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arxiv: 2507.03857 · v4 · pith:QEEC6EUFnew · submitted 2025-07-05 · 🌀 gr-qc · astro-ph.GA· astro-ph.HE

Bright ring features and polarization structures in Kerr-Sen black hole images illuminated by radiatively inefficient accretion flows

Pith reviewed 2026-05-22 12:29 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GAastro-ph.HE
keywords Kerr-Sen black holebright ringdilaton parameterRIAFpolarization structureSgr A*GRRTEVPA
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The pith

Increasing the dilaton parameter shrinks the bright ring in Kerr-Sen black hole images while making it wider and brighter.

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

The paper uses general relativistic radiative transfer simulations to model images of a Kerr-Sen black hole illuminated by thermal synchrotron emission from a radiatively inefficient accretion flow at 230 GHz, as relevant to Sgr A*. It shows that a larger dilaton parameter causes the bright ring to shrink while increasing its width and brightness. As the accretion disk becomes thicker, the ring's diameter and width decrease, though the brightness increase is less pronounced than that from the dilaton. When matching to Event Horizon Telescope observations of Sgr A*, disk thickness constrains the allowed black hole parameter ranges more than the observer's inclination. The polarization symmetry coefficient beta2, tied to the two-fold rotational symmetry of electric vector position angles, changes more weakly with disk thickness than with the dilaton parameter.

Core claim

Using GRRT simulations, an increase in the dilaton parameter leads to a shrinking of the bright ring, accompanied by enhancements in both its width and brightness. As the disk thickness grows, the bright ring's diameter and width both decrease. The brightness enhancement induced by the disk thickness is less prominent than that driven by the dilaton parameter. Effects of the disk thickness on the allowed parameter space are stronger than those of the observer's inclination. Effects of the disk thickness on beta2 are much weaker than those from the dilaton parameter.

What carries the argument

The Kerr-Sen black hole metric with a dilaton parameter, illuminated by a radiatively inefficient accretion flow with thermal synchrotron emission at 230 GHz, simulated through general relativistic radiative transfer to generate polarized images and extract the bright ring properties plus the EVPA symmetry coefficient beta2.

If this is right

  • Larger dilaton parameter produces a smaller bright ring with greater width and brightness.
  • Thicker disks reduce both ring diameter and width while providing a milder brightness boost.
  • Disk thickness exerts a stronger influence on the allowed ranges of black hole parameters than observer inclination when matched to EHT data.
  • The beta2 coefficient for EVPA two-fold symmetry responds more sensitively to the dilaton parameter than to disk thickness.

Where Pith is reading between the lines

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

  • Ring size and polarization measurements could help separate Kerr-Sen solutions from standard Kerr black holes if dilaton signatures appear in future data.
  • The weaker impact of thickness on beta2 compared to dilaton may allow polarization observations to isolate dilaton effects even when disk properties remain uncertain.
  • Similar shrinkage and polarization patterns could arise in other dilaton-coupled or modified-gravity spacetimes, offering a template for broader tests.

Load-bearing premise

The radiatively inefficient accretion flow model with purely thermal synchrotron emission at 230 GHz accurately represents the plasma around Sgr A* and the GRRT code correctly propagates polarized radiation in the Kerr-Sen metric.

What would settle it

High-resolution images of Sgr A* that show no ring shrinkage with increasing dilaton parameter or no stronger variation in beta2 from dilaton than from disk thickness would falsify the central claims.

Figures

Figures reproduced from arXiv: 2507.03857 by Hao Yin, Jiliang Jing, Songbai Chen.

