Light deflection and gravitational lensing effects in acoustic black-bounce spacetime

A. R. Soares; C. F. S. Pereira; H. Belich; M. V. de S. Silva; R. L. L. Vit\'oria

REVIEW 1 major objections 5 minor 2 cited by

Light deflection and gravitational lensing effects in acoustic black-bounce spacetime

T0 review · 1 major / 5 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 This paper derives analytic light-deflection formulas for the acoustic black-bounce spacetime and shows that its strong-field image separation can reach 0.24 of the shadow boundary, far above Schwarzschild's 0.001.

desk verdict The analytic lensing formulas for the acoustic black-bounce are mostly consistent and the limits check out, but Table I—which carries the paper's main observational claim—contradicts the paper's own equations. read the letter →

arxiv 2505.12577 v3 pith:EEGOJNA5 submitted 2025-05-18 gr-qc

C. F. S. Pereira , A. R. Soares , M. V. de S. Silva , R. L. L. Vitória , H. Belich This is my paper · ORCID

classification gr-qc

keywords acousticblack-bouncegravitationallensinglightdeflectionstrong-fieldlimitphotonsphererelativisticimagesEinsteinring

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 sets out to show that the acoustic black-bounce spacetime, a Simpson-Visser regularized version of an acoustic black hole, has a distinctive gravitational-lensing signature. It derives analytic formulas for the deflection of light in the weak-field limit and a logarithmic strong-field deflection near the photon sphere, then converts them into standard observables: the angular separation of relativistic images, the relative magnification of those images, and the Einstein ring radius. The paper's central quantitative claim is that when the throat radius $a$ is close to $\sqrt{|q|}$, the normalized separation $s/\theta_\infty$ reaches about 0.24, orders of magnitude above the Schwarzschild value of roughly $10^{-3}$. The formulas reduce to the acoustic black hole when $a\to0$ and to the Ellis-Bronnikov wormhole when $q\to0$, which the authors use as consistency checks.

What carries the argument

The carrying mechanism is the Bozza-Tsukamoto strong-field expansion of the deflection integral. Writing $z=1-\rho_0/\rho$, the integral's integrand $G(z,\rho_0)$ is expanded near $z=0$ as $G(z,\rho_0)\simeq \Lambda_1(\rho_0)z+\Lambda_2(\rho_0)z^2$. The vanishing of $\Lambda_1$ at the photon sphere forces a logarithmic divergence, and the coefficient $\Lambda_2(\rho_m)=4\sqrt{3|q|}+4a^4/\sqrt{3|q|}-8a^2$ controls the prefactor in Eq. (46). This expansion is the bridge from the metric functions to the observables $s$ and $\tilde{r}$, and it is also what makes the $a\to0$ limit reproduce the acoustic black hole.

What would settle it

Numerically integrate the full deflection integral around Eq. (37) without replacing $G(z,\rho_0)$ by $\Lambda_1 z+\Lambda_2 z^2$ for the Table I values, particularly $a/\sqrt{q}=0.99$; if the exact deflection and the resulting $s/\theta_\infty$ disagree with Eq. (46) plus the regular part, the predicted enhancement is an artifact of the truncation rather than a property of the acoustic black-bounce metric.

Watch

Extended reading notes

Core claim

The central claim is that the acoustic black-bounce metric with $f(\rho)=1-q^2/(\rho^2+a^2)^2$ and $\Sigma^2(\rho)=\rho^2+a^2$ produces a strong-field deflection whose divergent part near the photon sphere is $$\$\Delta$\phi_D = -\frac12\sqrt{\frac{\sqrt{3}|q|}{\sqrt{3}|q|-$a^{2}$}}\,\log\left(\frac{\$\beta$}{$3^{{3/4}}$\sqrt{|q|/2}}-1\right)+\mathrm{const},$$ with a prefactor that grows as $a/\sqrt{q}\to1$. Together with a numerically evaluated regular part and the Bozza observable construction, this gives a normalized image separation $s/\theta_\infty$ that rises from $7.59\times10^{-7}$ at $a/\sqrt{q}=0.10$ to $2.42\times10^{-1}$ at $a/\sqrt{q}=0.99$, while the Schwarzschild value is about $10^{-3}$. In the weak-field regime the paper obtains $\delta\phi \simeq \pi a^2/(4\beta^2) + 15\pi q^2/(16\beta^4) + 9\pi a^4/(64\beta^4) + 19\pi a^2 q^2/(64\beta^6) + 1545\pi q^4/(1024\beta^8)$, reducing to the Ellis-Bronnikov wormhole when $q\to0$ and to the acoustic black hole when $a\to0$. The conclusion is that strong-field lensing could distinguish the acoustic black-bounce spacetime from Schwarzschild, whereas the acoustic black hole alone would be far harder to resolve.

