Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Interferometric signature of higher-order images in a parametrized framework

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Higher-order black hole images imprint a staircase in the interferometric visibility whose step heights and spacings encode deviations from Schwarzschild geometry.

desk verdict Section VI's perturbative formulas are wrong; the numerical part may survive, but the paper's analytic parameter dependencies need correction. read the letter →

arxiv 2508.03615 v2 pith:PD2R6R6U submitted 2025-08-05 gr-qc astro-ph.HE

Fabiano Feleppa , Fabio Aratore , Valerio Bozza This is my paper
classification gr-qcastro-ph.HE MSC 83C5783C10 PACS 04.70.-s95.30.Sf98.62.Sb
keywords strongdeflectionlimithigher-orderimagesphotonringvisibilityfunctionstaircasepatternparametrizedblack-holemetricsinterferometricsignaturetestsofgeneralrelativity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the faint, repeatedly looped images of a compact source around a black hole, the higher-order images, leave a measurable step-like signature in the interferometric visibility function. Within a five-parameter continued-fraction description of static, spherically symmetric black hole metrics, the paper shows that the height of each step and the period of the interference fringes depend on the metric-deviation parameters. It derives first-order analytic formulas connecting those parameters to the observables and verifies the behavior numerically for Schwarzschild, Reissner-Nordström, and dilaton black holes. If correct, this gives a direct route from interferometric measurements to constraints on how much a real black hole's metric departs from general relativity.

What carries the argument

The load-bearing object is the strong-deflection expansion of the deflection angle, $\Delta\varphi=-\bar a\log(\varepsilon/(\eta_O\eta_S))+\bar b$, whose logarithmic divergence near the photon sphere organizes the infinite sequence of higher-order images into exponentially spaced positions $\varepsilon_{n,\pm}=\eta_O\eta_S\exp[(\bar b\pm\varphi_S-(2n+1)\pi)/\bar a]$. Entering these positions into the image-sum expression for the complex visibility produces the staircase, with step heights $h_{n,\pm}=N(v)\varepsilon_{n,\pm}$ and fringe periodicities given by Eqs. (23)-(24). The metric side of the construction is a continued-fraction parametrization of static, spherically symmetric black holes truncated to the leading parameters $\epsilon, a_0, a_1, b_0, b_1$, whose photon-sphere radius $r_m$ and critical impact parameter $u_m$ feed the coefficients $\bar a$ and $\bar b$, and therefore every observable in the staircase.

What would settle it

Ray-trace a realistic extended, time-varying emission region around a Schwarzschild black hole, compute its visibility function with the same normalization, and test whether the staircase and the parameter dependencies of Eqs. (43)-(59) survive; if the steps are washed out or reproduced by a different metric with different parameters, the claimed mapping is not observable.

Watch

Extended reading notes

Core claim

The paper establishes that the strong-deflection higher-order image sequence, ordered by loop number $n$ and parity, generates a staircase in the normalized visibility amplitude whose first step is the $n=1$ positive-parity image, followed by the $n=1$ negative-parity image, the $n=2$ positive-parity image, and so on. The step heights are $h_{n,\pm}=N(v)\varepsilon_{n,\pm}$, and the fringe periodicities are $P_{n,+}=[\theta_m(2+\varepsilon_{n,+}+\varepsilon_{n,-})]^{-1}$ and $P_{n,-}=[\theta_m(2+\varepsilon_{n,-}+\varepsilon_{n+1,+})]^{-1}$, with $\varepsilon_{n,\pm}$ fixed by the strong-deflection coefficients $\bar a$ and $\bar b$. Because those coefficients are computed from the photon-sphere radius and critical impact parameter of the metric, any metric deviation shifts the staircase. The paper computes these shifts for the leading parameters $\epsilon, a_0, a_1, b_0, b_1$ and gives explicit first-order formulas for the dependence, so that the mapping from spacetime parameters to observable step heights and periodicities is systematic enough to serve as a template for testing general relativity.

Load-bearing premise

The load-bearing premise is that the visibility function of Eq. (19), built from a single static Gaussian source, correctly represents what a real compact source would produce interferometrically, and that the strong-deflection expansion remains accurate at $n=1$; the authors themselves warn that mixing geometric and emission effects can cause significant degeneracies.

Editorial extensions

If this is right

  • If the staircase is observed, the first few step heights locate $\varepsilon_{1,\pm}$, and the modulations between opposite-parity images fix the shadow angular scale $\theta_m$.
  • The explicit first-order formulas mean small metric deviations can be fitted linearly to visibility data without solving the full lens equation image by image.
  • Spacetimes with nearly identical primary images can still be told apart by higher-order steps, because step heights and periodicities respond differently to each metric parameter.
  • Parameter dependencies are largely monotonic within some families, so future high-signal observations could identify which coefficient is responsible for a detected anomaly.
  • The same machinery maps any spherically symmetric metric in the parametrized family onto a predicted staircase, making the framework a general template for model comparison.

