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REVIEW 3 major objections 6 minor 69 references

Shear viscosity leaves only a few microjansky imprints on black hole shadow images from stationary magnetized tori.

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2026-08-01 06:53 UTC pith:ZDACD73O

load-bearing objection First quantitative GRRT study of viscosity imprints on Schwarzschild torus shadows; the few-μJy claim is conditional on an 80% pressure perturbation that likely exceeds the linear regime. the 3 major comments →

arxiv 2607.21713 v1 pith:ZDACD73O submitted 2026-07-23 gr-qc astro-ph.HE

Viscosity effects on the shadow of a non-rotating black hole

Sayantani Lahiri , Sergio Gimeno-Soler , Alejandro Cruz-Osorio , Jos\'e A. Font , Ziri Younsi This is my paper
classification gr-qc astro-ph.HE
keywords black hole shadowshear viscosityrelativistic hydrodynamicsradiative transfersynchrotron emissionaccretion toriSchwarzschild black holemagnetized tori
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that shear viscosity in a magnetized accretion torus affects the 230 GHz black hole shadow image only weakly: the overall silhouette is unchanged and brightness differences stay at the level of a few microjansky. This matters because, if true, interpreting Event Horizon Telescope-style images of stationary disks does not require modeling viscous transport to first order; the shadow shape is robust to dissipative fluid effects. The paper also claims the largest viscosity signatures appear in strongly magnetized tori, at high viewing inclinations, and with larger values of the electron-temperature parameter R_high.

Core claim

The paper finds that introducing shear viscosity and spacetime-curvature terms as linear perturbations in stationary Komissarov-type tori changes the computed shadow images mainly in local pixel flux, not in shadow morphology. The curvature-coupled transport coefficient m2 suppresses synchrotron emission more broadly than the shear coefficient m1, which produces a more balanced redistribution of flux. Quantitatively, pixel-wise flux differences remain at a few microjansky, and two normalized image-comparison metrics show the strongest viscous-versus-inviscid differences for the most magnetized torus considered (beta_c = 10) and at high inclination angles.

What carries the argument

The central mechanism is a second-order causal gradient expansion of the shear viscosity tensor, with transport coefficients m1 (shear viscosity) and m2 (spacetime-curvature coupling), applied as a first-order perturbation to a stationary magnetized torus in Schwarzschild spacetime. The electron temperature is inferred from the perturbed pressure through an R–beta prescription, and thermal synchrotron radiative transfer produces the synthetic images. The coefficients are normalized by requiring the pressure correction to reach 80% of the zeroth-order pressure at the cusp, and the causality-preserving relaxation coefficient tau_2 is set to 1.

Load-bearing premise

The whole calculation assumes the viscous correction to the pressure can be treated as a small perturbation even though it is normalized to 80% of the zeroth-order pressure at the cusp; if the real plasma viscosity is much smaller, the predicted image differences would be even tinier and their spatial pattern could change.

What would settle it

Recompute the synthetic images with the transport coefficients m1 and m2 reduced by an order of magnitude (so the pressure correction at the cusp is about 8% rather than 80%) and check whether the pixel-wise flux differences scale linearly down to sub-microjansky levels or change sign; alternatively, adopt shear-viscosity values from a microphysical plasma model and redo the normalization.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If viscosity leaves only microjansky-level differences, shadow size and photon-ring morphology in stationary disk models at 230 GHz are effectively independent of shear viscosity.
  • Detecting viscosity from such images would require extremely high dynamic range; strongly magnetized, edge-on tori are the most promising targets.
  • The curvature-coupling term m2 dominates the thermodynamic impact, so identifying which sign of flux difference appears could help disentangle viscous transport channels.
  • Because the analysis is limited to stationary tori, fully dynamical accretion flows may exhibit larger viscosity imprints, as the authors themselves note.