Load-bearing premise

The strong-field predictions stand on the assumption that the light-bending function $G(z,\rho_0)$ is faithfully represented by its first two Taylor terms near the photon sphere, and the paper does not quantify the error from dropping the remainder, especially as $a/\sqrt{q}\to1$.

Editorial extensions

If this is right

If the acoustic black-bounce formulas are right, the normalized image separation $s/\theta_\infty$ spans about $7.6\times10^{-7}$ to $2.4\times10^{-1}$ as $a/\sqrt{q}$ runs from 0.10 to 0.99, so the throat parameter is in principle measurable from strong-field lensing.
The acoustic black hole alone gives $s/\theta_\infty\sim10^{-7}$, roughly four orders of magnitude below Schwarzschild, which means its relativistic images would be considerably harder to resolve.
In the $q\to0$ limit the weak-field deflection becomes the Ellis-Bronnikov wormhole result, and in the $a\to0$ limit it becomes the acoustic black hole result, so the paper's formulas interpolate between known limits.
The weak-field Einstein ring radius depends only on the throat radius $a$ and the distance ratios, not on the magnetic charge $q$, so measuring the ring would give a direct estimate of $a$.

Reading between the lines

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

If the predicted jump near $a/\sqrt{q}\simeq0.9$ is real, then a single resolved relativistic-image separation above about $10^{-2}$ times the shadow radius would already disfavor both Schwarzschild and the acoustic black hole, assuming the lens mass and distances are known.
A direct numerical evaluation of the full deflection integral, without replacing $G(z,\rho_0)$ by $\Lambda_1 z+\Lambda_2 z^2$, would settle whether the sharp enhancement at $a/\sqrt{q}\to1$ survives; the paper does not report that comparison.
The same observable pipeline could in principle be applied to a rotating or time-dependent acoustic black bounce or to other regularized metrics, an extension the paper leaves open.

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/Tsukamoto lensing calculation applied to the acoustic black-bounce metric. The analytic parts are mostly consistent, but the advertised strong-field enhancement rests on a table that does not agree with the paper's own equations. The conditional verdict is right, and I would put more weight on the Table I inconsistency than on the missing derivation details.

What's actually new: explicit weak-field series (Eq. 35) and strong-field divergent term (Eq. 46) for the ABB, plus the Bozza observables. The a→0 limit recovers the ABH result and q→0 recovers Ellis-Bronnikov, which is a real check. Eq. (50) follows algebraically from Eq. (46); I found no separate problem with the divergent coefficient. This is competent use of a standard formalism, with no fitting and no circularity.

Soft spots. First and most important: Table I cannot be reconciled with Eq. (50) and Fig. 3. Using Eq. (58) and Eq. (50) with Δφ_R ≈ 1.4 at a/√q = 0.90 gives s/θ∞ ≈ 3×10⁻⁵, about a hundred times smaller than the tabulated 3.1×10⁻³; the tabulated value would require Δφ_R ≈ 4.6. At a/√q = 0.99 the table implies Δφ_R ≈ 7.7, while Fig. 3 shows values around 1.5. So the claim that ABB lensing is distinguishable from Schwarzschild by large image separations is unsupported. The authors need to recompute Table I, or withdraw the claim. This is load-bearing, not cosmetic.

Second, Eq. (35) is asserted without derivation. The expansion mixes two small parameters (a and q), and the ordering of terms is not obvious; they should provide the series or a clear reference. Moderate.

Third, the regular part Eq. (47) is evaluated only numerically, with no method, error control, or convergence check, and the Bozza quadratic truncation G ≈ Λ₁z + Λ₂z² is not tested near a/√q → 1. Minor to moderate.

Finally, there is likely overlap with Ref. [105] from the same group; the paper should state what is new relative to that earlier work. Self-citation is not the issue; unacknowledged overlap is.

Who it's for: people working on black-bounce lensing or analog-gravity observables. The analytic formulas are probably worth having, but not until the table is fixed.

Recommendation: I would send this to peer review, asking the referee specifically to recompute Table I and check the weak-field expansion. If the table cannot be fixed, the observational claims fail and the paper reduces to a routine formula paper. Major revision before acceptance.

Referee Report

1 major / 5 minor

Summary. The paper computes the deflection of light and gravitational lensing observables for two spherically symmetric metrics: the acoustic black hole (ABH) and the acoustic black-bounce (ABB). In the weak-field limit it obtains a series expansion for the deflection angle, Eq. (35), which reduces to the ABH result (Eq. (16)) for a→0 and to the Ellis-Bronnikov result (Eq. (36)) for q→0. In the strong-field limit it applies the Bozza–Tsukamoto method, deriving an analytic divergent part, Eq. (46), and a numerically evaluated regular part, Eq. (47). It then constructs lensing observables: the angular separation s between the outermost relativistic image and the asymptotic image position θ∞, and the flux ratio ˜r. The central claim is that for a/√q near 1 the observable ratio s/θ∞ reaches 10^-3 to 10^-1, exceeding the Schwarzschild value (~10^-3) and therefore making the ABB potentially distinguishable from Schwarzschild.

