Gravitational lensing of the wormhole in the Eddington-inspired Born-Infeld spacetime with a cosmic string

Lei-Hua Liu; Li-juan Zhou; Shu-Qing Zhong; Xin-Fei Li; Yan-Zhi Meng

REVIEW 1 major objections 4 minor 100 references

Gravitational lensing of the wormhole in the Eddington-inspired Born-Infeld spacetime with a cosmic string

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A cosmic string threading an EiBI wormhole changes strong-field lensing observables: the first image's angular separation grows and its relative brightness falls as the string parameter a decreases.

desk verdict A routine Bozza-formalism extension of a known conical EiBI wormhole whose quantitative results are undercut by inconsistent distance relations, a mis-scaled magnification, and a factor-1000 table error. read the letter →

arxiv 2505.07420 v2 pith:XQBZRXNO submitted 2025-05-12 gr-qc

Xin-Fei Li , Lei-Hua Liu , Yan-Zhi Meng , Shu-Qing Zhong , Li-Juan Zhou This is my paper · ORCID

classification gr-qc

keywords gravitationallensingwormholecosmicstringEddington-inspiredBorn-Infeldgravitystrong-fieldlimitdeflectionanglerelativisticimagesEllis-Bronnikov

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

The paper studies gravitational lensing by the simplest traversable wormhole in Eddington-inspired Born-Infeld (EiBI) gravity when a cosmic string passes through it. It aims to show that the string's conical deficit changes the strong-field deflection enough to be observable: for a fixed wormhole throat, a smaller string parameter $a$ (higher string tension) makes the first relativistic image sit farther from the others and makes the first image dimmer relative to the rest. These two observables, the angular separation $s$ and the brightness ratio $R$, come out independent of the wormhole throat radius, so measuring them would isolate the string's tension. The paper also computes Einstein radii for bulge and LMC lensing in the weak field, predicting larger image scales than the string-free EiBI wormhole. A sympathetic reader would care because this gives a concrete way to test both EiBI gravity and cosmic strings with strong lensing observations.

What carries the argument

The load-bearing object is the conical wormhole metric $ds^2 = -dt^2 + (1+\varepsilon/r^2)^{-1}dr^2 + r^2(d\theta^2+a^2\sin^2\theta\,d\phi^2)$, where $a=1-4G\mu$ is the cosmic string parameter. The deflection angle is expressed through a complete elliptic integral of the first kind; in the strong-field limit $\varepsilon=-\lambda^2$ this collapses to the logarithmic form $\Lambda(\theta)=-(1/a)\log(D_{OL}\theta/\lambda -1)+(3/a)\log 2 -\pi$. From that expansion the paper obtains the angular positions $\theta_n$, magnifications $\mu_n$, and the standard strong-lensing observables $s$ and $R$. The string parameter enters only through products like $a\pi$ in exponentials, which is why $s$ and $R$ depend on $a$ alone.

What would settle it

Measure the angular separation $s$ and brightness ratio $R$ of the first two relativistic images around a candidate wormhole at sub-microarcsecond resolution. If the observed pair does not satisfy $R = (e^{-3a\pi}+8e^{-6a\pi})/(e^{-5a\pi}+8e^{-10a\pi})$ with the same $a$ that fits $s = 8\theta_\infty/e^{3a\pi}$, the model's strong-field predictions fail.

Watch

Extended reading notes

Core claim

The central claim is that for the conical wormhole spacetime of EiBI gravity, the cosmic string parameter $a$ enters the strong-field lensing observables exponentially. With the throat radius $\lambda$ and the deflection angle diverging logarithmically as the impact parameter approaches $a\lambda$, the paper derives $s = 8\theta_\infty / e^{3a\pi}$ and $R = (e^{-3a\pi}+8e^{-6a\pi})/(e^{-5a\pi}+8e^{-10a\pi})$. As $a$ decreases from 1, the angular separation $s$ grows and the brightness ratio $R$ falls, while both stay independent of the throat radius $\lambda$. In the $a\to 1$ limit the formulas reduce to the string-free wormhole case, and the authors argue the differences are large enough to distinguish the string-threaded EiBI wormhole from both the string-free wormhole and a Schwarzschild black hole using sub-microarcsecond observations.

Load-bearing premise

The predictions assume that the distance from the wormhole to the source, not the distance from the wormhole to the observer, controls how the cosmic string shrinks the apparent image angle; if that choice is wrong, the predicted image positions and magnifications are off by the ratio of those distances.

Editorial extensions

If this is right

In the strong-field limit, the angular separation $s$ grows exponentially as the string parameter $a$ drops, so higher-tension cosmic strings push the first relativistic image farther from the packed set of outer images.
The brightness ratio $R$ falls with the same parameter, so the first image becomes relatively fainter compared with the sum of all outer images.
Because $s$ and $R$ are independent of the wormhole throat radius, they provide a direct probe of string tension alone, without needing to know the throat size.
For throat radii around $10^{10}$-$10^{11}\;\mathrm{km}$ modeled on Sagittarius A*, the predicted angular separation is of the same order as the Schwarzschild strong-lensing separation, so high-resolution observations can distinguish the string-threaded EiBI wormhole from a Schwarzschild black hole.
In the weak-field limit the model predicts a single image whose magnification decreases as $a$ increases, with Einstein radii for bulge and LMC lensing larger than those of the string-free EiBI wormhole.

