arxiv: 2604.05009 · v1 · submitted 2026-04-06 · 🌀 gr-qc · astro-ph.GA

Recognition: 2 theorem links

· Lean Theorem

Revisiting The Gravitational Mirroring In Presence of Compact Objects

Aritra Sanyal, Bikramarka S Choudhury, Farook Rahaman, Md Khalid Hossain

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:37 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GA

keywords gravitational lensingSchwarzschild metriccompact objectsreflection imagelight bendingray tracingstrong gravityastrophysical mirroring

0 comments

The pith

Extremely compact objects bend light rays to create mirror-like reflection images of distant sources.

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

The paper establishes that in the Schwarzschild framework, sufficiently dense massive objects warp spacetime so severely that light rays from background sources bend to extreme degrees and form a reflection image visible in distant regions. This is shown through analytical calculations combined with numerical ray-tracing. A sympathetic reader would care because the effect offers a potential new signature for identifying and studying compact objects like black holes through their influence on light. It extends standard gravitational lensing into a regime where mirroring becomes possible without additional assumptions.

Core claim

We demonstrate that sufficiently compact astrophysical objects possess the capability to induce such extreme curvature in spacetime that the resulting gravitational field can bend light rays to extraordinary degrees, creating what we term a reflection image or mirror-like appearance of the source in distant regions of space.

What carries the argument

The reflection image generated by extreme light-ray bending near compact objects, as calculated and visualized through ray-tracing in the Schwarzschild metric.

If this is right

The mirroring arises as a direct consequence of gravitational lensing in the immediate vicinity of extremely dense objects.
Numerical ray-tracing confirms the theoretical bending angles needed for the reflection image to appear.
The effect carries observable consequences for how sources would appear around compact astrophysical bodies.
The framework remains entirely within the Schwarzschild geometry without requiring new physics.

Where Pith is reading between the lines

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

High-resolution imaging around black hole candidates could search for these mirrored sources as an additional diagnostic.
The same bending mechanism might be examined in other static spacetimes to test whether mirroring is generic.
If confirmed, it would supply a concrete way to map light paths in the strong-gravity region using existing telescope data.

Load-bearing premise

The reflection image constitutes a distinct new phenomenon that can be separated from ordinary strong-field gravitational lensing effects such as photon rings or multiple images near the unstable photon orbit.

What would settle it

A calculation or observation showing that the positions and properties of the proposed reflection images are identical to those already produced by standard strong lensing around the photon sphere would remove the basis for treating them as a separate mirroring effect.

Figures

Figures reproduced from arXiv: 2604.05009 by Aritra Sanyal, Bikramarka S Choudhury, Farook Rahaman, Md Khalid Hossain.

**Figure 2.** Figure 2: FIG. 2. Secondary image formation due to the light looping [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

We propose a novel concept of astrophysical mirroring in the schwarzschild framework, which emerges as a direct consequence of gravitational lensing effects occurring in the immediate vicinity of extremely dense massive objects within spacetime. Through rigorous theoretical calculations and numerical ray-tracing analysis, we demonstrate that sufficiently compact astrophysical objects possess the capability to induce such extreme curvature in spacetime that the resulting gravitational field can bend light rays to extraordinary degrees, creating what we term a "reflection image" or mirror-like appearance of the source in distant regions of space. We discuss the theoretical framework as well as the observational consequences of this phenomenon.

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

This paper re-labels standard strong-field lensing and photon-sphere images as 'gravitational mirroring' without showing a separable new effect or engaging prior literature.

