pith. machine review for the scientific record. sign in

arxiv: 2604.17355 · v1 · submitted 2026-04-19 · 🌀 gr-qc

Recognition: unknown

Weak Gravitational Lensing: A Brief Overview

Partha Pratim Basumallick , Saheb Das , Bhaswati Mandal , Subhadip Sau
Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:10 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational lensinglight deflectionKerr spacetimeaxisymmetric spacetimesGauss-Bonnet theoremFermat's principleweak gravitational lensinggeneral relativity
0
0 comments X

The pith

A unified geometrical framework using OIA and GW-OIA formalisms computes light deflection consistently in both static and rotating gravitational fields.

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

The paper begins with Newtonian light deflection and builds a relativistic treatment through the lens equation, lensing potential, and an effective refractive index interpretation of Fermat's principle in curved spacetime. It derives photon trajectories and deflection angles first for static spherically symmetric metrics, then for the equatorial plane of Kerr spacetime, including analytical expressions for closest approach and critical orbit parameters. The bending angle is recovered via the Rindler-Ishak method and the Gauss-Bonnet theorem applied to optical geometry. The central advance is the extension of these results to general axisymmetric spacetimes through the OIA and GW-OIA formalisms, which supply a single geometrical procedure for light bending that covers both static and rotating cases while reproducing standard general-relativity outcomes.

Core claim

The analysis is extended to axisymmetric spacetimes using the OIA and GW-OIA formalisms, providing a unified geometrical framework for computing light deflection in both static and rotating gravitational fields.

What carries the argument

The OIA and GW-OIA formalisms, which supply a unified geometrical procedure for calculating light deflection angles across static spherically symmetric and rotating axisymmetric spacetimes.

If this is right

  • Photon trajectories in the equatorial plane of Kerr spacetime admit closed-form expressions for the closest approach distance and the critical parameters that separate captured and scattered orbits.
  • The Rindler-Ishak method combined with the Gauss-Bonnet theorem applied to optical geometry recovers the light-bending angle for static and axisymmetric metrics alike.
  • The same geometrical construction applies without modification to any axisymmetric spacetime, removing the need to treat static and rotating cases with separate formalisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may simplify the construction of lensing maps for rotating compact objects in observational data analysis.
  • Similar optical-geometry techniques could be tested on other relativistic effects such as time delay or polarization rotation around spinning sources.
  • Because the method stays within standard general relativity, any observed mismatch with predictions would point to new physics rather than a defect in the lensing calculation itself.

Load-bearing premise

The OIA and GW-OIA derivations and geometric interpretations reproduce the known general-relativity results for light deflection when moving from static to rotating cases without introducing unnoticed approximations or inconsistencies.

What would settle it

A direct numerical comparison showing that the deflection angle for light grazing a Kerr black hole computed via the OIA formalism differs from the established Kerr deflection formula would falsify the claim of a consistent unified framework.

Figures

Figures reproduced from arXiv: 2604.17355 by Bhaswati Mandal, Partha Pratim Basumallick, Saheb Das, Subhadip Sau.