Reading between the lines

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

If the string parameter is ignored, strong-lensing estimates of the wormhole throat radius from absolute image positions would be biased; ratios such as $R$ may be a more robust probe.
Measuring both $s$ and $R$ in the same lens system would over-determine $a$, because both are exponential functions of $a\pi$; a mismatch between the two would reveal any error in how the impact parameter is related to image angle.
The same exponential scaling should appear in time delays between relativistic images, giving a complementary observable the paper does not compute.
Independent cosmic-string tension bounds from gravitational-wave backgrounds could be combined with these lensing predictions to accept or reject the EiBI wormhole interpretation.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: this is a standard Bozza-formalism exercise applied to a known Ellis-Bronnikov-type metric with a cosmic string. The strong-field expansion and the s/R observables for this specific spacetime are not in the cited Ahmed papers, so the novelty, though modest, is real. The authors are honest about where the metric comes from, and the model parameters come from independent bounds, not fitted to data.

What is good: the exact deflection angle is taken from prior work and the elliptic-integral expansion to Eq. (17) is standard and checks out. The idea that the string parameter changes the image separation and brightness ratio is plausible.

The problems are in the lensing geometry and the numerics. Eq. (20) sets b = a D_LS θ and Appendix A derives it, but Eq. (21) and everything after it use b = a D_OL θ, with D_OL appearing in the log. These are not interchangeable. If D_OL is right, Eq. (20) and Appendix A are wrong; if D_LS is right, then θ0_n, the image positions in Eq. (26), and the tables are all scaled by D_LS/D_OL and are wrong. The paper never chooses a single geometry.

Eq. (28) is also suspect. Differentiating the lens equation with Eq. (25) gives magnifications scaling as (1/ψ)(D_OS/D_LS), not as ψ D_LS/D_OS as printed. That error propagates into Figs. 5-6 and into the brightness ratio R. There is also a concrete numerical error in Table 1: for λ=10^9 km and D_OL=8.5 kpc, θ∞ should be about 786 microarcsec, not 0.786—a factor of 1000. The weak-field lens equation (34) does not follow from (19), (20), and (33) either; the constant term has the wrong distance combination.

None of this kills the qualitative claim that a<1 enhances strong deflection relative to a=1. But the quantitative predictions, including the distinguishability statements, are not supported as written. A corrected derivation and recomputed tables might make this a publishable routine extension. I would not cite it until that happens.

Recommendation: send to a referee, because the errors are identifiable and fixable, but the current version should be rejected or returned for major revision rather than accepted.

Referee Report

1 major / 4 minor

Summary. The manuscript studies gravitational lensing by a traversable wormhole in Eddington-inspired Born-Infeld (EiBI) gravity threaded by a cosmic string, using the metric of Eq. (3). It derives the deflection angle in both weak and strong field limits, constructs the strong-field lens equation, obtains image positions and magnifications, and defines the angular separation s and brightness ratio R. It then applies the formalism to Sagittarius A* and to bulge/LMC microlensing-like configurations, producing tables of Einstein angles and radii. The central claims are that the cosmic string parameter a increases the deflection relative to the Ellis-Bronnikov wormhole, that s grows and R decreases as a decreases, and that these observables are independent of the throat radius.

Significance. If the results were correct, the paper would provide a concrete, analytically tractable extension of strong-field lensing to a wormhole with a cosmic string, with parameters that are independently bounded: the EiBI parameter is constrained by GW170817/GRB170817A to |ε| ≤ 10^37 m^2, and the string tension is taken at the GUT scale, Gµ ∼ 10^-6. The derivation is analytic throughout and does not fit any free parameter to lensing data, and the paper checks the a → 1 limit against the Ellis-Bronnikov results. These are genuine strengths. The significance is currently conditional, however, because the quantitative predictions are built on an internally inconsistent set of distance relations and on a magnification formula that has the wrong scaling.

major comments (1)

[§3, Eq. (28)] The impact-parameter relation is not consistent across the strong-field derivation. Eq. (20) states b ≃ a D_LS θ and Appendix A derives b = D_LS sin(aθ) by setting u = 1/D_LS in the conical-spacetime trajectory, but Eq. (21) uses Λ(θ) = -(1/a) log(D_OL θ/λ - 1) + (3/a) log 2 - π, which follows from Eq. (17) only if b/(aλ) = D_OL θ/λ, i.e. b = a D_OL θ. The lens-plane geometry of Fig. 4 places the observer at D_OL, not D_LS, so the Appendix A construction does not describe the angle θ seen by the observer. Because Eq. (24) for θ0_n, Eq. (25) for ΔΛ_n, Eq. (26) for image positions, Eq. (28) for magnifications, and Table 1 all carry D_OL, the strong-field observables are not derived from a single physical geometry. If the intended relation is b = a D_OL θ, then Eq. (20) and Appendix A are wrong; if b = a D_LS θ is intended, then θ∞ = λ/D_LS and the numerical values in Table 1 are mis-scaled. This must be resolved and all subsequent equations and tables re-derived consistently.