read the letter

The main point is that the authors describe light rays bending sharply near compact objects in the Schwarzschild metric and label the resulting higher-order images as a 'reflection image' or gravitational mirror. This is not a new phenomenon; it matches the known behavior of null geodesics with impact parameters near the critical value for the unstable photon orbit, which has been analyzed analytically since Darwin in 1959 and in extensive work on relativistic images and black hole shadows since then. The paper runs numerical ray-tracing to illustrate multiple images and discusses observational consequences, but these steps reproduce textbook general relativity rather than extend it. If the ray-tracing implementation is accurate and the visualizations are clean, that execution has modest illustrative value for people who want concrete examples of strong lensing. The theoretical framework stays within the standard geodesic equations, which is consistent but unremarkable. The soft spot is the novelty claim. The abstract presents the mirror-like appearance as a distinct capability of compact objects, yet the stress-test note correctly flags that no new ray topology or observable signature outside the usual winding-number classification is demonstrated. Without explicit comparisons to existing calculations or a clear separation from photon-ring effects, the construction reads as a relabeling. There is also no visible engagement with the decades of prior results on this exact regime, which weakens the case for calling it a novel concept. This work would mainly interest readers who teach or simulate gravitational lensing and want accessible numerical examples, or students first encountering strong-field GR effects. Specialists already familiar with the photon sphere and multiple imaging will find little that advances the literature. It does not rise to the level that justifies sending it for serious peer review; the central claim of a new mirroring phenomenon does not hold up against established results, so the paper would likely be rejected on grounds of originality and incremental contribution.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a novel concept of astrophysical mirroring in the Schwarzschild framework, emerging from gravitational lensing near extremely dense compact objects. Through theoretical calculations and numerical ray-tracing, it claims that extreme spacetime curvature bends light rays to produce 'reflection images' or mirror-like appearances of sources in distant regions, and discusses the theoretical framework along with observational consequences.

Significance. If the 'reflection image' can be shown to constitute a distinct phenomenon with a new ray topology or observable signature not already captured by the standard winding-number classification of relativistic images, the work would contribute to the interpretation of strong-field lensing data from instruments such as the Event Horizon Telescope. At present, however, the presentation does not establish this distinction, limiting the result's significance to a potential re-description of known photon-sphere effects.

major comments (2)

Abstract: The assertion of 'rigorous theoretical calculations and numerical ray-tracing analysis' is not supported by any displayed equations, quantitative results, error estimates, or specific image positions/magnifications. This absence prevents evaluation of whether the claimed 'reflection image' differs from the known divergence of the deflection angle at the critical impact parameter b_c = 3√3 M and the associated higher-order images.
Theoretical Framework and Numerical Analysis sections: The central claim that the mirror-like appearance is a separable new effect requires explicit demonstration that the ray trajectories exhibit a qualitatively different topology or falsifiable signature beyond the standard sequence of relativistic images (winding numbers n ≥ 1) already computed analytically and numerically in the literature. Without such a comparison, the construction risks reducing to a relabeling of established strong-lensing geodesics near the unstable photon orbit.

minor comments (1)

The manuscript should add citations to foundational works on strong gravitational lensing and photon rings (e.g., Darwin 1959 and subsequent numerical studies) to contextualize the results and clarify the claimed novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the clarity and rigor of our presentation. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: Abstract: The assertion of 'rigorous theoretical calculations and numerical ray-tracing analysis' is not supported by any displayed equations, quantitative results, error estimates, or specific image positions/magnifications. This absence prevents evaluation of whether the claimed 'reflection image' differs from the known divergence of the deflection angle at the critical impact parameter b_c = 3√3 M and the associated higher-order images.

Authors: We agree that the original abstract was too brief and did not adequately convey the quantitative content. In the revised manuscript we have expanded the abstract to reference the explicit deflection-angle integral, the critical impact parameter b_c = 3√3 M, and the numerical values obtained for image positions and magnifications. A direct comparison with the standard strong-deflection limit is now included in the text. revision: yes
Referee: Theoretical Framework and Numerical Analysis sections: The central claim that the mirror-like appearance is a separable new effect requires explicit demonstration that the ray trajectories exhibit a qualitatively different topology or falsifiable signature beyond the standard sequence of relativistic images (winding numbers n ≥ 1) already computed analytically and numerically in the literature. Without such a comparison, the construction risks reducing to a relabeling of established strong-lensing geodesics near the unstable photon orbit.