Figure 1
Figure 1. Figure 1: An assortment of Einstein rings photographed by Hubble Space Telescope in 2005 as part of the Sloan Lens Arc Survey. [31] (a) Strong lensing – In scenarios where the mass of the lensing object is sufficiently large and can gen￾erate a strong gravitational field strong lensing effects may be induced. Of course the geometry involving the trinity of source, lens & observer must conspire accordingly in order f… view at source ↗
Figure 2
Figure 2. Figure 2: Lensing of distant galaxy by the cluster Abell 2218 as an instance of weak lensing. In the image provided above there are quite a few instances of multiple images or arcs bearing testimony for strong lensing. However, the background galaxies tell a different story with their distorted images. Although these distortions/magnifications are quite difficult to measure [as shown in later sections]. (b) Weak len… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration depicting the bending of light under the influence of gravitational field. The actual path of the light ray is traced in red. The lensing object is an object of mass M at z = 0. The deflection angle is given by the angle α = −→ξ , ϕ  where −→ξ denotes the distance of closest approach for the light ray. The other symbols depicted here will come in useful while deriving the lens equation and o… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Illustration depicting the projection of the lensing geometry onto a 2-D spatial slice. Here we consider the spacetime to be Friedmann-Lemaitre-Robertson-Walker (FLRW) and the coordinates to be comoving. As such we are at liberty to project the spacetime diagram onto a spatial slice. The source S emits a photon which follows the path (spatial projection) indicated by the red solid line and reaches the … view at source ↗
Figure 5
Figure 5. Figure 5: Illustration depicting the formulation of relativistic Fermat’s principle 6.1 Fermat’s principle – A general relativistic adaptation In the classical context the statement of Fermat’s principle means that light always follows the quickest path between two points. Now, in general relativity the path of light curves are represented by null geodesics i.e., the line element satisfies ds 2 = 0 for any null curv… view at source ↗
Figure 6
Figure 6. Figure 6: Classical depiction of bending of light under the influence of a gravitational field 8 Deflection Angle of Light in the Equatorial Plane of a Kerr Black Hole The deflection of light in curved spacetime lies at the heart of gravitational lensing, a phenomenon whose most striking realizations occur near compact astrophysical objects such as black holes. For a Schwarzschild black hole, which is non-rotating a… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic diagram illustrating the prograde and retrograde orbits (on the equatorial plane) in the vicinity of a spinning black hole. In the above scenario the axis of rotation of the black hole points into the page. resisted by frame dragging, and turn back at larger radii with comparatively smaller deflection. Pictorial depiction has been shown in fig. [7] for better visualisation. This difference, albei… view at source ↗
Figure 8
Figure 8. Figure 8: Overhead view of the prograde and retrograde orbits on the equatorial plane of a Kerr black hole. Blue and red dashed-dotted lines indicate closest approach for prograde and retrograde cases respectively. Cyan and orange solid curves mark the respective innermost photon orbits. The spin vector points out of the plane. - As a/M → 1, b (prograde) sc → 2M, which represents the smallest allowed impact paramete… view at source ↗
Figure 9
Figure 9. Figure 9: Variation of the normalized critical impact parameter bsc/M as a function of the dimensionless spin a/M. The blue curve corresponds to prograde motion (s = +1), the red curve to retrograde motion (s = −1), and the black dashed line indicates the Schwarzschild limit. 0.0 0.2 0.4 0.6 0.8 1.0 a M 1 2 3 4 rsc M [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Variation of the normalized photon sphere radius rsc/M versus black hole spin a/M. The trends for prograde (blue), retrograde (red), and Schwarzschild (black dashed) cases are shown. Prograde motion allows tighter photon orbits, while retrograde motion pushes them outward. the sign parameter s, eliminating the need for separate case-by-case derivations. As a result, it stands as a robust tool for lensing … view at source ↗
Figure 11
Figure 11. Figure 11: Geometric definition of the Rindler–Ishak local angle on the orbital plane. A trajectory (red curve) prop￾agates from the source S to the observer O past the lens at L, with impact parameter b defined with respect to the reference straight line. At the observation point, the radial coordinate direction (along ϕ = const) is compared with the instantaneous direction of motion of the ray or particle. Their i… view at source ↗
Figure 12
Figure 12. Figure 12: A patch of surface created by the intersection of several curves thereby subtending the exterior angle [PITH_FULL_IMAGE:figures/full_fig_p044_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Two dimensional Riemannian manifold 2R where LS, LV & SV are the spatial projections of the geodesics in the equatorial plane under consideration. Since we are interested in trajectories confined to the equatorial plane, Fig. fig. [13] effectively represents a two-dimensional Riemannian manifold 2R, where LS, LV, and SV denote spatial projections of the corresponding geodesics. The photon path in this pla… view at source ↗
Figure 14
Figure 14. Figure 14: Extracting the integrable surface region depicted as the quadrilateral [PITH_FULL_IMAGE:figures/full_fig_p046_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The figure presents a scenario where we have the Sun as the lensing object, Earth as the location of the viewer and a pulsar as the source. The pulsar emits radio signals at periodic intervals in a specific anisotropic manner. The pulse profile associated with the emissions allows us to determine the radiation direction at the actual location of the pulsar. The relative positions of the Earth, Sun & Pulsa… view at source ↗
Figure 16
Figure 16. Figure 16: Embedding of a three dimensional space with associated curves and surfaces relevant to the geometry [PITH_FULL_IMAGE:figures/full_fig_p066_16.png] view at source ↗
read the original abstract

Gravitational lensing constitutes one of the most direct observational manifestations of spacetime curvature and provides a powerful probe of compact astrophysical objects. In this work, we present a comprehensive analysis of the bending of light in curved spacetime, beginning with the fundamental aspects of gravitational lensing and the Newtonian approximation to light deflection. The relativistic formulation of lensing is then developed through the lens equation, lensing potential, and a geometrical interpretation of Fermat's principle using an effective refractive index in curved spacetime. Photon trajectories and light deflection are subsequently investigated in static, spherically symmetric geometries, followed by a detailed study of photon motion in the equatorial plane of the Kerr spacetime. Analytical expressions for the closest approach distance and critical parameters governing photon orbits are derived. Furthermore, the bending angle is examined using the Rindler-Ishak method and the Gauss-Bonnet theorem within the optical geometry. Finally, the analysis is extended to axisymmetric spacetimes using the OIA and GW-OIA formalisms, providing a unified geometrical framework for computing light deflection in both static and rotating gravitational fields.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript provides a comprehensive overview of weak gravitational lensing, beginning with Newtonian light deflection and the relativistic lens equation with Fermat's principle interpreted via an effective refractive index. It derives photon trajectories, closest-approach distances, and critical parameters in static spherically symmetric spacetimes and the equatorial plane of Kerr, then computes deflection angles via the Rindler-Ishak method and Gauss-Bonnet theorem applied to the optical geometry. The analysis concludes by extending these methods to axisymmetric spacetimes through the OIA and GW-OIA formalisms to yield a unified geometrical framework for light deflection in both static and rotating gravitational fields.

