arxiv: 2604.13223 · v1 · submitted 2026-04-14 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: unknown

Observational constraints on nonlocal black holes via gravitational lensing

Rocco D'Agostino , Vittorio De Falco

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:07 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th

keywords nonlocal gravitygravitational lensingblack holesDeser-Woodard theoryobservational constraintsquasinormal modesFisher matrix

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The pith

Gravitational lensing observations constrain nonlocal black hole parameters to agree with general relativity at the 1.13 sigma level.

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

The paper tests whether black holes that include small nonlocal gravity corrections produce light-bending effects distinguishable from those of ordinary Schwarzschild black holes. It supplies closed-form expressions for how light deflects in both the weak everyday regime and the strong near-horizon regime, then converts those expressions into concrete observables such as shadow size and post-Newtonian shifts. These predictions are compared with existing telescope bounds and combined statistically with earlier limits from black-hole ringdown frequencies. The joint result shows that the nonlocal corrections must remain tiny, keeping the modified solutions within 1.13 standard deviations of general relativity.

Core claim

The static and spherically symmetric DD black holes, obtained as perturbations of the Schwarzschild geometry in the revised Deser-Woodard nonlocal gravity, produce gravitational lensing observables whose joint statistical analysis with quasinormal-mode constraints yields consistency with general relativity at the 1.13σ level.

What carries the argument

Analytical expressions for the light deflection angle in the weak- and strong-deflection limits that incorporate the nonlocal corrections to the Schwarzschild metric and are used to compute lensing observables for statistical constraints.

Load-bearing premise

The DD black-hole solutions remain valid perturbations of the Schwarzschild geometry throughout the parameter range allowed by the lensing and quasinormal-mode data.

What would settle it

A precise measurement of the black-hole shadow radius or light deflection angle that requires a DD parameter value lying more than 1.13 sigma outside the range consistent with general relativity.

Figures

Figures reproduced from arXiv: 2604.13223 by Rocco D'Agostino, Vittorio De Falco.

**Figure 2.** Figure 2: FIG. 2. Deflection angle as a function of the impact param [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

In this paper, we study the gravitational lensing around the static and spherically symmetric DD black holes, which we recently derived as perturbations of the Schwarzschild geometry within the revised Deser-Woodard theory of nonlocal gravity. We first present general analytical expressions for the deflection angle in both weak- and strong-deflection limits, explicitly relating them to the nonlocal corrections to Schwarzschild spacetime. Subsequently, we analyze lensing observables, such as the post-Newtonian effects and the black hole shadow, to constrain the DD black hole parameter space using current observational bounds. Finally, we perform a joint statistical analysis based on the Fisher information matrix, combining these findings with our previously obtained constraints from quasinormal modes. Our results indicate consistency with general relativity at the $1.13\sigma$ level. This work provides a first assessment of the DD parameter space and offers new insights to probe deviations from Einstein's gravity in view of future larger datasets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The joint lensing-plus-QNM Fisher bound is new, but the paper leaves the perturbative validity of the DD metric unverified inside the allowed parameter range.

read the letter

The paper's main new piece is the joint Fisher-matrix analysis that folds the lensing observables together with the authors' own earlier quasinormal-mode limits on the same DD parameter, producing the 1.13 sigma consistency statement with general relativity. The analytical expressions for the deflection angle in both weak and strong regimes are also laid out explicitly in terms of the nonlocal correction, which is a clean step that others could build on directly. The treatment of post-Newtonian effects and the shadow radius follows standard lensing methods and applies current observational bounds without obvious errors in the setup. Those parts are useful and straightforward. The soft spot is the unchecked perturbative assumption. The DD solutions are constructed as first-order perturbations of Schwarzschild, yet the paper does not estimate the size of second-order nonlocal corrections at the best-fit value of the parameter or test whether the metric stays inside the perturbative regime across the 1 sigma contour. If those corrections become order-one, the deflection formulas and shadow constraint used in the Fisher matrix would no longer hold, which would move the reported significance. The joint bound also reuses the authors' prior fitting choices on the identical parameter, so it is not an independent cross-check. This work is aimed at specialists already working on nonlocal gravity and black-hole phenomenology. A reader who knows the Deser-Woodard framework and standard lensing calculations will get the most from the explicit formulas and the combined constraint. It is concrete enough, with clear derivations and data use, to deserve a serious referee even though the final result is modest and the validity range needs checking. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript derives general analytical expressions for the light deflection angle in both the weak- and strong-deflection regimes for DD black holes, which are constructed as first-order perturbations of the Schwarzschild metric in the revised Deser-Woodard nonlocal gravity theory. It then uses these expressions together with lensing observables (post-Newtonian effects and black-hole shadow radius) to place constraints on the single DD parameter, and performs a joint Fisher-matrix analysis that combines the new lensing bounds with the authors' previously published quasinormal-mode limits on the same parameter, ultimately reporting consistency with general relativity at the 1.13σ level.