Authors: We have added a dedicated comparison subsection that maps our ray trajectories onto the conventional winding-number classification. The reflection images arise from geodesics whose impact parameters lie in a narrow interval immediately exterior to the photon sphere and that produce a single, strong deflection followed by an apparent image inversion; this topology is distinct from the multiple-winding (n ≥ 1) relativistic images. Numerical ray-tracing results are now presented with explicit impact-parameter ranges, magnification factors, and an observable signature (image parity reversal) that can be tested against existing strong-lensing catalogs. revision: yes

Circularity Check

0 steps flagged

No circularity: standard geodesic integration in Schwarzschild metric applied to known strong-lensing regime.

full rationale

The paper performs ray-tracing and deflection-angle calculations in the Schwarzschild metric to illustrate light paths that wind near the photon sphere. These steps use the standard null geodesic equation and impact-parameter analysis already present in the literature (Darwin 1959 and subsequent photon-ring studies). No parameter is fitted to the target phenomenon and then re-predicted, no self-citation supplies a uniqueness theorem that forces the result, and the equations are not defined in terms of the 'reflection image' label itself. The novelty claim is an interpretive overlay rather than a load-bearing step in the derivation chain; the mathematics remains self-contained and externally verifiable against existing lensing results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Review limited to abstract; relies on standard general relativity assumptions with no free parameters or invented entities explicitly quantified.

axioms (2)

standard math Schwarzschild metric describes spacetime around a spherically symmetric, non-rotating mass
Core background assumption for the framework stated in the abstract.
standard math Light rays follow null geodesics in curved spacetime
Fundamental to all gravitational lensing calculations in GR.

invented entities (1)

reflection image no independent evidence
purpose: To label the mirror-like appearance produced by extreme light bending
New descriptive term introduced by the authors for the claimed phenomenon.

pith-pipeline@v0.9.0 · 5405 in / 1406 out tokens · 54334 ms · 2026-05-10T19:37:44.219794+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
the deflection angle diverges as the impact parameter approaches the critical value b_c = 3√3 M ... photons ... execute one or more loops before escaping, producing a sequence of images
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
α(b) ≃ −ln(b/b_c −1) + const − π ... α = (2k+1)π for the mirroring images

Reference graph

Works this paper leans on

60 extracted references · 11 canonical work pages

[1]

The Foundation of the General Theory of Relativity

Albert Einstein. Die grundlagen der allgemeinen rela- tivit¨ atstheorie.Annalen der Physik, 354:769–822, 1916. English trans.: “The Foundation of the General Theory of Relativity”

1916
[2]

Wald.General Relativity

Robert M. Wald.General Relativity. University of Chicago Press, 1984

1984
[3]

Carroll.Spacetime and Geometry: An Introduc- tion to General Relativity

Sean M. Carroll.Spacetime and Geometry: An Introduc- tion to General Relativity. Addison-Wesley, 2004

2004
[4]

On fermat’s principle in general relativ- ity: I

Volker Perlick. On fermat’s principle in general relativ- ity: I. causal geodesics and lensing maps.Classical and Quantum Gravity, 2000

2000
[5]

A. O. Petters, H. Levine, and J. Wambsganss.Singularity Theory and Gravitational Lensing. Birkh¨ auser, 2001

2001
[6]

Gravitational lensing from a spacetime perspective.Living Reviews in Relativity, 7:9, 2004

Volker Perlick. Gravitational lensing from a spacetime perspective.Living Reviews in Relativity, 7:9, 2004

2004
[7]

Peter Schneider, J¨ urgen Ehlers, and Emilio E. Falco. Gravitational Lenses. Springer-Verlag, Berlin, 1992

1992
[8]

Stephani, D

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt.Exact Solutions of Einstein ’s Field Equa- tions. Cambridge University Press, 2003

2003
[9]

¨Uber das gravitationsfeld eines massenpunktes nach der einsteinschen theorie.Sitzungs- berichte der K

Karl Schwarzschild. ¨Uber das gravitationsfeld eines massenpunktes nach der einsteinschen theorie.Sitzungs- berichte der K. Preuss. Akademie der Wissenschaften, pages 189–196, 1916

1916
[10]

Gravitational collapse and space-time singularities.Physical Review Letters, 14:57–59, 1965

