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Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing

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Pith's one-line read The geodesics method of computing gravitational lensing angles is a special case of the Gauss-Bonnet method.

desk verdict A correct but incremental paper that re-derives a known equivalence and overstates 'derivation' by importing the geodesic trajectory from the very method it claims to derive. read the letter →

arxiv 1908.05592 v2 pith:ESEUU36W submitted 2019-08-15 gr-qc

Zonghai Li , Tao Zhou This is my paper · ORCID
classification gr-qc MSC 83C1053C2283C57
keywords gravitationallensingdeflectionangleGauss-BonnettheoremgeodesiccurvaturegeodesicsmethodGibbons-WernerKerr-Newmanspacetimeopticalmetric
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

Gravitational deflection angles are usually computed either by integrating the geodesic equation for the light ray or by applying the Gauss-Bonnet theorem to a curved optical surface. This paper argues that for asymptotically flat spacetimes these two routes are not independent: the geodesics method is the special case of the Gauss-Bonnet method in which the background space is chosen to be Euclidean, so the Gaussian-curvature term vanishes and the entire deflection angle comes from the geodesic curvature of the particle's trajectory. The paper proves this by showing that the geodesic-curvature integral reduces exactly to the standard boundary formula $\alpha=[\arctan(dy/dx)]_{x\to-\infty}^{x\to\infty}$. It then computes the second-order deflection angle in Kerr-Newman spacetime by three versions of the Gauss-Bonnet method and obtains the same expression, supporting the claimed equivalence. A sympathetic reader would care because the result unifies two apparently different pictures of lensing: one global and topological, the other local and ray-based.

What carries the argument

The load-bearing object is the Gauss-Bonnet theorem applied to a two-dimensional lens region with boundary $\partial D=\gamma_g\cup C_1\cup C_2\cup C_3$, together with the geodesic curvature $k_g$ of the particle ray. The theorem packages the deflection angle as $\alpha=-\iint_D K\,dS+\int_S^O k_g\,d\sigma$; the paper's Case 3 sets $K=0$ by choosing a Euclidean background, so the surviving boundary integral $\int k_g\,d\sigma$ is the whole story. In Euclidean coordinates the geodesic curvature of a graph $y=y(x)$ is $k_g=y''/(1+y'^2)^{3/2}$, and integrating it yields the arctangent difference that defines the geodesics-method angle. The machinery shows that the split between area curvature and boundary curvature is background-dependent but their sum is not.

What would settle it

Evaluate Eq. (6) on the exact numerical null geodesic of a Schwarzschild black hole in the strong-deflection regime, where the ray winds around the lens and cannot be written as a single-valued function $y(x)$. If the boundary arctangent difference then fails to match the deflection angle read from asymptotic tangent directions, the claimed equivalence holds only for weak, single-valued rays.

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Extended reading notes

Core claim

In an asymptotically flat setting, choose the Euclidean plane as the lens background and apply the Gauss-Bonnet theorem to the region bounded by the particle ray $\gamma_g$ and three large coordinate segments. Since the background is flat, $K=0$, and Eq. (3) becomes $\alpha=\lim_{R\to\infty}\int_S^O k_g(\gamma_g)\,d\sigma$. Writing the ray as $y=y(x)$ and using the Euclidean geodesic curvature $k_g=y''/(1+y'^2)^{3/2}$, this integral evaluates to $[\arctan(dy/dx)]_{x\to-\infty}^{x\to\infty}$, which is exactly the formula used in the geodesics method. The paper concludes that the geodesics method is a special case of the Gibbons-Werner method, with the deflection attributed entirely to geodesic curvature rather than Gaussian curvature; more generally, the two contributions can be reshuffled by changing the asymptotically Euclidean background while keeping the total angle fixed. The Kerr-Newman example gives $\alpha=4m/b-4am/b^2+3\pi(5m^2-q^2)/(4b^2)$ by all three routes.

Load-bearing premise

The argument takes the particle ray $\gamma_g$ as an input rather than deriving it inside the Gauss-Bonnet framework, so the claimed derivation still relies on the geodesics method to supply the trajectory on which the curvature integral is evaluated.