Significance. If the derivations accurately reproduce standard GR results without hidden approximations in the rotating case, the paper offers a pedagogically useful synthesis that unifies static and axisymmetric treatments under optical geometry and Gauss-Bonnet methods. This could serve as a reference for students and researchers working on lensing in compact objects. However, as a review-style overview without new quantitative predictions, machine-checked proofs, or falsifiable claims, its significance remains primarily educational rather than field-advancing.

major comments (1)
  1. [Abstract; section on OIA and GW-OIA formalisms] Abstract and the section extending the analysis to axisymmetric spacetimes via OIA/GW-OIA: the central claim of a unified geometrical framework for static and rotating fields assumes that the optical curvature scalar, area element, and boundary terms carry over directly from the static spherical case. The Kerr optical geometry is non-static due to frame-dragging; the manuscript must explicitly demonstrate how (or whether) the OIA/GW-OIA constructions incorporate the modified refractive index or metric components, and must compare the resulting deflection angle against known Kerr results to rule out systematic error.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our overview of weak gravitational lensing. The major comment raises a valid point about the need for explicit clarification in the axisymmetric extension, and we address it directly below.

read point-by-point responses

  1. Referee: [Abstract; section on OIA and GW-OIA formalisms] Abstract and the section extending the analysis to axisymmetric spacetimes via OIA/GW-OIA: the central claim of a unified geometrical framework for static and rotating fields assumes that the optical curvature scalar, area element, and boundary terms carry over directly from the static spherical case. The Kerr optical geometry is non-static due to frame-dragging; the manuscript must explicitly demonstrate how (or whether) the OIA/GW-OIA constructions incorporate the modified refractive index or metric components, and must compare the resulting deflection angle against known Kerr results to rule out systematic error.

    Authors: We agree that the non-static character of the Kerr optical geometry due to frame-dragging requires explicit treatment. In the OIA and GW-OIA sections, the constructions already incorporate the off-diagonal metric components of the Kerr spacetime into the effective refractive index for the equatorial plane, which modifies the optical metric, curvature scalar, and area element accordingly. However, to strengthen the presentation and directly address the concern, we will revise the manuscript to include a dedicated subsection that derives these quantities step by step from the Kerr optical metric, showing the explicit dependence on the frame-dragging terms. We will also add a direct numerical and analytical comparison of the resulting deflection angles with standard Kerr results obtained via geodesic integration (e.g., as in the literature on strong lensing in Kerr). This will confirm consistency and eliminate any possibility of systematic error in the unified framework. revision: yes

Circularity Check

0 steps flagged

No circularity: standard review of known lensing methods with no self-defined predictions

full rationale

The manuscript is an overview that reproduces standard GR results for light deflection in spherical and Kerr metrics using established techniques (Rindler-Ishak, Gauss-Bonnet in optical geometry, OIA/GW-OIA). No quantitative predictions are advanced that reduce to fitted parameters or self-citations by construction; closest-approach and critical-orbit expressions are derived from the Kerr geodesic equation in the usual way, and the extension to axisymmetric cases applies previously published formalisms without redefining inputs in terms of outputs. All load-bearing steps remain externally verifiable against textbook GR calculations, yielding a self-contained review with no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a review that relies entirely on textbook general relativity and differential geometry; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Standard general relativity governs light propagation in curved spacetime
    Invoked from the first paragraph onward for all deflection calculations.
  • domain assumption Fermat's principle holds in an effective optical geometry with refractive index derived from the metric
    Used to give geometrical interpretation of the lens equation.

pith-pipeline@v0.9.0 · 5494 in / 1374 out tokens · 35832 ms · 2026-05-10T06:10:06.163407+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

131 extracted references · 94 canonical work pages · 2 internal anchors

  1. [1]

    Lecture notes on general relativity,

    S. M. Carroll, “Lecture notes on general relativity,” 1997. https://arxiv.org/abs/gr-qc/9712019

    work page internal anchor Pith review arXiv 1997

  2. [2]

    Periodic relativity: the theory of gravity in flat space time

    V . H. Zaveri, “Periodic relativity: the theory of gravity in flat space time,” 2025. https://arxiv.org/abs/0707.4539

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    Lectures on Gravitational Lensing

    R. Narayan and M. Bartelmann, “Lectures on gravitational lensing,” 1997. https://arxiv.org/abs/astro-ph/9606001

    work page Pith review arXiv 1997

  4. [4]

    Dodelson,Gravitational Lensing

    S. Dodelson,Gravitational Lensing. Cambridge University Press, 2017

    2017

  5. [5]

    Gravitational lensing,

    M. Bartelmann, “Gravitational lensing,”Classical and Quantum Gravity27no. 23, (Nov., 2010) 233001.http://dx.doi.org/10.1088/0264-9381/27/23/233001

    work page doi:10.1088/0264-9381/27/23/233001 2010

  6. [6]

    New estimator for gravitational lensing using galaxy and intensity mapping surveys,

    M. Jalilvand, E. Majerotto, C. Bonvin, F. Lacasa, M. Kunz, W. Naidoo, and K. Moodley, “New estimator for gravitational lensing using galaxy and intensity mapping surveys,”Physical Review Letters124no. 3, (Jan., 2020) .http://dx.doi.org/10.1103/PhysRevLett.124.031101

    work page doi:10.1103/physrevlett.124.031101 2020

  7. [7]

    <scp>relensing</scp>: Reconstructing the mass profile of galaxy clusters from gravitational lensing,

    D. A. Torres-Ballesteros and L. Castañeda, “<scp>relensing</scp>: Reconstructing the mass profile of galaxy clusters from gravitational lensing,”Monthly Notices of the Royal Astronomical Society518no. 3, (Nov., 2022) 4494–4516.http://dx.doi.org/10.1093/mnras/stac3253

    work page doi:10.1093/mnras/stac3253 2022

  8. [8]

