pith. machine review for the scientific record. sign in

arxiv: 2604.11652 · v1 · submitted 2026-04-13 · 🌀 gr-qc

Recognition: unknown

Optical Appearance of Scalarized Kerr-Newman Black Holes with Multiple Light Rings

Yiqian Chen , Li Li , Peng Wang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:47 UTC · model grok-4.3

classification 🌀 gr-qc
keywords scalarized black holesKerr-Newman black holeslight ringsphoton shellcritical curveblack hole shadowhigher-order imagesray tracing
0
0 comments X

The pith

Scalarized Kerr-Newman black holes can develop an extra inner photon shell that creates a second critical curve and new crescent-shaped higher-order images.

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

The paper investigates the visual appearance of rotating scalarized black holes in Einstein-Maxwell-scalar gravity when viewed through a thin accretion disk. It shows that these objects, unlike ordinary Kerr black holes, can possess multiple light rings including an additional inner photon shell outside the event horizon. This extra shell produces an inner critical curve inside the familiar outer one, which varies from absent to partial to fully closed depending on parameters and viewing angle. The inner shell also creates new higher-order images in the space between the curves, some displaying crescent morphologies rather than the round shapes seen in Kerr cases. A reader would care because these image features could act as observable markers that distinguish scalarized black holes from standard ones in high-resolution astronomy.

Core claim

Unlike the Kerr case governed by a single outer photon shell and critical curve, scalarized Kerr-Newman black holes in the Einstein-Maxwell-scalar theory with exponential coupling can develop an additional inner photon shell outside the event horizon. This shell produces an inner critical curve inside the usual outer one, which may be absent, partial, or closed depending on black hole parameters and observer inclination, and it generates new higher-order images between the two curves, some with crescent-like shapes distinct from the nearly circular higher-order images of Kerr black holes.

What carries the argument

Classification of the black holes into six types based on the number and properties of equatorial light rings, obtained by solving for null geodesics, which then determines the critical curves and the ray-traced images from the thin disk.

If this is right

  • The region between the inner and outer critical curves contains additional higher-order photon images not present in Kerr black holes.
  • Some of the new higher-order images take on crescent morphologies instead of nearly circular shapes.
  • The inner critical curve can appear absent, partial, or closed, altering the overall shadow structure depending on spin, charge, scalar coupling, and inclination.
  • These multiple-ring configurations enrich the possible optical appearances beyond the single-shell Kerr case.

Where Pith is reading between the lines

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

  • Future very-long-baseline interferometry could target the space between critical curves to search for scalar hair signatures.
  • The crescent images might require separate modeling to avoid confusion with disk inhomogeneities or other lensing effects.
  • Extending the ray tracing beyond equatorial geodesics would likely reveal even more varied image structures for inclined observers.

Load-bearing premise

The accretion disk is geometrically and optically thin, allowing its light to be ray-traced without significant self-obscuration or thick-disk effects.

What would settle it

High-resolution images of a black hole expected to be scalarized that show only a single critical curve with no inner structure or crescent higher-order images would contradict the prediction.

Figures

Figures reproduced from arXiv: 2604.11652 by Li Li, Peng Wang, Yiqian Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. Radial effective potential in the slow-rotation approximation for three representative classes of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Critical curves for several values of the spin [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Critical curves for several values of spin [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Critical curves for several values of spin [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots for a scalarized KN black hole of Type I ( [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plots for scalarized KN black holes of Type II1 (Upper: [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots for scalarized KN black holes of Type III1 (Upper: [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Plots for two scalarized KN black holes of Type III2, admitting two prograde and two retrograde [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Observed images of scalarized KN black holes with [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

This study investigates the optical appearance of rotating scalarized Kerr-Newman black holes in the Einstein-Maxwell-scalar theory with exponential coupling. By analyzing equatorial null geodesics, these black holes are classified into six types according to the number and properties of their light rings. Combining slow-rotation analysis with full numerical ray tracing, we investigate images of black holes illuminated by a geometrically and optically thin accretion disk. Unlike the Kerr case, where the image is typically governed by a single outer photon shell and a single critical curve, scalarized Kerr-Newman black holes can develop an additional inner photon shell outside the event horizon. This extra shell gives rise to an inner critical curve inside the usual outer one, which may be absent, partial, or closed depending on the black hole parameters and the observer's inclination. Moreover, it generates new higher-order images in the region between the two critical curves, some of which exhibit crescent-like morphologies distinct from the nearly circular higher-order images familiar from Kerr black holes. These features enrich the optical appearance of scalarized black holes and could serve as distinctive observational signatures in future high-resolution observations.

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 examines the optical appearance of scalarized Kerr-Newman black holes in Einstein-Maxwell-scalar theory with exponential coupling. Equatorial null geodesic analysis is used to classify these solutions into six types according to the number and location of light rings. Slow-rotation approximations combined with full numerical ray tracing of a geometrically and optically thin accretion disk are then employed to generate images, showing that an additional inner photon shell (outside the horizon) produces an inner critical curve inside the outer one. This leads to new higher-order images between the curves, some with crescent morphologies distinct from those in Kerr spacetime, depending on parameters and observer inclination.

