Photon Deflection and Magnification in Kalb-Ramond Black Holes with Topological String Configurations

Ahmad Al-Badawi; Faizuddin Ahmed; \.Izzet Sakall{\i}

REVIEW 4 major objections 3 minor 1 cited by

Photon Deflection and Magnification in Kalb-Ramond Black Holes with Topological String Configurations

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper derives two-term formulas for photon deflection and magnification around Schwarzschild-like black holes in Kalb-Ramond gravity with cosmic-string or cloud-of-strings topological configurations, and shows how Lorentz-violation…

desk verdict Weak-field lensing formulas in this paper are standard and roughly correct, but the strong-field Bozza section contains demonstrable algebra errors that break the paper's central claims. read the letter →

arxiv 2507.22673 v1 pith:XITSMX6U submitted 2025-07-30 gr-qc hep-th

Faizuddin Ahmed , İzzet Sakall{i} , Ahmad Al-Badawi This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C1083D05

keywords gravitationallensingphotondeflectionmagnificationKalb-RamondgravitycosmicstringcloudofstringsLorentzviolationstrong-field

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 claims that in Kalb-Ramond (KR) gravity, a Schwarzschild-like black hole that is either pierced by a cosmic string (with parameter $\beta=1-4G\mu$) or threaded by a cloud of strings (with parameter $\alpha$) deflects passing starlight according to a two-term formula: $\alpha \approx 4M/(\gamma\delta\beta)+15\pi M^2/(4\gamma^2\delta^2\beta^2)$ in the cosmic-string case, and the same expression with $\eta=1/(1-\ell)-\alpha$ in the cloud case, where $\ell$ is the KR Lorentz-violation parameter. The paper derives these results both by a perturbative expansion of the null geodesic equation and by an exact elliptic-integral evaluation, and it shows how the modified deflection shifts the critical curves and magnifications in a way that could distinguish the exotic spacetimes from ordinary Schwarzschild. If the paper is right, precision gravitational lensing, from Solar System tests to galactic-center observations, becomes a concrete probe of Lorentz violation and string-topology parameters whose allowed ranges differ by orders of magnitude.

What carries the argument

The load-bearing device is the null-geodesic orbit equation written in the inverse radial coordinate $u=1/r$ for each metric. After reducing the SKRCS line element to the equatorial plane, the orbit obeys $\frac{d^2u}{d\phi^2}+\beta^2\delta u=3M\beta u^2$, with homogeneous solution $u_0=(1/\gamma)\cos(\beta\sqrt{\delta}\,\phi)$; for SKRCoS the same equation holds with $\beta^2\delta$ replaced by $\eta$. The paper feeds this equation through a two-step machinery: a perturbative expansion $u=u_0+\varepsilon u_1+\varepsilon^2 u_2$ with $\varepsilon=M/\gamma$ yields the asymptotic deflection $\alpha=2\psi$ after the shift $\phi=\pi/2+\psi$, while factorization of the cubic polynomial in the exact equation yields incomplete elliptic integrals of the first kind and hence an expression valid across all impact parameters. The strong-field section applies a standard singular-integral regularization near the photon sphere to extract $b_1$ and $b_m$.

What would settle it

Numerically integrate the null geodesic equations for the exact SKRCS metric (2.5) across a range of impact parameters down to the photon sphere, and compare the resulting deflection angles with Eq. (2.26) and with the logarithmic law (6.16) using Table III; if the slope of $\alpha$ versus $\log(b/b_m-1)$ deviates from $b_1$ for the cosmic-string case, the strong-field reduction in Section VI is falsified, while agreement of the weak-field branch would support Eq. (2.26).