Reading between the lines

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

  • The parameter-to-observable map is in principle invertible: a measured first-step height and fringe periodicity can be converted into constraints on $\epsilon, a_0, a_1, b_0, b_1$, but the inversion will be partially degenerate because $P_{1,+}$ is dominated by the shadow scale while $h_{1,+}$ carries the remaining parameter dependence.
  • A direct test of the proposal is to feed the same metric through rigorous ray-traced images with realistic extended, time-variable emission prescriptions; this would tell whether the single-Gaussian staircase is a clean observable or a template that astrophysical complexity obscures.
  • The same staircase logic should carry over to rotating metrics, where image positions are no longer collinear and the step structure gains a position-angle dependence, making the test applicable to real astrophysical black hole candidates.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper applies the Rezzolla-Zhidenko parametrization of static, spherically symmetric black-hole metrics to the strong-deflection-limit formalism for higher-order images, and studies how the deformation parameters (epsilon, a0, a1, b0, b1) affect the interferometric visibility function of a compact source. The main quantities of interest are the photon-sphere radius r_m, the minimum impact parameter u_m, the strong-deflection coefficients abar and bbar, the step heights h_{n,pm} of the staircase visibility pattern, and the modulation periodicities P_{n,pm}. The paper presents numerical scans of these quantities (Figs. 1-7) and then proposes first-order perturbative formulas for all of them in Section VI, claiming that these formulas reproduce the qualitative behavior of the full numerics. The central advertised result is a computable mapping from the RZ parameters to the observable step heights and periodicities, which could in principle be inverted against future interferometric data.

Significance. If correct, the paper would extend the interferometric photon-ring formalism of Aratore and Bozza (Ref. [74]) and Johnson et al. (Ref. [40]) to the model-independent Rezzolla-Zhidenko framework, providing analytic parameter dependencies for the staircase features of the visibility function. The paper is clearly organized, uses a standard strong-deflection formalism, and its numerical survey of parameter space is a useful reference. However, the analytic first-order results in Section VI, which are the principal new content, contain load-bearing errors: Eqs. (43)-(44) are incorrect expansions of the photon-sphere radius and minimum impact parameter. Because the downstream formulas for step heights and periodicities are built on these quantities, the claimed analytic parameter mapping is not established. The numerical figures may be correct, but the abstract and Section VII advertise the parameter dependencies that are precisely the part that is wrong as written.

major comments (2)
  1. [Section VI, Eqs. (43)-(44)] Direct first-order expansion of Eq. (3) using the truncated metric (32) yields r_m = 3M - (4/9)M(a0 + 5epsilon + a1) and u_m = 3sqrt(3)M - (2sqrt(3)/9)M(3a0 + 6epsilon + 2a1), not the expressions in Eqs. (43)-(44). The printed formulas omit the a1 dependence in r_m, and the a0 coefficient in u_m is wrong by a factor of three (it should be 2sqrt(3)/3, not 2sqrt(3)). This is not a higher-order truncation artifact: for the paper's own Reissner-Nordstroem example with q = 0.8 (a0 = epsilon = 0.25), Eq. (44) gives u_m approximately 2.60M, whereas Eq. (5) gives u_m approximately 4.55M; the corrected first-order expansion gives approximately 4.33M, close to the exact value. Since Eqs. (45)-(59) inherit these errors, the analytic parameter dependencies for step heights and periodicities claimed in Section VI and in the abstract are not established, and the closing statement that these formulas reproduce the full numerical results is not credible as written.
  2. [Section VI] All perturbative results in Section VI are stated without derivation; the single sentence 'By perturbatively solving Eq. (3) and subsequently calculating the minimum impact parameter using Eq. (5)' is insufficient for the claimed first-order expressions, especially since those expressions are not correct. The authors should present the expansion for at least one quantity (e.g., r_m) so that the error in Eqs. (43)-(44) becomes transparent, and they should verify the resulting h_{1,+} and P_{1,+} against the full numerical evaluation of Section V for a representative set of parameter values. Without such a derivation and cross-check, the analytic part of the paper cannot be reproduced or trusted.
minor comments (5)
  1. [Section IV, Eq. (36)] The display for a0 = epsilon is garbled in the typeset text; it should read a0 = epsilon = 2/(1 + sqrt(1 - q^2)) - 1, and the sentence 'see the last expression in Eq. (35)' should refer to Eq. (36).
  2. [Section VI, Eq. (54)] Eq. (54) writes h_{n,+} = epsilon_{n,+}, while Eq. (22) defines h_{n,pm} = N(v) epsilon_{n,pm}; if the perturbative section sets N(v) = 1 (as in the caption of Fig. 6), this normalization should be stated explicitly before Eq. (54).
  3. [References] Reference [87] is dated 2007, but the cited article appears in Phys. Rev. D 109, 064064 (2024); the year should be corrected.
  4. [Fig. 6 caption] The second panel of Fig. 6 refers to 'orange curves (b0 = 0.3)', but the legend lists b0 = -0.3, 0, 0.3; please check the color assignment and make the caption consistent with the legend.
  5. [Sections III and V] The symbol u is used both for the impact parameter (Eqs. (5)-(7)) and for the interferometric baseline spatial frequency (Section III); this double use is confusing and should be eliminated, for instance by renaming the baseline coordinate.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found: the RZ-parameter-to-visibility mapping is computed from explicit strong-deflection equations, with no fitted input being renamed as a prediction.