Significance. If the central claims hold, the work identifies potentially distinctive morphological features in black hole images arising from scalarization, which could serve as observational discriminants in future high-resolution data. The combination of analytic classification of light rings with numerical ray tracing provides concrete, falsifiable predictions for image structure. The numerical approach to geodesic integration is a methodological strength that enables exploration of non-Kerr effects beyond analytic limits.

major comments (2)
  1. [Equatorial null geodesic analysis and numerical results sections] The central claim that scalarized solutions with multiple light rings produce observable inner critical curves and crescent-like higher-order images presupposes that these backgrounds are stable. No linear stability analysis against perturbations is reported for the specific parameter sets (spin, charge, and coupling strength) where the inner photon shell appears outside the horizon (see the classification in the equatorial geodesic section and the numerical examples in the ray-tracing results). Unstable modes would render the predicted morphologies transient and unobservable, making this a load-bearing omission for the physical relevance of the six-type classification and image features.
  2. [Ray-tracing and image morphology discussion] The ray-tracing computations assume a geometrically and optically thin disk with no self-obscuration. While this is standard, the interaction with the inner photon shell (which generates images between the two critical curves) requires explicit checks that the thin-disk approximation remains valid for the reported inclinations and parameters; otherwise the crescent morphologies could be altered or suppressed (see the image generation procedure and the discussion of higher-order images).
minor comments (2)
  1. [Figure captions and § on light rings] Notation for the critical curves and photon shells should be made consistent between the geodesic analysis and the image figures to avoid ambiguity when comparing inner and outer features.
  2. [Numerical methods subsection] A brief statement on the range of coupling parameters explored and any convergence tests for the numerical integrator would strengthen the reproducibility of the six-type classification.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below, indicating revisions where appropriate.

read point-by-point responses

  1. Referee: The central claim that scalarized solutions with multiple light rings produce observable inner critical curves and crescent-like higher-order images presupposes that these backgrounds are stable. No linear stability analysis against perturbations is reported for the specific parameter sets (spin, charge, and coupling strength) where the inner photon shell appears outside the horizon. Unstable modes would render the predicted morphologies transient and unobservable, making this a load-bearing omission for the physical relevance of the six-type classification and image features.

    Authors: We agree that stability is crucial for the physical observability of the predicted image features. The present work focuses on classifying light rings via equatorial null geodesics and computing ray-traced images assuming the scalarized solutions exist. A full linear stability analysis for the multi-light-ring parameter sets would require solving the perturbation equations in the Einstein-Maxwell-scalar theory and is beyond the scope of this manuscript. Related literature indicates stable branches for moderate coupling strengths and charges, and our examples lie in those regimes. We will add a note in the revised manuscript acknowledging this limitation and identifying stability analysis as future work. revision: partial

  2. Referee: The ray-tracing computations assume a geometrically and optically thin disk with no self-obscuration. While this is standard, the interaction with the inner photon shell (which generates images between the two critical curves) requires explicit checks that the thin-disk approximation remains valid for the reported inclinations and parameters; otherwise the crescent morphologies could be altered or suppressed.

    Authors: The geometrically and optically thin disk is a standard assumption to highlight spacetime effects. For the inclinations and parameters in our figures, the disk scale height is negligible relative to photon orbit radii, so self-obscuration does not suppress the inter-curve images. We performed additional test ray tracings with small finite thickness and confirmed the crescent morphologies persist. We will include a short justification and these checks in the revised image-generation section. revision: yes

standing simulated objections not resolved
  • No linear stability analysis is provided for the specific parameter sets exhibiting an inner photon shell outside the horizon.

Circularity Check

0 steps flagged

No circularity: numerical geodesic classification and ray-tracing in a given metric family

full rationale

The paper classifies scalarized Kerr-Newman solutions into six types by direct integration of equatorial null geodesics and performs ray-tracing of a thin accretion disk. These steps are computational applications of the geodesic equation in a pre-specified metric; no parameters are fitted to the resulting images, no target observable is used to define the input metric, and no uniqueness theorem or ansatz is smuggled via self-citation. The derivation chain is therefore self-contained and does not reduce any claimed optical feature to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of scalarized rotating charged black-hole solutions in Einstein-Maxwell-scalar theory with exponential coupling, the validity of null geodesic equations in that spacetime, and the thin-disk illumination model. No explicit free parameters or new entities are introduced beyond the theory itself.

axioms (2)
  • standard math Null geodesics follow the standard geodesic equation in the given metric
    Implicit in the equatorial null geodesic analysis described in the abstract.
  • domain assumption The accretion disk is geometrically and optically thin
    Stated in the abstract as the illumination model used for image construction.

pith-pipeline@v0.9.0 · 5493 in / 1476 out tokens · 56626 ms · 2026-05-10T15:47:27.859562+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

87 extracted references · 85 canonical work pages

  1. [1]

    2019, ApJL, 875, L1, doi: 10.3847/2041-8213/ab0ec7 Event Horizon Telescope Collaboration, Akiyama, K.,

    Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole.Astrophys. J. Lett., 875:L1, 2019.arXiv:1906.11238,doi:10.3847/2041-8213/ab0ec7

    work page doi:10.3847/2041-8213/ab0ec7 2019

  2. [2]

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

    Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. IV. Imaging the Central Su- permassive Black Hole.Astrophys. J. Lett., 875(1):L4, 2019.arXiv:1906.11241,doi:10.3847/ 2041-8213/ab0e85

    work page Pith review arXiv 2019

  3. [3]

    G., Yuan, W., Macri, L

    Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. V. Physical Origin of the Asym- metric Ring.Astrophys. J. Lett., 875(1):L5, 2019.arXiv:1906.11242,doi:10.3847/2041-8213/ ab0f43

    work page doi:10.3847/2041-8213/ 2019

  4. [4]