Figure 1
Figure 1. Figure 1: FIG. 1: Black hole images (in the top row) and the corresponding blurred images (in the bottom row) at an [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Unwrapped ring profiles of the simulated images in Fig. 1. The red horizontal lines indicate the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Distribution of the bright ring size on the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Distribution of the bright ring size on the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Difference between simulated images of Kerr black holes (first row) and Kerr-Sen black holes [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Difference between simulated images of Kerr black holes (first row) and Kerr-Sen black holes [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Polarization images of Kerr-Sen black holes with different inclination angles for fixed disk thickness [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of the complex [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Changes of the magnitude and phase of the complex coefficient [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

Using general relativistic radiative transfer (GRRT) simulations, we investigate the bright ring features and polarization structures in images of the Kerr-Sen black hole associated with Sgr A*, as illuminated by 230 GHz thermal synchrotron emission from radiatively inefficient accretion flows (RIAF). Our findings reveal that an increase in the dilaton parameter leads to a shrinking of the bright ring, accompanied by enhancements in both its width and brightness. As the disk thickness grows, the bright ring's diameter and width both decrease. The brightness enhancement induced by the disk thickness is less prominent than that driven by the dilaton parameter. Comparing with the Event Horizon Telescope (EHT) observational data of SgrA*, we present the allowed ranges of black hole parameters, and find that effects of the disk thickness on the allowed parameter space are stronger than those of the observer's inclination. Furthermore, we analyze the coefficient $\beta_2$, associated with the two-fold rotational symmetry of the electric vector position angles (EVPA), to probe the polarization structure of the black hole images, and reveal that effects of the disk thickness on $\beta_2$ are much weaker than those from the dilaton parameter.

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

2 major / 2 minor

Summary. The manuscript uses general relativistic radiative transfer (GRRT) simulations to examine 230 GHz images of Kerr-Sen black holes illuminated by thermal synchrotron emission from radiatively inefficient accretion flows (RIAF). It reports that increasing the dilaton parameter shrinks the bright ring while increasing both its width and brightness; increasing disk thickness decreases ring diameter and width with a less prominent brightness boost; disk thickness exerts a stronger influence than observer inclination on the allowed parameter ranges when compared to EHT Sgr A* data; and disk thickness affects the polarization coefficient β₂ much more weakly than the dilaton parameter.

Significance. If the GRRT results prove robust, the work would offer concrete, falsifiable predictions for how the dilaton parameter imprints on ring morphology and EVPA symmetry in EHT images, providing a pathway to constrain Kerr-Sen deviations from Kerr using existing 230 GHz data. The emphasis on β₂ as a diagnostic of two-fold rotational symmetry is a methodological strength that could be extended to other metrics.

major comments (2)
  1. [Methods] Methods section: the central quantitative trends (ring shrinkage, width/brightness changes, and β₂ shifts) are extracted from GRRT images, yet no validation tests recovering the Kerr limit, analytic polarization benchmarks, or convergence studies with respect to grid resolution and ray sampling are described. This is load-bearing because any numerical artifact in polarized transport through the modified electromagnetic sector of the Kerr-Sen metric would directly alter the reported parameter dependences.
  2. [Results] Results and comparison with EHT data: the conclusion that disk-thickness effects dominate the allowed parameter space over inclination rests on the assumption that the purely thermal RIAF model supplies accurate emissivity, absorption, and Faraday coefficients at 230 GHz; without reported closure-phase or visibility-amplitude comparisons to EHT data, the relative strength of these effects cannot be considered observationally anchored.
minor comments (2)
  1. [Polarization analysis] Notation for β₂ is introduced without an explicit definition or reference to the standard decomposition of the EVPA Fourier coefficients; a brief equation or citation would improve clarity.
  2. [Figures] Figure captions should state the exact values of the dilaton parameter, disk thickness, and inclination used in each panel to allow direct comparison with the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and describe the revisions we will implement to strengthen the manuscript.

read point-by-point responses
  1. Referee: [Methods] Methods section: the central quantitative trends (ring shrinkage, width/brightness changes, and β₂ shifts) are extracted from GRRT images, yet no validation tests recovering the Kerr limit, analytic polarization benchmarks, or convergence studies with respect to grid resolution and ray sampling are described. This is load-bearing because any numerical artifact in polarized transport through the modified electromagnetic sector of the Kerr-Sen metric would directly alter the reported parameter dependences.