full rationale

The derivation chain is not circular. The paper starts from the RZ metric (Eqs. 28-33), solves the photon-sphere condition (Eq. 3), computes the minimum impact parameter (Eq. 5) and the strong-deflection coefficients (Eqs. 11-15), then uses the image positions (Eq. 17) inside the visibility sum (Eq. 19) to obtain step heights and periodicities via Eqs. (22)-(24). No parameter is fitted to the staircase or to the periodicities; the claimed dependencies are explicit functions of the metric parameters, and no target observable is used to define the metric coefficients. The perturbative formulas in Section VI are expansions of the same equations, not fits. The use of the authors' prior Ref. [74] for the visibility formalism is a normal self-citation that is restated in the paper rather than an unverified black box; it does not make the central claim tautological. The Section VII caveat about degeneracies between geometric and emission effects is a physical modeling limitation, not a circular step. The skeptic's algebraic objection to Eqs. (43)-(44) concerns correctness, not circularity: even if those first-order expansions are wrong, the derivation would be erroneous, not equivalent to its inputs by construction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the strong deflection formalism, the RZ parametrized metric, and an idealized Gaussian source model. No entities are invented. The RZ parameters and source geometry are chosen by hand for the parametric study, but they are not fitted to data, so they count as free parameters rather than fitted values.

free parameters (6)
  • epsilon (RZ horizon-deviation parameter) = scanned over range [-0.6, 1.0]
    Chosen by hand to scan deviations of the horizon radius from the Schwarzschild radius; no data fitting.
  • a0 (RZ asymptotic parameter) = values in {-1, 0, 0.3, 1}
    Chosen by hand to explore metric deviations at infinity; no data fitting.
  • a1 (RZ near-horizon continued-fraction coefficient) = values in {-0.2, -0.1, 0, 0.1, 0.2}
    Chosen by hand to explore near-horizon deviations; no data fitting.
  • b0 (RZ asymptotic parameter for N(r)) = values in {-1, -0.3, -0.1, 0, 0.3, 0.5, 1}
    Chosen by hand to explore deviations in the N(r) metric function; no data fitting.
  • b1 (RZ near-horizon continued-fraction coefficient for N(r)) = values in {-0.2, -0.1, 0, 0.1, 0.2}
    Chosen by hand to explore near-horizon deviations in N(r); no data fitting.
  • source geometry (rO, rS, Delta rS, phiS, Delta phiS, Delta varthetaS, I0, M) = rO = 16.8 Mpc, rS = 20M, Delta rS = M, phiS = 45 degrees, Delta phiS = Delta rS / rS, I0 arbitrary (normalized), M…
    Illustrative values chosen for the visibility plots (Figs. 5-7); the central parametric dependencies are reported at fixed source geometry.
assumptions (6)
  • domain assumption The spacetime is static, spherically symmetric, and asymptotically flat (Eqs. (1)-(2)).
    Underlies the entire strong deflection formalism; excludes rotation, which the paper defers to future work.
  • domain assumption The photon sphere equation (3) admits a largest positive root rm for all parameter choices considered.
    Needed to define the strong deflection expansion; parameter ranges are restricted in Section IV A to cases where this holds.
  • domain assumption The strong deflection limit expansion (epsilon much less than 1) is accurate even for the lowest-order images n = 1.
    Stated in Section II ('typically serves as a very accurate approximation even for n = 1'); supports the physical relevance of the computed h1,+ and P1,+.
  • domain assumption The source is a static, Gaussian, axially-centered compact source, so images lie along a single sky axis and the visibility factorizes as in Eq. (19).
    Section III; the staircase signature and its step heights and periodicities are derived under this emission model, later flagged as a degeneracy source in Section VII via Ref. [94].
  • ad hoc to paper The Rezzolla-Zhidenko parametrization truncated at order (a1, b1) is sufficient to capture the metric deviations relevant for the computed lensing observables.
    Section IV; the paper restricts to leading-order coefficients and asserts that additional coefficients 'would certainly improve accuracy' without quantifying the truncation error for the observables.
  • domain assumption The theoretical constraints of Refs. [75, 91, 92] (single outermost non-degenerate horizon, nonvanishing B(r)) delimit the valid parameter ranges shown in the plots.
    Section IV A; used to justify why curves terminate; relies on prior literature rather than an independent derivation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Interferometric signature of higher-order images in a parametrized framework." pith.science (2026). https://pith.science/paper/PD2R6R6U