Where Pith is reading between the lines

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

  • If the normalization were lowered to a microphysically motivated viscosity (for instance, from plasma kinetic theory), the predicted few-microjansky differences would likely shrink further, making viscosity even harder to observe.
  • The 80% pressure-correction normalization places the 'first-order' perturbation near or beyond its regime of validity; a fully nonlinear viscous solution could produce qualitatively different local emission features.
  • A direct consequence for parameter-estimation pipelines is that fixing viscous transport coefficients to zero for stationary torus images is a safe approximation at current sensitivity.
  • The same perturbative framework could be applied to other dissipative effects, such as heat conduction, to test whether they also leave only small imprints on shadows.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies how shear viscosity, implemented through first- and second-order transport coefficients in a causal relativistic hydrodynamics framework, affects synthetic 230 GHz images of a Schwarzschild black hole shadow illuminated by stationary magnetized tori. The fluid configurations are built by linear perturbation of Komissarov-type tori, with the shear-viscosity and spacetime-curvature coefficients m1 and m2 calibrated so that the first-order pressure correction reaches 80% of the zeroth-order pressure at the cusp. The authors compute GRRT images with thermal synchrotron emission and a plasma-β-dependent electron-proton temperature ratio, varying the magnetization parameter β_c, inclination angle, R_high, and the viscous coefficients. They compare viscous and inviscid images using pixel-wise flux differences and two metrics (1−NCC and MSE), finding that viscosity and curvature leave only modest imprints, with differences at the few-μJy level, largest for strongly magnetized tori and at higher inclinations and R_high. The paper explicitly notes the limitation to stationary tori and suggests dynamical systems may show larger effects.

Significance. If the result holds, it provides a useful negative result for the interpretation of EHT images: for stationary, non-rotating thick-disk models, viscous transport does not significantly alter the shadow morphology. The work combines a somewhat novel second-order viscous torus model with a standard GRRT pipeline (BHOSS) and performs a systematic parameter study, which is a strength. The image differences are genuine outputs of the radiative-transfer calculation and are not used to fit the viscosity parameters, so the central quantitative statement is not circular. However, the paper's main claim is conditional on an arbitrary normalization (|p^(1)|=0.8 p^(0)) that calls the perturbative treatment into question, and the cross-β_c comparisons are confounded by renormalizing m1 and m2 for each magnetization. These issues must be addressed before the 'modest imprints / few μJy' conclusion can be considered robust.

major comments (3)
  1. [§II.B, Table I, §IV] The transport coefficients m1 and m2 are fixed by requiring |p^(1)|=0.8 p^(0) at the cusp. This is not a small perturbation: the first-order correction is comparable to the background, and the expected size of neglected second-order terms is O((0.8)^2)≈0.64 of p^(1). The statement in §IV that the 'perturbative treatment remains valid' is therefore unsupported. Since m1 and m2 are calibrated to this large amplitude, the 'few μJy' conclusion is set by an arbitrary normalization rather than by a physical viscosity. Please provide a convergence check (e.g., repeat with |p^(1)|/p^(0)=0.1, 0.4, 0.8 and verify linear scaling of ΔS) and, if possible, a microphysical estimate or range for η.
  2. [§II.B and §IV.C] The cross-β_c comparison is confounded: for each β_c the coefficients m1 and m2 are renormalized to keep |p^(1)|=0.8 p^(0) at the cusp, and Table I shows m1 and m2 grow with magnetization (e.g., m1=0.0646 at β_c=10^3 vs 0.115 at β_c=1). Thus the stronger ΔS for strongly magnetized tori in Figs. 5 and 7 does not isolate the magnetization effect; it also reflects a larger viscosity coefficient. A fixed-(m1,m2) comparison, or an explicit demonstration that ΔS scales linearly with m_i, is needed before claiming that magnetization amplifies the viscosity imprint.
  3. [§II.A, Eq. (3)] The choice τ2=1 is stated 'without loss of generality,' but τ2 multiplies the second-order term D(2ησ^μν)> in Eq. (3). With η=λ m1 this term contributes at the same order as the leading viscous term, and its relative weight depends on τ2 and on the local gradients. Since m1 is fitted to the 80% pressure condition, changing τ2 changes the partition of the viscosity tensor and hence the image differences. Please demonstrate the rescaling freedom explicitly or treat τ2 as a parameter and quote the sensitivity of ΔS to it.
minor comments (6)
  1. [§IV.D] Typographical errors: 'inidividual' should be 'individual', and 'colmuns' should be 'columns'.
  2. [§III] After Eq. (10), 'obervers' should be 'observers'.
  3. [§II.A, Eq. (8)] Equation (8) appears malformed: '⃗ α(r, θ)⃗∇(r,θ)p(1) − c(r, θ) = 0' — the vector notation and arguments are unclear. Please rewrite the operator and coefficient explicitly.
  4. [Fig. 7] The left column is labeled β_c=10^1 in the figure while the text says β_c=10; please make the notation consistent.
  5. [Reference [54]] Reference [54] is incomplete (journal, volume, and pages only, no authors or title); please supply the full citation.
  6. [§V] Minor grammar: 'Both metric show' should be 'Both metrics show.' Also consider stating whether the few-μJy differences are above or below typical EHT calibration uncertainties, to frame the observational relevance.