Watch

Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that the combined Lorentz-violation and topological-string parameters enter the weak-field deflection only through a composite multiplicative factor. In the cosmic-string case the null-geodesic orbit equation reduces to $\frac{d^2u}{d\phi^2}+\beta^2\delta u=3M\beta u^2$, whose second-order solution gives $\alpha \approx \frac{4M}{\gamma\delta\beta}+\frac{15\pi M^2}{4\gamma^2\delta^2\beta^2}$; in the cloud-of-strings case the same calculation with $\eta=\frac{1}{1-\ell}-\alpha$ gives $\alpha \approx \frac{4M}{\gamma\eta}+\frac{15\pi M^2}{4\gamma^2\eta^2}$. The exact elliptic-integral expressions reproduce the same limits and reduce to Schwarzschild when $\ell=0$ together with $\beta=1$ or $\alpha=0$. The magnification analysis then shows that the modified Einstein-angle squared $\bar{\theta}_\xi^2$ shifts the tangential and radial magnification components, with $\mu_{\rm rad}$ diverging at the critical curve, and the strong-field section extracts the photon-sphere radius $r_m=3M/(\kappa\xi)$, the critical impact parameter $b_m$, and the logarithmic coefficient $b_1$ tabulated for both configurations.

Load-bearing premise

For the strong-field results, the load-bearing assumption is that a single spherical metric form $ds^2=-A(r)dt^2+A(r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ with $A=\xi-2M/(\kappa r)$ covers both configurations; the SKRCS line element (2.5) actually has $\beta^2\sin^2\theta\,d\phi^2$ in its angular part, so that reduction does not hold for the cosmic-string case and the Table III strong-field coefficients for SKRCS do not follow from the true metric.

Editorial extensions

If this is right

If the deflection formula in Eq. (2.26) is correct, the leading-order deflection for the cosmic-string case is exactly Schwarzschild's $4M/\gamma$ term with the mass rescaled to $M_{\rm eff}=M/(\delta\beta)$, so a single deflection measurement can only fix the product $\delta\beta$ unless the second-order term is also measured.
The ratio of the second-order term to the first-order term is $(15\pi M)/(16\gamma\delta\beta)$, a parameter-free shape relation that a precise deflection measurement could isolate.
The radial magnification $\mu_{\rm rad}$ diverges where the image angle equals the modified Einstein angle; locating that critical curve in lensed systems would measure $\bar{\theta}_\xi$ and hence constrain $(\delta\beta)^{-1}$ or $\eta^{-1}$.
In the cloud-of-strings case the same formula with $\eta=\frac{1}{1-\ell}-\alpha$ means that lensing alone cannot separate Lorentz violation from the cloud density without an independent bound on $\alpha$.
The strong-field coefficient $b_1$ for the SKRCS case scales as $\beta^{-1}(1-\ell)^{3/2}$, so the logarithmic divergence amplitude depends inversely on the cosmic-string parameter $\beta$ and grows with the Lorentz-violation parameter $\ell$.

Reading between the lines

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

Inference: because both the first- and second-order terms in Eq. (2.26) are proportional to powers of $M/(\delta\beta)$, weak-field deflection observations alone cannot distinguish a Kalb-Ramond/cosmic-string modification from a Schwarzschild black hole of a different mass; the magnification critical-curve position is the same degeneracy in disguise.
Inference: the Section VI argument for the cosmic-string case assumes the angular part $r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ even though Eq. (2.5) has $\beta^2\sin^2\theta\,d\phi^2$, so the strong-field $b_1$ and $b_m$ values in Table III for SKRCS should be re-derived from the full metric; the weak-field and magnification results do not depend on that step.
Inference: a natural testable extension is to compute the deflection to third order in $M/\gamma$ for both metrics; if the $(\delta\beta)^{-n}$ multiplicative structure persists, the formulas can be promoted to a general resummation, and if it does not, the second-order formula sets the range of validity of the perturbation series.
Inference: comparing the exact elliptic-integral expression with standard Schwarzschild lensing at intermediate impact parameters would quantify how large the deviations actually are under the galactic-center bounds $-0.185<\ell<0.061$, providing a concrete observational target for future lensing surveys.