@misc{pith2026250803615,
  author       = {Pith},
  title        = {Pith review of: Interferometric signature of higher-order images in a parametrized framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PD2R6R6U}},
  note         = {Machine review of arXiv:2508.03615}
}
read the original abstract

This paper investigates gravitational lensing in the strong deflection limit, focusing particularly on higher-order images produced near compact objects such as black holes and their observable impact through the visibility function. Employing a robust parametrization framework proposed by Rezzolla and Zhidenko, the study systematically explores deviations from the Schwarzschild metric. A detailed theoretical analysis of interferometric observables is provided, highlighting how higher-order images imprint distinctive, measurable patterns in the visibility function, notably characterized by a staircase-like structure. By parametrically varying metric coefficients, the analysis reveals clear dependencies between spacetime deviations and key observational signatures, specifically the step heights and periodicities in the interferometric visibility. The results enhance the theoretical groundwork for interpreting data from advanced interferometric observations, potentially enabling precise tests of general relativity and the discrimination among alternative gravitational theories.

Figures

Figures reproduced from arXiv: 2508.03615 by the authors.

Figure 1
Figure 1. Behavior of the photon sphere radius rm as a function of ϵ, across different values of a0 and a1. a0=-1, a1=-0.1 a0=-1, a1=0 a0=-1, a1=0.1 a0=0, a1=-0.1 a0=0, a1=0 a0=0, a1=0.1 a0=1, a1=-0.1 a0=1, a1=0 a0=1, a1=0.1 -0.5 0.0 0.5 1.0 4 6 8 10 12 ϵ um/m [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Behavior of the minimum impact parameter [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Behavior of the strong deflection coefficient [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Complex visibility function arising from higher [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: Periodicities of the modulations in the visibility, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Observational constraints on nonlocal black holes via gravitational lensing

    gr-qc 2026-04 unverdicted novelty 5.0 of 10

    Nonlocal black holes remain consistent with general relativity at the 1.13-sigma level after joint lensing and quasinormal-mode constraints.

Reference graph

Works this paper leans on

94 extracted references · 79 canonical work pages · cited by 1 Pith paper

  1. [74]

    Aratore and V

    F. Aratore and V. Bozza, Decoding a black hole metric from the interferometric pattern of the relativistic images of a compact source, Journal of Cosmology and Astropar- ticle Physics 10 (2021) 054

    work page 2021

  2. [40]

    M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong, S. Hadar, D. Kapec, R. Narayan, A. Chael, C. F. Gam- mie, P. Galison, et al., Universal interferometric signa- tures of a black hole’s photon ring, Science Advances6, 12 (2020)

    work page 2020

  3. [1]

    Weinberg,Gravitation and cosmology: principles and applications of the general theory of relativity(John Wi- ley and Sons, New York, 1972)

    S. Weinberg,Gravitation and cosmology: principles and applications of the general theory of relativity(John Wi- ley and Sons, New York, 1972)

    1972

  4. [2]

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

    1973

  5. [3]

    Schneider, J

    P. Schneider, J. Ehlers, and E. Falco, Gravitational Lenses (Springer-Verlag, Berlin Heidelberg, 1992)

    1992

  6. [4]

    R.NarayanandM.Bartelmann, Lectures on gravitational lensing, arXiv:astro-ph/9606001v2

    arXiv

  7. [5]

    Wambsganss, Gravitational lensing in astronomy, Liv- ing Reviews in Relativity1, 12 (1998)

    J. Wambsganss, Gravitational lensing in astronomy, Liv- ing Reviews in Relativity1, 12 (1998)

    1998

  8. [6]

    Bartelmann and P

    M. Bartelmann and P. Schneider, Weak gravitational lensing, Physics Reports340, 291–472 (2001)

    2001

Show all 94 references
  1. [7]

    Dodelson,Gravitational Lensing (Cambridge Univer- sity Press, 2017)

    S. Dodelson,Gravitational Lensing (Cambridge Univer- sity Press, 2017)

    2017

  2. [8]

    Meneghetti, Introduction to Gravitational Lensing (Springer International Publishing, 2021)