Significance. The formalism used is standard and the weak-field expansion is carefully derived and cross-checked against known limits. The paper's strong-field divergent part is obtained in closed form, which is a useful addition. If the observable predictions were correct, they would provide a concrete way to differentiate the acoustic black-bounce from Schwarzschild in strong-field lensing. The main strength is that the derivations are analytical and do not rely on numerical fitting. However, as detailed in the major comment, the key observable table is inconsistent with the paper's own equations, and the headline claim is therefore not supported by the manuscript as written.

major comments (1)

[V. LENS EQUATION AND OBSERVABLES, Table I and Fig. 7] Table I is internally inconsistent with the paper's own formulas. Combining Eq. (58) with Eqs. (50) and (51) and using the values of Δϕ_R shown in Fig. 3 (Δϕ_R ≈ 1.4 at a/√q = 0.90 and ≈ 1.5 at a/√q = 0.99) yields s/θ∞ ≈ 3 × 10^−5 and ≈ 7 × 10^−5, respectively, whereas Table I lists 3.1 × 10^−3 and 2.42 × 10^−1. To reproduce the tabulated entries, Eq. (50) would require Δϕ_R ≈ 4.6 at a/√q = 0.90 and Δϕ_R ≈ 7.7 at a/√q = 0.99, far outside the range of Fig. 3. The paper's conclusion in Sections V and VI that the ABB angular separation can exceed the Schwarzschild value (~10^−3) rests entirely on this table and is therefore unsupported. With the correct Δϕ_R, the ABB s/θ∞ remains below ~10^−4 for all a/√q < 1, i.e., smaller than the Schwarzschild value.

minor comments (5)

[Fig. 1 and text] The axis label in Fig. 1 reads 'q /β' but the text correctly refers to the ratio p|q|/β; the square-root notation should be used consistently in both the figure and the text.
[Section V, text near Eq. (58)] There is a typographical error: 'Bozaa's point of view' should read 'Bozza's point of view'.
[Eqs. (19) and (39)] The strong-field observation is based on the truncation G(z,ρ0) ≃ Λ1 z + Λ2 z^2. The authors do not quantify the neglected higher-order terms or discuss the accuracy of this truncation as a/√q approaches 1; a brief comment on the validity of the expansion would be useful.
[Eq. (50)] The definition of ¯b in Eq. (50) is algebraically equivalent to the constant term in Eq. (46), but the equivalence is not transparent. A short explanatory sentence would improve readability.
[Fig. 7] The y-axis label '(s/θ∞)×10^-2' is difficult to interpret; a logarithmic scale would make the comparison between SBH, ABH, and ABB much clearer.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the deflection and observables follow from the ABB metric via the standard Bozza–Tsukamoto expansion; the same-author citations are literature-survey only and not load-bearing.

full rationale

The derivation is self-contained against the input metric. The ABB line element (Eq. (29)) is the external input from Cañate (Ref. [29]); the geodesic quadrature (Eqs. (32)–(34)) and the weak-field expansion (Eq. (35)) are obtained by direct series expansion, with the limiting checks a→0 and q→0 reproducing the ABH and Ellis–Bronnikov results of Refs. [124,125]. The strong-field divergent part (Eq. (46)) is obtained from the Bozza–Tsukamoto expansion (Eqs. (37)–(43)); the constants in Eq. (46) are algebraic functions of q and a, with no parameter fitted to any deflection datum. The observables (Eqs. (58)–(59)) are the standard Bozza inversion of the deflection coefficients ¯a and ¯b, so the predicted s/θ∞ is not an input renamed as an output. The self-citations [38,41,44,45,105] occur only in the introductory survey of black-bounce scenarios and are not used to justify the deflection calculation or the observable formulas, so they are not load-bearing. The reader-flagged disagreement between Table I and Eqs. (58)+(50) together with Fig. 3 is an internal numerical-consistency problem (the tabulated s/θ∞ values do not correspond to the Δϕ_R plotted), not a circularity: if anything, the table is not forced by the equations, which is the opposite of a self-fulfilling prediction. Score 2 reflects the presence of several author self-citations in the reference list; no circular reduction was found.