Share X Bluesky LinkedIn Reddit HN

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

Briefly: this is a routine lensing calculation for two modified Schwarzschild metrics. The weak-field deflection angles (Eqs 2.26, 4.30) and the magnification analysis are standard perturbative and elliptic-integral material, and they look broadly correct. Those sections are worth something as a catalog entry. The strong-field Bozza section (Section VI) is not correct, and the paper advertises its coefficients as a central result, so the paper cannot be accepted in its current form.

The algebra errors are concrete. From the photon-sphere condition (6.3) with h(r)=ξ−2M/(κr), the radius is r_m=3M/(κξ) and h(r_m)=ξ/3, which gives b_m=3√3M/(κξ^{3/2}). Equation (6.5) instead gives 3√3M/(κ√(ξ(ξ−2/3))), which for Schwarzschild reduces to 9M instead of 3√3M. A'(x) for A(x)=ξ−2Mx/κ is simply −2M/κ, while Eq (6.20) reports a term with κ^4ξ^4/M^3 that is dimensionally wrong. The logarithmic coefficient b1 in Eq (6.22) carries a factor of M, so it has units of mass; the standard b1 is dimensionless and for Schwarzschild equals 1, not 4.60M. These are not typos in a marginal section; they are the strong-field results in Table III.

There is also a structural problem: Section VI proceeds from the unified metric (6.1) with C(r)=r^2, but the SKRCS line element (2.5) has angular part β^2r^2 sin^2θ dφ^2. One can rescale φ to bring it to the form C=r^2, but the physical deflection angle and impact parameter in the conical geometry then require careful bookkeeping. The paper does not do that. Additionally, Eq (2.30) disagrees in sign with Eq (2.15), and the weak-field derivation later uses the correct sign in the elliptic integrals; this looks like a typo, but it should be fixed.

Who would get value from this paper? A reader collecting lensing templates for exotic black holes might look at the weak-field formulas, but they would have to re-derive the strong-field part themselves. The paper deserves a reworking, not a referee round in its current form. I would recommend that the editor either send it back for major revision with a clear list of these errors, or desk reject and invite resubmission after the strong-field section is redone. If I were the editor, I would not send it out for review until the authors have corrected Section VI and rechecked the conical treatment.

Referee Report

4 major / 3 minor

Summary. The paper derives gravitational lensing observables for two modified Schwarzschild-like black hole spacetimes: the Kalb-Ramond spacetime pierced by a cosmic string (SKRCS, eq. 2.5) and the Kalb-Ramond spacetime with a cloud of strings (SKRCoS, eq. 4.5). For each geometry it presents weak-field photon deflection angles via perturbative and elliptic-integral methods, tangential and radial magnification formulas, and a strong-field analysis based on the Bozza formalism. The claimed central results are the deflection formulas (2.26) and (4.30), the corresponding magnification expressions, and the strong-field logarithmic divergence coefficients and critical impact parameters in Table III.

Significance. If the derived formulas were correct, the paper would provide concrete, testable predictions connecting Lorentz-violation and topological-string parameters to gravitational lensing observables. The weak-field perturbative expansion and the elliptic-integral representation follow standard techniques, and the parameters ℓ, β, α are imported from the metric and from prior observational bounds rather than fitted, so the framework is not circular. However, the strong-field section contains direct algebraic errors and an invalid metric reduction, and the tabulated strong-field coefficients cannot be used as they stand. The paper's main new strong-field claims therefore do not survive scrutiny.

major comments (4)