    2022, ApJL, 930, L12, doi: 10.3847/2041-8213/ac6674

    Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way.Astrophys. J. Lett., 930(2):L12, 2022. doi:10.3847/2041-8213/ac6674

    work page doi:10.3847/2041-8213/ac6674 2022

  5. [5]

    First Sagittarius A* Event Horizon Telescope Results

    Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole.Astrophys. J. Lett., 930(2):L14, 2022.doi:10.3847/ 2041-8213/ac6429

    2022

  6. [6]

    Gralla, Daniel E

    Samuel E. Gralla, Daniel E. Holz, and Robert M. Wald. Black Hole Shadows, Photon Rings, and Lensing Rings.Phys. Rev. D, 100(2):024018, 2019.arXiv:1906.00873,doi:10.1103/PhysRevD.100.024018

    work page doi:10.1103/physrevd.100.024018 2019

  7. [7]

    Measuring the Kerr spin parameter of regular black holes from their shadow.JCAP, 01:041, 2014.arXiv:1309.1606,doi:10.1088/1475-7516/2014/01/041

    Zilong Li and Cosimo Bambi. Measuring the Kerr spin parameter of regular black holes from their shadow.JCAP, 01:041, 2014.arXiv:1309.1606,doi:10.1088/1475-7516/2014/01/041

    work page doi:10.1088/1475-7516/2014/01/041 2014

  8. [8]

    Constraining the spin and the deformation parameters from the black hole shadow.JCAP, 06:043, 2014.arXiv:1403.0371,doi:10.1088/1475-7516/2014/ 06/043

    Naoki Tsukamoto, Zilong Li, and Cosimo Bambi. Constraining the spin and the deformation parameters from the black hole shadow.JCAP, 06:043, 2014.arXiv:1403.0371,doi:10.1088/1475-7516/2014/ 06/043

    work page doi:10.1088/1475-7516/2014/ 2014

  9. [9]

    Rahul Kumar and Sushant G. Ghosh. Black Hole Parameter Estimation from Its Shadow.Astrophys. J., 892:78, 2020.arXiv:1811.01260,doi:10.3847/1538-4357/ab77b0

    work page doi:10.3847/1538-4357/ab77b0 2020

  10. [10]

    Testing the rotational na- ture of the supermassive object M87* from the circularity and size of its first image.Phys

    Cosimo Bambi, Katherine Freese, Sunny Vagnozzi, and Luca Visinelli. Testing the rotational na- ture of the supermassive object M87* from the circularity and size of its first image.Phys. Rev. D, 100(4):044057, 2019.arXiv:1904.12983,doi:10.1103/PhysRevD.100.044057

    work page doi:10.1103/physrevd.100.044057 2019

  11. [11]

    Misba Afrin, Sunny Vagnozzi, and Sushant G. Ghosh. Tests of Loop Quantum Gravity from the Event Horizon Telescope Results of Sgr A*.Astrophys. J., 944(2):149, 2023.arXiv:2209.12584, doi:10.3847/1538-4357/acb334

    work page doi:10.3847/1538-4357/acb334 2023

  12. [12]

    Dynamical features and shadows of quantum Schwarzschild black hole in effective field theories of gravity.Eur

    Zi-Liang Wang and Emmanuele Battista. Dynamical features and shadows of quantum Schwarzschild black hole in effective field theories of gravity.Eur. Phys. J. C, 85(3):304, 2025.arXiv:2501.14516, doi:10.1140/epjc/s10052-025-13833-7

    work page doi:10.1140/epjc/s10052-025-13833-7 2025

  13. [13]

    The Polarized Image of a Synchrotron-emitting Ring of Gas Orbiting a Black 26 Hole.Astrophys

    Kazunori Akiyama et al. The Polarized Image of a Synchrotron-emitting Ring of Gas Orbiting a Black 26 Hole.Astrophys. J., 912(1):35, 2021.arXiv:2105.01804,doi:10.3847/1538-4357/abf117

    work page doi:10.3847/1538-4357/abf117 2021

  14. [14]

    Zachary Gelles, Elizabeth Himwich, Daniel C. M. Palumbo, and Michael D. Johnson. Polarized image of equatorial emission in the Kerr geometry.Phys. Rev. D, 104(4):044060, 2021.arXiv:2105.09440, doi:10.1103/PhysRevD.104.044060

    work page doi:10.1103/physrevd.104.044060 2021

  15. [16]

    Polarized image of equa- torial emission in horizonless spacetimes: Traversable wormholes.Phys

    Valentin Delijski, Galin Gyulchev, Petya Nedkova, and Stoytcho Yazadjiev. Polarized image of equa- torial emission in horizonless spacetimes: Traversable wormholes.Phys. Rev. D, 106(10):104024, 2022. arXiv:2206.09455,doi:10.1103/PhysRevD.106.104024

    work page doi:10.1103/physrevd.106.104024 2022

  16. [17]

    Polarized Image of a Rotating Black Hole in Scalar–Tensor–Vector–Gravity Theory.Astrophys

    Xin Qin, Songbai Chen, Zelin Zhang, and Jiliang Jing. Polarized Image of a Rotating Black Hole in Scalar–Tensor–Vector–Gravity Theory.Astrophys. J., 938(1):2, 2022.arXiv:2207.12034,doi: 10.3847/1538-4357/ac8f49

    work page doi:10.3847/1538-4357/ac8f49 2022

  17. [18]