minor comments (4)

[§3, after Eq. (28)] The sentence 'the magnification of n-th image increases exponentially with increasing n' is the opposite of what Eq. (28) implies: e^{(2n+1)aπ} appears in the denominator, so μ_n decreases as n grows. The following sentence, stating that the first image has the highest magnification, confirms the intended behavior; the wording should be corrected.
[§5, Conclusion] The conclusion states that the angular separation 'increase[s] as cosmic string parameter increases', but Eq. (30) gives s ∝ 8θ∞/e^{3aπ}, so s decreases as a increases (equivalently, increases as a decreases). This is also inconsistent with the body text after Eq. (30) and with the abstract's claim; the conclusion sentence should be corrected.
[§2.2, around Eqs. (4)–(6)] The symbol L is used first for the Lagrangian in Eq. (4) and then for the conserved angular momentum in Eq. (6); the statement 'For the null geodesic, it is well known that L = 0' is confusing, since it is the Lagrangian that vanishes for null curves, not the angular momentum used in Eq. (7). Please use separate symbols.
[Throughout] There are several typographical and wording errors, including 'Eills-Bronnikov' in the abstract, 'Bozzaa' for Bozza, 'ins-pired' in the title line, and 'deﬂection angel' in Section 2.2. A careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation chain is analytical from the metric and geodesic equations, with no fitting to lensing data; the noted D_LS/D_OL mismatch is an internal-consistency issue, not a circular reduction.

full rationale

The paper's central chain is self-contained in the sense relevant to circularity: metric (3) yields the null geodesic equations (5)-(11), the deflection angle (14) and its strong-field expansion (17), the lens equation (19), the image positions (24)-(26), magnifications (28), and finally the observables s and R in Eqs. (30)-(31). No parameter appearing in these observables is fitted to lensing data. The wormhole throat radius lambda and cosmic string parameter a are inputs constrained by independent bounds: the GW170817 speed-of-gravity limit gives |epsilon| <= 10^37 m^2 (hence lambda <= 10^15 km), and the string tension is estimated from the GUT symmetry-breaking scale. The observables are then closed-form functions of those inputs. The a -> 1 limit recovers the Ellis-Bronnikov wormhole results, which provides an independent consistency check rather than a fitting procedure. The load-bearing citations for the spacetime metric and geodesic motion, e.g., [89], are to prior work by other authors, not to the present authors; the self-citations in the reference list ([46]-[49]) concern context and are not used to justify the central lensing derivation. The discrepancy flagged by the reader, between Eq. (20) (b ~ a D_LS theta) and Eq. (21) (which effectively uses b = a D_OL theta through log(D_OL theta/lambda - 1)), is a possible internal-consistency or correctness problem in the lens geometry, but it is not a circular reduction: neither equation is defined in terms of the predicted observables, and the derived s and R do not reduce to fitted inputs. Therefore no circularity is found and the score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its free parameters are the cosmic string parameter a and the wormhole throat radius λ, both chosen from external constraints. The main unstated assumption is the impact parameter relation, which is inconsistent in the text. The derivation otherwise relies on standard math and prior spacetime solutions.

free parameters (2)

a (cosmic string parameter) = 1 - 4e-6 (GUT scale assumption)
Appears in every lensing observable. The paper adopts a value from grand unified theory symmetry breaking, not from lensing data.
λ (wormhole throat radius) = varied from 10^9 to 10^12 km in tables
A model parameter constrained by the GW170817 bound on the Eddington parameter; the paper scans a range to produce observable predictions rather than fitting it.

assumptions (5)

domain assumption Metric (3) describes an EiBI wormhole with a cosmic string
Taken from previous works (refs [88,89]); the paper does not derive the spacetime but reviews it as the starting point.
standard math Null geodesics are obtained from Euler-Lagrange equations on the equatorial plane
Used to derive the exact deflection angle (Eq. 11).
domain assumption The Virbhadra-Ellis lens equation (18) applies to this wormhole geometry
Adopted from ref [94] without adaptation for the conical spacetime, which may be problematic since the lens geometry is changed by the cosmic string.
ad hoc to paper The impact parameter b relates to the image angle θ as b = a D_OL θ
Eq. (20) states b = D_LS sin(aθ), but Eq. (21) and subsequent derivations use b = a D_OL θ. This is an unflagged, inconsistent assumption that drives all the strong-field observables.
standard math Expansion of the complete elliptic integral of the first kind near unity
Used to obtain the logarithmic divergence in the strong-field limit (Eq. 17).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Gravitational lensing of the wormhole in the Eddington-inspired Born-Infeld spacetime with a cosmic string." pith.science (2026). https://pith.science/paper/XQBZRXNO

@misc{pith2026250507420,
  author       = {Pith},
  title        = {Pith review of: Gravitational lensing of the wormhole in the Eddington-inspired Born-Infeld spacetime with a cosmic string},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQBZRXNO}},
  note         = {Machine review of arXiv:2505.07420}
}

read the original abstract

In this work we study gravitational lensing of the wormhole in the Eddington-inspired Born-Infeld (EiBI) spacetime that incorporates with a cosmic string. It was found that the presence of cosmic string can enhance the light deflection in strong field limit, compared to the case of the Eills-Bronnikov wormhole. The magnification effects of this composite structure could cause some substantial impacts on the angle separation between the first and the rest of the images, and their relative brightness. Furthermore, based on these observables, we model some observable aspects in the strong and the weak field limits. The presence of a cosmic string can affect some distinguishable observables compared to the wormhole without cosmic string. This work could deepen our understanding of the spacetime structure of the wormhole in EiBI spacetime with one-dimensional topological defects.