[VI, Eq. (6.5)] Equation (6.5) for the critical impact parameter is inconsistent with the photon-sphere condition stated just above it. With h(r)=ξ−2M/(κr), Eq. (6.3) gives r_m=3M/(κξ) and h(r_m)=ξ/3, hence b_m=r_m/√h(r_m)=3√3M/(κξ^{3/2}). Equation (6.5), 3√3M/[κ√(ξ(ξ−2/3))], does not reduce to 3√3M in the Schwarzschild limit ξ=κ=1 but to 9M. The b_m entries in Table III therefore follow from an incorrect formula.
[VI, Eq. (6.20)] Direct differentiation of A(x)=h(1/x)=ξ−2Mx/κ gives A′(x)=−2M/κ, so at the photon sphere A′_m=−2M/κ. Equation (6.20) instead states A′_m=−2κ⁴ξ⁴/(81M³), which has the wrong sign, the wrong parametric dependence, and dimensions of inverse mass cubed rather than inverse mass. Since A′_m enters R(0,x_m) and therefore b1 and b2, all strong-field coefficients derived from it are invalid.
[VI, Eq. (6.22)] The coefficient b1 in the strong-field expansion α(b)=−b1 log(b/b_m−1)+b2 must be dimensionless, but Eq. (6.22) contains an explicit factor of M and has dimensions of mass. In the Schwarzschild limit ξ=1, κ=1 it gives b1≈4.60M instead of the standard dimensionless value 1. This is not a normalization convention: the argument of the logarithm is already dimensionless, and b1 multiplies a pure number.
[VI, Eq. (6.1)] The unified strong-field treatment assumes the metric (6.1) with C(r)=r². The SKRCS line element (2.5) has g_φφ=β²r² sin²θ, so it is not of the assumed form with periodic φ∈[0,2π). The identities C′=2r and C″=2 used in Eqs. (6.11)–(6.12) do not apply to the SKRCS geometry, and the Table III entries for SKRCS do not follow. A conical-coordinate treatment gives r_m=3M/(βδ) and b_m=3√3M/(βδ^{3/2}), which do not match the tabulated values. Even for SKRCoS, where C(r)=r² is valid, the algebraic inconsistencies in Eqs. (6.5) and (6.20) remain.

minor comments (3)

[III, Eqs. (3.7)–(3.8)] The magnification formulas treat ū²_ξ as θ-independent, but Eq. (3.4) contains a θ-dependent correction term. If that correction is intended only at leading order, the derivative of ū²_ξ should be neglected explicitly; if it is retained, the radial magnification formula must include the extra derivative contribution.
[VI, Eq. (6.10)] The expansion coefficients β1 and β2 reuse the symbol β already used for the cosmic-string parameter, which makes the strong-field equations confusing, especially in the SKRCS case.
[Figures 3–8] The captions state |M|=1 M_sun while the equations treat M as a length; please state the unit convention used in the plots or convert M to geometric units consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lensing predictions are derived from the stated SKRCS and SKRCoS metrics via standard perturbative, elliptic-integral, and Bozza methods; the Section VI inconsistencies are algebraic or geometric errors, not input-output circularity.

full rationale

No circular step is present. The deflection-angle results, such as Eq. (2.26), follow from the null-geodesic equation (2.14) through the perturbative solution of Eqs. (2.15)-(2.22), and the SKRCoS counterpart Eq. (4.30) follows from Eq. (4.20). No parameter is fitted to data and no prediction is defined in terms of a fitted constant; the parameters ℓ, β, and α are independent inputs carried through the derivation. The magnification formulas are standard lens-equation manipulations using the derived deflection angles, so they also are not circular. The strong-field section uses the standard Bozza formalism and derives coefficients from the stated forms h(r)=ξ−2M/(κr) and C(r)=r²; the fact that C(r)=r² is not valid for the SKRCS metric, and the algebraic errors in Eqs. (6.5), (6.20), and (6.22), are correctness defects rather than circular reasoning. The results do not reduce to their inputs by construction; they are inconsistent with the actual SKRCS angular metric. Self-citations [34-36] are prior related lensing studies, but they are not used as a uniqueness theorem or as the load-bearing justification for any central formula, so they do not constitute circularity. The derivation chain is self-contained relative to the metrics and standard methods, even where it is mathematically flawed.