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

The calculation rests on metric models from Refs. [29,123] and on the standard Bozza/Tsukamoto strong-field expansion. The only parameters are q and a, both taken from the literature; nothing is fitted to data. No new entities are introduced. The main unverified inputs are the physical viability of the source matter and the numerical evaluation of the regular part.

free parameters (2)

q
Magnetic charge in the gravitational interpretation and fluid density parameter in the acoustic context; model input from Refs. [108,123], not fitted in this paper. All deflection and observable expressions depend on it.
a
Black-bounce throat radius; model input from Ref. [29], not fitted in this paper. It controls the Ellis-Bronnikov contribution and the deviation from Schwarzschild in the observables.

assumptions (5)

domain assumption The metrics in Eqs. (10) and (29) are exact solutions of the stated gravitational theories, namely EsGB theory for the acoustic black hole and phantom scalar plus nonlinear electrodynamics for the acoustic black-bounce.
Taken from Refs. [123] and [29]; the paper does not verify the field equations or the energy conditions of the source matter.
standard math Null geodesics in a static spherically symmetric spacetime are governed by the effective potential and conserved quantities in Section II, Eqs. (4)-(9).
Standard general relativity; no derivation of the geodesic formalism beyond the variational statement is given.
domain assumption The Bozza/Tsukamoto strong-field expansion applies: G(z,r_0) near the photon sphere is dominated by the linear and quadratic terms Lambda_1 z + Lambda_2 z^2, so higher-order terms can be dropped in the divergent part.
Invoked at Eqs. (19) and (39); the error from this truncation is not quantified for the acoustic black-bounce metric.
domain assumption Weak-field expansion in small q and a is valid: the deflection is computed by expanding the integrand and impact parameter to finite order, assuming the photon passes far from the lens.
Used in Eqs. (16) and (35); the actual dimensionless small parameters are not stated precisely.
domain assumption Asymptotic flatness holds and the observer and source are at large distance, so the deflection is twice the integral from the turning point to infinity.
Used throughout the weak- and strong-field calculations; standard for lensing but implicit in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Light deflection and gravitational lensing effects in acoustic black-bounce spacetime." pith.science (2026). https://pith.science/paper/EEGOJNA5

@misc{pith2026250512577,
  author       = {Pith},
  title        = {Pith review of: Light deflection and gravitational lensing effects in acoustic black-bounce spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EEGOJNA5}},
  note         = {Machine review of arXiv:2505.12577}
}

read the original abstract

In the present work, we analyze the gravitational deflection for a light beam in the weak and strong field regimes for the gravitational analogue geometry of an acoustic black hole (ABH) and acoustic black-bounce (ABB). Motivationally, the first spacetime arises as an exact solution of the field equations for gravitational black holes (BHs) in an Einstein-scalar-Gauss-Bonnet theory (EsGB) \cite{3}. In contrast, the second model arises from the combination of phantom scalar field and nonlinear electrodynamics in general relativity (GR) \cite{INTRO24}. We construct analytical expressions for the angular deflection of light in both limits and, from them, analyze the construction of the observables, which allow us to relate theoretical models to observational data. We compare these observables and show how much they differ from those obtained in the Schwarzschild solution.

Figures

Figures reproduced from arXiv: 2505.12577 by the authors.

**Figure 2.** FIG. 2. Angular deflection in seconds of arc for some values [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** FIG. 4. Deviation of light as a function of the impact param [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5. Deviation of light as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 3.** Figure 3: FIG. 3. Regular part of the integration of angular deviation. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]

**Figure 6.** Figure 6: FIG. 6. Light angular deflection diagram. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 8.** Figure 8: FIG. 8. Plot of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 7.** Figure 7: FIG. 7. Behavior of the ratio [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Continue with ORCID to comment.

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

Photon Propagation and Black Hole Imaging in Kruglov Nonlinear Electrodynamics
gr-qc 2026-04 unverdicted novelty 5.0 of 10

In Kruglov's Born-Infeld-type nonlinear electrodynamics, the effective photon geometry around a charged black hole produces q-dependent shifts in light deflection, shadow radius, and accretion disk images, including s...
Strong field gravitational lensing of particles by a black-bounce-Schwarzschild black hole
gr-qc 2026-02 accept novelty 5.0 of 10

For a black-bounce-Schwarzschild black hole, the paper derives the strong-deflection lensing observables for massive particles and quantifies how they differ from photon lensing.

Reference graph

Works this paper leans on

126 extracted references · 57 canonical work pages · cited by 2 Pith papers

[105]

He, Yi Xie, C

G. He, Yi Xie, C. Jiang and W. Lin, Phys.Rev.D1106, 064008 (2024)

work page 2024
[1]

Schwarzschild, About the gravitational field of a mass point according to Einstein’s theory, in Berlin

K. Schwarzschild, About the gravitational field of a mass point according to Einstein’s theory, in Berlin. Session Reports,18(1916)

1916
[2]

Penrose, Phys

R. Penrose, Phys. Rev. Lett.14, 57-59 (1965); S. W. Hawking and G. F. R. Ellis, Cambridge University Press, 2023

1965
[3]