Circularity Check

0 steps flagged

No significant circularity: the image differences are genuine forward-model outputs, not inputs renamed as predictions.

full rationale

The central chain is: (i) take stationary magnetized torus models from the authors' previous work [51], with transport coefficients m1 and m2 fixed by the arbitrary normalization |p^(1)| = 0.8 p^(0) at the cusp (Section II.B, Table I); (ii) map the resulting plasma state to electron temperature via the R-beta prescription; (iii) solve GRRT with BHOS to obtain synthetic 230 GHz images; (iv) compare viscous and non-viscous images pixel-by-pixel. The image differences Delta-S are not equal by construction to the fitted pressure correction or to m1/m2: they depend nonlinearly on synchrotron emissivity, optical depth, electron temperature, geometry, and the radiative-transfer solution. Equation (13), Theta_e proportional to gamma(1+Delta p)/(gamma+Delta p), is an algebraic rewriting of the temperature prescription, not a hidden identification of the image with the perturbation. The accretion-rate normalization to reproduce the M87* total flux only sets the overall brightness scale; the claimed few-muJy differences are computed differences between two self-consistently normalized images. The self-citations [51] and [55] supply the background viscous-torus construction and the second-order transport expression, but the new imaging calculation is an independent forward computation and no uniqueness theorem or ansatz is smuggled in via self-citation. The 80% pressure-correction calibration is a legitimate free-parameter and perturbative-validity concern -- |p^(1)| is not small compared to p^(0), so the first-order truncation may be unreliable -- but that is a robustness/correctness issue, not a circular reduction. No step of the derivation reduces to its own input by definition, so the circularity score is 0.

Axiom & Free-Parameter Ledger

6 free parameters · 8 axioms · 1 invented entities

The central imaging claim rests on the torus model from the authors' prior work [51], which itself relies on several physical assumptions about the plasma (toroidal field, cold plasma, polytropic EOS, R-β temperature law). The most consequential free parameters are m1, m2, and τ2 that set the viscosity amplitude: they are chosen to produce an 80% pressure correction at the cusp, not derived from microphysics. No genuinely new entities are introduced beyond the curvature coupling already present in [55].