Editorial extensions

If this is right

  • In asymptotically flat lensing, any asymptotically Euclidean background can be used: the Gaussian-curvature and geodesic-curvature contributions may shift, but the total deflection angle stays the same.
  • The geodesics method is not a logically separate technique; it is the Euclidean-background limit of the Gauss-Bonnet method, so its geometric content is carried by the geodesic curvature of the ray.
  • The flat-space route $\alpha=\int k_g(\gamma_g)\,d\sigma$ provides a direct one-line formula for the deflection angle whenever the ray is known as a graph $y(x)$.
  • For Kerr-Newman spacetime at second post-Minkowskian order, the three computational routes agree, giving $\alpha=4m/b-4am/b^2+3\pi(5m^2-q^2)/(4b^2)$, which is evidence that the equivalence holds in practice.

Reading between the lines

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

  • Extension, not stated in the paper: because the background is arbitrary, one could engineer backgrounds that distribute the angle between $\alpha_{\mathrm{Gauss}}$ and $\alpha_{\mathrm{geod}}$ in ways that simplify strong-field or finite-distance calculations; the invariance of the sum is a testable computational strategy.
  • Testable extension: use the flat-space formula with an exact numerical null ray for strong deflection, where the ray is no longer a single-valued graph $y(x)$; if the $\arctan(y')$ boundary values fail to reproduce the asymptotic deflection, the equivalence is limited to weak lensing, a restriction the paper does not discuss.
  • The example imports the ray $y_1(x)$ from the geodesic solution, so the paper's demonstration of equivalence at the level of formulas still presupposes the geodesics method for the trajectory; deriving the trajectory entirely inside the Gauss-Bonnet framework would make the unification complete.
  • One could test the split invariance numerically: compute $\alpha_{\mathrm{Gauss}}$ and $\alpha_{\mathrm{geod}}$ on two different asymptotically Euclidean backgrounds for the same spacetime and verify that the sum is identical order by order.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper claims to demonstrate the equivalence of the Gibbons-Werner (GW) method, based on the Gauss-Bonnet theorem, and the standard geodesics method for gravitational deflection in asymptotically flat spacetimes. For a Euclidean lens background (K=0), the Gauss-Bonnet formula reduces to α = ∫_S^O k_g(γ_g) dσ (Eq. (6)), which is then evaluated for a curve y=y(x) to give α = [arctan(dy/dx)] evaluated from x=−∞ to x=+∞ (Eq. (11)), identified with the geodesics-method formula. The paper then computes the second-order deflection angle in Kerr-Newman spacetime by three variants—Werner's osculating-Riemannian method, the generalized optical metric method with nonzero geodesic curvature, and the geodesics method—and obtains the same result α = 4m/b − 4am/b^2 + 3π(5m^2−q^2)/4b^2. The stated conclusion is that the geodesics method is a special case of the GW framework.

Significance. Within its scope, the paper gives a clean and correct elementary derivation of the deflection-angle identity that underlies the geodesics method, and the Kerr-Newman calculation is a useful concrete demonstration that the different GW-type decompositions yield consistent second-order results. The paper honestly cites earlier first- and second-order equivalence results. The main novelty—the K=0 special case—is, however, a short curve-geometry identity rather than an independent derivation of the light trajectory; the significance is therefore conditional on how the claim of 'derivation' is framed. If the claim is read as an equivalence of angle formulas for a given trajectory, the paper succeeds; if it is read as deriving the full geodesics method without external input of the orbit, it overstates its case.

major comments (2)
  1. [Sec. II.C, Eqs. (6)-(11)] The argument leading to Eq. (11) is a purely geometric identity: for any sufficiently regular curve y = y(x) in the Euclidean plane, the integral of its geodesic curvature equals the total change of its tangent angle, [arctan(dy/dx)] from x→−∞ to x→+∞. This identity contains no dynamical information and gives no procedure for determining y(x). In the geodesics method, y(x) is obtained by solving the null geodesic equation in the spacetime metric, whereas nothing in the Gauss-Bonnet setup of Sec. II supplies that orbit. The statement in the Abstract and Sec. II.C that the geodesics method 'can be derived' with the Gibbons-Werner method should therefore be restated as an equivalence of the deflection-angle formula for a given trajectory, or supplemented with a GB-based derivation of the trajectory itself.
  2. [Sec. III.A and Appendix A] In the Kerr-Newman example the boundary curve is imported from the geodesics method: y1(x) in Eq. (25) is the first-order geodesic orbit and the second-order orbit in Eq. (A3) is taken from Ref. [51]. The integrals in Eqs. (24), (33), and (34) are all evaluated along that same ray. The agreement among the three methods is therefore partly by construction: they are three post-processing formulas applied to a single geodesic trajectory, and an error in the imported ray would produce the same wrong angle in all three. Please state this explicitly and qualify the claim that the example demonstrates equivalence of the methods.
minor comments (4)
  1. [Sec. II.B] The sentence 'Since the lens L is excluded in the domain D, χ(D)=1' is ambiguous: if the lens were excised as an interior hole, the Euler characteristic would be 0. Please state explicitly that D is the simply connected region on the side of γg away from the lens.
  2. [Below Eq. (16)] The word 'Remannian' should be 'Riemannian'.
  3. [Eqs. (6) and (11)] The line element is denoted dσ in Eq. (6) but dl in Eq. (11); please use a consistent notation for the arc length along γg.
  4. [Sec. IV] The concluding formula [∫_S^O k_g(γ_g)dσ]|_{Euclidean} = [−∫∫_D KdS]|_{Optical} is introduced as a summary; please indicate explicitly that it follows from the preceding equations rather than being a new assumption.