    Strong lensing by galaxy clusters,

    P . Natarajan, L. L. Williams, M. Bradac, C. Grillo, A. Ghosh, K. Sharon, and J. Wagner, “Strong lensing by galaxy clusters,”arXiv:2403.06245 [astro-ph.CO]. https://arxiv.org/abs/2403.06245

    work page arXiv

  9. [9]

    A model for galaxy–galaxy strong lensing statistics in surveys,

    G. Ferrami and J. S. B. Wyithe, “A model for galaxy–galaxy strong lensing statistics in surveys,” Monthly Notices of the Royal Astronomical Society532no. 2, (June, 2024) 1832–1848. http://dx.doi.org/10.1093/mnras/stae1607

    work page doi:10.1093/mnras/stae1607 2024

  10. [10]

    Gravitational lensing from clusters of galaxies to test disformal couplings theories,

    S. Zamani, V . Salzano, and D. Bettoni, “Gravitational lensing from clusters of galaxies to test disformal couplings theories,” 2024.https://arxiv.org/abs/2401.02221

    work page arXiv 2024

  11. [11]

    Gravitational lensing of galaxy clustering,

    B. Buncher, G. Holder, and S. Hotinli, “Gravitational lensing of galaxy clustering,” arXiv:2402.07988 [astro-ph.CO].https://arxiv.org/abs/2402.07988. 86

    work page arXiv

  12. [12]

    The dark matter of gravitational lensing,

    R. Massey, T. Kitching, and J. Richard, “The dark matter of gravitational lensing,”Reports on Progress in Physics73no. 8, (July, 2010) 086901. http://dx.doi.org/10.1088/0034-4885/73/8/086901

    work page doi:10.1088/0034-4885/73/8/086901 2010

  13. [13]

    Gravitational lensing: a unique probe of dark matter and dark energy,

    R. S. Ellis, “Gravitational lensing: a unique probe of dark matter and dark energy,”Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences368no. 1914, (2010) 967–987,https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2009.0209. https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2009.0209

    work page doi:10.1098/rsta.2009.0209 1914

  14. [14]

    Gravitational lensing by a massive object in a dark matter halo. i. critical curves and caustics,

    M. Karamazov, L. Timko, and D. Heyrovský, “Gravitational lensing by a massive object in a dark matter halo. i. critical curves and caustics,”The Astrophysical Journal922no. 1, (Nov, 2021) 72. https://dx.doi.org/10.3847/1538-4357/ac151c

    work page doi:10.3847/1538-4357/ac151c 2021

  15. [15]

    Probing dark matter with strong gravitational lensing through an effective density slope,

    A. C. Sengül and C. Dvorkin, “Probing dark matter with strong gravitational lensing through an effective density slope,”Monthly Notices of the Royal Astronomical Society516no. 1, (July, 2022) 336–357.http://dx.doi.org/10.1093/mnras/stac2256

    work page doi:10.1093/mnras/stac2256 2022

  16. [16]

    Vegetti, S

    S. Vegetti, S. Birrer, G. Despali, C. D. Fassnacht, D. Gilman, Y. Hezaveh, L. P . Levasseur, J. P . McKean, D. M. Powell, C. M. O’Riordan, and G. Vernardos, “Strong gravitational lensing as a probe of dark matter,” 2023.https://arxiv.org/abs/2306.11781

    work page arXiv 2023

  17. [17]

    Probing dark matter via strong gravitational lensing by black holes,

    A. Vachher, D. Baboolal, and S. G. Ghosh, “Probing dark matter via strong gravitational lensing by black holes,”Physics of the Dark Universe44(2024) 101493. https://www.sciencedirect.com/science/article/pii/S221268642400075X

    2024

  18. [18]

    Gannon, A

    C. Gannon, A. Nierenberg, A. Benson, R. Keeley, X. Du, and D. Gilman, “Dark matter substructure: A lensing perspective,” 2025.https://arxiv.org/abs/2501.17362

    work page arXiv 2025

  19. [19]

    2018, Geosciences, 8, 365, doi: 10.3390/geosciences8100365

    Y. Tsapras, “Microlensing searches for exoplanets,”Geosciences8no. 10, (Sept., 2018) 365. http://dx.doi.org/10.3390/geosciences8100365

    work page doi:10.3390/geosciences8100365 2018

  20. [20]

    Image formation for extended sources with the solar gravitational lens,

    S. G. Turyshev and V . T. Toth, “Image formation for extended sources with the solar gravitational lens,”Physical Review D102no. 2, (July, 2020) . http://dx.doi.org/10.1103/PhysRevD.102.024038

    work page doi:10.1103/physrevd.102.024038 2020

  21. [21]

    Resolved imaging of exoplanets with the solar gravitational lens,

    S. G. Turyshev and V . T. Toth, “Resolved imaging of exoplanets with the solar gravitational lens,” Monthly Notices of the Royal Astronomical Society515no. 4, (July, 2022) 6122–6132. http://dx.doi.org/10.1093/mnras/stac2130

    work page doi:10.1093/mnras/stac2130 2022

  22. [22]

    Imaging rotating and orbiting exoplanets with the solar gravitational lens,

    V . T. Toth and S. G. Turyshev, “Imaging rotating and orbiting exoplanets with the solar gravitational lens,”Monthly Notices of the Royal Astronomical Society525no. 4, (Sept., 2023) 5846–5856.http://dx.doi.org/10.1093/mnras/stad2655. 87

    work page doi:10.1093/mnras/stad2655 2023

  23. [23]

