal reduction and finite distance deflection We reconsider the Schwarzschild spacetime with mass parameter M > 0in standard coordinates( t, r, θ, ϕ)and signature (−+,+,+). In...

work page
[7]

Claudel, K

C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, The Geometry of photon surfaces, J. Math. Phys.42, 818 (2001), arXiv:gr- qc/0005050

work page arXiv 2001
[8]

Perlick, Gravitational lensing from a spacetime perspective, Living Rev

V. Perlick, Gravitational lensing from a spacetime perspective, Living Rev. Rel.7, 9 (2004)

work page 2004
[9]

P. V. P. Cunha and C. A. R. Herdeiro, Shadows and strong gravitational lensing: a brief review, Gen. Rel. Grav.50, 42 (2018), arXiv:1801.00860 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[10]

S. L. Adler and K. S. Virbhadra, Cosmological constant corrections to the photon sphere and black hole shadow radii, Gen. Rel. Grav.54, 93 (2022), arXiv:2205.04628 [gr-qc]

work page arXiv 2022
[11]

K. S. Virbhadra and G. F. R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D62, 084003 (2000), arXiv:astro-ph/9904193

work page internal anchor Pith review Pith/arXiv arXiv 2000
[12]

Time Delay in Black Hole Gravitational Lensing as a Distance Estimator

V. Bozza and L. Mancini, Time delay in black hole gravitational lensing as a distance estimator, Gen. Rel. Grav.36, 435 (2004), arXiv:gr-qc/0305007

work page internal anchor Pith review Pith/arXiv arXiv 2004
[13]

K. S. Virbhadra, Relativistic images of Schwarzschild black hole lensing, Phys. Rev. D79, 083004 (2009), arXiv:0810.2109 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[14]

Gravitational Lensing by Black Holes

V. Bozza, Gravitational Lensing by Black Holes, Gen. Rel. Grav.42, 2269 (2010), arXiv:0911.2187 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[15]

K. S. Virbhadra and G. F. R. Ellis, Gravitational lensing by naked singularities, Phys. Rev. D65, 103004 (2002)

work page 2002
[16]

K. S. Virbhadra and C. R. Keeton, Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities, Phys. Rev. D77, 124014 (2008), arXiv:0710.2333 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[17]

K. S. Virbhadra, Distortions of images of Schwarzschild lensing, Phys. Rev. D106, 064038 (2022), arXiv:2204.01879 [gr-qc]

work page arXiv 2022
[18]

K. S. Virbhadra, Conservation of distortion of gravitationally lensed images, Phys. Rev. D109, 124004 (2024), arXiv:2402.17190 [gr-qc]

work page arXiv 2024
[19]

Strong gravitational lensing for black hole with scalar charge in massive gravity

R. Zhang, J. Jing, and S. Chen, Strong gravitational lensing for black holes with scalar charge in massive gravity, Phys. Rev. D 95, 064054 (2017), arXiv:1805.02330 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[20]

Strong Gravitational Lensing by a Charged Kiselev Black Hole

M. Azreg-A ¨ ınou, S. Bahamonde, and M. Jamil, Strong Gravitational Lensing by a Charged Kiselev Black Hole, Eur. Phys. J. C 77, 414 (2017), arXiv:1701.02239 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[21]

Gravitational lensing in the strong field limit

V. Bozza, Gravitational lensing in the strong field limit, Phys. Rev. D66, 103001 (2002), arXiv:gr-qc/0208075

work page internal anchor Pith review Pith/arXiv arXiv 2002
[22]

Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime

N. Tsukamoto, Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime, Phys. Rev. D95, 064035 (2017), arXiv:1612.08251 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[23]

G. W. Gibbons and M. C. Werner, Applications of the Gauss-Bonnet theorem to gravitational lensing, Class. Quant. Grav.25, 235009 (2008), arXiv:0807.0854 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[24]

C. B. Allendoerfer and A. Weil, The gauss-bonnet theorem for riemannian polyhedra, Transactions of the American Mathematical Society53, 101 (1943)

work page 1943
[25]

Takizawa, T

K. Takizawa, T. Ono, and H. Asada, Gravitational deflection angle of light: Definition by an observer and its application to an asymptotically nonflat spacetime, Phys. Rev. D101, 104032 (2020), arXiv:2001.03290 [gr-qc]

work page arXiv 2020
[26]

Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem

A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada, Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem, Phys. Rev. D94, 084015 (2016), arXiv:1604.08308 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[27]

Light Deflection and Gauss-Bonnet Theorem: Definition of Total Deflection Angle and its Applications

H. Arakida, Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications, Gen. Rel. Grav. 50, 48 (2018), arXiv:1708.04011 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[28]

M. C. Werner, Gravitational lensing in the Kerr-Randers optical geometry, Gen. Rel. Grav.44, 3047 (2012), arXiv:1205.3876 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[29]

G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, and M. C. Werner, Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry, Phys. Rev. D79, 044022 (2009), arXiv:0811.2877 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[30]