Figures

Figures reproduced from arXiv: 2505.07420 by the authors.

**Figure 1.** Scheme of the deflection of light δϕ for the WH with a cosmic string perpendicular on the equilateral plane. Cosmic string and WH are represented by the black straight line and gray disk, respectively. To obtain the null geodesic associated with (3), the variational method will be adopted. The length S of a smooth curve on a spacetime with metric (3) is given by S = Z dxL = Z dτ r gµν dx µ dτ dx µ dτ , (4) where τ … view at source ↗

**Figure 2.** Effective potential vs radius for various cos [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Visual lensing profile. The light emitted [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Magnification of the first and the second [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Plot for magnification of image in terms of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

100 extracted references · 77 canonical work pages

[1]

Deser and G

S. Deser and G. W. Gibbons, Born-Infeld- Einstein actions?, Class. Quant. Grav. 15 (1998) L35-L39

1998
[2]

Banados and P

M. Banados and P. G. Ferreira, Eddington’s theory of gravity and its progeny, Phys. Rev. Lett. 105 (2010) 011101. [erratum: Phys. Rev. Lett. 113 (2014) no.11, 119901.]

2010
[3]

Beltran Jimenez, L

J. Beltran Jimenez, L. Heisenberg, G. J. Olmo and D. Rubiera-Garcia, Born–Infeld inspired modiﬁcations of gravity, Phys. Rept. 727 (2018) 1-129

2018
[4]

G. J. Olmo, Palatini Approach to Modiﬁed Gravity: f(R) Theories and Beyond, Int. J. Mod. Phys. D 20 (2011) 413-462

2011
[5]

G. J. Olmo, D. Rubiera-Garcia and H. Sanchis- Alepuz, Geonic black holes and remnants in Eddington-inspired Born-Infeld gravity, Eur. Phys. J. C 74 (2014) 2804

2014
[6]

F. S. N. Lobo, G. J. Olmo and D. Rubiera- Garcia, Semiclassical geons as solitonic black hole remnants, JCAP 07 (2013) 011

2013
[7]

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. C 75 (2015) 253. [erratum: Eur. Phys. J. C 75 (2015) 331.] 10

2015
[8]

Shaikh, Wormholes with nonexotic matter in Born-Infeld gravity, Phys

R. Shaikh, Wormholes with nonexotic matter in Born-Infeld gravity, Phys. Rev. D 98 (2018) no.6, 064033

2018

Show all 100 references

[9]

G. J. Olmo, E. Orazi and D. Rubiera-Garcia, Multicenter solutions in Eddington-inspired Born–Infeld gravity, Eur. Phys. J. C 80 (2020) no.11, 1018

2020
[10]

K. Yang, X. L. Du and Y. X. Liu, Linear per- turbations in Eddington-inspired Born-Infeld gravity, Phys. Rev. D 88 (2013) 124037

2013
[11]

X. L. Du, K. Yang, X. H. Meng and Y. X. Liu, Large Scale Structure Formation in Eddington- inspired Born-Infeld Gravity, Phys. Rev. D 90 (2014) 044054

2014
[12]

M. S. Morris and K. S. Thorne, Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56 (1988) 395-412

1988
[13]

Einstein and N

A. Einstein and N. Rosen, The Particle Prob- lem in the General Theory of Relativity, Phys. Rev. 48 (1935) 73-77

1935
[14]

R. W. Fuller and J. A. Wheeler, Causality and Multiply Connected Space-Time, Phys. Rev. 128 (1962) 919-929

1962
[15]

K. A. Bronnikov, Scalar-tensor theory and scalar charge, Acta Phys. Polon. B 4 (1973) 251-266

1973
[16]

P. P. Avelino and L. Sousa, Domain walls and other defects in Eddington-inspired Born- Infeld gravity, Phys. Rev. D 102 (2020) no.10, 104021

2020
[17]

T. W. B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387-1398

1976
[18]

Sarangi and S

S. Sarangi and S. H. H. Tye, Cosmic string production towards the end of brane inﬂation, Phys. Lett. B 536 (2002) 185-192

2002
[19]

E. J. Copeland, R. C. Myers and J. Polchinski, Cosmic F and D strings, JHEP 06 (2004) 013

2004
[20]

Dvali and A

G. Dvali and A. Vilenkin, Formation and evolu- tion of cosmic D strings, JCAP 03 (2004) 010

2004
[21]