free parameters (6)
  • m1 (shear viscosity transport coefficient) = Table I values (e.g., 6.46e-2, 7.07e-2, 1.15e-1 for m2=0 models)
    Set by imposing |p^(1)| = 0.8 p^(0) at the cusp, not derived from microphysics.
  • m2 (spacetime-curvature transport coefficient) = Table I values (e.g., 1.48e-2, 1.61e-2, 2.62e-2 for m1=0 models)
    Set by the same 80% pressure-correction normalization.
  • tau_2 (second-order relaxation coefficient) = 1
    Set to 1 'without loss of generality' (Sec. II.A); the value affects the magnitude of the causal term and is not observationally constrained.
  • l0 (constant specific angular momentum) = 3.8
    Fixed model parameter from [51]; not varied.
  • Ws (surface potential) = -0.04
    Fixed model parameter from [51]; yields a disk slightly overflowing its Roche lobe.
  • Rlow and Rhigh (electron-proton temperature ratio parameters) = Rlow=1; Rhigh varied (1,10,20)
    Parameters of the R-β temperature prescription; Rlow is fixed, Rhigh scanned.
axioms (8)
  • domain assumption Schwarzschild spacetime with c=G=1
    All GRRT and torus calculations use this metric; no spin.
  • domain assumption Gradient expansion for the viscous stress tensor, Eq. (3), with coefficients η, τ2, κ2
    Taken from second-order causal (Müller-Israel-Stewart) hydrodynamics; the κ2 curvature term is from [55].
  • domain assumption Purely toroidal magnetic field, b^μ=(bt,0,0,bϕ)
    Assumed in the torus construction; no poloidal component.
  • ad hoc to paper Cold plasma approximation e≈ρ
    Stated in Sec. II.A; reduces the equation of state and affects temperature.
  • domain assumption Polytropic equations of state p=K e^γ and p_m=K_m L^{γ-1} e^γ with γ=5/3
    Adopted for gas and magnetic pressure, following [44,45,51].
  • domain assumption R-β electron temperature prescription, Eq. (12)
    Empirical EHT-motivated relation between proton and electron temperatures; introduces Rlow,Rhigh.
  • domain assumption Thermal Maxwell-Jüttner electron distribution and thermal synchrotron emission
    Emission/absorption coefficients from [59]; neglects non-thermal electrons.
  • ad hoc to paper Linear perturbation w=w0+w^(1) for viscous effects
    Shear viscosity is treated as a small perturbation, despite the 80% normalization chosen later.
invented entities (1)
  • Spacetime-curvature transport term κ2 u^α u^β R^{α<μν>β} no independent evidence
    purpose: Introduces dependence of the viscous stress tensor on spacetime curvature in equilibrium tori
    Term from [55] (same research group); no independent observational or experimental handle; its coefficient m2 is normalized arbitrarily via the 80% condition.

pith-pipeline@v1.3.0-alltime-deepseek · 16922 in / 14301 out tokens · 127378 ms · 2026-08-01T06:53:28.142861+00:00 · methodology

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read the original abstract

We study the effect of shear viscosity in stationary magnetized accretion tori on synthetic images of non-rotating black hole shadows. Shear viscosity and spacetime-curvature contributions are introduced perturbatively in the tori through first and second-order transport coefficients within a second-order causal theory of non-ideal relativistic hydrodynamics. Synthetic black hole shadow images at 230\,GHz are obtained via general relativistic radiative transfer computations assuming thermal synchrotron emission and for a wide range of plasma magnetization parameters, viewing inclination angles, electron-temperature prescriptions, and viscosity parameters. A comparative pixel-by-pixel analysis using two normalized metrics shows that the largest image differences occur for strongly magnetized tori. While shear viscosity induces only minor changes in the overall shadow morphology, its combined effects with spacetime curvature are more evident in localized modifications of the synchrotron emission and pixel-wise flux distribution. These effects become increasingly pronounced at higher inclination angles, with the largest brightness differences between viscous and non-viscous configurations occurring for larger values of the electron-temperature parameter. Overall, our results indicate that shear viscosity and spacetime curvature leave only modest imprints on black hole shadow images produced by stationary thick disks, with differences remaining at the level of a few $\mu$Jy. Since our analysis is limited to stationary tori, the effects of shear viscosity might however be more significant in fully dynamical accretion systems.

Figures

Figures reproduced from arXiv: 2607.21713 by Alejandro Cruz-Osorio, Jos\'e A. Font, Sayantani Lahiri, Sergio Gimeno-Soler, Ziri Younsi.

Figure 1
Figure 1. Figure 1: FIG. 1. Pressure, in logarithmic scale, as a function of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Distribution of the absolute value of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Black hole shadows at 230 GHz for three inclination [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: illustrates the shadow images when varying the plasma magnetization parameter βc and the shear viscos￾ity with non-vanishing mi . In this case we fix Rhigh = 1 and the inclination angle to ι = 45. The shadow im￾ages in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pixel-to-pixel image comparison of the inclination viewing angle [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

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Reference graph

Works this paper leans on

69 extracted references · 46 linked inside Pith

  1. [1]

    Going Merry

    The third row shows the differences between the viscous and non-viscous solutions. proaching side towards the black hole. This behavior in- dicates that shear viscosity and curvature-induced mod- ifications of the synchrotron emission become more visi- ble in near edge-on viewing geometries. This implies that with increasing viewing angles, the synchrotro...