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

The central formulas rest on three model parameters imported from prior literature and on the unproven assumption that the SKRCS and SKRCoS line elements are valid solutions. No free parameters are fitted to data in this paper, so the circularity burden is low. The strong-field section further assumes a unified spherical metric that contradicts the SKRCS angular part.

free parameters (3)

ℓ (Kalb-Ramond Lorentz-violation parameter) = Shapiro bound -6.1e-13≤ℓ≤2.8e-14; Sgr A* bound -0.18502<ℓ<0.06093
Imported from the KR metric and prior constraints; not fitted in this paper, but every deflection and magnification formula depends on it.
β (cosmic-string parameter, 1-4Gμ) = None quoted
Imported from the cosmic-string metric; no observational bound is given in the paper.
α (cloud-of-strings parameter) = Restricted to 0<α<1
Imported from the cloud-of-strings metric; no observational bound is derived.

assumptions (4)

domain assumption The KR black-hole metric f(r)=1/(1-ℓ)-2M/r and its combinations with cosmic string (β) and cloud of strings (α) are valid solutions of the extended action
Invoked through Eqs (2.1)-(2.6) and (4.1)-(4.6); metric forms are imported from refs [9,16,20,21], not derived here.
standard math Null geodesics are obtained from the standard Lagrangian/Euler-Lagrange equations in the given metrics
Used in Sections II and IV, Eqs (2.7)-(2.15) and (4.7)-(4.20).
domain assumption Bozza's strong-field formalism is applicable without modification to the unified metric with C(r)=r²
Section VI, Eqs (6.6)-(6.18); the assumption fails for the SKRCS metric because its angular part contains β²sin²θ.
domain assumption Quoted Solar System and Sgr A* bounds on ℓ apply to the KR metric used here
Table I and Section II cite refs [6,38]; no re-derivation or consistency check is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Photon Deflection and Magnification in Kalb-Ramond Black Holes with Topological String Configurations." pith.science (2026). https://pith.science/paper/XITSMX6U

@misc{pith2026250722673,
  author       = {Pith},
  title        = {Pith review of: Photon Deflection and Magnification in Kalb-Ramond Black Holes with Topological String Configurations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XITSMX6U}},
  note         = {Machine review of arXiv:2507.22673}
}

read the original abstract

This theoretical investigation examines gravitational lensing phenomena in Schwarzschild-like black holes (BHs) within Kalb-Ramond (KR) gravity frameworks incorporating cosmic string (CS) and cloud of strings (CoS) topological configurations. We develop comprehensive analytical methodologies for investigating photon deflection angles and magnification characteristics in both Schwarzschild-like BHs in KR gravity pierced by CSs (SKRCS) and Schwarzschild-like BHs in KR gravity with CoS (SKRCoS) spacetime geometries through dual approaches: perturbative expansions yielding approximate solutions and exact elliptic integral formulations providing complete mathematical descriptions across parameter spaces. For CS configurations characterized by Lorentz violation (LV) and CS parameters, deflection angles exhibit systematic modifications through composite geometric factors, while CoS geometries demonstrate distinct deflection characteristics incorporating additional topological parameters that fundamentally alter light propagation dynamics. Magnification analysis reveals distinctive critical curve positioning modifications and amplitude scaling relationships enabling observational discrimination between exotic BH scenarios and conventional spacetime geometries. Strong field analysis utilizing established mathematical frameworks establishes logarithmic divergence coefficients characterizing fundamental scaling behavior in photon sphere proximity regimes. Observational constraints from Solar System precision tests restrict LV parameters within stringent bounds, while galactic-scale observations permit expanded parameter ranges for CS and CoS configurations.

Figures

Figures reproduced from arXiv: 2507.22673 by the authors.

**Figure 1.** FIG. 1: (a) Gravitational lensing visualization depicting how light from distant galaxies is distorted by an intervening massive [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** FIG. 2: (a) Hubble’s observations illustrate how the gravity of a foreground BH bends and deflects the light from a distant star [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** demonstrates the systematic influence of LV parameter variations on magnification characteristics within the strong CS regime β = 0.1, revealing profound modifications to conventional gravitational lensing behavior. The observationally constrained scenarios in panels (a) and (b) exhibit substantially different critical curve positioning and magnification amplitude scaling compared to theoretical exploration regimes … view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: illustrates the fundamental transition regime where CS effects become negligible (β = 1), effectively reducing the SKRCS configuration to pure KR gravity modifications, thereby providing essential baseline comparisons for understanding CS contributions to gravitational…