R. M. Wald, General Relativity, The University of Chicago Press, Chicago (1984)

1984
[4]

D’ Inverno, Introducing Einstein’s Relativity, Oxford University Press, New York (1998)

R. D’ Inverno, Introducing Einstein’s Relativity, Oxford University Press, New York (1998)

1998
[5]

J. M. Bardeen, Non-singular general relativistic grav- itational collapse, in Proceedings of the International Conference GR5, (Tbilisi, U.S.S.R) 1968

1968
[6]

E. M. Rodrigues and M. V. de S. Silva, J. Cosmol. As- tropart. Phys.6025 (2018)

2018
[7]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Phys. Lett. B4931490- 152 (2000)

2000

Show all 126 references

[8]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Phys. Rev. Lett.80, 5056- 5059 (1998), [arXiv:gr-qc/9911046 [gr-qc]]

1998 arXiv
[9]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Phys. Lett. B464, 25 (1999), [arXiv:hep-th/9911174 [hep-th]]

1999 arXiv
[10]

K. A. Bronnikov, Phys. Rev. Lett.85, 4641 (2000)

2000
[11]

K. A. Bronnikov, Phys. Rev. D63, 044005 (2001), [arXiv:gr-qc/0006014 [gr-qc]]

2001 arXiv
[12]

Dymnikova, Class

I. Dymnikova, Class. Quant. Grav.21, 4417-4429 (2004), [arXiv:gr-qc/0407072 [gr-qc]]

2004 arXiv
[13]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Gen. Rel. Grav.37, 635 (2005), [arXiv:hep-th/0403229 [hep-th]]

2005 arXiv
[14]

S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006), [arXiv:gr-qc/0506126 [gr-qc]]

2006 arXiv
[15]

Uchikata, S

N. Uchikata, S. Yoshida and T. Futamase, Phys. Rev. D86, 084025 (2012), [arXiv:1209.3567 [gr-qc]]

2012 arXiv
[16]

Balart and E

L. Balart and E. C. Vagenas, Phys. Rev. D90, no.12, 124045 (2014), [arXiv:1408.0306 [gr-qc]]

2014 arXiv
[17]

Balart and E

L. Balart and E. C. Vagenas, Phys. Lett. B730, 14-17 (2014), [arXiv:1401.2136 [gr-qc]]

2014 arXiv
[18]

Culetu, Int

H. Culetu, Int. J. Mod. Phys. D24, no.09, 1542001 (2015). 11

2015
[19]

K. A. Bronnikov, Phys. Rev. D96, no.12, 128501 (2017), [arXiv:1712.04342 [gr-qc]]

2017 arXiv
[20]

Ponce de Leon, Phys

J. Ponce de Leon, Phys. Rev. D95, no.12, 124015 (2017), [arXiv:1706.03454 [gr-qc]]

2017 arXiv
[21]

M. E. Rodrigues, E. L. B. Junior and M. V. de S. Silva, JCAP02, 059 (2018)

2018
[22]

Simpson and M

A. Simpson and M. Visser, JCAP02042 (2019)

2019
[23]

K. A. Bronnikov and R. K. Walia, Phys. Rev. D105, no.4, 044039 (2022)

2022
[24]

Huang and J

H. Huang and J. Yang, Phys. Rev. D100, no.12, 124063 (2019)

2019
[25]

Simpson, P

A. Simpson, P. Martin-Moruno and M. Visser, Class. Quant. Grav.36, no.14, 145007 (2019)

2019
[26]

F. S. N. Lobo, M. E. Rodrigues, M. V. d. S. Silva, A. Simpson and M. Visser, Phys. Rev. D103, no.8, 084052 (2021)

2021
[27]

Franzin, S

E. Franzin, S. Liberati, J. Mazza, A. Simpson and M. Visser, JCAP07, 036 (2021)

2021
[28]

Mazza, E

J. Mazza, E. Franzin and S. Liberati, JCAP04, 082 (2021)

2021
[29]

Ca˜ nate, Phys

P. Ca˜ nate, Phys. Rev. D106, no.2, 024031 (2022)

2022
[30]

M. E. Rodrigues and M. V. d. S. Silva, Phys. Rev. D 106, no.8, 084016 (2022)

2022
[31]

K. A. Bronnikov, M. E. Rodrigues and M. V. de S. Silva, Phys. Rev. D108, no.2, 024065 (2023)

2023
[32]

A. Lima, G. Alencar, R. N. Costa Filho and R. R. Landim, Gen. Relativ. Gravit.55, 108 (2023)

2023
[33]

A. M. Lima, G. M. de Alencar Filho and J. S. Furtado Neto, Symmetry15, 150 (2023)

2023
[34]

M. E. Rodrigues and M. V. de S. Silva, Phys. Rev. D 107, 044064 (2023)

2023
[35]