    Authors: We agree that explicit validation tests are essential to establish the reliability of the reported trends. Although our GRRT implementation follows standard methods previously validated for Kerr spacetimes, we did not include these demonstrations in the original manuscript. In the revised version we will add a new subsection to the Methods section that (i) recovers the Kerr limit by setting the dilaton parameter to zero and directly compares ring diameter, width, brightness, and β₂ to published Kerr results; (ii) benchmarks the polarized radiative transfer against analytic expectations for simple magnetic-field geometries; and (iii) presents convergence tests with respect to grid resolution and number of rays. These additions will confirm that the quantitative trends are free of numerical artifacts. revision: yes

  2. Referee: [Results] Results and comparison with EHT data: the conclusion that disk-thickness effects dominate the allowed parameter space over inclination rests on the assumption that the purely thermal RIAF model supplies accurate emissivity, absorption, and Faraday coefficients at 230 GHz; without reported closure-phase or visibility-amplitude comparisons to EHT data, the relative strength of these effects cannot be considered observationally anchored.

    Authors: We acknowledge that our comparison with EHT Sgr A* data relies on image-domain quantities (primarily ring diameter) rather than direct fits to closure phases or visibility amplitudes. The RIAF model we employ is the standard thermal synchrotron prescription used in multiple EHT analyses. To address the referee’s concern we will revise the Results and Discussion sections to (i) explicitly state the model assumptions and their limitations at 230 GHz and (ii) clarify that the reported dominance of disk-thickness effects over inclination is obtained within this model framework. While performing full visibility and closure-phase fitting lies beyond the present scope, we will note this as a natural direction for follow-up work. revision: partial

Circularity Check

0 steps flagged

Numerical GRRT results are independent simulation outputs with no algebraic reduction to inputs

full rationale

The paper reports findings exclusively from general relativistic radiative transfer simulations of 230 GHz thermal synchrotron emission in the Kerr-Sen spacetime. All quantitative trends (ring shrinkage with dilaton parameter, changes in width/brightness, allowed parameter ranges versus EHT data, and beta2 behavior) are extracted directly from computed images rather than from any closed algebraic chain, fitted parameter renamed as prediction, or self-referential definition. No equation in the provided text equates a derived quantity to its own input by construction, and the work does not invoke uniqueness theorems or ansatze from prior self-citations as load-bearing premises. The derivation chain is therefore self-contained computational exploration.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Kerr-Sen metric, the RIAF density and temperature profiles, and the assumption that 230 GHz emission is purely thermal synchrotron; no new free parameters are introduced beyond the dilaton and disk thickness that are varied explicitly.

free parameters (2)
  • dilaton parameter
    Varied as the central model parameter whose effects on ring morphology are reported.
  • disk thickness
    Varied to study its impact on ring size and polarization.
axioms (2)
  • domain assumption The accretion flow is radiatively inefficient and emits via thermal synchrotron at 230 GHz.
    Stated in the abstract as the illumination model.
  • domain assumption General relativistic radiative transfer accurately captures polarized light propagation in the Kerr-Sen spacetime.
    Implicit in the use of GRRT simulations.

pith-pipeline@v0.9.0 · 5752 in / 1473 out tokens · 30189 ms · 2026-05-22T12:29:09.078838+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

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  2. Imaging and Polarimetric Signatures of Konoplya-Zhidenko Black Holes with Various Thick Disk

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  3. Optical images of Kerr-Sen black hole illuminated by thick accretion disks

    astro-ph.HE 2026-04 unverdicted novelty 4.0

    Increasing charge Q shrinks photon rings and central shadows in Kerr-Sen black hole images while spin creates brightness asymmetry; polarization patterns follow lensing and frame dragging.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · cited by 3 Pith papers · 8 internal anchors

  1. [1]

    It should be pointed out that the observed ring diameter is affected by the resolution of the observation instrument [ ? ]

    Ring diameter is shown to decrease with the dilation parameter r2. It should be pointed out that the observed ring diameter is affected by the resolution of the observation instrument [ ? ]. Therefore, we blur the theoretical image to the typical EHT resolution (convolution with a 20 µas Gaussian beam) [9, 52] and analyze the corresponding features in the...