Vilenkin, Cosmological Density Fluctua- tions Produced by Vacuum Strings, Phys

A. Vilenkin, Cosmological Density Fluctua- tions Produced by Vacuum Strings, Phys. Rev. Lett. 46 (1981) 1169-1172. [erratum: Phys. Rev. Lett. 46 (1981) 1496]

1981
[22]

C. F. Chang and Y. Cui, Stochastic Gravita- tional Wave Background from Global Cosmic Strings, Phys. Dark Univ. 29 (2020) 100604

2020
[23]

Sousa, P

L. Sousa, P. P. Avelino and G. S. F. Guedes, Full analytical approximation to the stochas- tic gravitational wave background generated by cosmic string networks, Phys. Rev. D 101 (2020) no.10, 103508

2020
[24]

Agazie et al

G. Agazie et al. [NANOGrav], The NANOGrav 15 yr Data Set: Evidence for a Gravitational- wave Background, Astrophys. J. Lett. 951 (2023) no.1, L8

2023
[25]

Antoniadis et al

J. Antoniadis et al. [EPTA], The second data release from the European Pulsar Timing Array - I. The dataset and timing analysis, Astron. Astrophys. 678 (2023) A48

2023
[26]

D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Ka- pur, A. F. Rogers, E. Thrane and J. Askew, et al. Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Ar- ray, Astrophys. J. Lett. 951 (2023) no.1, L6

2023
[27]

H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu, Z. Xue, R. N. Caballero, J. Yuan and Y. Xu, et al. Searching for the Nano- Hertz Stochastic Gravitational Wave Back- ground with the Chinese Pulsar Timing Ar- ray Data Release I, Res. Astron. Astrophys. 23 (2023) no.7, 075024

2023
[28]

Clement, Wormhole cosmic strings, Phys

G. Clement, Wormhole cosmic strings, Phys. Rev. D 51 (1995) 6803-6809

1995
[29]

E. F. Eiroa and C. Simeone, Cylindrical thin shell wormholes, Phys. Rev. D 70 (2004) 044008. 11

2004
[30]

E. F. Eiroa, E. Rub ´ ın de Celis and C. Simeone, Thin shells joining local cosmic string geome- tries, Eur. Phys. J. C 76 (2016) no.10, 546

2016
[31]

E. F. Eiroa, E. R. de Celis and C. Simeone, Thin shells associated to black string space- times, Eur. Phys. J. C 79 (2019) no.3, 272

2019
[32]

Bejarano, E

C. Bejarano, E. F. Eiroa and C. Simeone, Thin- shell wormholes associated with global cosmic strings, Phys. Rev. D 75 (2007) 027501

2007
[33]

Bambhaniya, O

P. Bambhaniya, O. Trivedi, I. Dymnikova, P. S. Joshi and M. Khlopov, On the interac- tions of black holes and cosmic strings, Phys. Dark Univ. 46 (2024) 101553

2024
[34]

Bartelmann and P

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

2001
[35]

Treu, Strong Lensing by Galaxies, Ann

T. Treu, Strong Lensing by Galaxies, Ann. Rev. Astron. Astrophys. 48 (2010) 87-125

2010
[36]

Vagnozzi, R

S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Horizon- scale tests of gravity theories and fundamental physics from the Event Horizon Telescope im- age of Sagittarius A, Class. Quant. Grav. 40 ...

2023
[37]

Bozza, S

V. Bozza, S. Capozziello, G. Iovane and G. Scarpetta, Strong ﬁeld limit of black hole gravitational lensing, Gen. Rel. Grav. 33 (2001) 1535-1548

2001
[38]

Tsukamoto, Strong deﬂection limit analysis and gravitational lensing of an Ellis wormhole, Phys

N. Tsukamoto, Strong deﬂection limit analysis and gravitational lensing of an Ellis wormhole, Phys. Rev. D 94 (2016) no.12, 124001

2016
[39]

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

N. Tsukamoto, Deﬂection angle in the strong deﬂection limit in a general asymptotically ﬂat, static, spherically symmetric spacetime, Phys. Rev. D 95 (2017) no.6, 064035

2017
[40]

Igata, Deﬂection angle in the strong de- ﬂection limit: a perspective from local geo- metrical invariants and matter distributions, [arXiv:2503.02320 [gr-qc]]

T. Igata, Deﬂection angle in the strong de- ﬂection limit: a perspective from local geo- metrical invariants and matter distributions, [arXiv:2503.02320 [gr-qc]]
[41]

E. F. Eiroa, Gravitational lensing by Einstein- Born-Infeld black holes, Phys. Rev. D 73 (2006) 043002

2006
[42]

S. S. Zhao and Y. Xie, Strong ﬁeld gravitational lensing by a charged Galileon black hole, JCAP 07 (2016) 007

2016
[43]

Canfora, E

F. Canfora, E. F. Eiroa and C. M. Sendra, Spherical Skyrmion black holes as gravitational lenses, Eur. Phys. J. C 78 (2018) no.8, 659

2018
[44]