T. Ono, A. Ishihara, and H. Asada, Gravitomagnetic bending angle of light with finite-distance corrections in stationary axisymmetric spacetimes, Phys. Rev. D96, 104037 (2017), arXiv:1704.05615 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[31]

Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit

A. Ishihara, Y. Suzuki, T. Ono, and H. Asada, Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit, Phys. Rev. D95, 044017 (2017), arXiv:1612.04044 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[32]

T. Ono, A. Ishihara, and H. Asada, Deflection angle of light for an observer and source at finite distance from a rotating wormhole, Phys. Rev. D98, 044047 (2018), arXiv:1806.05360 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[33]

G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Gravitational lensing in a non-uniform plasma, Mon. Not. Roy. Astron. Soc.404, 1790 (2010), arXiv:1006.2321 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[34]

Atamurotov, A

F. Atamurotov, A. Abdujabbarov, and W.-B. Han, Effect of plasma on gravitational lensing by a Schwarzschild black hole immersed in perfect fluid dark matter, Phys. Rev. D104, 084015 (2021). 31

work page 2021
[35]

Alloqulov, F

M. Alloqulov, F. Atamurotov, A. Abdujabbarov, B. Ahmedov, and N. Juraeva, Shadow and gravitational weak lensing for quantum improved charged black hole in plasma, Chin. Phys. C48, 115116 (2024)

work page 2024
[36]

Gravitational lensing of massive particles in Reissner-Nordstr\"om spacetime

X. Pang and J. Jia, Gravitational lensing of massive particles in Reissner-Nordstr¨ om spacetime, Class. Quant. Grav.36, 065012 (2019), arXiv:1806.04719 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[37]

Z. Li, G. He, and T. Zhou, Gravitational deflection of relativistic massive particles by wormholes, Phys. Rev. D101, 044001 (2020), arXiv:1908.01647 [gr-qc]

work page arXiv 2020
[38]

Li and J

Z. Li and J. Jia, Kerr-Newman-Jacobi geometry and the deflection of charged massive particles, Phys. Rev. D104, 044061 (2021), arXiv:2108.05273 [gr-qc]

work page arXiv 2021
[39]

Alloqulov, H

M. Alloqulov, H. Chakrabarty, D. Malafarina, B. Ahmedov, and A. Abdujabbarov, Gravitational lensing of neutrinos in parametrized black hole spacetimes, JCAP02, 070, arXiv:2408.12916 [gr-qc]

work page arXiv
[40]

E. F. Eiroa and C. M. Sendra, Gravitational lensing by a regular black hole, Class. Quant. Grav.28, 085008 (2011), arXiv:1011.2455 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[41]

Shaikh, P

R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Strong gravitational lensing by wormholes, JCAP07, 028, [Erratum: JCAP 12, E01 (2023)], arXiv:1905.06932 [gr-qc]

work page arXiv 2023
[42]

J. R. Nascimento, A. Y. Petrov, P. J. Porfirio, and A. R. Soares, Gravitational lensing in black-bounce spacetimes, Phys. Rev. D 102, 044021 (2020), arXiv:2005.13096 [gr-qc]

work page arXiv 2020
[43]

Y. Guo, C. Lan, and Y.-G. Miao, Bounce corrections to gravitational lensing, quasinormal spectral stability, and gray-body factors of Reissner-Nordstr¨ om black holes, Phys. Rev. D106, 124052 (2022), arXiv:2201.02971 [gr-qc]

work page arXiv 2022
[44]

S.-W. Wei, K. Yang, and Y.-X. Liu, Black hole solution and strong gravitational lensing in Eddington-inspired Born–Infeld gravity, Eur. Phys. J. C75, 253 (2015), [Erratum: Eur.Phys.J.C 75, 331 (2015)], arXiv:1405.2178 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[45]

Kuang, Z.-Y

X.-M. Kuang, Z.-Y. Tang, B. Wang, and A. Wang, Constraining a modified gravity theory in strong gravitational lensing and black hole shadow observations, Phys. Rev. D106, 064012 (2022), arXiv:2206.05878 [gr-qc]

work page arXiv 2022
[46]

S. Hui, B. Mu, and P. Wang, Gravitational Lensing by Black Holes in Einstein-nonlinear Electrodynamic Theories with Multiple Photon Spheres (2025), arXiv:2507.13048 [gr-qc]

work page arXiv 2025
[47]

Lu, X.-Y

Y. Lu, X.-Y. Pan, M.-Y. Lai, and Q.-h. Wang, Finite-distance gravitational lensing of a global monopole in a Schwarzschild–de Sitter spacetime, Phys. Rev. D112, 064051 (2025), arXiv:2504.00777 [gr-qc]

work page arXiv 2025
[48]

Lambiase, R

G. Lambiase, R. C. Pantig, and A. ¨Ovg¨ un, Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data, EPL148, 49001 (2024), arXiv:2408.09620 [gr-qc]

work page arXiv 2024
[49]

Kumar, S

R. Kumar, S. G. Ghosh, and A. Wang, Shadow cast and deflection of light by charged rotating regular black holes, Phys. Rev. D 100, 124024 (2019), arXiv:1912.05154 [gr-qc]

work page arXiv 2019
[50]