**Figure 5.** Figure 5: represents the extreme CS parameter regime (β = 10) where topological defect contributions systematically suppress LV effects, demonstrating the complex interplay between these two fundamental modifications to gravitational lensing phenomena. The observationally constr…

**Figure 6.** Figure 6: demonstrates the systematic influence of LV parameter variations within the minimal CoS regime (α = 0.01), revealing fundamental modifications to conventional gravitational lensing behavior under conditions where CoS contributions remain negligible compared to LV effec…

**Figure 7.** Figure 7: illustrates the intermediate CoS regime (α = 0.3) where string cloud contributions begin to substantially influence the magnification characteristics, demonstrating the complex interplay between LV and CoS effects in determining observable lensing signatures. The obser…

**Figure 8.** Figure 8: represents the extreme CoS parameter regime (α = 0.95) where string cloud contributions achieve near-maximal values within the theoretical constraint α < 1, demonstrating the dramatic influence of strong CoS fields on gravitational lensing magnification characteristics…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics
gr-qc 2025-08 reject novelty 3.0 of 10

A parameter scan of geodesic, thermal, shadow, and lensing observables for Kalb-Ramond ModMax black holes, with sign errors in the temperature, specific heat, and deflection angle formulas.

Reference graph

Works this paper leans on

73 extracted references · 58 canonical work pages · cited by 1 Pith paper

[1]

Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley & Sons, New York, 1972)

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley & Sons, New York, 1972)

work page 1972
[2]

S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity(Addison Wesley, San Francisco, 2004)

2004
[3]

Clifton, P

T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513, 1-189 (2012)

2012
[4]

V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989)

1989
[5]

Colladay and V

D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998)

1998
[6]

V. A. Kostelecky and N. Russell, Rev. Mod. Phys. 83, 11 (2011)

work page 2011
[7]

Altschul, Q

B. Altschul, Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D 81, 065028 (2010)

work page 2010
[8]

L. A. Lessa, J. E. G. Silva, R. V. Maluf and C. A. S. Almeida, Eur. Phys. J. C 80, no.4, 335 (2020)

work page 2020

Show all 73 references

[9]

K. Yang, Y. Z. Chen, Z. Q. Duan and J. Y. Zhao, Phys. Rev. D 108, 124004 (2023). 25

2023
[10]

Ednaldo L. B. Junior, Jos´ e Tarciso S. S. Junior, Francisco S. N. Lobo, Manuel E. Rodrigues, Diego Rubiera-Garcia, Lu ´ ıs F. Dias da Silva, Henrique A. Vieira, Phys. Rev. D 110, 024077 (2024)

2024
[11]

al-Badawi, S

A. al-Badawi, S. Shaymatov and I. Sakallı, Eur. Phys. J. C 84, no.8, 825 (2024). [arXiv:2408.09228 [gr-qc]]

2024 arXiv
[12]

Mangut, H

M. Mangut, H. G¨ ursel and˙I. Sakallı, Chin. Phys. C 49, no.6, 065106 (2025)

2025
[13]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects(Cambridge University Press, Cambridge, 1994)

1994
[14]

M. B. Hindmarsh and T. W. B. Kibble, Rept. Prog. Phys. 58, 477-562 (1995)

1995
[15]

Vachaspati, Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons(Cambridge University Press, Cambridge, 2006)

T. Vachaspati, Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons(Cambridge University Press, Cambridge, 2006)

2006
[16]

Aryal, L

M. Aryal, L. H. Ford and A. Vilenkin, Phys. Rev. D 34, 2263 (1986)

1986
[17]

Linde, Phys

A. Linde, Phys. Rev. D 49, 748-754 (1994)

1994
[18]

P. S. Letelier, Phys. Rev. D 20, 1294 (1979)

1979
[19]