    Circular orbits and polarized images of charged particles orbiting a Kerr black hole with a weak magnetic field.Phys

    Tsai-Chen Lee, Zezhou Hu, Minyong Guo, and Bin Chen. Circular orbits and polarized images of charged particles orbiting a Kerr black hole with a weak magnetic field.Phys. Rev. D, 108(2):024008, 2023.arXiv:2211.04143,doi:10.1103/PhysRevD.108.024008

    work page doi:10.1103/physrevd.108.024008 2023

  18. [19]

    Polarized image of synchrotron radiations of hotspots in Schwarzschilld- Melvin black hole spacetime

    Hu Zhu and Minyong Guo. Polarized image of synchrotron radiations of hotspots in Schwarzschilld- Melvin black hole spacetime. 5 2022.arXiv:2205.04777

    work page arXiv 2022

  19. [20]

    Polarization distribution in the image of a synchrotron emitting ring around a regular black hole.Sci

    Xueyao Liu, Songbai Chen, and Jiliang Jing. Polarization distribution in the image of a synchrotron emitting ring around a regular black hole.Sci. China Phys. Mech. Astron., 65(12):120411, 2022. arXiv:2205.00391,doi:10.1007/s11433-022-1946-2

    work page doi:10.1007/s11433-022-1946-2 2022

  20. [21]

    Polarized images of synchrotron radiations in curved spacetime.Eur

    Zezhou Hu, Yehui Hou, Haopeng Yan, Minyong Guo, and Bin Chen. Polarized images of synchrotron radiations in curved spacetime.Eur. Phys. J. C, 82(12):1166, 2022.arXiv:2203.02908,doi:10.1140/ epjc/s10052-022-11144-9

    work page arXiv 2022

  21. [22]

    Polarized image of a rotating black hole surrounded by a cold dark matter halo.Eur

    Xin Qin, Songbai Chen, Zelin Zhang, and Jiliang Jing. Polarized image of a rotating black hole surrounded by a cold dark matter halo.Eur. Phys. J. C, 83(2):159, 2023.arXiv:2301.01551,doi: 10.1140/epjc/s10052-023-11300-9

    work page doi:10.1140/epjc/s10052-023-11300-9 2023

  22. [23]

    Polarized image of equa- torial emission in horizonless spacetimes: Naked singularities.Phys

    Valentin Deliyski, Galin Gyulchev, Petya Nedkova, and Stoytcho Yazadjiev. Polarized image of equa- torial emission in horizonless spacetimes: Naked singularities.Phys. Rev. D, 108(10):104049, 2023. arXiv:2303.14756,doi:10.1103/PhysRevD.108.104049

    work page doi:10.1103/physrevd.108.104049 2023

  23. [24]

    A ring-like accretion structure in M87 connecting its black hole and jet.Nature, 616(7958):686–690, 2023.arXiv:2304.13252,doi:10.1038/s41586-023-05843-w

    Ru-Sen Lu et al. A ring-like accretion structure in M87 connecting its black hole and jet.Nature, 616(7958):686–690, 2023.arXiv:2304.13252,doi:10.1038/s41586-023-05843-w

    work page doi:10.1038/s41586-023-05843-w 2023

  24. [25]

    Evan Papoutsis, Michi Baub¨ ock, Dominic Chang, and Charles F. Gammie. Jets and Rings in Images of Spinning Black Holes.Astrophys. J., 944(1):55, 2023.arXiv:2212.06281,doi:10.3847/1538-4357/ acafe3

    work page doi:10.3847/1538-4357/ 2023

  25. [26]

    R., Zanjani, M

    Zhenyu Zhang, Yehui Hou, Minyong Guo, and Bin Chen. Imaging thick accretion disks and jets surrounding black holes.JCAP, 05:032, 2024.arXiv:2401.14794,doi:10.1088/1475-7516/2024/ 27 05/032

    work page doi:10.1088/1475-7516/2024/ 2024

  26. [27]

    Detection of orbital motions near the last stable circular orbit of the massive black hole sgra.Astronomy & Astrophysics, 618:L10, 2018

    Roberto Abuter, A Amorim, M Baub¨ ock, JP Berger, H Bonnet, W Brandner, Y Cl´ enet, V Coud´ e Du Foresto, PT de Zeeuw, C Deen, et al. Detection of orbital motions near the last stable circular orbit of the massive black hole sgra.Astronomy & Astrophysics, 618:L10, 2018

    2018

  27. [28]

    Baub¨ ock et al

    M. Baub¨ ock et al. Modeling the orbital motion of Sgr A*’s near-infrared flares.Astron. Astrophys., 635:A143, 2020.arXiv:2002.08374,doi:10.1051/0004-6361/201937233

    work page doi:10.1051/0004-6361/201937233 2020

  28. [29]

    Orbital motion near Sagittarius A* - Constraints from polarimetric ALMA observations.Astron

    Maciek Wielgus, Monika Moscibrodzka, Jesse Vos, Zachary Gelles, Ivan Marti-Vidal, Joseph Farah, Nicola Marchili, Ciriaco Goddi, and Hugo Messias. Orbital motion near Sagittarius A* - Constraints from polarimetric ALMA observations.Astron. Astrophys., 665:L6, 2022.arXiv:2209.09926,doi: 10.1051/0004-6361/202244493

    work page doi:10.1051/0004-6361/202244493 2022

  29. [30]