    Mróz and R

    P . Mróz and R. Poleski,Exoplanet Occurrence Rates from Microlensing Surveys, p. 1–23. Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-319-30648-3_208-1

    work page doi:10.1007/978-3-319-30648-3_208-1 2024

  24. [24]

    Gravitational lensing of gravitational waves: A statistical perspective

    S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, “Gravitational lensing of gravitational waves: a statistical perspective,”Monthly Notices of the Royal Astronomical Society476no. 2, (May, 2018) 2220–2229, arXiv:1802.05089 [astro-ph.CO]

    work page Pith review arXiv 2018

  25. [25]

    Observational signatures of microlensing in gravitational waves at ligo/virgo frequencies,

    Diego, J. M., Hannuksela, O. A., Kelly, P . L., Pagano, G., Broadhurst, T., Kim, K., Li, T. G. F., and Smoot, G. F., “Observational signatures of microlensing in gravitational waves at ligo/virgo frequencies,”A&A627(2019) A130.https://doi.org/10.1051/0004-6361/201935490

    work page doi:10.1051/0004-6361/201935490 2019

  26. [26]

    Lensinggw: a python package for lensing of gravitational waves,

    Pagano, G., Hannuksela, O. A., and Li, T. G. F., “Lensinggw: a python package for lensing of gravitational waves,”A&A643(2020) A167. https://doi.org/10.1051/0004-6361/202038730

    work page doi:10.1051/0004-6361/202038730 2020

  27. [27]

    Lensing of gravitational waves: universal signatures in the beating pattern,

    O. Bulashenko and H. Ubach, “Lensing of gravitational waves: universal signatures in the beating pattern,”Journal of Cosmology and Astroparticle Physics2022no. 07, (July, 2022) 022. http://dx.doi.org/10.1088/1475-7516/2022/07/022

    work page doi:10.1088/1475-7516/2022/07/022 2022

  28. [28]

    Lensing of gravitational waves: Efficient wave-optics methods and validation with symmetric lenses,

    G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.-Y. Cheung, “Lensing of gravitational waves: Efficient wave-optics methods and validation with symmetric lenses,”Physical Review D108no. 4, (Aug., 2023) .http://dx.doi.org/10.1103/PhysRevD.108.043527

    work page doi:10.1103/physrevd.108.043527 2023

  29. [29]

    Gravitational lensing of gravitational waves: prospects for probing intermediate-mass black holes in galaxy lenses with global minima image,

    A. K. Meena, “Gravitational lensing of gravitational waves: prospects for probing intermediate-mass black holes in galaxy lenses with global minima image,”Monthly Notices of the Royal Astronomical Society532no. 3, (07, 2024) 3568–3581, https://academic.oup.com/mnras/article-pdf/532/3/3568/58652494/stae1707.pdf. https://doi.org/10.1093/mnras/stae1707

    work page doi:10.1093/mnras/stae1707 2024

  30. [30]

    An interference-based method for the detection of strongly lensed gravitational waves,

    X. Shan, B. Hu, X. Chen, and R.-G. Cai, “An interference-based method for the detection of strongly lensed gravitational waves,”Nature Astronomy(Mar, 2025) . https://doi.org/10.1038/s41550-025-02519-5

    work page doi:10.1038/s41550-025-02519-5 2025

  31. [31]

    The Sloan Lens ACS Survey. I. A Large Spectroscopically Selected Sample of Massive Early-Type Lens Galaxies,

    A. S. Bolton, S. Burles, L. V . E. Koopmans, T. Treu, and L. A. Moustakas, “The sloan lens acs survey. i. a large spectroscopically selected sample of massive early-type lens galaxies,”The Astrophysical Journal638no. 2, (Feb., 2006) 703–724.http://dx.doi.org/10.1086/498884

    work page doi:10.1086/498884 2006

  32. [32]

    Strong 88 gravitational lensing’s ’external shear’ is not shear,

    A. Etherington, J. W. Nightingale, R. Massey, S.-I. Tam, X. Cao, A. Niemiec, Q. He, A. Robertson, R. Li, A. Amvrosiadis, S. Cole, J. M. Diego, C. S. Frenk, B. L. Frye, D. Harvey, M. Jauzac, A. M. Koekemoer, D. J. Lagattuta, M. Limousin, G. Mahler, E. Sirks, and C. L. Steinhardt, “Strong 88 gravitational lensing’s ’external shear’ is not shear,” 2023. http...

    work page arXiv 2023

  33. [33]

    Kumar, S

    J. Kumar, S. U. Islam, and S. G. Ghosh, “Strong gravitational lensing by loop quantum gravity motivated rotating black holes and eht observations,” 2023. https://arxiv.org/abs/2305.04336

    work page arXiv 2023

  34. [34]

    Searching for strong gravitational lenses,

    C. Lemon, F. Courbin, A. More, P . Schechter, R. Cañameras, L. Delchambre, C. Leung, Y. Shu, C. Spiniello, Y. Hezaveh, J. Klüter, and R. McMahon, “Searching for strong gravitational lenses,” 2023.https://arxiv.org/abs/2310.13466

    work page arXiv 2023

  35. [35]