    2023

  2. [2]

    Akiyama, et al., Astrophys

    Event Horizon Telescope Collaboration, K. Akiyama, et al., Astrophys. J. Lett.875, L1 (2019)

    2019

  3. [3]

    Akiyama, et al., Astrophys

    Event Horizon Telescope Collaboration, K. Akiyama, et al., Astrophys. J. Lett.875, L5 (2019)

    2019

  4. [4]

    Akiyama, et al., Astron

    Event Horizon Telescope Collaboration, K. Akiyama, et al., Astron. Astrophys.693, A265 (2025)

    2025

  5. [5]

    Akiyama, et al., Astrophys

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

    2022

  6. [6]

    Akiyama, et al., Astrophys

    Event Horizon Telescope Collaboration, K. Akiyama, et al., Astrophys. J. Lett.930, L16 (2022)

    2022

  7. [7]

    J. P. Luminet, Astronomy and Astrophysics75, 228 (1979)

    1979

  8. [8]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Phys. Rev. D 100, 024018 (2019), arXiv:1906.00873 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  9. [9]

    Abdujabbarov, M

    A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, Phys. Rev. D93, 104004 (2016), arXiv:1604.03809 [gr-qc]

    Pith/arXiv arXiv 2016

  10. [10]

    Falcke, F

    H. Falcke, F. Melia, and E. Agol, Astrophys. J. Lett. 528, L13 (2000), arXiv:astro-ph/9912263

    Pith/arXiv arXiv 2000

  11. [11]

    Narayan and J

    R. Narayan and J. E. McClintock, (2013), arXiv:1312.6698 [astro-ph.HE]

    Pith/arXiv arXiv 2013

  12. [12]

    Grenzebach, V

    A. Grenzebach, V. Perlick, and C. L¨ ammerzahl, Phys. Rev. D89, 124004 (2014), arXiv:1403.5234 [gr-qc]

    Pith/arXiv arXiv 2014

  13. [13]

    Vagnozzi and L

    S. Vagnozzi and L. Visinelli, Phys. Rev. D100, 024020 (2019), arXiv:1905.12421 [gr-qc]

    Pith/arXiv arXiv 2019

  14. [14]

    Amarilla and E

    L. Amarilla and E. F. Eiroa, Phys. Rev. D85, 064019 (2012), arXiv:1112.6349 [gr-qc]

    Pith/arXiv arXiv 2012

  15. [15]

    Perlick, O

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D92, 104031 (2015), arXiv:1507.04217 [gr- qc]

    Pith/arXiv arXiv 2015

  16. [16]

    Gyulchev, P

    G. Gyulchev, P. Nedkova, T. Vetsov, and S. Yazad- jiev, European Physical Journal C81, 885 (2021), arXiv:2106.14697 [gr-qc]

    Pith/arXiv arXiv 2021

  17. [17]

    R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C80, 1049 (2020), arXiv:2003.01188 [gr-qc]

    Pith/arXiv arXiv 2020

  18. [18]

    P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. Lett.123, 011101 (2019), arXiv:1904.09997 [gr-qc]

    Pith/arXiv arXiv 2019

  19. [19]

    P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz, and E. Radu, Phys. Lett. B768, 373 (2017), arXiv:1701.00079 [gr-qc]

    Pith/arXiv arXiv 2017

  20. [20]

    R. A. Konoplya, Phys. Lett. B795, 1 (2019), arXiv:1905.00064 [gr-qc]

    Pith/arXiv arXiv 2019

  21. [21]

    Gyulchev, P

    G. Gyulchev, P. Nedkova, T. Vetsov, and S. Yazadjiev, Phys. Rev. D100, 024055 (2019), arXiv:1905.05273 [gr- qc]

    Pith/arXiv arXiv 2019

  22. [22]

    Deliyski, G

    V. Deliyski, G. Gyulchev, P. Nedkova, and S. Yazadjiev, Phys. Rev. D111, 064068 (2025), arXiv:2401.14092 [gr- qc]

    Pith/arXiv arXiv 2025

  23. [23]

    Deliyski, G

    V. Deliyski, G. Gyulchev, P. Nedkova, and S. Yazadjiev, Phys. Rev. D106, 104024 (2022), arXiv:2206.09455 [gr- qc]

    Pith/arXiv arXiv 2022

  24. [24]