Circularity Check

0 steps flagged · score 1.0 of 10

Core equivalence identity is non-circular; only a minor, non-load-bearing self-citation appears.

full rationale

The chain leading to the claimed equivalence is: (i) the GB theorem applied to a Euclidean lens region yields Eq. (6), alpha = integral of k_g dsigma; (ii) for a graph y=y(x) in Euclidean plane, Eqs. (8)-(10) give k_g = y''/(1+y'^2)^(3/2); (iii) the integral telescopes to [arctan(dy/dx)] from x=-infinity to x=+infinity (Eq. 11), which is exactly the geodesics-method formula. This is a genuine identity derived from the GB theorem and the standard geodesic-curvature definition; it does not presuppose the geodesics-method deflection formula. The Kerr-Newman illustration imports the light orbit from the geodesic solution (Eq. 25; Appendix A, citing Ref. [51]). Thus the three calculations in Sec. III are all applied to one common trajectory, so their agreement is a consistency check rather than an independent derivation of the orbit. However, the central equivalence claim concerns the angle formula, not the production of the orbit, so this does not make the derivation circular. The Introduction cites the authors' own Ref. [56] for the fact that second-order equivalence was previously shown, but the paper then provides its own derivation; this citation is not load-bearing. No fitted parameters are relabeled as predictions, and no self-citation chain forbids alternatives. I therefore find no significant circularity; the only minor point is the non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Gauss-Bonnet theorem (standard math), the asymptotic-flatness and domain assumptions, and the import of the geodesic trajectory from the geodesic method (which is the subject of the claimed equivalence). No new free parameters or invented entities are introduced.

assumptions (6)
  • standard math Gauss-Bonnet theorem (Eq 1)
    The foundation of the Gibbons-Werner method; used to derive Eq (3).
  • domain assumption The lens region D is asymptotically Euclidean and the source/observer lie in the asymptotic region
    Needed for the boundary terms in Eq (2) (k_g(C_i)=0, exterior angles) to simplify; stated in Sec II.B.
  • domain assumption The lens singularity can be excluded from D while keeping chi(D)=1
    Asserted in Sec II.B without specifying the small boundary; the Gaussian curvature is singular at the lens, so a regulator is implicit.
  • domain assumption The physical light ray is given by the geodesic solution y1(x) and y(x) (Eq (25), Appendix A, from Ref [51])
    Used as the integration boundary in all three methods; not derived within the GB framework in this paper.
  • standard math Werner's osculating Riemannian construction makes the Finsler geodesic a geodesic of the osculating metric
    Imported from Ref [14]; needed for Sec III.A.
  • domain assumption The metric expansion Eq (12) in harmonic coordinates is valid to second order
    Basis for the Kerr-Newman example; from Refs [66,67].

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Pith. "Pith review of Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing." pith.science (2026). https://pith.science/paper/ESEUU36W

@misc{pith2026190805592,
  author       = {Pith},
  title        = {Pith review of: Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ESEUU36W}},
  note         = {Machine review of arXiv:1908.05592}
}
read the original abstract

The Gibbons-Werner method where the Gauss-Bonnet theorem is applied to study the gravitational deflection angle has received much attention recently. In this paper, we study the equivalence of the Gibbons-Werner method to the standard geodesics method, and it is shown that the geodesics method can be derived with the Gibbons-Werner method, for asymptotically flat case. In the geodesics method, the gravitational deflection angle of particle depends entirely on the geodesic curvature of the particle ray in the Euclidean space. The gravitational deflection of light in Kerr-Newman spacetime is calculated by different technologies under the Gibbons-Werner framework, as an intuitive example to show the equivalence.