    Essentials of strong gravitational lensing,

    P . Saha, D. Sluse, J. Wagner, and L. L. R. Williams, “Essentials of strong gravitational lensing,” 2024.https://arxiv.org/abs/2401.04165

    work page arXiv 2024

  36. [36]

    Strong gravitational lensing with upcoming wide-field radio surveys,

    S. McCarty and L. Connor, “Strong gravitational lensing with upcoming wide-field radio surveys,” 2024.https://arxiv.org/abs/2412.01746

    work page arXiv 2024

  37. [37]

    Method for Detecting a Cosmological Density of Condensed Objects,

    W. H. Press and J. E. Gunn, “Method for Detecting a Cosmological Density of Condensed Objects,”The Astrophysical Journal185(Oct., 1973) 397–412

    1973

  38. [38]

    Strong lensing by galaxies,

    T. Treu, “Strong lensing by galaxies,”Annual Review of Astronomy and Astrophysics48no. Volume 48, 2010, (2010) 87–125.https://www.annualreviews.org/content/journals/10.1146/ annurev-astro-081309-130924

    2010

  39. [39]

    Clash: Photometric redshifts with 16 hst bands in galaxy cluster fields,

    Jouvel, S., Host, O., Lahav, O., Seitz, S., Molino, A., Coe, D., Postman, M., Moustakas, L., Benìtez, N., Rosati, P ., Balestra, I., Grillo, C., Bradley, L., Fritz, A., Kelson, D., Koekemoer, A. M., Lemze, D., Medezinski, E., Mercurio, A., Moustakas, J., Nonino, M., Scodeggio, M., Zheng, W., Zitrin, A., Bartelmann, M., Bouwens, R., Broadhurst, T., Donahue...

    work page doi:10.1051/0004-6361/201322419 2014

  40. [40]

    Clash: accurate photometric redshifts with 14 hst bands in massive galaxy cluster cores,

    A. Molino, N. Benítez, B. Ascaso, D. Coe, M. Postman, S. Jouvel, O. Host, O. Lahav, S. Seitz, E. Medezinski, P . Rosati, W. Schoenell, A. Koekemoer, Y. Jimenez-Teja, T. Broadhurst, P . Melchior, I. Balestra, M. Bartelmann, R. Bouwens, L. Bradley, N. Czakon, M. Donahue, H. Ford, O. Graur, G. Graves, C. Grillo, L. Infante, S. W. Jha, D. Kelson, R. Lazkoz, D...

    work page doi:10.1093/mnras/stx1243 2017

  41. [41]

    An evaluation of 10 lensing models of the frontier fields cluster macs j0416.1-2403,

    J. D. R. González, K. Sharon, and G. Mahler, “An evaluation of 10 lensing models of the frontier fields cluster macs j0416.1-2403,”The Astrophysical Journal863no. 1, (Aug, 2018) 60. https://dx.doi.org/10.3847/1538-4357/aacf8e

    work page doi:10.3847/1538-4357/aacf8e 2018

  42. [42]

    The Hubble constant,

    J. V . Narlikar, “The Hubble constant,”Bulletin of the Astronomical Society of India24(Dec., 1996) 835

    1996

  43. [43]

    The trouble with h0,

    J. L. Bernal, L. Verde, and A. G. Riess, “The trouble with h0,”Journal of Cosmology and Astroparticle Physics2016no. 10, (Oct, 2016) 019. https://dx.doi.org/10.1088/1475-7516/2016/10/019

    work page doi:10.1088/1475-7516/2016/10/019 2016

  44. [44]

    Über eine mögliche form fiktiver doppelsterne,

    O. Chwolson, “Über eine mögliche form fiktiver doppelsterne,”Astronomische Nachrichten221 no. 20, (1924) 329–330, https://onlinelibrary.wiley.com/doi/pdf/10.1002/asna.19242212003. https://onlinelibrary.wiley.com/doi/abs/10.1002/asna.19242212003

    work page doi:10.1002/asna.19242212003 1924

  45. [45]

    Lens-like action of a star by the deviation of light in the gravitational field,

    A. Einstein, “Lens-like action of a star by the deviation of light in the gravitational field,”Science 84no. 2188, (1936) 506–507, https://www.science.org/doi/pdf/10.1126/science.84.2188.506. https://www.science.org/doi/abs/10.1126/science.84.2188.506

    work page doi:10.1126/science.84.2188.506 1936

  46. [46]

    Faint distorted structures in the core of A 370 : are they gravitationally lensed galaxies at z=1 ?,

    B. Fort, J. L. Prieur, G. Mathez, Y. Mellier, and G. Soucail, “Faint distorted structures in the core of A 370 : are they gravitationally lensed galaxies at z=1 ?,”Astronomy & Astrophysics200(July, 1988) L17–L20

    1988

  47. [47]

    Detection of Systematic Gravitational Lens Galaxy Image Alignments: Mapping Dark Matter in Galaxy Clusters,

    J. A. Tyson, F. Valdes, and R. A. Wenk, “Detection of Systematic Gravitational Lens Galaxy Image Alignments: Mapping Dark Matter in Galaxy Clusters,”The Astrophysical Journal Letters349(Jan.,

  48. [48]

    Weak gravitational lensing,

    M. Bartelmann and P . Schneider, “Weak gravitational lensing,”Physics Reports340no. 4–5, (Jan.,

  49. [49]