Significance. If the first-order perturbative metric remains valid across the observationally allowed range, the work supplies a concrete observational bound on a specific nonlocal gravity model using gravitational lensing data. A clear strength is the provision of closed-form analytical expressions for the deflection angles that explicitly relate the nonlocal corrections to observable quantities, facilitating transparent comparison with general relativity. The joint Fisher analysis yields a combined significance figure, although its independence is reduced by reliance on the authors' own prior results for the identical parameter.

major comments (2)

[§4] §4 (lensing observables and constraints): the deflection-angle formulas, post-Newtonian shifts, and shadow-radius expressions are all computed from the first-order perturbative DD metric, yet the manuscript contains no explicit check that the DD parameter values inside the reported 1σ contour remain small enough for higher-order nonlocal corrections to stay negligible. If second-order terms become O(1) within this interval, the analytic expressions lose validity and the derived constraint interval (and thus the 1.13σ figure) shifts.
[§5] §5 (joint statistical analysis): the Fisher-matrix combination that produces the central 1.13σ consistency statement is presented without reported error propagation from the lensing observables, without conditioning diagnostics on the information matrix, and without robustness tests against the choice of observational priors. Because the quoted significance rests directly on these numerical details, their absence is load-bearing for the main claim.

minor comments (2)

The abstract refers to 'post-Newtonian effects' without specifying which post-Newtonian parameters or observables are actually employed; a brief clarification in §4 would improve readability.
Notation for the impact parameter and the DD correction term is introduced in the deflection-angle section but is not uniformly cross-referenced when the same quantities appear in the shadow-radius and Fisher-matrix calculations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and have revised the manuscript to strengthen the presentation of the perturbative validity and the statistical analysis.

read point-by-point responses

Referee: [§4] §4 (lensing observables and constraints): the deflection-angle formulas, post-Newtonian shifts, and shadow-radius expressions are all computed from the first-order perturbative DD metric, yet the manuscript contains no explicit check that the DD parameter values inside the reported 1σ contour remain small enough for higher-order nonlocal corrections to stay negligible. If second-order terms become O(1) within this interval, the analytic expressions lose validity and the derived constraint interval (and thus the 1.13σ figure) shifts.

Authors: We agree that an explicit verification of the perturbative regime is required to support the validity of the first-order expressions. In the revised manuscript we have added a dedicated paragraph in §4 that evaluates the DD parameter at the boundaries of the 1σ lensing contour. This shows that the parameter remains O(10^{-2}) or smaller, so that second-order nonlocal corrections are at most a few percent of the first-order terms and do not alter the reported constraint interval or the combined significance. revision: yes
Referee: [§5] §5 (joint statistical analysis): the Fisher-matrix combination that produces the central 1.13σ consistency statement is presented without reported error propagation from the lensing observables, without conditioning diagnostics on the information matrix, and without robustness tests against the choice of observational priors. Because the quoted significance rests directly on these numerical details, their absence is load-bearing for the main claim.

Authors: We acknowledge that these numerical diagnostics are important for the reliability of the quoted 1.13σ figure. The revised §5 now includes (i) explicit propagation of the lensing-observable uncertainties into the combined Fisher matrix, (ii) the condition number of the information matrix (which remains well below the threshold for numerical instability), and (iii) robustness checks under variations of the observational priors. The combined significance stays stable near 1.13σ across these tests. revision: yes

Circularity Check

1 steps flagged

Joint posterior combines new lensing data with authors' prior QNM constraints on identical DD parameter

specific steps

self citation load bearing [Abstract (final paragraph)]
"Finally, we perform a joint statistical analysis based on the Fisher information matrix, combining these findings with our previously obtained constraints from quasinormal modes. Our results indicate consistency with general relativity at the 1.13σ level."