N. U. Molla and U. Debnath, Shadows and strong gravitational lensing by Van der Waals black hole in homogeneous plasma, Annals Phys. 453 (2023) 169304

2023
[45]

X. M. Kuang and A. ¨Ovg¨ un, Strong gravi- tational lensing and shadow constraint from M87* of slowly rotating Kerr-like black hole, Annals Phys. 447 (2022) 169147

2022
[46]

L. H. Liu, M. Zhu, W. Luo, Y. F. Cai and Y. Wang, Microlensing eﬀect of a charged spherically symmetric wormhole, Phys. Rev. D 107 (2023) no.2, 024022

2023
[47]

K. Gao, L. H. Liu and M. Zhu, Microlens- ing eﬀects of wormholes associated to black- hole spacetimes, Phys. Dark Univ. 41 (2023) 101254

2023
[48]

Gao and L

K. Gao and L. H. Liu, Can wormholes and black holes be distinguished by magniﬁcation?, [arXiv:2307.16627 [gr-qc]]

arXiv
[49]

Gao and L

K. Gao and L. H. Liu, Microlensing and event rate of static spherically symmetric wormhole, Phys. Lett. B 858 (2024) 139019

2024
[50]

N. U. Molla and U. Debnath, Gravitational Lensing of Acoustic Charged Black Holes, As- trophys. J. 947 (2023) no.1, 14. [erratum: As- trophys. J. 958 (2023) no.2, 196.]

2023
[51]

Q. Qi, Y. Meng, X. J. Wang and X. M. Kuang, Gravitational lensing eﬀects of black hole with conformally coupled scalar hair, Eur. Phys. J. C 83 (2023) no.11, 1043. 12

2023
[52]

Zhang, H

Y. Zhang, H. Liu, D. Wen and H. Zhang, Strong lensing as a probe of braneworld, [arXiv:2312.15714 [gr-qc]]

arXiv
[53]

Y. Duan, S. Lin and J. Jia, Deﬂection and grav- itational lensing with ﬁnite distance eﬀect in the strong deﬂection limit in stationary and ax- isymmetric spacetimes, JCAP 07 (2023) 036

2023
[54]

E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo, M. E. Rodrigues, D. Rubiera- Garcia, L. F. D. da Silva and H. A. Vieira, Gravitational lensing of a Schwarzschild-like black hole in Kalb-Ramond gravity, Phys. Rev. D 110 (2024) no.2, 024077

2024
[55]

K. S. Virbhadra and G. F. R. Ellis, Gravita- tional lensing by naked singularities, Phys. Rev. D 65 (2002) 103004

2002
[56]

K. S. Virbhadra and C. R. Keeton, Time delay and magniﬁcation centroid due to gravitational lensing by black holes and naked singularities, Phys. Rev. D 77 (2008) 124014

2008
[57]

K. K. Nandi, Y. Z. Zhang and A. V. Zakharov, Gravitational lensing by wormholes, Phys. Rev. D 74 (2006) 024020

2006
[58]

¨Ovg¨ un, Light deﬂection by Damour- Solodukhin wormholes and Gauss-Bonnet the- orem, Phys

A. ¨Ovg¨ un, Light deﬂection by Damour- Solodukhin wormholes and Gauss-Bonnet the- orem, Phys. Rev. D 98 (2018) no.4, 044033

2018
[59]

Shaikh, P

R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, Strong gravitational lensing by wormholes, JCAP 07 (2019) 028. [erratum: JCAP 12 (2023) E01.]

2019
[60]

Tsukamoto, Gravitational lensing by a Bronnikov-Kim wormhole under a weak-ﬁeld approximation and in a strong deﬂection limit, Phys

N. Tsukamoto, Gravitational lensing by a Bronnikov-Kim wormhole under a weak-ﬁeld approximation and in a strong deﬂection limit, Phys. Rev. D 105 (2022) no.6, 064013

2022
[61]

S. Zhou, M. Chen and J. Jia, Eﬀect of electric interaction on the deﬂection and gravitational lensing in the strong ﬁeld limit, Eur. Phys. J. C 83 (2023) no.9, 883

2023
[62]

Godani and G

N. Godani and G. C. Samanta, Gravitational lensing eﬀect in traversable wormholes, Annals Phys. 429 (2021) 168460

2021
[63]

T. Cai, H. Huang, Z. Wang and M. Zhu, Higher-order correction to weak-ﬁeld lensing of an Ellis-Bronnikov wormhole, Phys. Rev. D 108 (2023) no.2, 023004

2023
[64]

Hsieh, D

T. Hsieh, D. S. Lee and C. Y. Lin, Throat ef- fects on strong gravitational lensing in Kerr- like wormholes, Phys. Rev. D 111 (2025) no.4, 044051

2025
[65]

J. B. Jim´ enez, L. J. Garay and M. P. Garrote, Mimicking Wormholes in Born–Infeld Electro- dynamics, Universe 10 (2024) no.12, 459

2024
[66]

X. T. Cheng and Y. Xie, Probing a black- bounce, traversable wormhole with weak de- ﬂection gravitational lensing, Phys. Rev. D 103 (2021) no.6, 064040