J. C. Fabris, E. L. B. Junior and M. E. Rodrigues, Eur. Phys. J. C83: 884 (2023)

2023
[36]

E. L. B. Junior and M. E. Rodrigues, Gen. Rel. Grav. 55, 8 (2023), arXiv:2203.03629

2023 arXiv
[37]

Lima, G Alencar and D

A. Lima, G Alencar and D. S. C. G´ omez, Phys. Rev. D 109, 064038 (2024)

2024
[38]

C. F. S. Pereira, D. C. Rodrigues, J. C. Fabris and M. E. Rodrigues, Phys. Rev. D109, 044011 (2024)

2024
[39]

J. T. S. S Junior, F. S. N. Lobo and M. E. Rodrigues, Eur. Phys. J. C84, 557 (2024)

2024
[40]

E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo et al. Eur. Phys. J. C84, 1190 (2024)

2024
[41]

C. F. S. Pereira, ´E. L. Martins, D. C. Rodrigues, J. C. Fabris and M. E. Rodrigues, Class. Quantum Grav.42 015001 (2024), [arXiv:2405.07455 [gr-qc]]

2024 arXiv
[42]

Alencar, K

G. Alencar, K. A. Bronnikov, M. E. Rodrigues, D. S´ aez- Chill´ on G´ omez and M. V. de S. Silva, Eur. Phys. J. C 84, no.7, 745 (2024) [arXiv:2403.12897 [gr-qc]]

2024 arXiv
[43]

M. V. de S. Silva, T. M. Crispim, G. Alencar, R. R. Landim and M. E. Rodrigues, [arXiv:2502.19186 [gr- qc]]
[44]

Carlos F. S. Pereira, Denis C. Rodrigues, Marcos V. de S. Silva, J´ ulio C. Fabris, Manuel E. Rodrigues and H. Belich, Phys. Rev. D111, 084025 (2025)

2025
[45]

C. F. S. Pereira, M. V. de S. Silva, H. Belich, D. C. Rodrigues, J. C. Fabris and M. E. Rodrigues, Phys. Rev. D111, 124005 (2025)

2025
[46]

M. E. Rodrigues and M. V. de S. Silva, Class. Quantum Grav42, 055005 (2025)

2025
[47]

Einstein, Science84, 506 (1936)

A. Einstein, Science84, 506 (1936)

1936
[48]

Liebes, Jr., Phys

S. Liebes, Jr., Phys. Rev.133, B835 (1964)

1964
[49]

Darwin, Proc

C. Darwin, Proc. R. Soc. A249, 180 (1959);263, 39 (1961)

1959
[50]

Atkinson, Astron

R. Atkinson, Astron. J.70, 517 (1965)

1965
[51]

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

2000
[52]

then developed a technique for obtaining the equa- tions in the strong field regime that was later improved by Tsukamoto [53]. Since then, the phenomenology of gravitational lenses has been explored in a wide variety of scenarios due to its possibility of measurement, contexts...
[53]

Bozza, Phys

V. Bozza, Phys. Rev. D66, 103001 (2002)

2002
[54]

Tsukamoto, Phys.Rev

N. Tsukamoto, Phys.Rev. D95, 064035 (2017)

2017
[55]

K. S. Virbhadra, D. Narasimha, and S. M. Chitre, As- tron.Astrophys.3371-8 (1998)

1998
[56]

E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D66, 024010 (2002)

2002
[57]

V.Bozza, Phys. Rev. D67, 103006 (2003)

2003
[58]

V´ azquez and E.P

S. V´ azquez and E.P. Esteban, Nuovo Cim.119B, 489 (2004)

2004
[59]

E. F. Eiroa and D. F. Torres, Phys. Rev. D69, 063004 (2004)

2004
[60]

Bozza, F

V. Bozza, F. De Luca, G. Scarpetta and M. Sereno, Phys. Rev. D72, 083003 (2005)

2005
[61]

Bozza, F

V. Bozza, F. De Luca and G. Scarpetta, Phys. Rev. D 74, 063001 (2006)

2006
[62]

K. S. Virbhadra, Phys. Rev.D79083004 (2009)

2009
[63]

A. B. Aazami, C. R. Keeton and A. O. Petters, J. Math. Phys.52, 092502 (2011)

2011
[64]

A. B. Aazami, C. R. Keeton and A. O. Petters, J. Math. Phys.52, 102501 (2011)

2011
[65]

Tsukamoto and Y

N. Tsukamoto and Y. Gong, Phys. Rev. D95, 064034 (2017)

2017
[66]

K. S. Virbhadra, Phys. Rev. D106, 064038 (2022)

2022
[67]

Afrin, S

M. Afrin, S. Vagnozzi, S. G. Ghosh, Astrophys. J.944 149 (2023)

2023
[68]