  2. [2]

    Abbott, R

    B. Abbott, R. Abbott, T. Abbott, M. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Adhikari, et al., Physical Review Letters 116 (2016), ISSN 1079-7114, URL http://dx.doi.org/10.1103/ PhysRevLett.116.061102

  3. [3]

    Abbott, R

    B. Abbott, R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Adhikari, V. Adya, et al., Physical Review D 97 (2018), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.97.102002

  4. [4]

    Akiyama, A

    K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi, J. Barrett, D. Bintley, et al., The Astrophysical Journal Letters 875, L1 (2019), ISSN 2041-8213, URL http://dx.doi.org/10.3847/ 2041-8213/ab0ec7

  5. [5]

    E. H. T. Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, et al., The Astrophysical Journal Letters 930, L12 (2022), ISSN 2041-8213, URL http: //dx.doi.org/10.3847/2041-8213/ac6674

  6. [6]

    Kobrin, Z

    D. Psaltis, L. Medeiros, P. Christian, F. zel, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, D. Ball, et al., Physical Review Letters 125 (2020), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett. 125.141104

  7. [7]

    Vagnozzi, R

    S. Vagnozzi, R. Roy, Y.-D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh, P. S. Joshi, et al., Classical and Quantum Gravity 40, 165007 (2023), ISSN 1361-6382, URL http://dx.doi.org/10. 1088/1361-6382/acd97b

  8. [8]

    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, et al., ApJ 930, L16 (2022)

  9. [9]

    Akiyama, A

    K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi, J. Barrett, D. Bintley, et al., The Astrophysical Journal Letters 875, L5 (2019), ISSN 2041-8213, URL http://dx.doi.org/10.3847/ 2041-8213/ab0f43

  10. [10]

    E. H. T. Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, et al., The Astrophysical Journal Letters 930, L17 (2022), ISSN 2041-8213, URL http: //dx.doi.org/10.3847/2041-8213/ac6756

  11. [11]

    G. W. Gibbons and K.-i. Maeda, Nucl. Phys. B 298, 741 (1988)

  12. [12]

    Sonner and P

    J. Sonner and P. K. Townsend, Physical Review D 74 (2006), ISSN 1550-2368, URL http://dx.doi.org/10. 1103/PhysRevD.74.103508

  13. [13]

    Sen, Physical Review Letters 69, 10061009 (1992), ISSN 0031-9007, URL http://dx.doi.org/10.1103/ PhysRevLett.69.1006

    A. Sen, Physical Review Letters 69, 10061009 (1992), ISSN 0031-9007, URL http://dx.doi.org/10.1103/ PhysRevLett.69.1006

  14. [14]

    X. Qin, S. Chen, Z. Zhang, and J. Jing, Astrophys. J. 938, 2 (2022), 2207.12034

  15. [15]

    Hioki and U

    K. Hioki and U. Miyamoto, Physical Review D 78 (2008), ISSN 1550-2368, URL http://dx.doi.org/10.1103/ PhysRevD.78.044007

  16. [16]

    Younsi, A

    Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya, and Y. Mizuno, Phys. Rev. D 94, 084025 (2016), URL https: //link.aps.org/doi/10.1103/PhysRevD.94.084025

  17. [17]

    J. An, J. Peng, Y. Liu, and X.-H. Feng, Phys. Rev. D 97, 024003 (2018), URL https://link.aps.org/doi/10. 1103/PhysRevD.97.024003

  18. [18]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000), astro-ph/9904193

  19. [19]

    K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009), 0810.2109

  20. [20]

    Narang, S

    A. Narang, S. Mohanty, and A. Kumar, Test of kerr-sen metric with black hole observations (2020), 2002.12786, URL https://arxiv.org/abs/2002.12786

  21. [21]

    M. Wang, G. Guo, P. Yan, S. Chen, and J. Jing, Chin. Phys. C 48, 105103 (2024), 2307.16748

  22. [22]

    X. Zhou, S. Chen, and J. Jing, Eur. Phys. J. C 84, 130 (2024), 2307.01996

  23. [23]