    M. Meneghetti, Introduction to Gravitational Lensing (Springer International Publishing, 2021)

    2021

  3. [9]

    Walsh, R

    D. Walsh, R. F. Carswell, and R. J. Weymann, 0957 + 561 A, B: twin quasistellar objects or gravitational lens?, Nature 279, 381–384 (1979)

    1979

  4. [10]

    Soucail, Y

    G. Soucail, Y. Mellier, B. Fort, J. P. Picat, A blue ring- likestructureinthecenteroftheA370clusterofgalaxies, Astronomy & Astrophysics172, L14–L16 (1987)

    1987

  5. [11]

    Bozza, S

    V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Strong field limit of black hole gravitational lensing, Gen- eral Relativity and Gravitation33, 1535–1548 (2001)

    2001

  6. [12]

    Darwin, The gravity field of a particle, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences249, 180 (1959)

    C. Darwin, The gravity field of a particle, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences249, 180 (1959)

    1959

  7. [13]

    R. d’E. Atkinson, On light tracks near a very massive star, Astronomical Journal70, 517 (1965)

    1965

  8. [14]

    Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics75, 228–235 (1979)

    J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics75, 228–235 (1979)

    1979

  9. [15]

    H. C. Ohanian, The black hole as a gravitational “lens”, American Journal of Physics55, 428–432 (1987)

    1987

  10. [16]

    Bozza, Gravitational lensing in the strong field limit, Physical Review D66, 103001 (2002)

    V. Bozza, Gravitational lensing in the strong field limit, Physical Review D66, 103001 (2002)

    2002

  11. [17]

    Bozza, Quasiequatorial gravitational lensing by spin- ning black holes in the strong field limit, Physical Review D 67, 103006 (2003)

    V. Bozza, Quasiequatorial gravitational lensing by spin- ning black holes in the strong field limit, Physical Review D 67, 103006 (2003)

    2003

  12. [18]

    K. S. Virbhadra and G. F. R. Ellis, Schwarzschild black hole lensing, Physical Review D62, 084003 (2000)

    2000

  13. [19]

    Frittelli, T

    S. Frittelli, T. P. Kling, and E. T. Newman, Spacetime perspective of Schwarzschild lensing, Physical Review D 61, 064021 (2000)

    2000

  14. [20]

    Perlick, Exact gravitational lens equation in spheri- cally symmetric and static spacetimes, Physical Review D 69, 064017 (2004)

    V. Perlick, Exact gravitational lens equation in spheri- cally symmetric and static spacetimes, Physical Review D 69, 064017 (2004)

    2004

  15. [21]

    Claudel, K

    C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, The geometry of photon surfaces, Journal of Mathematical Phyics 42, 818–838 (2001)

    2001

  16. [22]

    Hasse and V

    W. Hasse and V. Perlick, Gravitational lensing in spher- ically symmetric static spacetimes with centrifugal force reversal, General Relativity and Gravitation34, 415–433 (2002)

    2002

  17. [23]

    Perlick, Gravitational lensing from a spacetime per- spective, Living Reviews in Relativity7, 9 (2004)

    V. Perlick, Gravitational lensing from a spacetime per- spective, Living Reviews in Relativity7, 9 (2004)

    2004

  18. [24]

    S. V. Iyer and A. O. Petters, Light’s bending angle due to black holes: from the photon sphere to infinity, General Relativity and Gravitation39, 1563–1582 (2007)

    2007

  19. [25]

    K. S. Virbhadra and C. R. Keeton, Time delay and mag- nification centroid due to gravitational lensing by black holes and naked singularities, Physical Review D 77, 124014 (2008)

    2008

  20. [26]

    G. S. Bisnovatyi-Kogan and O. Yu. Tsupko, Strong grav- itational lensing by Schwarzschild black holes, Astro- physics 51, 99–111 (2008)

    2008

  21. [27]

    N.MukherjeeandA.S.Majumdar, Rotatingbrane-world black hole lensing in the strong deflection limit, Gravita- tion and Cosmology15, 263–272 (2009)

    2009

  22. [28]

    Tarasenko, Reconstruction of a compact object mo- tion in the vicinity of a black hole by its electromagnetic radiation, Physical Review D81, 123005 (2010)

    A. Tarasenko, Reconstruction of a compact object mo- tion in the vicinity of a black hole by its electromagnetic radiation, Physical Review D81, 123005 (2010)

    2010

  23. [29]

    E. F. Eiroa and C. M. Sendra, Gravitational lensing by a regular black hole, Classical and Quantum Gravity28, 085008 (2011)

    2011

  24. [30]

    Wei, Yu.-X

    S.-W. Wei, Yu.-X. Liu, C.-E. Fu, and K. Yang, Strong field limit analysis of gravitational lensing in Kerr-Taub- NUT spacetime, Journal of Cosmology and Astroparticle Physics, 10 (2012) 053