Vagnozzi, R

S. Vagnozzi, R. Roy, Y.-D. Tsai, L. Visinelli, et al., ,Class. Quant. Grav.40165007 (2023)

2023
[69]

A. A. A. Filho, J. R. Nascimento, A. Y. Petrov and P. J. Porf´ ırio, [arXiv:2404.04176 [gr-qc]]
[70]

Igata, M

T. Igata, M. Omamiuda and Y. Takamori, arXiv:2502.11755 [gr-qc] (2025)

2025 arXiv
[71]

Takahisa Igata, arXiv:2503.02320 [gr-qc] (2025)

2025
[72]

Chetouani and G

L. Chetouani and G. Cl´ ement, Gen. Relativ. Gravit.16, 111 (1984)

1984
[73]

K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov, Phys. Rev. D74, 024020 (2006)

2006
[74]

T. K. Dey and S. Sen, Mod. Phys. Lett. A,23, 953 (2008)

2008
[75]

Abe, Astrophys

F. Abe, Astrophys. J.725, 787 (2010)

2010
[77]

G. W. Gibbons and M. Vyska, Class. Quant. Grav.29, 065016 (2012)

2012
[78]

Tsukamoto, T

N. Tsukamoto, T. Harada, and K. Yajima, Phys. Rev. D86, 104062 (2012)

2012
[79]

K. K. Nandi, A. A. Potapov, R. N. Izmailov, A. Tamang, and J. C. Evans, Phys. Rev. D93, 104044 (2016)

2016
[80]

Tsukamoto, Phys

N. Tsukamoto, Phys. Rev. D95, 084021 (2017)

2017
[81]

Tsukamoto and T

N. Tsukamoto and T. Harada, Phys. Rev. D95, 024030 (2017)

2017
[82]

Shaikh, P

R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, JCAP 1907, 028 (2019)

2019
[83]

Cheng and J

H. Cheng and J. Man, Class. Quantum. Grav.28, 015001 (2011)

2011
[84]

Sharif and S

M. Sharif and S. Iftikhar, Adv. High Energy Phys.2015, 854264 (2015)

2015
[85]

Man and H

J. Man and H. Cheng, Phys. Rev. D92, 024004 (2015)

2015
[86]

Furtado, J

C. Furtado, J. R. Nascimento, A. Y. Petrov, P. J. Por- f´ ırio and A. R. Soares, Phys. Rev. D103(2021) no.4, 12 044047

2021
[87]

A. R. Soares, R. L. L. Vit´ oria and C. F. S. Pereira, Eur. Phys. J. C83(10), 903 (2023)

2023
[88]

Bhadra, Phys

A. Bhadra, Phys. Rev. D67, 103009 (2003)

2003
[89]

E. F . Eiroa, Phys. Rev. D73, 043002 (2006)

2006
[90]

Sarkar and A

K. Sarkar and A. Bhadra, Class. Quantum. Grav.23, 6101 (2006)

2006
[91]

Mukherjee and A

N. Mukherjee and A. S. Majumdar, Gen. Relativ. Gravit.39, 583 (2007)

2007
[92]

G. N. Gyulchev and S. S. Yazadjiev, Phys. Rev. D75, 023006 (2007)

2007
[93]

Chen and J

S. Chen and J. Jing, Phys. Rev. D80, 024036 (2009)

2009
[94]

Sotani and U

H. Sotani and U. Miyamoto, Phys. Rev. D92, 044052 (2015)

2015
[95]

S. W. Wei, K. Yang, and Y. X. Liu, Eur. Phys. J. C75 253 (2015) [Erratum: Eur. Phys. J. C (2015) 331]

2015
[96]

Shaikh and S

R. Shaikh and S. Kar, Phys. Rev. D96, 044037 (2017)

2017
[97]

E. F. Eiroa and C. M. Sendra, Class. Quantum. Grav. 28, 085008 (2011)

2011
[98]

E. F. Eiroa and C. M. Sendra, Phys. Rev. D88, 103007 (2013)

2013
[99]

Bianchi, M

E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M Haggard and C. Rovelli, Class. Quantum. Grav.35, 225003 (2018)

2018
[100]

J. R. Nascimento, A. Y. Petrov, P. J. Porfirio and A. R. Soares, Phys. Rev. D102(2020) no.4, 044021 [arXiv:2005.13096 [gr-qc]]

2020 arXiv
[101]

Tsukamoto, Phys

N. Tsukamoto, Phys. Rev. D104, 064022 (2021)

2021
[102]

Tsukamoto, Phys

N. Tsukamoto, Phys. Rev. D103(2), 024033 (2021)

2021
[103]

Ghosh and A

S. Ghosh and A. Bhattacharyya, J. Cosmol. Astropart. Phys.11, 006, (2022)

2022
[104]