Stachel, Phys

J. Stachel, Phys. Rev. D 26, 1281 (1982)

1982
[20]

Liu and X

Y. Liu and X. Zhang, Chin. Phys. C 47, no.12, 125103 (2023)

2023
[21]

J. R. Nascimento, A. Yu. Petrov, P. J. Porfirio and A. R. Soares, Phys. Rev. D 99, 064049 (2019)

2019
[22]

Schneider, J

P. Schneider, J. Ehlers and E. E. Falco, Gravitational Lenses (Springer-Verlag, New York, 1992)

1992
[23]

A. O. Petters, H. Levine and J. Wambsganss, Singularity Theory and Gravitational Lensing(Birkhauser Boston, Cambridge, 2001)

2001
[24]

Chandrasekhar, The Mathematical Theory of Black Holes(Oxford University Press, Oxford, 1992)

S. Chandrasekhar, The Mathematical Theory of Black Holes(Oxford University Press, Oxford, 1992)

1992
[25]

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

2014
[26]

Sucu and ˙I

E. Sucu and ˙I. Sakall.ı, Phys. Rev. D 111, no.6, 064049 (2025)

2025
[27]

K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000)

2000
[28]

Bozza, Phys

V. Bozza, Phys. Rev. D 66, 103001 (2002)

2002
[29]

Bozza, Phys

V. Bozza, Phys. Rev. D 67, 103006 (2003)

2003
[30]

R. D. Blandford and R. Narayan, Ann. Rev. Astron. Astrophys. 30, 311-358 (1992)

1992
[31]

Bartelmann and P

M. Bartelmann and P. Schneider, Phys. Rept. 340, 291-472 (2001)

2001
[32]

C. S. Kochanek, in Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro, eds. G. Meylan, P. Jetzer and P. North (Springer-Verlag, Berlin, 2006), pp. 91-268

2006
[33]

Treu, Ann

T. Treu, Ann. Rev. Astron. Astrophys. 48, 87-125 (2010)

2010
[34]

Ahmed, A

F. Ahmed, A. Al-Badawi and ˙I. Sakallı, Phys. Dark Univ. 49, 101988 (2025)

2025
[35]

Ahmed, ˙I

F. Ahmed, ˙I. Sakallı and A. Al-Badawi, Phys. Lett. B 864, 139448 (2025)

2025
[36]

Ahmed, A

F. Ahmed, A. Al-Badawi and I. Sakallı, Eur. Phys. J. C 85, no.5, 554 (2025)

2025
[37]

The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, no.1, L1 (2019)

2019
[38]

Abuter et al

R. Abuter et al. [GRA VITY], Astron. Astrophys.615, L15 (2018)

2018
[39]

Do et al., Science 365, no.6454, 664-666 (2019)

T. Do et al., Science 365, no.6454, 664-666 (2019)

2019
[40]

LSST Science Collaboration, arXiv preprint (2009)

2009
[41]

Euclid Collaboration, ESA/SRE(2011)12 (2011)

2011
[42]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists(Springer-Verlag, Berlin, 1971)

1971
[43]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, New York, 2007)

2007
[44]

Capozziello and M

S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167-321 (2011)

2011
[45]

T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451-497 (2010)

2010
[46]

E. F. Eiroa, Phys. Rev. D 73, 043002 (2006)

2006
[47]

Bhadra, Phys

A. Bhadra, Phys. Rev. D 67, 103009 (2003)

2003
[48]

C. Liu, L. Tang and J. Jing, Int. J. Mod. Phys. D 31, no.06, 2250041 (2022)

2022
[49]

A. F. Zakharov, A. A. Nucita, F. DePaolis and G. Ingrosso, New Astronomy 10, 479 (2005)

2005
[50]

Available at:https://webbtelescope

NASA, ESA, Ann Feild (STScI), Frank Summers (STScI), Webb Space Telescope Archive (2018). Available at:https://webbtelescope. org/contents/media/images/2018/23/4149-Image

2018
[51]