    2012

  25. [31]

    G. Li, Y. Zhang, L. Zhang, Z. Feng, and X. Zu, Strong gravitational lensing in the Einstein-Proca theory, Inter- national Journal of Theoretical Physics 54, 1245–1252 (2015)

    2015

  26. [32]

    Alhamzawi and R

    A. Alhamzawi and R. Alhamzawi, Gravitational lensing in the strong field limit by modified gravity, General Rel- ativity and Gravitation48, 167 (2016)

    2016

  27. [33]

    Tsukamoto, Strong deflection limit analysis and grav- itational lensing of an Ellis wormhole, Physical Review D 94, 124001 (2016)

    N. Tsukamoto, Strong deflection limit analysis and grav- itational lensing of an Ellis wormhole, Physical Review D 94, 124001 (2016)

    2016

  28. [34]

    G. F. Aldi and V. Bozza, Relativistic iron lines in ac- cretion disks: The contribution of higher order images in the strong deflection limit, Journal of Cosmology and Astroparticle Physics 02 (2017) 033

    2017

  29. [35]

    D.-C. Dai, D. Stojkovic, and G. D. Starkman, Strong lensing constraints on modified gravity models, Physical Review D 98 124027 (2018)

    2018

  30. [36]

    Kuang, Z.-Y

    X.-M. Kuang, Z.-Y. Tang, B. Wang, and A. Wang, Con- straining a modified gravity theory in strong gravita- tional lensing and black hole shadow observations, Phys- ical Review D106, 064012 (2022)

    2022

  31. [37]

    Aratore and V

    F. Aratore and V. Bozza, Analytical perturbations of rel- ativisticimagesinKerrspace-time, JournalofCosmology and Astroparticle Physics 07 (2024) 033

    2024

  32. [38]

    Guo, M.-H

    M.-Y. Guo, M.-H. Wu, H. Guo, X.M. Kuang, and F.- Y. Liu, Strong gravitational lensing effects around rotat- ing regular black holes, Physics Letters B860, 139211 (2025)

    2025

  33. [39]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Black hole shadows, photon rings, and lensing rings, Physical Re- view D 100, 024018 (2019)

    2019

  34. [41]

    S. E. Gralla and A. Lupsasca, Observable shape of black hole photon rings, Physical Review D102, 124003 (2020)

    2020

  35. [42]

    S. E. Gralla, A. Lupsasca, D. P. Marrone, The shape of the black hole photon ring: A precise test of strong- field 12 general relativity, Physical Review D102, 124004 (2020)

    2020

  36. [43]

    S. E. Gralla and A. Lupsasca, Lensing by Kerr black holes, Physical Review D101, 044031 (2020)

    2020

  37. [44]

    Wielgus, Photon rings of spherically symmetric black holes and robust tests of non-Kerr metrics, Physical Re- view D 104, 124058 (2021)

    M. Wielgus, Photon rings of spherically symmetric black holes and robust tests of non-Kerr metrics, Physical Re- view D 104, 124058 (2021)

    2021

  38. [45]

    A. E. Broderick, P. Tiede, D. W. Pesce, and R. Gold, Measuring spin from relative photon ring sizes, The As- trophysical Journal 927, 6 (2022)

    2022

  39. [46]

    Ayzenberg, Testing gravity with black hole shadow subrings, Class

    D. Ayzenberg, Testing gravity with black hole shadow subrings, Class. Quant. Grav.39, 105009 (2022)

    2022

  40. [47]

    Guerrero, G

    M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, and D. S.- C. Gómez, Multiring images of thin accretion disk of a regular naked compact object, Physical Review D106, 044070 (2022)

    2022

  41. [48]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Analytical study of higher-order ring images of the accretion disk around a black hole, Physical Review D 105, 064040 (2022)

    2022

  42. [49]

    O. Yu. Tsupko, Shape of higher-order images of equa- torial emission rings around a Schwarzschild black hole: Analytical description with polar curves, Physical Re- view D 106, 064033 (2022)

    2022

  43. [50]

    Eichhorn, A

    A. Eichhorn, A. Held, and P.-V. Johannsen, Universal signatures of singularity-resolving physics in photon rings of black holes and horizonless objects, Journal of Cosmol- ogy and Astroparticle Physics 01 (2023) 043

    2023

  44. [51]

    A. E. Broderick, K. Salehi, and B. Georgiev, Shadow im- plications: What does measuring the photon ring imply for gravity?, The Astrophysical Journal958, 114 (2023)

    2023

  45. [52]

    Kocherlakota, L

    P. Kocherlakota, L. Rezzolla, R. Roy, and M. Wielgus, Hotspots and Photon Rings in Schwarzschild Black Hole Spacetimes, Mon. Not. R. Astron. Soc.531, 3606 (2024)