    Images and flares of geodesic hot spots around a Kerr black hole,

    Jiewei Huang, Zhenyu Zhang, Minyong Guo, and Bin Chen. Images and flares of geodesic hot spots around a Kerr black hole.Phys. Rev. D, 109(12):124062, 2024.arXiv:2402.16293,doi:10.1103/ PhysRevD.109.124062

    work page arXiv 2024

  30. [31]

    Leonardo Amarilla and Ernesto F. Eiroa. Shadow of a rotating braneworld black hole.Phys. Rev. D, 85:064019, 2012.arXiv:1112.6349,doi:10.1103/PhysRevD.85.064019

    work page doi:10.1103/physrevd.85.064019 2012

  31. [32]

    Observational signature of a near-extremal Kerr-Sen black hole in the heterotic string theory.Phys

    Minyong Guo, Shupeng Song, and Haopeng Yan. Observational signature of a near-extremal Kerr-Sen black hole in the heterotic string theory.Phys. Rev. D, 101(2):024055, 2020.arXiv:1911.04796, doi:10.1103/PhysRevD.101.024055

    work page doi:10.1103/physrevd.101.024055 2020

  32. [33]

    Shadows and deflection angle of charged and slowly rotating black holes in Einstein-Æther theory.Phys

    Tao Zhu, Qiang Wu, Mubasher Jamil, and Kimet Jusufi. Shadows and deflection angle of charged and slowly rotating black holes in Einstein-Æther theory.Phys. Rev. D, 100(4):044055, 2019.arXiv: 1906.05673,doi:10.1103/PhysRevD.100.044055

    work page doi:10.1103/physrevd.100.044055 2019

  33. [34]

    Ghosh, and Anzhong Wang

    Rahul Kumar, Sushant G. Ghosh, and Anzhong Wang. Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes.Phys. Rev. D, 101(10):104001, 2020.arXiv:2001.00460, doi:10.1103/PhysRevD.101.104001

    work page doi:10.1103/physrevd.101.104001 2020

  34. [35]

    Mayerson, Bart Ripperda, Jordy Davelaar, H´ ector Olivares, Thomas Hertog, and Bert Vercnocke

    Fabio Bacchini, Daniel R. Mayerson, Bart Ripperda, Jordy Davelaar, H´ ector Olivares, Thomas Hertog, and Bert Vercnocke. Fuzzball Shadows: Emergent Horizons from Microstructure.Phys. Rev. Lett., 127(17):171601, 2021.arXiv:2103.12075,doi:10.1103/PhysRevLett.127.171601

    work page doi:10.1103/physrevlett.127.171601 2021

  35. [36]

    Pedro V. P. Cunha, Jos´ e A. Font, Carlos Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, and Miguel Zilh˜ ao. Lensing and dynamics of ultracompact bosonic stars.Phys. Rev. D, 96(10):104040, 2017. arXiv:1709.06118,doi:10.1103/PhysRevD.96.104040

    work page doi:10.1103/physrevd.96.104040 2017

  36. [37]

    Pedro V. P. Cunha and Carlos A. R. Herdeiro. Shadows and strong gravitational lensing: a brief review. Gen. Rel. Grav., 50(4):42, 2018.arXiv:1801.00860,doi:10.1007/s10714-018-2361-9

    work page doi:10.1007/s10714-018-2361-9 2018

  37. [38]

    A novel gravitational lensing feature by wormholes.Phys

    Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar. A novel gravitational lensing feature by wormholes.Phys. Lett. B, 789:270–275, 2019. [Erratum: Phys.Lett.B 791, 422–423 (2019)]. arXiv:1811.08245,doi:10.1016/j.physletb.2018.12.030

    work page doi:10.1016/j.physletb.2018.12.030 2019

  38. [39]

    Analytical approach to strong gravitational lensing from ultracompact objects.Phys

    Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar. Analytical approach to strong gravitational lensing from ultracompact objects.Phys. Rev. D, 99(10):104040, 2019.arXiv:1903. 28 08211,doi:10.1103/PhysRevD.99.104040

    work page doi:10.1103/physrevd.99.104040 2019

  39. [40]

    Strong gravitational lensing by wormholes.JCAP, 07:028, 2019.arXiv:1905.06932,doi:10.1088/1475-7516/2019/07/028

    Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar. Strong gravitational lensing by wormholes.JCAP, 07:028, 2019.arXiv:1905.06932,doi:10.1088/1475-7516/2019/07/028

    work page doi:10.1088/1475-7516/2019/07/028 2019

  40. [41]

    Reflection-asymmetric worm- holes and their double shadows.Phys

    Maciek Wielgus, Jiri Horak, Frederic Vincent, and Marek Abramowicz. Reflection-asymmetric worm- holes and their double shadows.Phys. Rev. D, 102(8):084044, 2020.arXiv:2008.10130,doi: 10.1103/PhysRevD.102.084044

    work page doi:10.1103/physrevd.102.084044 2020

  41. [42]

    Observational signature and additional photon rings of an asymmetric thin-shell wormhole.Phys

    Jun Peng, Minyong Guo, and Xing-Hui Feng. Observational signature and additional photon rings of an asymmetric thin-shell wormhole.Phys. Rev. D, 104(12):124010, 2021.arXiv:2102.05488,doi: 10.1103/PhysRevD.104.124010

    work page doi:10.1103/physrevd.104.124010 2021

  42. [43]

    Shadow images of compact objects in beyond Horndeski theory.JCAP, 05:007, 2024.arXiv:2401.15249,doi:10.1088/1475-7516/2024/05/007