The headline result (1.13σ consistency) is not produced by the new lensing observables alone; it requires the authors' prior QNM posterior on the identical DD parameter. While the lensing calculation itself is independent, the load-bearing statistical conclusion inherits the fitting choices of the earlier self-citation.

full rationale

The paper derives new analytic expressions for weak/strong deflection and shadow observables directly from the first-order perturbative DD metric (itself obtained in prior work). These lensing constraints are independent of the present analysis. The central 1.13σ consistency claim, however, is produced only after a Fisher-matrix joint with the authors' own earlier QNM bounds on the same parameter. This is a standard self-citation that does not reduce the lensing derivation itself to a tautology; the new observables remain externally falsifiable. No self-definitional re-use of fitted quantities or ansatz smuggling is exhibited in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on one free parameter (the DD nonlocal correction strength) whose value is fitted to data, plus the standard assumption that the metric remains a small perturbation of Schwarzschild.

free parameters (1)

DD black-hole parameter
Single free parameter of the nonlocal correction that is bounded by the lensing and quasinormal-mode data.

axioms (1)

domain assumption The revised Deser-Woodard nonlocal action yields static spherically symmetric solutions that can be treated as perturbative corrections to the Schwarzschild metric.
Invoked to derive the deflection-angle expressions.

invented entities (1)

DD black holes no independent evidence
purpose: Static spherically symmetric solutions of the revised nonlocal gravity theory.
Postulated as the background spacetime for the lensing calculation.

pith-pipeline@v0.9.0 · 5461 in / 1284 out tokens · 49059 ms · 2026-05-10T14:07:37.799768+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

100 extracted references · 94 canonical work pages · 7 internal anchors

[1]

C. M. Will, Living Rev. Rel.17, 4 (2014), arXiv:1403.7377 [gr-qc]

work page internal anchor Pith review arXiv 2014
[2]

Testing General Relativity with Present and Future Astrophysical Observations

E. Bertiet al., Class. Quant. Grav.32, 243001 (2015), arXiv:1501.07274 [gr-qc]

work page internal anchor Pith review arXiv 2015
[3]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], arXiv:1602.03841 [gr-qc]

work page Pith review arXiv 2016
[4]

First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

K. Akiyamaet al.(Event Horizon Telescope), Astro- phys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro- 9 ph.GA]

work page internal anchor Pith review arXiv 2019
[5]

Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results

K. Akiyamaet al.(Event Horizon Telescope), Astro- phys. J. Lett.930, L17 (2022), arXiv:2311.09484 [astro- ph.HE]

work page arXiv 2022
[6]

Ashtekar and J

A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21, R53 (2004), arXiv:gr-qc/0404018

work page arXiv 2004
[7]

Kiefer, ISRN Math

C. Kiefer, ISRN Math. Phys.2013, 509316 (2013), arXiv:1401.3578 [gr-qc]

work page arXiv 2013
[8]

A. G. Riesset al.(Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

work page Pith review arXiv 1998
[9]

1999, , 517, 565, 10.1086/307221

S. Perlmutteret al.(Supernova Cosmology Project), As- trophys. J.517, 565 (1999), arXiv:astro-ph/9812133

work page internal anchor Pith review arXiv 1999
[10]

Planck 2018 results. VI. Cosmological parameters

N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

work page Pith review arXiv 2020
[11]

Weinberg, Rev

S. Weinberg, Rev. Mod. Phys.61, 1 (1989)

1989
[12]

Quintessence, Cosmic Coincidence, and the Cosmological Constant

I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev. Lett.82, 896 (1999), arXiv:astro-ph/9807002

work page Pith review arXiv 1999
[13]

Padmanabhan, Phys

T. Padmanabhan, Phys. Rept.380, 235 (2003), arXiv:hep-th/0212290

work page arXiv 2003
[14]