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Figure 1
Figure 1. FIG. 1. The region [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Phase-plane formulation of weak gravitational deflection in static spherical spacetimes

    gr-qc 2026-08 accept novelty 6.0 of 10

    The Schwarzschild weak-deflection angle is derived to all orders from a phase-plane amplitude cubic, giving an explicit coefficient formula with convergence radius at the photon sphere.

  2. Semi-Analytic Trajectory Analysis of Light in Generic Static Spacetimes

    gr-qc 2025-07 conditional novelty 4.0 of 10

    The authors present a generic weak-field lensing framework built from three known semi-analytic methods and apply it to a scalar hairy Reissner-Nordstrom black hole, recovering the standard deflection with q^2 = Q^2+Q_s^2.

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Works this paper leans on

71 extracted references · 66 canonical work pages · cited by 2 Pith papers

  1. [51]

    Bhadra, K

    A. Bhadra, K. Sarkar, and K. K. Nandi, Phys. Rev. D 75, 123004 (2007)

    work page 2007

  2. [56]

    X. Liu, N. Yang, and J. Jia, Classical Quantum Gravity 33, 175014 (2016). 7

    work page 2016

  3. [1]

    (3) becomes α =− lim R→∞ ∫∫ D KdS

    Case 1: K⁄= 0, andkg(γg) = 0 In this case, the particle ray γg is a spatial geodesic in (M, ˆgij), and Eq. (3) becomes α =− lim R→∞ ∫∫ D KdS. (4) Indeed, this is the original consideration of Gibbons and Werner [11, 14] and for convenience we shall call it the narrow Gibbons-Werner method. In fact, many studies fall into this category. For light deflection...

    work page

  4. [2]

    (3) can be written as α =αGauss +αgeod, (5) where αGauss =− lim R→∞ ∫∫ D KdS, αgeod = lim R→∞ ∫ O S kg(γg)dσ

    Case 2: K⁄= 0, andkg(γg)⁄= 0 Now, the particle ray is not geodesic in a curved space, and Eq. (3) can be written as α =αGauss +αgeod, (5) where αGauss =− lim R→∞ ∫∫ D KdS, αgeod = lim R→∞ ∫ O S kg(γg)dσ. In Refs. [42–44], Ono et al. considered the so-called general- ized optical metric space as the lens background, and used Eq. (5) to study the deflection ...

    work page

  5. [3]

    (3) arrives at α = lim R→∞ ∫ O S kg(γg)dσ

    Case 3: K = 0, andkg(γg)⁄= 0 In this case, we assume that M is Euclidean space, and Eq. (3) arrives at α = lim R→∞ ∫ O S kg(γg)dσ. (6) To our best knowledge, Eq. (6) has not been considered yet, and next it will be proved that this result is the same with the expression in the geodesics method. The line element of a three-dimensional Euclidean space is dl...

    work page

  6. [4]

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

    1920

  7. [5]

    C. M. Will, Classical Quantum Gravity 32, 124001 (2015)

    work page 2015

  8. [6]

    Hoekstra, M

    H. Hoekstra, M. Bartelmann, H. Dahle, H. Israel, M. Limousin, and M. Meneghetti, Space Sci. Rev. 177, 75 (2013)

    work page 2013

Show all 71 references
  1. [7]

    M. M. Brouwer et al. , Mon. Not. R. Astron. Soc. 481, 5189 (2018)

    2018

  2. [8]

    Bellagamba et al

    F. Bellagamba et al. , Mon. Not. R. Astron. Soc. 484, 1598 (2019)

    2019

  3. [9]

    R. A. Vanderveld, M. J. Mortonson, W. Hu, and T. Eifler, Phys. Rev. D 85, 103518 (2012)

    2012

  4. [10]

    H. J. He and Z. Zhang, J. Cosmol. Astropart. Phys. 08 (2017) 036

    2017

  5. [11]

    S. Cao, G. Covone, and Z. H. Zhu, Astrophys. J.755, 31 (2012)

    2012

  6. [12]