    2024

  46. [53]

    Aratore, O

    F. Aratore, O. Yu. Tsupko, and V. Perlick, Constraining spherically symmetric metrics by the gap between photon rings, Physical Review D109, 124057 (2024)

    2024

  47. [54]

    O.Yu.TsupkoandG.S.Bisnovatyi-Kogan, Gravitational lensing in plasma: Relativistic images at homogeneous plasma, Physical Review D87, 124009 (2013)

    2013

  48. [55]

    Feleppa, V

    F. Feleppa, V. Bozza, and O. Yu. Tsupko, Strong deflec- tion limit analysis of black hole lensing in inhomogeneous plasma, Physical Review D110, 064031 (2024)

    2024

  49. [56]

    Feleppa, V

    F. Feleppa, V. Bozza, and O. Yu. Tsupko, Strong deflec- tion of massive particles in spherically symmetric space- times, Physical Review D111, 044018 (2024)

    2024

  50. [57]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, et al., First M87 Event Horizon Telescope re- sults. I. The shadow of the supermassive black hole, The Astrophysical Journal Letters875, L1 (2019)

    2019

  51. [58]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, etal., FirstM87EventHorizonTelescoperesults.II. Array and instrumentation, The Astrophysical Journal Letters 875, L2 (2019)

    2019

  52. [59]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, et al., First M87 Event Horizon Telescope results. III. Data processing and calibration, The Astrophysical Journal Letters 875, L3 (2019)

    2019

  53. [60]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, et al., First M87 Event Horizon Telescope re- sults. IV. Imaging the central supermassive black hole, The Astrophysical Journal Letters875, L4 (2019)

    2019

  54. [61]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, etal., FirstM87EventHorizonTelescoperesults.V. Physicaloriginoftheasymmetricring, TheAstrophysical Journal Letters 875, L5 (2019)

    2019

  55. [62]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, et al., First M87 Event Horizon Telescope results. VI. The shadow and mass of the central black hole, The Astrophysical Journal Letters875, L6 (2019)

    2019

  56. [63]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, D. Ball, et al., First Sagittarius A* Event Horizon Telescope results. I. The shadow of the supermassive black hole in the center of th...

    2022

  57. [64]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W.Alef, J.C.Algaba, R.Anantua, K.Asada, R. Azulay, et al., First Sagittarius A* Event Horizon Tele- scope results. II. EHT and multiwavelength observations, dataprocessing, andcalibration, TheAstrophysicalJour- nal...

    2022

  58. [65]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W.Alef, J.C.Algaba, R.Anantua, K.Asada, R. Azulay, et al., First Sagittarius A* Event Horizon Tele- scope results. III. Imaging of the Galactic center super- massive black hole, The Astrophysical Journal Letters 93...

    2022

  59. [66]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, et al., First Sagittarius A* Event Horizon Telescope results. IV. Variability, morphology, and black hole mass, The Astrophysical Journal Letters930, L15 (2022)

    2022

  60. [67]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, et al., First Sagittarius A* Event Horizon Telescope results. V. Testing astrophysical models of the Galactic center black hole, The Astrophysical Journal Let...

    2022

  61. [68]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W.Alef, J.C.Algaba, R.Anantua, K.Asada, R. Azulay, et al., First Sagittarius A* Event Horizon Tele- scope results. VI. Testing the black hole metric, The As- trophysical Journal Letters930, L17 (2022)

    2022

  62. [69]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, et al., First M87 Event Horizon Telescope Results. VII. Polarization of the Ring, The Astrophysical Journal Letters 910, L12 (2021)

    2021

  63. [70]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, et al., First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon, The Astrophysical Journal Letters 910, L13 (2021)

    2021

  64. [71]

    Tamburini, B

    F. Tamburini, B. Thidé, and M. Della Valle, Measure- ment of the spin of the M87 black hole from its observed twisted light, Monthly Notices of the Royal Astronomical Society: Letters 492, 1, L22–L27 (2020)

    2020

  65. [72]

    Tamburini, F

    F. Tamburini, F. Feleppa, and B. Thidé, Twisted light, a new tool for general relativity and beyond — Revealing the properties of rotating black holes with the vorticity 13 of light —, International Journal of Modern Physics D 30, 14, 2142017 (2021)

    2021

  66. [73]

    Tamburini, F

    F. Tamburini, F. Feleppa, B. Thidé, and I. Licata, Kerr- spacetime geometric optics for vortex beams, Physical Review D 104, 013718 (2021)

    2021

  67. [75]

    Rezzolla and A

    L. Rezzolla and A. Zhidenko, New parametrization for spherically symmetric black holes in metric theories of gravity, Physical Review D90, 084009 (2014)