    Cruz-Osorio, S

    A. Cruz-Osorio, S. Gimeno-Soler, J. A. Font, M. De Lau- rentis, and S. Mendoza, Phys. Rev. D103, 124009 (2021), arXiv:2102.10150 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  25. [25]

    Olivares, Z

    H. Olivares, Z. Younsi, C. M. Fromm, M. De Lauren- tis, O. Porth, Y. Mizuno, H. Falcke, M. Kramer, and L. Rezzolla, MNRAS497, 521 (2020), arXiv:1809.08682 [gr-qc]

    Pith/arXiv arXiv 2020

  26. [27]

    Moriyama, A

    K. Moriyama, A. Cruz-Osorio, Y. Mizuno, I. K. Dihingia, and A. Uniyal, Astron. Astrophys.694, A135 (2025), arXiv:2501.08720 [astro-ph.HE]

    Pith/arXiv arXiv 2025

  27. [28]

    X. Wang, Z. Zhao, X.-X. Zeng, and X.-Y. Wang, Phys. Rev. D111, 084054 (2025), arXiv:2501.08287 [gr-qc]

    Pith/arXiv arXiv 2025

  28. [29]

    Younsi, A

    Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya, and Y. Mizuno, Phys. Rev. D94, 084025 (2016), arXiv:1607.05767 [gr-qc]

    Pith/arXiv arXiv 2016

  29. [30]

    Younsi, D

    Z. Younsi, D. Psaltis, and F. ¨Ozel, Astrophys. J.942, 47 (2023), arXiv:2111.01752 [astro-ph.HE]

    Pith/arXiv arXiv 2023

  30. [31]

    Chatterjee, Z

    K. Chatterjee, Z. Younsi, P. Kocherlakota, and R. Narayan, The Astrophysical Journal Letters991, L58 (2025), arXiv:2310.20043 [gr-qc]

    arXiv 2025

  31. [32]

    Kocherlakotaet al.(Event Horizon Telescope), Phys

    P. Kocherlakotaet al.(Event Horizon Telescope), Phys. Rev. D103, 104047 (2021), arXiv:2105.09343 [gr-qc]

    Pith/arXiv arXiv 2021

  32. [33]

    Chatterjee, P

    K. Chatterjee, P. Kocherlakota, Z. Younsi, and R. Narayan, arXiv e-prints , arXiv:2310.20040 (2023), arXiv:2310.20040 [gr-qc]

    Pith/arXiv arXiv 2023

  33. [34]

    Johannsen and D

    T. Johannsen and D. Psaltis, Astrophys. J.718, 446 (2010), arXiv:1005.1931 [astro-ph.HE]

    Pith/arXiv arXiv 2010

  34. [35]

    Akiyama, A

    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., Astrophys. J. Lett.875, L6 (2019)

    2019

  35. [36]

    Akiyama, et al., Astrophys

    Event Horizon Telescope Collaboration, K. Akiyama, et al., Astrophys. J. Lett.930, L17 (2022)

    2022

  36. [37]

    Frank, A

    J. Frank, A. King, and D. J. Raine,Accretion Power in Astrophysics: Third Edition(2002)

    2002

  37. [38]

    L. G. Fishbone and V. Moncrief, Astrophys. J.207, 962- 976 (1976)

    1976

  38. [39]

    Kozlowski, M

    M. Kozlowski, M. Jaroszynski, and M. A. Abramowicz, Astronomy and Astrophysics63, 209 (1978)

    1978

  39. [40]

    Abramowicz, M

    M. Abramowicz, M. Jaroszynski, and M. Sikora, Astron- omy and Astrophysics63, 221 (1978)

    1978

  40. [41]

    Uniyal, I

    A. Uniyal, I. K. Dihingia, and Y. Mizuno, Astrophys. J. 970, 172 (2024), arXiv:2406.16309 [astro-ph.HE]

    Pith/arXiv arXiv 2024

  41. [42]

    J. A. Font and F. Daigne, Mon. Not. R. Astron. Soc.334, 383-400 (2002), arXiv:astro-ph/0203403 [astro-ph]

    Pith/arXiv arXiv 2002

  42. [43]