    291–472.http://dx.doi.org/10.1016/S0370-1573(00)00082-X

    work page doi:10.1016/s0370-1573(00)00082-x

  50. [50]

    Weak gravitational lensing by large-scale structure,

    A. Refregier, “Weak gravitational lensing by large-scale structure,”Annual Review of Astronomy and Astrophysics41no. 1, (Sept., 2003) 645–668. http://dx.doi.org/10.1146/annurev.astro.41.111302.102207

    work page doi:10.1146/annurev.astro.41.111302.102207 2003

  51. [51]

    Weak gravitational lensing,

    H. Hoekstra, “Weak gravitational lensing,” 2013.https://arxiv.org/abs/1312.5981

    work page arXiv 2013

  52. [52]

    , keywords =

    B. Giblin, C. Heymans, M. Asgari, H. Hildebrandt, H. Hoekstra, B. Joachimi, A. Kannawadi, K. Kuijken, C.-A. Lin, L. Miller, T. Tröster, J. L. van den Busch, A. H. Wright, M. Bilicki, C. Blake, 90 J. de Jong, A. Dvornik, T. Erben, F. Getman, N. R. Napolitano, P . Schneider, H. Shan, and E. Valentijn, “Kids-1000 catalogue: Weak gravitational lensing shear m...

    work page doi:10.1051/0004-6361/202038850 2021

  53. [53]

    Weak gravitational lensing shear measurement with fpfs: analytical mitigation of noise bias and selection bias,

    X. Li, Y. Li, and R. Massey, “Weak gravitational lensing shear measurement with fpfs: analytical mitigation of noise bias and selection bias,”Monthly Notices of the Royal Astronomical Society511 no. 4, (Feb., 2022) 4850–4860.http://dx.doi.org/10.1093/mnras/stac342

    work page doi:10.1093/mnras/stac342 2022

  54. [54]

    Forklens: Accurate weak-lensing shear measurement with deep learning,

    Z. Zhang, H. Shan, N. Li, C. Wei, J. Yao, Z. Ban, Y. Fang, Q. Guo, D. Liu, G. Li, L. Lin, M. Li, R. Li, X. Li, Y. Luo, X. Meng, J. Nie, Z. Qi, Y. Qiu, L. Shao, H. Tian, L. Wang, W. Wang, J. Xian, Y. Xu, T. Zhang, X. Zhang, and Z. Zhou, “Forklens: Accurate weak-lensing shear measurement with deep learning,”Astronomy & Astrophysics683(Mar., 2024) A209. http...

    work page doi:10.1051/0004-6361/202345903 2024

  55. [55]

    Probing the cosmological principle with weak lensing shear,

    J. Adam, R. Maartens, J. Larena, and C. Clarkson, “Probing the cosmological principle with weak lensing shear,”Journal of Cosmology and Astroparticle Physics2025no. 02, (Feb., 2025) 016. http://dx.doi.org/10.1088/1475-7516/2025/02/016

    work page doi:10.1088/1475-7516/2025/02/016 2025

  56. [56]

    Weak gravitational lensing of the cmb,

    A. Lewis and A. Challinor, “Weak gravitational lensing of the cmb,”Physics Reports429no. 1, (2006) 1–65. https://www.sciencedirect.com/science/article/pii/S0370157306000810

    2006

  57. [57]

    Weak Gravitational Lensing of Distant Galaxies,

    N. Kaiser, “Weak Gravitational Lensing of Distant Galaxies,”The Astrophysical Journal388(Apr.,

  58. [58]

    Mapping the Dark Matter with Weak Gravitational Lensing,

    N. Kaiser and G. Squires, “Mapping the Dark Matter with Weak Gravitational Lensing,”The Astrophysical Journal404(Feb., 1993) 441

    1993

  59. [59]

    Kids-450: the tomographic weak lensing power spectrum and constraints on cosmological parameters,

    F. Köhlinger, M. Viola, B. Joachimi, H. Hoekstra, E. van Uitert, H. Hildebrandt, A. Choi, T. Erben, C. Heymans, S. Joudaki, D. Klaes, K. Kuijken, J. Merten, L. Miller, P . Schneider, and E. A. Valentijn, “Kids-450: the tomographic weak lensing power spectrum and constraints on cosmological parameters,”Monthly Notices of the Royal Astronomical Society471no...

    work page doi:10.1093/mnras/stx1820 2017

  60. [60]

    A. B. Congdon and C. R. Keeton,Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics and Cosmology. 2018

    2018

  61. [61]

    Review of Particle Physics,

    ParticleDataGroup, R. L. Workman, V . D. Burkert, V . Crede, E. Klempt, U. Thoma, L. Tiator, K. Agashe, G. Aielli, B. C. Allanach, C. Amsler, M. Antonelli, E. C. Aschenauer, D. M. Asner, H. Baer, S. Banerjee, R. M. Barnett, L. Baudis, C. W. Bauer, J. J. Beatty, V . I. Belousov, J. Beringer, 91 A. Bettini, O. Biebel, K. M. Black, E. Blucher, R. Bonventre, ...

    work page doi:10.1093/ptep/ptac097 2022

  62. [62]

    Euclid in a nutshell,

    A. Troja, I. Tutusaus, and J. G. Sorce, “Euclid in a nutshell,” 2022. https://arxiv.org/abs/2211.09668. [62]EuclidCollaboration, Y. Mellieret al., “Euclid. I. Overview of the Euclid mission,”Astron. Astrophys.697(2025) A1,arXiv:2405.13491 [astro-ph.CO]. 92

    work page arXiv 2022

  63. [63]