    Huterer and D

    D. Huterer and D. L. Shafer, Rep. Prog. Phys. 81, 016901 (2018)

    2018

  7. [13]

    Jung and C

    S. Jung and C. S. Shin, Phys. Rev. Lett. 122, 041103 (2019)

    2019

  8. [14]

    G. W. Gibbons and M. C. Werner, Classical Quantum Gravity 25, 235009 (2008)

    2008

  9. [15]

    G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, and M. C. Werner, Phys. Rev. D79, 044022 (2009)

    2009

  10. [16]

    G. W. Gibbons and C. M. Warnick, Phys. Rev. D 79, 064031 (2009)

    2009

  11. [17]

    M. C. Werner, Gen. Relativ. Gravit. 44, 3047 (2012)

    2012

  12. [18]

    Caponio, A

    E. Caponio, A. V . Germinario, and M. Sanchez, J. Geom. Anal. 26, 791 (2016)

    2016

  13. [19]

    Jusufi, Eur

    K. Jusufi, Eur. Phys. J. C 76, 332 (2016)

    2016

  14. [20]

    Jusufi, Int

    K. Jusufi, Int. J. Geom. Methods Mod. Phys. 14, 1750137 (2017)

    2017

  15. [21]

    Jusufi, A

    K. Jusufi, A. ¨Ovg¨un, and A. Banerjee, Phys. Rev. D96, 084036 (2017)

    2017

  16. [22]

    Jusufi, Int

    K. Jusufi, Int. J. Geom. Methods Mod. Phys. 14, 1750179 (2017)

    2017

  17. [23]

    Jusufi, ˙I

    K. Jusufi, ˙I. Sakallı, and A. ¨Ovg¨un, Phys. Rev. D 96, 024040 (2017)

    2017

  18. [24]

    Jusufi, M

    K. Jusufi, M. C. Werner, A. Banerjee, and A. ¨Ovg¨un, Phys. Rev. D 95, 104012 (2017)

    2017

  19. [25]

    Jusufi, A

    K. Jusufi, A. ¨Ovg¨un, J. Saavedra, Y . Vasquez, and P. A. Gonza- lez, Phys. Rev. D 97, 124024 (2018)

    2018

  20. [26]

    Jusufi, N

    K. Jusufi, N. Sarkar, F. Rahaman, A. Banerjee, and S. Hansraj, Eur. Phys. J. C 78, 349 (2018)

    2018

  21. [27]

    Jusufi and A

    K. Jusufi and A. ¨Ovg¨un, Phys. Rev. D 97, 024042 (2018)

    2018

  22. [28]

    Jusufi, Phys

    K. Jusufi, Phys. Rev. D 98, 044016 (2018)

    2018

  23. [29]

    Jusufi and A

    K. Jusufi and A. ¨Ovg¨un, Phys. Rev. D 97, 064030 (2018)

    2018

  24. [30]

    Jusufi, F

    K. Jusufi, F. Rahaman, and A. Banerjee, Ann. Phys. (Amster- dam) 389, 219 (2018)

    2018

  25. [31]

    ¨Ovg¨un, K

    A. ¨Ovg¨un, K. Jusufi, and ˙I. Sakallı, Ann. Phys. (Amsterdam) 399, 193 (2018)

    2018

  26. [32]

    ¨Ovg¨un, K

    A. ¨Ovg¨un, K. Jusufi, and ˙I. Sakallı, Phys. Rev. D 99, 024042 (2019)

    2019

  27. [33]

    Jusufi and A

    K. Jusufi and A. ¨Ovg¨un, Int. J. Geom. Methods Mod. Phys. 16, 1950116 (2019)

    2019

  28. [34]

    T. Zhu, Q. Wu, M. Jamil, and K. Jusufi, Phys. Rev. D 100, 044055 (2019)

    2019

  29. [35]

    Sakallı and A

    ˙I. Sakallı and A. ¨Ovg¨un, Europhys. Lett. 118, 60006 (2017)

    2017

  30. [36]

    ¨Ovg¨un, ˙I

    A. ¨Ovg¨un, ˙I. Sakallı, and J. Saavedra, J. Cosmol. Astropart. Phys. 10 (2018) 041

    2018

  31. [37]

    Arakida, Gen

    H. Arakida, Gen. Relativ. Gravit. 50, 48 (2018)

    2018

  32. [38]

    ¨Ovg¨un, Phys

    A. ¨Ovg¨un, Phys. Rev. D 98, 044033 (2018)

    2018

  33. [39]