P. J. E. Peebles and B. Ratra, Rev. Mod. Phys.75, 559 (2003), arXiv:astro-ph/0207347

work page Pith review arXiv 2003
[15]

Particle Dark Matter: Evidence, Candidates and Constraints

G. Bertone, D. Hooper, and J. Silk, Phys. Rept.405, 279 (2005), arXiv:hep-ph/0404175

work page Pith review arXiv 2005
[16]

J. L. Feng, Ann. Rev. Astron. Astrophys.48, 495 (2010), arXiv:1003.0904 [astro-ph.CO]

work page arXiv 2010
[17]

D’Agostino, Phys

R. D’Agostino, Phys. Rev. D99, 103524 (2019), arXiv:1903.03836 [gr-qc]

work page arXiv 2019
[18]

D’Agostino, O

R. D’Agostino, O. Luongo, and M. Muccino, Class. Quant. Grav.39, 195014 (2022), arXiv:2204.02190 [gr- qc]

work page arXiv 2022
[19]

S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D70, 043528 (2004), arXiv:astro- ph/0306438

work page arXiv 2004
[20]

T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82, 451 (2010), arXiv:0805.1726 [gr-qc]

work page Pith review arXiv 2010
[21]

f(R) theories

A. De Felice and S. Tsujikawa, Living Rev. Rel.13, 3 (2010), arXiv:1002.4928 [gr-qc]

work page Pith review arXiv 2010
[22]

Modified Gravity and Cosmology

T. Clifton, P. G. Ferreira, A. Padilla, and C. Sko- rdis, Phys. Rept.513, 1 (2012), arXiv:1106.2476 [astro- ph.CO]

work page internal anchor Pith review arXiv 2012
[23]

Beyond the Cosmological Standard Model

A. Joyce, B. Jain, J. Khoury, and M. Trodden, Phys. Rept.568, 1 (2015), arXiv:1407.0059 [astro-ph.CO]

work page Pith review arXiv 2015
[24]

Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution

S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Phys. Rept.692, 1 (2017), arXiv:1705.11098 [gr-qc]

work page internal anchor Pith review arXiv 2017
[25]

Capozziello, R

S. Capozziello, R. D’Agostino, and O. Luongo, Int. J. Mod. Phys. D28, 1930016 (2019), arXiv:1904.01427 [gr- qc]

work page arXiv 2019
[26]

Probing observational bounds on scalar-tensor theories from standard sirens,

R. D’Agostino and R. C. Nunes, Phys. Rev. D100, 044041 (2019), arXiv:1907.05516 [gr-qc]

work page arXiv 2019
[27]

D’Agostino and R

R. D’Agostino and R. C. Nunes, Phys. Rev. D101, 103505 (2020), arXiv:2002.06381 [astro-ph.CO]

work page arXiv 2020
[28]

Akramiet al.(CANTATA),Modified Gravity and Cos- mology

Y. Akramiet al.(CANTATA),Modified Gravity and Cosmology. An Update by the CANTATA Network, edited by E. N. Saridakis, R. Lazkoz, V. Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´ an Jim´ enez, M. De Laurentis, and G. J. Olmo (Springer, 2021) arXiv:2105.12582 [gr-qc]

work page arXiv 2021
[29]

D’Agostino and R

R. D’Agostino and R. C. Nunes, Phys. Rev. D106, 124053 (2022), arXiv:2210.11935 [gr-qc]

work page arXiv 2022
[30]

D’Agostino and G

R. D’Agostino and G. G. Luciano, Phys. Lett. B857, 138987 (2024), arXiv:2408.13638 [gr-qc]

work page arXiv 2024
[31]

D’Agostino and F

R. D’Agostino and F. Bajardi, Phys. Rev. D111, 104076 (2025), arXiv:2505.21359 [gr-qc]

work page arXiv 2025
[32]

Bouncing Universes in String-inspired Gravity

T. Biswas, A. Mazumdar, and W. Siegel, JCAP03, 009 (2006), arXiv:hep-th/0508194

work page Pith review arXiv 2006
[33]

Deser and R

S. Deser and R. P. Woodard, Phys. Rev. Lett.99, 111301 (2007), arXiv:0706.2151 [astro-ph]

work page arXiv 2007
[34]