Kumar, S

R. Kumar, S. G. Ghosh, and A. Wang, Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes, Phys. Rev. D101, 104001 (2020), arXiv:2001.00460 [gr-qc]

work page arXiv 2020
[51]

Capozziello, E

S. Capozziello, E. Battista, and S. De Bianchi, Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes, Phys. Rev. D112, 044009 (2025), arXiv:2507.08431 [gr-qc]

work page arXiv 2025
[52]

Ono and H

T. Ono and H. Asada, The effects of finite distance on the gravitational deflection angle of light, Universe5, 218 (2019), arXiv:1906.02414 [gr-qc]

work page arXiv 2019
[53]

Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A unified treatment

G. Crisnejo and E. Gallo, Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A unified treatment, Phys. Rev. D97, 124016 (2018), arXiv:1804.05473 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[54]

The 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. D76, 043006 (2007), arXiv:0709.2948 [astro-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[55]

Ka¸ sık¸ cı and C

O. Ka¸ sık¸ cı and C. Deliduman, Gravitational Lensing in Weyl Gravity, Phys. Rev. D100, 024019 (2019), arXiv:1812.01076 [gr-qc]

work page arXiv 2019
[56]

Huang and Z

Y. Huang and Z. Cao, Finite-distance gravitational deflection of massive particles by a rotating black hole in loop quantum gravity, Eur. Phys. J. C83, 80 (2023), arXiv:2212.04254 [gr-qc]

work page arXiv 2023
[57]

Li and J

Z. Li and J. Jia, The finite-distance gravitational deflection of massive particles in stationary spacetime: a Jacobi metric approach, Eur. Phys. J. C80, 157 (2020), arXiv:1912.05194 [gr-qc]

work page arXiv 2020
[58]

Z. Li, G. Zhang, and A. ¨Ovg¨ un, Circular Orbit of a Particle and Weak Gravitational Lensing, Phys. Rev. D101, 124058 (2020), arXiv:2006.13047 [gr-qc]

work page arXiv 2020
[59]

Huang, B

Y. Huang, B. Sun, and Z. Cao, Extending Gibbons-Werner method to bound orbits of massive particles, Phys. Rev. D107, 104046 (2023), arXiv:2212.04251 [gr-qc]

work page arXiv 2023
[60]

Li, Gravitational lensing using Werner’s method in Cartesian-like coordinates, Phys

Z. Li, Gravitational lensing using Werner’s method in Cartesian-like coordinates, Phys. Rev. D111, 084017 (2025), arXiv:2404.19658 [gr-qc]

work page arXiv 2025
[61]

Chern, An elementary proof of the existence of isothermal parameters on a surface, Proceedings of the American Mathematical Society6, 771 (1955)

S.-s. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proceedings of the American Mathematical Society6, 771 (1955)

work page 1955
[62]

D. M. Deturck and J. L. Kazdan, Some regularity theorems in riemannian geometry, Annales scientifiques de l’ ´Ecole normale sup´ erieure14, 249 (1981)

work page 1981
[63]

R. M. Wald,General Relativity(Chicago Univ. Pr., Chicago, USA, 1984)

work page 1984
[64]

G. W. Gibbons, The Jacobi-metric for timelike geodesics in static spacetimes, Class. Quant. Grav.33, 025004 (2016), arXiv:1508.06755 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[65]

Perlick,Ray Optics, Fermat’s Principle, and Applications to General Relativity, Lecture Notes in Physics Monographs (Springer Berlin Heidelberg, 2003)

V. Perlick,Ray Optics, Fermat’s Principle, and Applications to General Relativity, Lecture Notes in Physics Monographs (Springer Berlin Heidelberg, 2003)

work page 2003
[66]

Chern, A simple intrinsic proof of the gauss-bonnet formula for closed riemannian manifolds, The Annals of Mathematics 45, 747 (1944)

S.-S. Chern, A simple intrinsic proof of the gauss-bonnet formula for closed riemannian manifolds, The Annals of Mathematics 45, 747 (1944)

work page 1944
[67]

Klingenberg,A course in differential geometry, Vol

W. Klingenberg,A course in differential geometry, Vol. 51 (Springer Science & Business Media, 2013). 32

work page 2013
[68]

M. P. Do Carmo,Differential geometry of curves and surfaces: revised and updated second edition(Courier Dover Publications, 2016)

work page 2016
[69]

Korn,Mathematische Abhandlungen Hermann Amandus Schwarz(Springer Berlin Heidelberg, 1914) pp

A. Korn,Mathematische Abhandlungen Hermann Amandus Schwarz(Springer Berlin Heidelberg, 1914) pp. 215–229

work page 1914
[70]

Lytchak, On conformal planes of finite area, Mathematische Zeitschrift305, 10.1007/s00209-023-03381-9 (2023)

A. Lytchak, On conformal planes of finite area, Mathematische Zeitschrift305, 10.1007/s00209-023-03381-9 (2023)

work page doi:10.1007/s00209-023-03381-9 2023