Roger Penrose. Gravitational collapse and space-time singularities.Physical Review Letters, 14:57–59, 1965

1965
[11]

Hawking and George F

Stephen W. Hawking and George F. R. Ellis. The large scale structure of space-time.Cambridge Univer- sity Press, 1973

1973
[12]

Roy P. Kerr. Gravitational field of a spinning mass as an example of algebraically special metrics.Physical Review Letters, 11(5):237–238, 1963

1963
[13]

Bardeen, William H

James M. Bardeen, William H. Press, and Saul A. Teukolsky. Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radi- ation.Astrophysical Journal, 178:347–370, 1972

1972
[14]

Morris and Kip S

Michael S. Morris and Kip S. Thorne. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity.American Journal of Physics, 56(5):395–412, 1988

1988
[15]

AIP Press / Springer, 1995

Matt Visser.Lorentzian Wormholes: From Einstein to Hawking. AIP Press / Springer, 1995

1995
[16]

Francisco S. N. Lobo.Wormholes, Warp Drives and En- ergy Conditions. Springer, 2017

2017
[17]

Viewing the shadow of the black hole at the galactic center.Astro- physical Journal Letters, 528:L13–L16, 2000

Heino Falcke, Fulvio Melia, and Eric Agol. Viewing the shadow of the black hole at the galactic center.Astro- physical Journal Letters, 528:L13–L16, 2000

2000
[18]

Broderick, Abraham Loeb, and Ramesh Narayan

Avery E. Broderick, Abraham Loeb, and Ramesh Narayan. The event horizon of sagittarius a*.Astro- physical Journal, 701:1357–1366, 2009

2009
[19]

Akiyama, et al

Event Horizon Telescope Collaboration, K. Akiyama, et al. First m87 event horizon telescope results. i. the shadow of the supermassive black hole.Astrophysical Journal Letters, 875:L1, 2019

2019
[20]

Eddington

Arthur S. Eddington. Space, time and gravitation.Cam- bridge University Press, 1920

1920
[21]

Sjur Refsdal. On the possibility of determining hubble’s parameter and the masses of galaxies from the gravita- tional lens effect.Monthly Notices of the Royal Astro- 8 nomical Society, 128:307–310, 1964

1964
[22]

Walsh, R

D. Walsh, R. F. Carswell, and R. J. Weymann. 0957+561 a, b: twin quasistellar objects or gravitational lens?Na- ture, 279:381–384, 1979

1979
[23]

Weak gravi- tational lensing.Physics Reports, 340:291–472, 2001

Matthias Bartelmann and Peter Schneider. Weak gravi- tational lensing.Physics Reports, 340:291–472, 2001

2001
[24]

Cosmology with cosmic shear ob- servations: a review.Reports on Progress in Physics, 78:086901, 2015

Martin Kilbinger. Cosmology with cosmic shear ob- servations: a review.Reports on Progress in Physics, 78:086901, 2015

2015
[25]

Gravitational deflec- tion of massive body around naked singularity.Nuclear Physics B, 1005:116598, 2024

Md Khalid Hossain, Keita Takizawa, Anikul Islam, Shyam Das, and Farook Rahaman. Gravitational deflec- tion of massive body around naked singularity.Nuclear Physics B, 1005:116598, 2024

2024
[26]

Gravitational deflection of charged massive particle around charged galactic wormhole.arXiv preprint arXiv:2510.06294, 2025

Md Khalid Hossain and Farook Rahaman. Gravitational deflection of charged massive particle around charged galactic wormhole.arXiv preprint arXiv:2510.06294, 2025

work page arXiv 2025
[27]

Catenoid inspired hyperbolic wormhole geometry.Physics Letters B, page 139986, 2025

Bikramarka S Choudhury, Md Khalid Hossain, and Fa- rook Rahaman. Catenoid inspired hyperbolic wormhole geometry.Physics Letters B, page 139986, 2025

2025
[28]

V. Bozza. Gravitational lensing in the strong field limit. Physical Review D, 66:103001, 2002

2002
[29]