Battista and G

E. Battista and G. Esposito, Eur. Phys. J. C 82, no.12, 1088 (2022)

2022
[52]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions(Cambridge University Press, Cambridge, 2010)

2010
[53]

Bozza and L

V. Bozza and L. Mancini, Gen. Relativ. Gravit. 39, no.7, 1041-1063 (2007)

2007
[54]

K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002)

2002
[55]

Fei and F

Q. Fei and F. Shen, Nucl. Phys. B 1005, 116590 (2024)

2024
[56]

Fei, Universe 10, no.6, 251 (2024)

Q. Fei, Universe 10, no.6, 251 (2024)

2024
[57]

R. A. Safonova, M. C. Loupias and E. E. Falco, Mod. Phys. Lett. A 16, no.3, 153-162 (2001)

2001
[58]

Jaini, A

A. Jaini, A. T. Deller, Y. Wang, E. Lenc and M. Glowacki, Publ. Astron. Soc. Austral. 42, e060 (2025)

2025
[59]

K. Xu, Y. P. Jing, H. Gao, X. Luo and M. Li, Astrophys. J. 973, no.2, 102 (2024)

2024
[60]

Available at: https://science

NASA, ESA, Ann Feild (STScI), Frank Summers (STScI), Hubble Space Telescope Archive (2000). Available at: https://science. nasa.gov/asset/hubble/the-gravitational-field-of-the-cluster-acts-like-a-zoom-lens/

2000
[61]

J. P. S. Lemos and V. T. Zanchin, Phys. Rev. D 54, 3840 (1996)

1996
[62]

Bakala, A

P. Bakala, A. Bakalov´ a, R. L. Placa, M. Falanga and L. Stella, Astron. Astrophys. 673, A164 (2023)

2023
[63]

Emparan and H

R. Emparan and H. S. Reall, Living Rev. Rel. 11, 6 (2008)

2008
[64]

Padmanabhan, Rept

T. Padmanabhan, Rept. Prog. Phys. 73, 046901 (2010)

2010
[65]

D. J. Bacon, A. R. Refregier and R. S. Ellis, Mon. Not. Roy. Astron. Soc. 318, 625-640 (2000)

2000
[66]

J. A. Tyson, G. P. Kochanski and I. P. Dell’Antonio, Astrophys. J. Lett. 498, no.2, L107-L110 (1998)

1998
[67]

Darwin, Proc

C. Darwin, Proc. Roy. Soc. Lond. A 249, no.1257, 180-194 (1959)

1959
[68]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973). 26

1973
[69]

Frittelli, T

S. Frittelli, T. P. Kling and E. T. Newman, Phys. Rev. D 61, 064021 (2000)

2000
[70]

K. S. Virbhadra, D. Narasimha and S. M. Chitre, Astron. Astrophys. 337, 1-8 (1998)

1998
[71]

E. F. Eiroa, G. E. Romero and D. F. Torres, Phys. Rev. 66 66, 024010 (2002)

2002
[72]

A. S. Majumdar and N. Mukherjee, Int. J. Mod. Phys. D 14, no.7, 1095-1129 (2005)

2005
[73]

Whisker, Phys

R. Whisker, Phys. Rev. D 71, 064004 (2005)

2005

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley & Sons, New York, 1972)

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(John Wiley & Sons, New York, 1972)

work page 1972

[2] [2]

S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity(Addison Wesley, San Francisco, 2004)

2004

[3] [3]

Clifton, P

T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513, 1-189 (2012)

2012

[4] [4]

V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989)

1989

[5] [5]

Colladay and V

D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998)

1998

[6] [6]

V. A. Kostelecky and N. Russell, Rev. Mod. Phys. 83, 11 (2011)

work page 2011

[7] [7]

Altschul, Q

B. Altschul, Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D 81, 065028 (2010)

work page 2010

[8] [8]

L. A. Lessa, J. E. G. Silva, R. V. Maluf and C. A. S. Almeida, Eur. Phys. J. C 80, no.4, 335 (2020)

work page 2020