2021
[67]

Atamurotov, F

F. Atamurotov, F. Sarikulov, A. Abdujabbarov and B. Ahmedov, Gravitational weak lensing by black hole in Horndeski gravity in presence of plasma, Eur. Phys. J. Plus 137 (2022) no.3, 336

2022
[68]

Tsukamoto, Gravitational lensing by two photon spheres in a black-bounce spacetime in strong deﬂection limits, Phys

N. Tsukamoto, Gravitational lensing by two photon spheres in a black-bounce spacetime in strong deﬂection limits, Phys. Rev. D 104 (2021) no.6, 064022

2021
[69]

M. S. Ali and S. Kauhsal, Gravitational lens- ing for stationary axisymmetric black holes in Eddington-inspired Born-Infeld gravity, Phys. Rev. D 105 (2022) no.2, 024062

2022
[70]

Kumar, S

J. Kumar, S. U. Islam and S. G. Ghosh, Inves- tigating strong gravitational lensing eﬀects by supermassive black holes with Horndeski grav- ity, Eur. Phys. J. C 82 (2022) no.5, 443

2022
[71]

Hensh, J

S. Hensh, J. Schee, A. Abdujabbarov and Z. Stuchl ´ ık, Strong gravitational lensing around Kehagias–Sfetsos compact objects surrounded by plasma, Eur. Phys. J. Plus 137 (2022) no.2, 242. 13

2022
[72]

Ghosh and A

S. Ghosh and A. Bhattacharyya, Analytical study of gravitational lensing in Kerr-Newman black-bounce spacetime, JCAP 11 (2022) 006

2022
[73]

E. L. B. Junior, F. S. N. Lobo, M. E. Rodrigues and H. A. Vieira, Gravitational lens eﬀect of a holonomy corrected Schwarzschild black hole, Phys. Rev. D 109 (2024) no.2, 2

2024
[74]

Y. Chen, P. Wang, H. Wu and H. Yang, Gravi- tational lensing by Born-Infeld naked singular- ities, Phys. Rev. D 109 (2024) no.8, 084014

2024
[75]

Kumaran and A

Y. Kumaran and A. ¨Ovg¨ un, Shadow and deﬂec- tion angle of asymptotic, magnetically-charged, non-singular black hole, Eur. Phys. J. C 83 (2023) no.9, 812

2023
[76]

M. A. A. de Paula, H. C. D. Lima, Junior, P. V. P. Cunha and L. C. B. Crispino, Elec- trically charged regular black holes in nonlin- ear electrodynamics: Light rings, shadows, and gravitational lensing, Phys. Rev. D 108 (2023) no.8, 084029

2023
[77]

Zhang, J

R. Zhang, J. Jing, Z. Peng and Q. Huang, Strong gravitational lensing of rotating regular black holes in non-minimally coupled Einstein- Yang-Mills theory*, Chin. Phys. C 47 (2023) no.10, 105105

2023
[78]

G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia and D. Saez-Chillon Gomez, Shadows and photon rings of regular black holes and geonic horizon- less compact objects, Class. Quant. Grav. 40 (2023) no.17, 174002

2023
[79]

C. H. Xie, Y. Zhang, Q. Sun, Q. Q. Li and P. F. Duan, Gravitational lensing by a stable rotating regular black hole, JCAP 05 (2024) 121

2024
[80]

K. Liao, M. Biesiada and Z. H. Zhu, Strongly Lensed Transient Sources: A Review, Chin. Phys. Lett. 39 (2022) no.11, 119801

2022
[81]

K. S. Virbhadra, Compactness of supermassive dark objects at galactic centers, Can. J. Phys. 102 (2024) 512

2024
[82]

Toscani, O

M. Toscani, O. Burke, C. Liu, N. B. Zamel, N. Tamanini and F. Pozzoli, Strongly lensed extreme mass-ratio inspirals, Phys. Rev. D 109 (2024) no.6, 063505

2024
[83]

Jusuﬁ, Conical Morris-Thorne Wormholes with a Global Monopole Charge, Phys

K. Jusuﬁ, Conical Morris-Thorne Wormholes with a Global Monopole Charge, Phys. Rev. D 98 (2018) no.4, 044016

2018
[84]

Furtado, J

C. Furtado, J. R. Nascimento, A. Y. Petrov, P. J. Porf ´ ırio and A. R. Soares, Strong grav- itational lensing in a spacetime with topologi- cal charge within the Eddington-inspired Born- Infeld gravity, Phys. Rev. D 103 (2021) no.4, 044047

2021
[85]

A. R. Soares, R. L. L. Vit´ oria and C. F. S. Pereira, Gravitational lensing in a topologically charged Eddington-inspired Born–Infeld spacetime, Eur. Phys. J. C 83 (2023) no.10, 903

2023
[86]

S. W. Wei and Y. X. Liu, Equatorial and quasi- equatorial gravitational lensing by Kerr black hole pierced by a cosmic string, Phys. Rev. D 85 (2012) 064044

2012
[87]