    2014

  68. [76]

    Johannsen and D

    T. Johannsen and D. Psaltis, Metric for rapidly spinning black holes suitable for strong-field tests of the no-hair theorem, Physical Review D83, 124015 (2011)

    2011

  69. [77]

    Cardoso, P

    V. Cardoso, P. Pani, J. Rico, On generic parametriza- tions of spinning black-hole geometries, Physical Review D 89, 064007 (2014)

    2014

  70. [78]

    Shaymatov, B

    S. Shaymatov, B. Ahmedov, M. De Laurentis, M. Jamil, Q. Wu, A. Wang, and M. Azreg-Aïnou, On the Param- eters of the Spherically Symmetric Parameterized Rez- zolla–Zhidenko Spacetime through Solar System Tests, the Orbit of the S2 Star about Sgr A*, and Quasiperiodic Oscillatio...

    2023

  71. [79]

    Toshmatov, B

    B. Toshmatov, B. Ahmedov, Tidal forces in parametrized spacetime: Rezzolla-Zhidenko parametrization, Physical Review D 108, 084035 (2023)

    2023

  72. [80]

    Alloqulov, H

    M. Alloqulov, H. Chakrabarty, D. Malafarina, B. Ahme- dov and A. Abdujabbarov, Gravitational lensing of neu- trinos in parametrized black hole spacetimes, Journal of Cosmology and Astroparticle Physics 02 (2025) 070

    2025

  73. [81]

    Moriyama, A

    K. Moriyama, A. Cruz-Osorio, Y. Mizuno, I. K. Dihingia and A. Uniyal, Black hole accretion and radiation vari- ability in general relativistic magnetohydrodynamic sim- ulations with Rezzolla–Zhidenko spacetime, Astronomy & Astrophysics 694, A135 (2025)

    2025

  74. [82]

    Bozza and G

    V. Bozza and G. Scarpetta, Strong deflection limit of black hole gravitational lensing with arbitrary source dis- tances, Physical Review D76, 083008 (2007)

    2007

  75. [83]

    V. I. Dokuchaev and N. O. Nazarova, Event Horizon Im- age within Black Hole Shadow, Journal of Experimental and Theoretical Physics128, 4 (2019)

    2019

  76. [84]

    V. I. Dokuchaev and N. O. Nazarova, Visible Shapes of Black Holes M87* and SgrA*, Universe6, 9 (2020)

    2020

  77. [85]

    V. I. Dokuchaev and N. O. Nazarova, Silhouettes of in- visible black holes, Physics Uspekhi63, 6 (2020)

    2020

  78. [86]

    Chael, M

    A. Chael, M. D. Johnson, and A. Lupsasca, Observing the Inner Shadow of a Black Hole: A Direct View of the Event Horizon, The Astrophysical Journal918, 1 (2021)

    2021

  79. [87]

    Kocherlakota, L

    P. Kocherlakota, L. Rezzolla, R. Roy, and M. Wielgus, Prospects for future experimental tests of gravity with black hole imaging: Spherical symmetry, Physical Re- view D 109, 064064 (2007)

    2007

  80. [88]

    Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Annalen der Physik 355, 106 (1916)

    H. Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Annalen der Physik 355, 106 (1916)

    1916

  81. [89]

    G. Nordström, On the energy of the gravitational field in Einstein’s theory, Verhandelingen der Koninklijke Ned- erlandse Akademie van Wetenschappen, Afdeling Natu- urkunde 26, 1201 (1918)

    1918

  82. [90]

    García, D

    A. García, D. Galtsov, and O. Kechkin, Class of Sta- tionary Axisymmetric Solutions of the Einstein-Maxwell- Dilaton-Axion Field Equations, Physical Review Letters 74, 1276 (1995)

    1995

  83. [91]

    Kocherlakota, L

    P. Kocherlakota, L. Rezzolla, Accurate mapping of spher- ically symmetric black holes in a parametrized frame- work, Physical Review D102, 064058 (2020)

    2020

  84. [92]

    Kocherlakota, L

    P. Kocherlakota, L. Rezzolla, Comment on the Analyt- ical Bounds in the Rezzolla-Zhidenko Parametrization, arXiv:2206.03146v1 (2022)

    2022 arXiv

  85. [93]

    Konoplya, L

    R. Konoplya, L. Rezzolla, and A. Zhidenko, General parametrization of axisymmetric black holes in metric theories of gravity, Physical Review D93, 064015 (2016)

    2016

  86. [94]

    Kocherlakota, L

    P. Kocherlakota, L. Rezzolla, Distinguishing gravita- tional and emission physics in black hole imaging: spher- ical symmetry, Mon. Not. R. Astron. Soc.513, 1 (2022)

    2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.