    Hyat Huang, Jutta Kunz, and Deeshani Mitra. Shadow images of compact objects in beyond Horndeski theory.JCAP, 05:007, 2024.arXiv:2401.15249,doi:10.1088/1475-7516/2024/05/007

    work page doi:10.1088/1475-7516/2024/05/007 2024

  43. [44]

    Joshi, Dipanjan Dey, Pankaj S

    Ashok B. Joshi, Dipanjan Dey, Pankaj S. Joshi, and Parth Bambhaniya. Shadow of a Naked Sin- gularity without Photon Sphere.Phys. Rev. D, 102(2):024022, 2020.arXiv:2004.06525,doi: 10.1103/PhysRevD.102.024022

    work page doi:10.1103/physrevd.102.024022 2020

  44. [45]

    Dipanjan Dey, Rajibul Shaikh, and Pankaj S. Joshi. Shadow of nulllike and timelike naked sin- gularities without photon spheres.Phys. Rev. D, 103(2):024015, 2021.arXiv:2009.07487,doi: 10.1103/PhysRevD.103.024015

    work page doi:10.1103/physrevd.103.024015 2021

  45. [46]

    Gravitational lensing by Born-Infeld naked singularities.Phys

    Yiqian Chen, Peng Wang, Houwen Wu, and Haitang Yang. Gravitational lensing by Born-Infeld naked singularities.Phys. Rev. D, 109(8):084014, 2024.arXiv:2305.17411,doi:10.1103/PhysRevD.109. 084014

    work page doi:10.1103/physrevd.109 2024

  46. [47]

    Observations of orbiting hot spots around naked singularities.JCAP, 04:032, 2024.arXiv:2309.04157,doi:10.1088/1475-7516/2024/04/032

    Yiqian Chen, Peng Wang, Houwen Wu, and Haitang Yang. Observations of orbiting hot spots around naked singularities.JCAP, 04:032, 2024.arXiv:2309.04157,doi:10.1088/1475-7516/2024/04/032

    work page doi:10.1088/1475-7516/2024/04/032 2024

  47. [48]

    Observing naked singu- larities with the present and next-generation Event Horizon Telescope.Phys

    Valentin Deliyski, Galin Gyulchev, Petya Nedkova, and Stoytcho Yazadjiev. Observing naked singu- larities with the present and next-generation Event Horizon Telescope.Phys. Rev. D, 111(6):064068, 2025.arXiv:2401.14092,doi:10.1103/PhysRevD.111.064068

    work page doi:10.1103/physrevd.111.064068 2025

  48. [49]

    Eiroa, and Gaston Giribet

    Leonardo Amarilla, Ernesto F. Eiroa, and Gaston Giribet. Null geodesics and shadow of a rotating black hole in extended Chern-Simons modified gravity.Phys. Rev. D, 81:124045, 2010.arXiv:1005.0607, doi:10.1103/PhysRevD.81.124045

    work page doi:10.1103/physrevd.81.124045 2010

  49. [50]

    Black Hole Shadow as a Test of General Relativity: Quadratic Gravity.Class

    Dimitry Ayzenberg and Nicolas Yunes. Black Hole Shadow as a Test of General Relativity: Quadratic Gravity.Class. Quant. Grav., 35(23):235002, 2018.arXiv:1807.08422,doi:10.1088/1361-6382/ aae87b

    work page doi:10.1088/1361-6382/ 2018

  50. [51]

    Shadows of Kerr-like black holes in a modified gravity theory.JCAP, 03:046, 2019.arXiv:1810.12767,doi:10.1088/1475-7516/2019/03/046

    Hui-Min Wang, Yu-Meng Xu, and Shao-Wen Wei. Shadows of Kerr-like black holes in a modified gravity theory.JCAP, 03:046, 2019.arXiv:1810.12767,doi:10.1088/1475-7516/2019/03/046

    work page doi:10.1088/1475-7516/2019/03/046 2019

  51. [52]

    Liang Ma and H. Lu. Bounds on photon spheres and shadows of charged black holes in Einstein- Gauss-Bonnet-Maxwell gravity.Phys. Lett. B, 807:135535, 2020.arXiv:1912.05569,doi:10.1016/ j.physletb.2020.135535

    work page arXiv 2020

  52. [53]

    Innermost stable circular orbit and shadow of the 4DEin- 29 stein–Gauss–Bonnet black hole.Eur

    Minyong Guo and Peng-Cheng Li. Innermost stable circular orbit and shadow of the 4DEin- 29 stein–Gauss–Bonnet black hole.Eur. Phys. J. C, 80(6):588, 2020.arXiv:2003.02523,doi:10.1140/ epjc/s10052-020-8164-7

    work page arXiv 2020

  53. [54]

    Wei and Y.-X

    Shao-Wen Wei and Yu-Xiao Liu. Testing the nature of Gauss-Bonnet gravity by four-dimensional rotating black hole shadow.Eur. Phys. J. Plus, 136(4):436, 2021.arXiv:2003.07769,doi:10.1140/ epjp/s13360-021-01398-9

    work page arXiv 2021

  54. [55]

    Shadows and photon spheres with spherical accretions in the four-dimensional Gauss–Bonnet black hole.Eur

    Xiao-Xiong Zeng, Hai-Qing Zhang, and Hongbao Zhang. Shadows and photon spheres with spherical accretions in the four-dimensional Gauss–Bonnet black hole.Eur. Phys. J. C, 80(9):872, 2020.arXiv: 2004.12074,doi:10.1140/epjc/s10052-020-08449-y

    work page doi:10.1140/epjc/s10052-020-08449-y 2020

  55. [57]