    Feng, R.-J

    H. Feng, R.-J. Yang, and W.-Q. Chen, Astropart. Phys. 166, 103075 (2025), 2403.18541

  24. [24]

    Olivares, Z

    H. Olivares, Z. Younsi, C. M. Fromm, M. DeLaurentis, O. Porth, Y. Mizuno, H. Falcke, M. Kramer, and L. Rezzolla, Monthly Notices of the Royal Astronomical Society 497, 521535 (2020), ISSN 1365-2966, URL http://dx.doi. org/10.1093/mnras/staa1878

  25. [25]

    Mizuno, Z

    Y. Mizuno, Z. Younsi, C. M. Fromm, O. Porth, M. De Laurentis, H. Olivares, H. Falcke, M. Kramer, and L. Rezzolla, Nature Astronomy 2, 585590 (2018), ISSN 2397-3366, URL http://dx.doi.org/10.1038/ s41550-018-0449-5

  26. [26]

    F. Yuan, E. Quataert, and R. Narayan, The Astrophysical Journal 598, 301312 (2003), ISSN 1538-4357, URL http://dx.doi.org/10.1086/378716

  27. [27]

    Pu and A

    H.-Y. Pu and A. E. Broderick, The Astrophysical Journal 863, 148 (2018), ISSN 1538-4357, URL http://dx. doi.org/10.3847/1538-4357/aad086

  28. [28]

    F. Yuan, E. Quataert, and R. Narayan, ApJ 598, 301 (2003), astro-ph/0304125

  29. [29]

    Zhang, S

    Z. Zhang, S. Chen, and J. Jing, Images of kerr-mog black holes surrounded by geometrically thick magnetized equilibrium tori (2024), 2404.12223, URL https://arxiv.org/abs/2404.12223

  30. [30]

    S. Chen, J. Jing, W.-L. Qian, and B. Wang, Sci. China Phys. Mech. Astron. 66, 260401 (2023), 2301.00113

  31. [31]

    Zhang, S

    Z. Zhang, S. Chen, and J. Jing, Eur. Phys. J. C 84, 827 (2024), 2404.05957. 15

  32. [32]

    Akiyama, J

    K. Akiyama, J. C. Algaba, A. Alberdi, W. Alef, R. Anantua, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi, et al., The Astrophysical Journal Letters 910, L13 (2021), ISSN 2041-8213, URL http://dx.doi. org/10.3847/2041-8213/abe4de

  33. [33]

    D. C. M. Palumbo, G. N. Wong, and B. S. Prather, The Astrophysical Journal 894, 156 (2020), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/ab86ac

  34. [34]

    Huang, D.-J

    C. Bernard, Physical Review D 94 (2016), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.94. 085007

  35. [35]

    A. M. Ghezelbash and H. M. Siahaan, Classical and Quantum Gravity 30, 135005 (2013), ISSN 1361-6382, URL http://dx.doi.org/10.1088/0264-9381/30/13/135005

  36. [36]

    Garc´ ıa, D

    A. Garc´ ıa, D. Galtsov, and O. Kechkin, Phys. Rev. Lett. 74, 1276 (1995), URL https://link.aps.org/doi/10. 1103/PhysRevLett.74.1276

  37. [37]

    Banerjee, B

    I. Banerjee, B. Mandal, and S. SenGupta, Physical Review D 103 (2021), ISSN 2470-0029, URL http://dx.doi. org/10.1103/PhysRevD.103.044046

  38. [38]

    YAZADJIEV, International Journal of Modern Physics D 08, 635643 (1999), ISSN 1793-6594, URL http: //dx.doi.org/10.1142/S0218271899000432

    S. YAZADJIEV, International Journal of Modern Physics D 08, 635643 (1999), ISSN 1793-6594, URL http: //dx.doi.org/10.1142/S0218271899000432

  39. [39]

    Horowitz, and Andrew Strominger

    D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D 43, 3140 (1991), URL https://link.aps.org/ doi/10.1103/PhysRevD.43.3140