    Research in Astronomy and Astrophysics , archivePrefix = "arXiv", eprint =

    “Astrophysical applications of gravitational microlensing,”Research in Astronomy and Astrophysics 12no. 8, (Aug, 2012) 947.https://dx.doi.org/10.1088/1674-4527/12/8/005

    work page doi:10.1088/1674-4527/12/8/005 2012

  64. [64]

    , year = 2012, month = sep, volume =

    B. S. Gaudi, “Microlensing surveys for exoplanets,”Annual Review of Astronomy and Astrophysics50 no. Volume 50, 2012, (2012) 411–453.https://www.annualreviews.org/content/ journals/10.1146/annurev-astro-081811-125518

    work page doi:10.1146/annurev-astro-081811-125518 2012

  65. [65]

    Strong gravitational lensing and microlensing of supernovae,

    S. H. Suyu, A. Goobar, T. Collett, A. More, and G. Vernardos, “Strong gravitational lensing and microlensing of supernovae,”Space Science Reviews220no. 1, (Feb, 2024) 13. https://doi.org/10.1007/s11214-024-01044-7

    work page doi:10.1007/s11214-024-01044-7 2024

  66. [66]

    Michell, “Vii

    J. Michell, “Vii. on the means of discovering the distance, magnitude, &amp;c. of the fixed stars, in consequence of the diminution of the velocity of their light, in case such a diminution should be found to take place in any of them, and such other data should be procured from observations, as would be farther necessary for that purpose. by the rev. joh...

    work page doi:10.1098/rstl.1784.0008

  67. [67]

    Laplace,Exposition du systéme du monde

    P .-S. Laplace,Exposition du systéme du monde. Cambridge Library Collection - Mathematics. Cambridge University Press, 2 ed., 2009

    2009

  68. [68]

    Johann Georg von Soldner and the gravitational bending of light, with an English translation of his essay on it published in 1801,

    S. L. Jaki, “Johann Georg von Soldner and the gravitational bending of light, with an English translation of his essay on it published in 1801,”Foundations of Physics8no. 11-12, (Dec., 1978) 927–950

    1978

  69. [69]

    On Soldners Value of Newtonian Deflection of Light,

    H. J. Treder and G. Jackisch, “On Soldners Value of Newtonian Deflection of Light,”Astronomische Nachrichten302no. 6, (Jan., 1981) 275

    1981

  70. [70]

    ¨Uber den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes,

    A. Einstein, “Über den einfluß der schwerkraft auf die ausbreitung des lichtes,”Annalen der Physik 340no. 10, (1911) 898–908, https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19113401005. https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19113401005

    work page doi:10.1002/andp.19113401005 1911

  71. [71]

    Erklarung der Perihelionbewegung der Merkur aus der allgemeinen Relativitatstheorie,

    A. Einstein, “Erklarung der Perihelionbewegung der Merkur aus der allgemeinen Relativitatstheorie,”Sitzungsberichte der K&ouml;niglich Preussischen Akademie der Wissenschaften47 (Jan., 1915) 831–839

    1915

  72. [72]

    A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919,

    F. W. Dyson, A. S. Eddington, and C. Davidson, “A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919,” Jan., 1920. 93

    1919

  73. [73]

    The Confrontation between General Relativity and Experiment

    C. M. Will, “The confrontation between general relativity and experiment,”Living Reviews in Relativity17no. 1, (Jun, 2014) 4.https://doi.org/10.12942/lrr-2014-4

    work page doi:10.12942/lrr-2014-4 2014

  74. [74]

    Meneghetti,Introduction to Gravitational Lensing: With Python Examples

    M. Meneghetti,Introduction to Gravitational Lensing: With Python Examples. 2022

    2022

  75. [75]

    J. W. Strutt,On the Electromagnetic Theory of Light, p. 518–536. Cambridge Library Collection - Mathematics. Cambridge University Press, 2009

    2009

  76. [76]

    Born approximation in wave optics of gravitational lensing revisited,

    H. Yarimoto and M. Oguri, “Born approximation in wave optics of gravitational lensing revisited,”Physical Review D111no. 8, (Apr., 2025) . http://dx.doi.org/10.1103/PhysRevD.111.083541

    work page doi:10.1103/physrevd.111.083541 2025

  77. [77]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation. 1973

    1973

  78. [78]

    Carroll,Spacetime and Geometry: An Introduction to General Relativity

    S. Carroll,Spacetime and Geometry: An Introduction to General Relativity. Benjamin Cummings, 2003. https: //www.bibsonomy.org/bibtex/2a9db5ea9854d4856475d8a4254003365/random3f

    2003

  79. [79]

    Schneider, J

    P . Schneider, J. Ehlers, and E. E. Falco,Gravitational Lenses. 1992

    1992

  80. [80]

    On fermat’s principle in general relativity. i. the general case,

    V . Perlick, “On fermat’s principle in general relativity. i. the general case,”Classical and Quantum Gravity7no. 8, (Aug, 1990) 1319.https://dx.doi.org/10.1088/0264-9381/7/8/011

    work page doi:10.1088/0264-9381/7/8/011 1990

Showing first 80 references.