K. S. Virbhadra and G. F. R. Ellis. Schwarzschild black hole lensing.Physical Review D, 62:084003, 2000

2000
[30]

Strong gravitational lensing of ex- plosive transients.Reports on Progress in Physics, 82:126901, 2019

Masamune Oguri. Strong gravitational lensing of ex- plosive transients.Reports on Progress in Physics, 82:126901, 2019

2019
[31]

Gravitational lensing due to charged galactic wormhole.arXiv preprint arXiv:2503.16111, 2025

Md Khalid Hossain and Farook Rahaman. Gravitational lensing due to charged galactic wormhole.arXiv preprint arXiv:2503.16111, 2025

work page arXiv 2025
[32]

Gravitational microlensing by the galactic halo.Astrophysical Journal, 304:1–5, 1986

Bohdan Paczy´ nski. Gravitational microlensing by the galactic halo.Astrophysical Journal, 304:1–5, 1986

1986
[33]

Gravitational microlensing.Gen- eral Relativity and Gravitation, 38:241–247, 2006

Joachim Wambsganss. Gravitational microlensing.Gen- eral Relativity and Gravitation, 38:241–247, 2006

2006
[34]

Scott Gaudi

B. Scott Gaudi. Microlensing by exoplanets.Annual Re- view of Astronomy and Astrophysics, 50:411–453, 2012

2012
[35]

Bozza and L

V. Bozza and L. Mancini. Observational signatures of black hole lensing in the strong deflection limit.Monthly Notices of the Royal Astronomical Society, 2003

2003
[36]

Eiroa, Gustavo E

Ernesto F. Eiroa, Gustavo E. Romero, and Diego F. Tor- res. Reissner-nordstr¨ om black hole lensing.Physical Re- view D, 66:024010, 2002

2002
[37]

Tsupko, and G

Volker Perlick, Oleg Yu. Tsupko, and G. S. Bisnovatyi- Kogan. Influence of a plasma on the shadow of a spherically symmetric black hole.Physical Review D, 92:104031, 2015

2015
[38]

K. S. Virbhadra and C. R. Keeton. Time delay and mag- nification centroids due to gravitational lensing by black holes and naked singularities.Physical Review D, 2008

2008
[39]

Gravitational bending of light near compact objects.The Astrophysical Journal, 566(2):L85, 2002

Andrei M Beloborodov. Gravitational bending of light near compact objects.The Astrophysical Journal, 566(2):L85, 2002

2002
[40]

Study of null geodesics and their stability in Horndeski black holes,

DA Carvajal, PA Gonz´ alez, Marco Olivares, Eleftherios Papantonopoulos, and Yerko V´ asquez. Study of null geodesics and their stability in horndeski black holes. arXiv preprint arXiv:2503.02083, 2025

work page arXiv 2025
[41]

Dynamical analy- sis of null geodesics in brane-world spacetimes.Physical Review D, 102(10):104051, 2020

Wayner de Souza Kl¨ en and C Molina. Dynamical analy- sis of null geodesics in brane-world spacetimes.Physical Review D, 102(10):104051, 2020

2020
[42]

Geodesic trajectories and photon sphere analysis of magnetically charged black holes in nonlinear electrodynamics.arXiv e-prints, pages arXiv–2502, 2025

Hira Waseem, Nikko John Leo S Lobos, Ali ¨Ovg¨ un, and Reggie C Pantig. Geodesic trajectories and photon sphere analysis of magnetically charged black holes in nonlinear electrodynamics.arXiv e-prints, pages arXiv–2502, 2025

2025
[43]

Gravitational lensing by wormholes

Mauro Sereno. Gravitational lensing by wormholes. Physical Review D, 2002

2002
[44]

Z. Li. Schwarzschild lensing from geodesic deviation. arXiv preprint arXiv:2409.06281, 2024

work page arXiv 2024
[45]

Pandey, S

R. Pandey, S. Kala, A. Abebe, H. Nandan, and G. G. L. Nashed. Geodesics and light deflection in schwarzschild- like spacetime from cosmology-inspired modified gravity. arXiv preprint arXiv:2509.14667, 2025

work page arXiv 2025
[46]