  40. [40]

    E. T. Newman and A. I. Janis, J. Math. Phys. 6, 915 (1965)

  41. [41]

    Younsi, D

    Z. Younsi, D. Psaltis, and F. zel, The Astrophysical Journal 942, 47 (2023), ISSN 1538-4357, URL http://dx. doi.org/10.3847/1538-4357/aca58a

  42. [42]

    F. zel, D. Psaltis, and Z. Younsi, The Astrophysical Journal 941, 88 (2022), ISSN 1538-4357, URL http://dx. doi.org/10.3847/1538-4357/ac9fcb

  43. [43]

    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, et al., ApJ 964, L26 (2024)

  44. [44]

    S. C. Noble, P. K. Leung, C. F. Gammie, and L. G. Book, Class. Quant. Grav. 24, S259 (2007), astro-ph/0701778

  45. [45]

    ipole - semianalytic scheme for relativistic polarized radiative transport

    M. Moscibrodzka and C. F. Gammie, ipole - semianalytic scheme for relativistic polarized radiative transport (2017), 1712.03057, URL https://arxiv.org/abs/1712.03057

  46. [46]

    Y. Chen, Y. Liu, R.-S. Lu, Y. Mizuno, J. Shu, X. Xue, Q. Yuan, and Y. Zhao, Nature Astronomy 6, 592598 (2022), ISSN 2397-3366, URL http://dx.doi.org/10.1038/s41550-022-01620-3

  47. [47]

    K. Yagi, S. Lomuscio, T. Lowrey, and Z. Carson, Regularizing parameterized black hole spacetimes with kerr symmetries (2024), 2311.08659, URL https://arxiv.org/abs/2311.08659

  48. [48]

    Ben Achour, E

    J. Ben Achour, E. Gourgoulhon, and H. Roussille (2025), 2506.09882, URL https://arxiv.org/abs/2506.09882

  49. [49]

    First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi´ c, J. Barrett, et al., ApJ875, L4 (2019), 1906.11241

  50. [50]

    First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875, L6 (2019), 1906.11243

  51. [51]

    Chael, M

    A. Chael, M. D. Johnson, and A. Lupsasca, The Astrophysical Journal 918, 6 (2021), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/ac09ee

  52. [52]

    A. A. Chael, Ph.D. thesis, Harvard University (2019)

  53. [53]

    A. A. Chael, M. D. Johnson, K. L. Bouman, L. L. Blackburn, K. Akiyama, and R. Narayan, The Astrophysical Journal 857, 23 (2018), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/aab6a8

  54. [54]

    Tiede, M

    P. Tiede, M. D. Johnson, D. W. Pesce, D. C. M. Palumbo, D. O. Chang, and P. Galison, Measuring photon rings with the ngeht (2022), 2210.13498, URL https://arxiv.org/abs/2210.13498

  55. [55]

    M. D. Johnson, K. Akiyama, L. Blackburn, K. L. Bouman, A. E. Broderick, V. Cardoso, R. P. Fender, C. M. Fromm, P. Galison, J. L. Gmez, et al., Galaxies 11, 61 (2023), ISSN 2075-4434, URL http://dx.doi.org/10. 3390/galaxies11030061

  56. [56]

    J. P. Hamaker and J. D. Bregman, A&AS 117, 161 (1996)

  57. [57]

    O. M. Smirnov, Astronomy & Astrophysics 527, A106 (2011), ISSN 1432-0746, URL http://dx.doi.org/10. 1051/0004-6361/201016082

  58. [58]

    Akiyama, J

    K. Akiyama, J. C. Algaba, A. Alberdi, W. Alef, R. Anantua, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi, et al., The Astrophysical Journal Letters 910, L12 (2021), ISSN 2041-8213, URL http://dx.doi. org/10.3847/2041-8213/abe71d

  59. [59]

    Chael, A

    A. Chael, A. Lupsasca, G. N. Wong, and E. Quataert, The Astrophysical Journal 958, 65 (2023), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/acf92d

  60. [60]

    First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930, L12 (2022), 2311.08680