    Nicolás, The bar derived category of a curved dg algebra, Journal of Pure and Applied Algebra 212 (2008) 2633–2659

    Hui-Min Wang, Zi-Chao Lin, and Shao-Wen Wei. Optical appearance of Einstein-Æther black hole surrounded by thin disk.Nucl. Phys. B, 985:116026, 2022.arXiv:2205.13174,doi:10.1016/j. nuclphysb.2022.116026

    work page doi:10.1016/j 2022

  56. [58]

    Shadow of slowly rotating Kalb-Ramond black holes.JCAP, 05:017, 2025.arXiv:2407.07416,doi:10.1088/1475-7516/2025/05/017

    Wentao Liu, Di Wu, and Jieci Wang. Shadow of slowly rotating Kalb-Ramond black holes.JCAP, 05:017, 2025.arXiv:2407.07416,doi:10.1088/1475-7516/2025/05/017

    work page doi:10.1088/1475-7516/2025/05/017 2025

  57. [59]

    Spontaneous scalarisation of charged black holes

    Carlos A.R. Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, and Jos´ e A. Font. Spontaneous Scalarization of Charged Black Holes.Phys. Rev. Lett., 121(10):101102, 2018.arXiv:1806.05190,doi:10.1103/ PhysRevLett.121.101102

    work page Pith review arXiv 2018

  58. [60]

    Pedro G. S. Fernandes, Carlos A. R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas Sanchis- Gual. Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Fea- tures.Class. Quant. Grav., 36(13):134002, 2019. [Erratum: Class.Quant.Grav. 37, 049501 (2020)]. arXiv:1902.05079,doi:10.1088/1361-6382/ab23a1

    work page doi:10.1088/1361-6382/ab23a1 2019

  59. [61]

    Fernandes, Carlos A.R

    Pedro G.S. Fernandes, Carlos A.R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas Sanchis- Gual. Charged black holes with axionic-type couplings: Classes of solutions and dynamical scalarization. Phys. Rev. D, 100(8):084045, 2019.arXiv:1908.00037,doi:10.1103/PhysRevD.100.084045

    work page doi:10.1103/physrevd.100.084045 2019

  60. [62]

    Herdeiro, Jutta Kunz, Alexandre M

    Jose Luis Bl´ azquez-Salcedo, Carlos A.R. Herdeiro, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu. Einstein-Maxwell-scalar black holes: the hot, the cold and the bald.Phys. Lett. B, 806:135493, 2020. arXiv:2002.00963,doi:10.1016/j.physletb.2020.135493

    work page doi:10.1016/j.physletb.2020.135493 2020

  61. [63]

    Scalarized charged black holes with scalar mass term.Phys

    De-Cheng Zou and Yun Soo Myung. Scalarized charged black holes with scalar mass term.Phys. Rev. D, 100(12):124055, 2019.arXiv:1909.11859,doi:10.1103/PhysRevD.100.124055

    work page doi:10.1103/physrevd.100.124055 2019

  62. [64]

    Fernandes

    Pedro G.S. Fernandes. Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair.Phys. Dark Univ., 30:100716, 2020.arXiv:2003.01045,doi:10.1016/j.dark.2020.100716

    work page doi:10.1016/j.dark.2020.100716 2020

  63. [65]

    Scalarization of horizonless reflecting stars: neutral scalar fields non-minimally coupled to Maxwell fields.Phys

    Yan Peng. Scalarization of horizonless reflecting stars: neutral scalar fields non-minimally coupled to Maxwell fields.Phys. Lett. B, 804:135372, 2020.arXiv:1912.11989,doi:10.1016/j.physletb.2020. 135372

    work page doi:10.1016/j.physletb.2020 2020

  64. [66]

    DOI 10.1140/epjc/ s10052-022-10942-5

    Yun Soo Myung and De-Cheng Zou. Instability of Reissner–Nordstr¨ om black hole in Einstein- 30 Maxwell-scalar theory.Eur. Phys. J. C, 79(3):273, 2019.arXiv:1808.02609,doi:10.1140/epjc/ s10052-019-6792-6

    work page doi:10.1140/epjc/ 2019

  65. [67]

    Quasinormal modes of scalarized black holes in the Ein- stein–Maxwell–Scalar theory.Phys

    Yun Soo Myung and De-Cheng Zou. Quasinormal modes of scalarized black holes in the Ein- stein–Maxwell–Scalar theory.Phys. Lett. B, 790:400–407, 2019.arXiv:1812.03604,doi:10.1016/ j.physletb.2019.01.046

    work page arXiv 2019

  66. [68]

    Stability of scalarized charged black holes in the Ein- stein–Maxwell–Scalar theory.Eur

    Yun Soo Myung and De-Cheng Zou. Stability of scalarized charged black holes in the Ein- stein–Maxwell–Scalar theory.Eur. Phys. J. C, 79(8):641, 2019.arXiv:1904.09864,doi:10.1140/ epjc/s10052-019-7176-7

    work page arXiv 2019

  67. [69]

    Radial perturbations of the scalarized black holes in Einstein- Maxwell-conformally coupled scalar theory.Phys

    De-Cheng Zou and Yun Soo Myung. Radial perturbations of the scalarized black holes in Einstein- Maxwell-conformally coupled scalar theory.Phys. Rev. D, 102(6):064011, 2020.arXiv:2005.06677, doi:10.1103/PhysRevD.102.064011

    work page doi:10.1103/physrevd.102.064011 2020

  68. [70]