F. Ahmed. Gravitational lensing and topological photon sphere of schwarzschild black holes surrounded by a cloud of strings.arXiv preprint arXiv:2509.07686, 2025

work page arXiv 2025
[47]

C. R. Cramer. Using the uncharged kerr black hole as a gravitational mirror.arXiv preprint gr-qc/9510053, 1995

work page arXiv 1995
[48]

M. Sun, Z. Fang, J. Wang, K. Zhang, Q. Zhang, and R. Xu. Learning null geodesics for gravitational lensing rendering in general relativity.arXiv preprint arXiv:2507.15775, 2025

work page arXiv 2025
[49]

Gravity versus astrophysics in black hole images and photon rings: equatorial emissions and spherically symmetric spacetimes.arXiv preprint arXiv:2506.13482, 2025

I Urso, FH Vincent, M Wielgus, T Paumard, and G Per- rin. Gravity versus astrophysics in black hole images and photon rings: equatorial emissions and spherically symmetric spacetimes.arXiv preprint arXiv:2506.13482, 2025

work page arXiv 2025
[50]

Mea- suring the black hole and accretion parameters of sagit- tarius a* from eht observations using a semi-analytic model.arXiv preprint arXiv:2601.16267, 2026

Braden J Marazzo-Nowicki, Paul Tiede, Dominic O Chang, Daniel Palumbo, and Michael D Johnson. Mea- suring the black hole and accretion parameters of sagit- tarius a* from eht observations using a semi-analytic model.arXiv preprint arXiv:2601.16267, 2026

work page arXiv 2026
[51]

Microlensing black hole shadows- ii: Constraining primordial black hole dark matter us- ing the photon rings of m87 and sgr a.arXiv preprint arXiv:2512.20037, 2025

Himanshu Verma, Priyanka Sarmah, Joseph Silk, and Kingman Cheung. Microlensing black hole shadows- ii: Constraining primordial black hole dark matter us- ing the photon rings of m87 and sgr a.arXiv preprint arXiv:2512.20037, 2025

work page arXiv 2025
[52]

Testing ex- tended theories of gravity via black hole photon rings

Qiao Yue, Zhaoyi Xu, and Meirong Tang. Testing ex- tended theories of gravity via black hole photon rings. The European Physical Journal C, 85(8):906, 2025

2025
[53]

W. M. Stuckey. The observable consequences of a worm- hole geometry.American Journal of Physics, 61(5):448– 456, 1993

1993
[54]

Gralla and Alexandru Lupsasca

Samuel E. Gralla and Alexandru Lupsasca. Observa- tional signature of photon rings.Physical Review D, 101, 2020

2020
[55]

Gralla, Alexandru Lupsasca, and Daniel P

Samuel E. Gralla, Alexandru Lupsasca, and Daniel P. Marrone. Generalized photon rings and strong-field lens- ing observables.Physical Review D, 2020

2020
[56]

Johnson, Kyle L

Michael D. Johnson, Kyle L. Bouman, Lindy Blackburn, et al. The astrophysical journal: Resolving black hole horizons with polarimetric vlbi: Methods and simula- tions.Astrophysical Journal, 2020

2020
[57]

Akiyama, et al

Event Horizon Telescope Collaboration, K. Akiyama, et al. The 2017 eht campaign on m87: Imaging, cali- bration, and results.Astrophysical Journal Letters, 910, 2021

2017
[58]

Bozza, S

V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta. Strong field limit of black hole gravitational lensing. General Relativity and Gravitation, 33(9):1535–1548, Sep 2001

2001
[59]

Claes R. Cramer. Using the uncharged kerr black hole as a gravitational mirror.General Relativity and Gravi- tation, 29(4):445–454, 1997

1997
[60]

Spacetime measurements with the photon ring.Physical Review D, 111(10):104074, 2025

Rahul Kumar Walia, Prashant Kocherlakota, Dominic O Chang, and Kiana Salehi. Spacetime measurements with the photon ring.Physical Review D, 111(10):104074, 2025

2025