Tsukamoto, Phys

N. Tsukamoto, Phys. Rev. D105(8), 084036 (2022)

2022
[106]

A. R. Soares, C. F. S. Pereira, R. L. L. Vit´ oria, M. V. de S. Silva and H. Belich, JCAP06034 (2025)

2025
[107]

Furtado, A

C. Furtado, A. L. A. Moreira, J. R. Nascimento, A. Y. Petrov and P. J. Porfirio, [arXiv:2504.19920 [gr- qc]]

arXiv
[108]

W. G. Unruh, Phys. Rev. Lett.46, 1351 (1981)

1981
[109]

Visser, Class

M. Visser, Class. Quantum Grav.15, 1767 (1998)

1998
[110]

Fischer and Matt Visser, Phys

Uwe R. Fischer and Matt Visser, Phys. Rev. Lett.88, 110201 (2002)

2002
[111]

Weinfurtner, E

S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011)

2011
[112]

C. L. Benone, L. C. B. Crispino, C. Herdeiro and E. Radu, Phys. Rev. D91, no.10, 104038 (2015)

2015
[113]

Steinhauer, Nature Phys.12, 959 (2016)

J. Steinhauer, Nature Phys.12, 959 (2016)

2016
[114]

Torres, S

T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford and S. Weinfurtner, Nature Phys.13, 833-836 (2017)

2017
[115]

Dardashti, S

R. Dardashti, S. Hartmann, K. P. Y. Th´ ebault and E. Winsberg, Stud. Hist. Phil. Sci. B67, 1-11 (2019)

2019
[116]

L. A. Oliveira, C. L. Benone, A. L. Almeida and L. C. B. Crispino, Int. J. Mod. Phys. D29, no.11, 2041018 (2020)

2020
[117]

Torres, S

T. Torres, S. Patrick, M. Richartz and S. Weinfurtner, Phys. Rev. Lett.125, no.1, 011301 (2020)

2020
[118]

Tien Hsieh, Da-Shin Lee, and Chi-Yong Lin, Phys. Rev. D103, 104063 (2021)

2021
[119]

Kunal Pal, Kuntal Pal and Tapobrata Sarkar, Universe, 8, 197 (2022)

2022
[120]

Quantum Grav.40215001 (2023)

Sang-Shin Baak, Satadal Datta and Uwe R Fischer, Class. Quantum Grav.40215001 (2023)

2023
[121]

Fischer, Phys

Kunal Pal and Uwe R. Fischer, Phys. Rev. D110, 116022 (2024)

2024
[122]

Malato Corrˆ ea, C

M. Malato Corrˆ ea, C. F. B. Macedo, R. Panosso Macedo and L. A. Oliveira, [arXiv:2504.00107 [gr-qc]]

arXiv
[123]

L. T. de Paula, P. H. C. Siqueira, R. Panosso Macedo and M. Richartz, Phys. Rev. D111, 104064 (2025)

2025
[124]

Ca˜ nate, J

P. Ca˜ nate, J. Sultana and D. Kazanas, Class. Quant. Grav.38, 125002 (2021)

2021
[125]

Nakajima and H

K. Nakajima and H. Asada, Phys. Rev. D85, 107501 (2012)

2012
[126]

Aounallah1, A

H. Aounallah1, A. R. Soares and R. L. L. Vit´ oria, Eur. Phys. J. C (2020)80447 (2020)

2020
[127]

Bozza, S

V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Gen. Rel. Grav.33, 1535 (2001)

2001

Pith tools

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

[1] [105]

He, Yi Xie, C

G. He, Yi Xie, C. Jiang and W. Lin, Phys.Rev.D1106, 064008 (2024)

work page 2024

[2] [1]

Schwarzschild, About the gravitational field of a mass point according to Einstein’s theory, in Berlin

K. Schwarzschild, About the gravitational field of a mass point according to Einstein’s theory, in Berlin. Session Reports,18(1916)

1916

[3] [2]

Penrose, Phys

R. Penrose, Phys. Rev. Lett.14, 57-59 (1965); S. W. Hawking and G. F. R. Ellis, Cambridge University Press, 2023

1965

[4] [3]

R. M. Wald, General Relativity, The University of Chicago Press, Chicago (1984)

1984

[5] [4]

D’ Inverno, Introducing Einstein’s Relativity, Oxford University Press, New York (1998)

R. D’ Inverno, Introducing Einstein’s Relativity, Oxford University Press, New York (1998)

1998

[6] [5]

J. M. Bardeen, Non-singular general relativistic grav- itational collapse, in Proceedings of the International Conference GR5, (Tbilisi, U.S.S.R) 1968

1968

[7] [6]

E. M. Rodrigues and M. V. de S. Silva, J. Cosmol. As- tropart. Phys.6025 (2018)

2018

[8] [7]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Phys. Lett. B4931490- 152 (2000)

2000