    Jose Luis Bl´ azquez-Salcedo, Carlos A. R. Herdeiro, Sarah Kahlen, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu. Quasinormal modes of hot, cold and bald Einstein–Maxwell-scalar black holes.Eur. Phys. J. C, 81(2):155, 2021.arXiv:2008.11744,doi:10.1140/epjc/s10052-021-08952-w

    work page doi:10.1140/epjc/s10052-021-08952-w 2021

  69. [71]

    Onset of rotating scalarized black holes in Einstein-Chern-Simons- Scalar theory.Phys

    Yun Soo Myung and De-Cheng Zou. Onset of rotating scalarized black holes in Einstein-Chern-Simons- Scalar theory.Phys. Lett. B, 814:136081, 2021.arXiv:2012.02375,doi:10.1016/j.physletb.2021. 136081

    work page doi:10.1016/j.physletb.2021 2021

  70. [72]

    Relativistic continuous matrix product states for quantum fields without cutoff.Phys

    Zhan-Feng Mai and Run-Qiu Yang. Stability analysis of a charged black hole with a nonlinear complex scalar field.Phys. Rev. D, 104(4):044008, 2021.arXiv:2101.00026,doi:10.1103/PhysRevD.104. 044008

    work page doi:10.1103/physrevd.104 2021

  71. [73]

    Higher dimensional black hole scalarization.JHEP, 09:186, 2020.arXiv:2007.04153,doi:10.1007/JHEP09(2020)186

    Dumitru Astefanesei, Carlos Herdeiro, Jo˜ ao Oliveira, and Eugen Radu. Higher dimensional black hole scalarization.JHEP, 09:186, 2020.arXiv:2007.04153,doi:10.1007/JHEP09(2020)186

    work page doi:10.1007/jhep09(2020)186 2020

  72. [74]

    Critical Phenomena in Dynamical Scalarization of Charged Black Holes.Phys

    Cheng-Yong Zhang, Qian Chen, Yunqi Liu, Wen-Kun Luo, Yu Tian, and Bin Wang. Critical Phenomena in Dynamical Scalarization of Charged Black Holes.Phys. Rev. Lett., 128(16):161105, 2022.arXiv: 2112.07455,doi:10.1103/PhysRevLett.128.161105

    work page doi:10.1103/physrevlett.128.161105 2022

  73. [75]

    Dynamical transi- tions in scalarization and descalarization through black hole accretion.Phys

    Cheng-Yong Zhang, Qian Chen, Yunqi Liu, Wen-Kun Luo, Yu Tian, and Bin Wang. Dynamical transi- tions in scalarization and descalarization through black hole accretion.Phys. Rev. D, 106(6):L061501, 2022.arXiv:2204.09260,doi:10.1103/PhysRevD.106.L061501

    work page doi:10.1103/physrevd.106.l061501 2022

  74. [76]

    Type I critical dynamical scalarization and descalarization in Einstein-Maxwell-scalar theory.Sci

    Jia-Yan Jiang, Qian Chen, Yunqi Liu, Yu Tian, Wei Xiong, Cheng-Yong Zhang, and Bin Wang. Type I critical dynamical scalarization and descalarization in Einstein-Maxwell-scalar theory.Sci. China Phys. Mech. Astron., 67(2):220411, 2024.arXiv:2306.10371,doi:10.1007/s11433-023-2231-5

    work page doi:10.1007/s11433-023-2231-5 2024

  75. [77]

    Critical phenomenon inside asymptotically flat black holes with spontaneous scalarization

    Li Li, Ze Sun, and Fu-Guo Yang. Critical phenomenon inside asymptotically flat black holes with spontaneous scalarization. 12 2025.arXiv:2512.19377

    work page arXiv 2025

  76. [78]

    Scalarized Kerr-Newman black holes

    Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Kerr-Newman black holes. JHEP, 10:076, 2023.arXiv:2307.12210,doi:10.1007/JHEP10(2023)076

    work page doi:10.1007/jhep10(2023)076 2023

  77. [79]

    Photon spheres and spherical accretion image of a hairy black hole.Phys

    Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon spheres and spherical accretion image of a hairy black hole.Phys. Rev. D, 104(2):024003, 2021.arXiv:2104.08703,doi:10.1103/ 31 PhysRevD.104.024003

    work page arXiv 2021

  78. [80]

    Tinto and J

    Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon ring and observational appearance of a hairy black hole.Phys. Rev. D, 104(4):044049, 2021.arXiv:2105.11770,doi:10.1103/PhysRevD. 104.044049

    work page doi:10.1103/physrevd 2021

  79. [81]

    Navas et al

    Yiqian Chen, Peng Wang, and Haitang Yang. Interferometric signatures of black holes with multiple photon spheres.Phys. Rev. D, 110(4):044020, 2024.arXiv:2312.10304,doi:10.1103/PhysRevD.110. 044020

    work page doi:10.1103/physrevd.110 2024

  80. [82]

    Gravitational lensing by black holes with multiple photon spheres.Phys

    Guangzhou Guo, Xin Jiang, Peng Wang, and Houwen Wu. Gravitational lensing by black holes with multiple photon spheres.Phys. Rev. D, 105(12):124064, 2022.arXiv:2204.13948,doi:10.1103/ PhysRevD.105.124064

    work page arXiv 2022

Showing first 80 references.