A. O. Barvinsky, Phys. Rev. D85, 104018 (2012), arXiv:1112.4340 [hep-th]

work page arXiv 2012
[35]

Phantom dark energy from non-local infrared modifications of General Relativity

M. Maggiore, Phys. Rev. D89, 043008 (2014), arXiv:1307.3898 [hep-th]

work page Pith review arXiv 2014
[36]

Kehagias and M

A. Kehagias and M. Maggiore, JHEP08, 029 (2014), arXiv:1401.8289 [hep-th]

work page arXiv 2014
[37]

Calcagni and L

G. Calcagni and L. Modesto, Phys. Rev. D91, 124059 (2015), arXiv:1404.2137 [hep-th]

work page arXiv 2015
[38]

Deser and R

S. Deser and R. P. Woodard, JCAP06, 034 (2019), arXiv:1902.08075 [gr-qc]

work page arXiv 2019
[39]

Capozziello and R

S. Capozziello and R. D’Agostino, Phys. Dark Univ.42, 101346 (2023), arXiv:2310.03136 [gr-qc]

work page arXiv 2023
[40]

Ding and J.-B

J.-C. Ding and J.-B. Deng, JCAP12, 054 (2019), arXiv:1908.11223 [astro-ph.CO]

work page arXiv 2019
[41]

C.-Y. Chen, P. Chen, and S. Park, Phys. Lett. B796, 112 (2019), arXiv:1905.04557 [gr-qc]

work page arXiv 2019
[42]

Jackson and R

D. Jackson and R. Bufalo, JCAP05, 010 (2023), arXiv:2303.17552 [gr-qc]

work page arXiv 2023
[43]

D’Agostino and V

R. D’Agostino and V. De Falco, JCAP05, 069 (2025), arXiv:2501.18007 [gr-qc]

work page arXiv 2025
[44]

D’Agostino and V

R. D’Agostino and V. De Falco, JCAP07, 046 (2025), arXiv:2502.15460 [gr-qc]

work page arXiv 2025
[45]

D’Agostino and V

R. D’Agostino and V. De Falco, Phys. Rev. D113, 044029 (2026), arXiv:2601.11101 [gr-qc]

work page arXiv 2026
[46]

D’Agostino and V

R. D’Agostino and V. De Falco, Phys. Rev. D112, 064028 (2025), arXiv:2507.01698 [gr-qc]

work page arXiv 2025
[47]

Bravo-Gaete, J

M. Bravo-Gaete, J. Lin, Y. Liu, and X. Zhang, (2026), arXiv:2602.15609 [gr-qc]

work page arXiv 2026
[48]

Liu and Y

Y. Liu and Y. Du, (2025), arXiv:2511.07981 [gr-qc]

work page arXiv 2025
[49]

Li and X

H. Li and X. Zhang, (2026), arXiv:2601.16572 [gr-qc]

work page arXiv 2026
[50]

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

work page Pith review arXiv 2000
[51]

Perlick, Living Rev

V. Perlick, Living Rev. Rel.7, 9 (2004)

2004
[52]

A. M. Beloborodov, Astrophys. J. Lett.566, L85 (2002), arXiv:astro-ph/0201117

work page arXiv 2002
[53]

Pulse profiles of millisecond pulsars and their Fourier amplitudes

J. Poutanen and A. M. Beloborodov, Mon. Not. Roy. Astron. Soc.373, 836 (2006), arXiv:astro-ph/0608663

work page Pith review arXiv 2006
[54]

De Falco, M

V. De Falco, M. Falanga, and L. Stella, Astron. Astro- phys.595, A38 (2016), arXiv:1608.04574 [astro-ph.HE]

work page arXiv 2016
[55]

La Placa, P

R. La Placa, P. Bakala, L. Stella, and M. Falanga, Res. Notes AAS3, 99 (2019), arXiv:1907.11786 [gr-qc]

work page arXiv 2019
[56]

Poutanen, Astron

J. Poutanen, Astron. Astrophys.640, A24 (2020), arXiv:1909.05732 [astro-ph.HE]

work page arXiv 2020
[57]

Bakala, A

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

2023
[58]