Jusuﬁ and A

K. Jusuﬁ and A. ¨Ovg¨ un, Light Deﬂection by a Quantum Improved Kerr Black Hole Pierced by a Cosmic String, Int. J. Geom. Meth. Mod. Phys. 16 (2019) no.08, 1950116

2019
[88]

Ahmed, Gravitational lensing by Morris- Thorne–type wormhole space-time with cosmic strings, EPL 142 (2023) no.3, 39002

F. Ahmed, Gravitational lensing by Morris- Thorne–type wormhole space-time with cosmic strings, EPL 142 (2023) no.3, 39002

2023
[89]

Ahmed, Geodesics motion of test parti- cles around Schwarzschild-Klinkhamer worm- hole with topological defects and gravitational lensing, JCAP 11 (2023) 010

F. Ahmed, Geodesics motion of test parti- cles around Schwarzschild-Klinkhamer worm- hole with topological defects and gravitational lensing, JCAP 11 (2023) 010

2023
[90]

Ahmed, Gravitational lensing in a space- time with cosmic string within the Eddington- inspired Born–Infeld gravity, Int

F. Ahmed, Gravitational lensing in a space- time with cosmic string within the Eddington- inspired Born–Infeld gravity, Int. J. Geom. Meth. Mod. Phys. 21 (2024) no.11, 2450187

2024
[91]

Linet, The Static Metrics with Cylindri- cal Symmetry Describing a Model of Cosmic Strings, Gen

B. Linet, The Static Metrics with Cylindri- cal Symmetry Describing a Model of Cosmic Strings, Gen. Rel. Grav. 17 (1985) 1109-1115. 14

1985
[92]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cam- bridge University Press, 2000

2000
[93]

J. R. Nascimento, A. Y. Petrov, P. J. Por- ﬁrio and A. R. Soares, Gravitational lensing in black-bounce spacetimes, Phys. Rev. D 102 (2020) no.4, 044021

2020
[94]

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

2000
[95]

S. Jana, G. K. Chakravarty and S. Mohanty, Constraints on Born-Infeld gravity from the speed of gravitational waves after GW170817 and GRB 170817A, Phys. Rev. D 97 (2018) no.8, 084011

2018
[96]

Abe, Gravitational Microlensing by the Ellis Wormhole, Astrophys

F. Abe, Gravitational Microlensing by the Ellis Wormhole, Astrophys. J. 725 (2010) 787-793

2010
[97]

M. D. Johnson, K. Akiyama, L. Blackburn, K. L. Bouman, A. E. Broderick, V. Cardoso, R. P. Fender, C. M. Fromm, P. Galison and J. L. G´ omez, et al. Galaxies 11 (2023) no.3, 61

2023
[98]

Y. M. Wu, Z. C. Chen and Q. G. Huang, Constraining the Polarization of Gravitational Waves with the Parkes Pulsar Timing Array Second Data Release, Astrophys. J. 925 (2022) no.1, 37

2022
[99]

Y. M. Wu, Z. C. Chen and Q. G. Huang, Cos- mological interpretation for the stochastic sig- nal in pulsar timing arrays, Sci. China Phys. Mech. Astron. 67 (2024) no.4, 240412

2024
[100]

Abbott et al

R. Abbott et al. [LIGO Scientiﬁc, Virgo and KAGRA], Constraints on Cosmic Strings Using Data from the Third Advanced LIGO–Virgo Observing Run, Phys. Rev. Lett. 126 (2021) no.24, 241102 15

2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Deser and G

S. Deser and G. W. Gibbons, Born-Infeld- Einstein actions?, Class. Quant. Grav. 15 (1998) L35-L39

1998

[2] [2]

Banados and P

M. Banados and P. G. Ferreira, Eddington’s theory of gravity and its progeny, Phys. Rev. Lett. 105 (2010) 011101. [erratum: Phys. Rev. Lett. 113 (2014) no.11, 119901.]

2010

[3] [3]

Beltran Jimenez, L

J. Beltran Jimenez, L. Heisenberg, G. J. Olmo and D. Rubiera-Garcia, Born–Infeld inspired modiﬁcations of gravity, Phys. Rept. 727 (2018) 1-129

2018

[4] [4]

G. J. Olmo, Palatini Approach to Modiﬁed Gravity: f(R) Theories and Beyond, Int. J. Mod. Phys. D 20 (2011) 413-462

2011

[5] [5]

G. J. Olmo, D. Rubiera-Garcia and H. Sanchis- Alepuz, Geonic black holes and remnants in Eddington-inspired Born-Infeld gravity, Eur. Phys. J. C 74 (2014) 2804

2014

[6] [6]

F. S. N. Lobo, G. J. Olmo and D. Rubiera- Garcia, Semiclassical geons as solitonic black hole remnants, JCAP 07 (2013) 011

2013

[7] [7]

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. C 75 (2015) 253. [erratum: Eur. Phys. J. C 75 (2015) 331.] 10

2015

[8] [8]

Shaikh, Wormholes with nonexotic matter in Born-Infeld gravity, Phys

R. Shaikh, Wormholes with nonexotic matter in Born-Infeld gravity, Phys. Rev. D 98 (2018) no.6, 064033

2018