    Goulart, Classical Quantum Gravity 35, 025012 (2018)

    P. Goulart, Classical Quantum Gravity 35, 025012 (2018)

    2018

  34. [40]

    Javed, R

    W. Javed, R. Babar, and A. ¨Ovg¨un, Phys. Rev. D 99, 084012 (2019)

    2019

  35. [41]

    ¨Ovg¨un, Phys

    A. ¨Ovg¨un, Phys. Rev. D 99, 104075 (2019)

    2019

  36. [42]

    de Leon and I

    K. de Leon and I. Vega, Phys. Rev. D 99, 124007 (2019)

    2019

  37. [43]

    Ishihara, Y

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

    2016

  38. [44]

    Ishihara, Y

    A. Ishihara, Y . Suzuki, T. Ono, and H. Asada, Phys. Rev. D95, 044017 (2017)

    2017

  39. [45]

    T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D 96, 104037 (2017)

    2017

  40. [46]

    T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D 98, 044047 (2018)

    2018

  41. [47]

    T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D 99, 124030 (2019)

    2019

  42. [48]

    Ono and H

    T. Ono and H. Asada, Universe 5, 218 (2019)

    2019

  43. [49]

    Accioly and S

    A. Accioly and S. Ragusa, Classical Quantum Gravity 19, 5429 (2002)

    2002

  44. [50]

    Accioly and R

    A. Accioly and R. Paszko, Phys. Rev. D 69, 107501 (2004)

    2004

  45. [52]

    O. Yu. Tsupko, Phys. Rev. D 89, 084075 (2014)

    2014

  46. [53]

    He and W

    G. He and W. Lin, Classical Quantum Gravity 33, 095007 (2016)

    2016

  47. [54]

    He and W

    G. He and W. Lin, Classical Quantum Gravity 34, 029401 (2017)

    2017

  48. [55]

    He and W

    G. He and W. Lin, Classical Quantum Gravity 34, 105006 (2017)

    2017

  49. [57]

    Pang and J

    X. Pang and J. Jia, Classical Quantum Gravity 36, 065012 (2019)

    2019

  50. [58]

    G. W. Gibbons, Classical Quantum Gravity 33, 025004 (2016)

    2016

  51. [59]

    Z. Li, G. He, and T. Zhou, Phys. Rev. D 101, 044001 (2020)

    2020

  52. [60]

    Chanda, G

    S. Chanda, G. W. Gibbons, P. Guha, P. Maraner, and M. C. Werner, J. Math. Phys.(N.Y .)60, 122501 (2019)

    2019

  53. [61]

    Crisnejo and E

    G. Crisnejo and E. Gallo, Phys. Rev. D 97, 124016 (2018)

    2018

  54. [62]

    Crisnejo, E

    G. Crisnejo, E. Gallo, and A. Rogers, Phys. Rev. D 99, 124001 (2019)

    2019

  55. [63]

    Crisnejo, E

    G. Crisnejo, E. Gallo, and J. R. Villanueva, Phys. Rev. D 100, 044006 (2019)

    2019

  56. [64]

    Jusufi, Phys

    K. Jusufi, Phys. Rev. D 98, 064017 (2018)

    2018

  57. [65]

    Jusufi, A

    K. Jusufi, A. Banerjee, G. Gyulchev, and M. Amir, Eur. Phys. J. C 79, 28 (2019)

    2019

  58. [66]

    Jusufi, arXiv:1906.12186

    K. Jusufi, arXiv:1906.12186

    1906 arXiv

  59. [67]

    Weinberg, Gravitation and Cosmology: Principles and Ap- plications of the General Theory of Relativity (Wiley, New York, 1972)

    S. Weinberg, Gravitation and Cosmology: Principles and Ap- plications of the General Theory of Relativity (Wiley, New York, 1972)

    1972

  60. [68]

    M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, New Jersey, 1976)

    1976

  61. [69]

    Lin and C

    W. Lin and C. Jiang, Phys. Rev. D 89, 087502 (2014)

    2014

  62. [70]

    B. Yang, C. Jiang, and W. Lin, Classical Quantum Gravity 36, 085010 (2019)

    2019

  63. [71]

    D. Bao, S. S. Chern, and Z. Shen, An Introduction to Remiann- Finsler Geometry (Springer, New York, 2002)

    2002

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