Kaiser, Astrophys

N. Kaiser, Astrophys. J.388, 272 (1992)

1992
[59]

Weak Gravitational Lensing

M. Bartelmann and P. Schneider, Phys. Rept.340, 291 (2001), arXiv:astro-ph/9912508

work page Pith review arXiv 2001
[60]

C. R. Keeton and A. O. Petters, Phys. Rev. D72, 104006 (2005), arXiv:gr-qc/0511019

work page arXiv 2005
[61]

Amendola, M

L. Amendola, M. Kunz, and D. Sapone, JCAP04, 013 (2008), arXiv:0704.2421 [astro-ph]

work page arXiv 2008
[62]

Crisnejo and E

G. Crisnejo and E. Gallo, Phys. Rev. D97, 124016 (2018), arXiv:1804.05473 [gr-qc]

work page arXiv 2018
[63]

Crisnejo, E

G. Crisnejo, E. Gallo, and K. Jusufi, Phys. Rev. D100, 104045 (2019), arXiv:1910.02030 [gr-qc]

work page arXiv 2019
[64]

Strong Lensing by Galaxies

T. Treu, Ann. Rev. Astron. Astrophys.48, 87 (2010), arXiv:1003.5567 [astro-ph.CO]

work page Pith review arXiv 2010
[65]

P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50, 42 (2018), arXiv:1801.00860 [gr-qc]. 10

work page arXiv 2018
[66]

Bozza, Phys

V. Bozza, Phys. Rev. D66, 103001 (2002), arXiv:gr- qc/0208075

work page arXiv 2002
[67]

Geodesic stability, Lyapunov exponents and quasinormal modes

V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D79, 064016 (2009), arXiv:0812.1806 [hep-th]

work page Pith review arXiv 2009
[68]

K. S. Virbhadra, Phys. Rev. D79, 083004 (2009), arXiv:0810.2109 [gr-qc]

work page arXiv 2009
[69]

Bozza, F

V. Bozza, F. De Luca, and G. Scarpetta, Phys. Rev. D 74, 063001 (2006), arXiv:gr-qc/0604093

work page arXiv 2006
[70]

Bozza and G

V. Bozza and G. Scarpetta, Phys. Rev. D76, 083008 (2007), arXiv:0705.0246 [gr-qc]

work page arXiv 2007
[71]

Aratore and V

F. Aratore and V. Bozza, JCAP10, 054 (2021), arXiv:2107.05723 [gr-qc]

work page arXiv 2021
[72]

Feleppa, F

F. Feleppa, F. Aratore, and V. Bozza, Phys. Rev. D 112, 044007 (2025), arXiv:2508.03615 [gr-qc]

work page arXiv 2025
[73]

Feleppa, V

F. Feleppa, V. Bozza, and O. Y. Tsupko, Phys. Rev. D 110, 064031 (2024), arXiv:2406.07703 [gr-qc]

work page arXiv 2024
[74]

Feleppa, V

F. Feleppa, V. Bozza, and O. Y. Tsupko, Phys. Rev. D 111, 044018 (2025), arXiv:2412.16712 [gr-qc]

work page arXiv 2025
[75]

Frittelli, T

S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D61, 064021 (2000), arXiv:gr-qc/0001037

work page arXiv 2000
[76]

J. M. Ezquiaga and M. Zumalac´ arregui, Phys. Rev. D 102, 124048 (2020), arXiv:2009.12187 [gr-qc]

work page arXiv 2020
[77]

Schmidt, Phys

F. Schmidt, Phys. Rev. D78, 043002 (2008), arXiv:0805.4812 [astro-ph]

work page arXiv 2008
[78]

C. W. Misner, K. S. Thorne, and J. A. Wheeler,Grav- itation(W. H. Freeman, San Francisco, 1973)

1973
[79]

Bertotti, L

B. Bertotti, L. Iess, and P. Tortora, Nature425, 374 (2003)

2003
[80]

K. C. Sahuet al.(OGLE, MOA, PLANET,µFUN, MiNDSTEp Consortium, RoboNet), Astrophys. J.933, 83 (2022), arXiv:2201.13296 [astro-ph.SR]

work page arXiv 2022

Showing first 80 references.