    Daigne and J

    F. Daigne and J. A. Font, Mon. Not. R. Astron. Soc.349, 841-868 (2004), arXiv:astro-ph/0311618 [astro-ph]

    Pith/arXiv arXiv 2004

  43. [44]

    M. A. Abramowicz and P. C. Fragile, Living Rev. Rel. 16, 1 (2013), arXiv:1104.5499 [astro-ph.HE]

    Pith/arXiv arXiv 2013

  44. [45]

    S. S. Komissarov, Monthly Notices of the Royal Astro- nomical Society368, 993 (2006), arXiv:astro-ph/0601678 [astro-ph]

    Pith/arXiv arXiv 2006

  45. [46]

    Gimeno-Soler and J

    S. Gimeno-Soler and J. A. Font, Astronomy and Astro- physics607, A68 (2017)

    2017

  46. [47]

    O. M. Pimentel, F. D. Lora-Clavijo, and G. A. Gonzalez, Astron. Astrophys.619, A57 (2018), arXiv:1808.07400 [astro-ph.HE]

    Pith/arXiv arXiv 2018

  47. [48]

    Gimeno-Soler, O

    S. Gimeno-Soler, O. M. Pimentel, F. D. Lora-Clavijo, A. Cruz-Osorio, and J. A. Font, Phys. Rev. D110, 023023 (2024), arXiv:2303.15867 [astro-ph.HE]

    Pith/arXiv arXiv 2024

  48. [49]

    Liska, A

    M. Liska, A. Tchekhovskoy, and E. Quataert, Monthly Notices of the Royal Astronomical Society494, 3656 (2020), arXiv:1809.04608 [astro-ph.HE]

    Pith/arXiv arXiv 2020

  49. [50]

    O. M. Pimentel Diaz, P. C. Fragile, F. D. Lora- Clavijo, B. Ierace, and D. Bollimpalli, Monthly No- tices of the Royal Astronomical Society505, 4278 (2021), arXiv:2105.11329 [astro-ph.HE]. 12

    Pith/arXiv arXiv 2021

  50. [51]

    Cruz-Osorio, S

    A. Cruz-Osorio, S. Gimeno-Soler, and J. A. Font, Mon. Not. R. Astron. Soc.492, 5730 (2020), arXiv:2001.09669 [gr-qc]

    Pith/arXiv arXiv 2020

  51. [52]

    Lahiri, S

    S. Lahiri, S. Gimeno-Soler, J. A. Font, and A. M. Mej ´ ıas, Phys. Rev. D103, 044034 (2021), arXiv:2012.06835 [gr- qc]

    Pith/arXiv arXiv 2021

  52. [53]

    Muller, Z

    I. Muller, Z. Phys.198, 329 (1967)

    1967

  53. [54]

    Israel, Annals Phys.100, 310 (1976)

    W. Israel, Annals Phys.100, 310 (1976)

    1976

  54. [55]

    Physica A: Statistical Mechanics and its Applications 106, 204 (1981)

    1981

  55. [56]

    Lahiri, Classical and Quantum Gravity37, 075010 (2020)

    S. Lahiri, Classical and Quantum Gravity37, 075010 (2020)

    2020

  56. [57]

    A. M. Anile,Relativistic fluids and magneto-fluids: With applications in astrophysics and plasma physics(Cam- bridge University Press, 2005)

    2005

  57. [58]

    Younsi, O

    Z. Younsi, O. Porth, Y. Mizuno, C. M. Fromm, and H. Olivares, IAU Symp.342, 9 (2020), arXiv:1907.09196 [astro-ph.HE]

    Pith/arXiv arXiv 2020

  58. [59]

    Younsi, K

    Z. Younsi, K. Wu, and S. V. Fuerst, Astron. Astrophys. 545, A13 (2012), arXiv:1207.4234 [astro-ph.HE]

    Pith/arXiv arXiv 2012

  59. [60]

    Pandya, Z

    A. Pandya, Z. Zhang, M. Chandra, and C. F. Gammie, Astrophys. J.822, 34 (2016), arXiv:1602.08749 [astro- ph.HE]

    Pith/arXiv arXiv 2016

  60. [61]

    Mo´ scibrodzka, H

    M. Mo´ scibrodzka, H. Falcke, and H. Shiokawa, Astron. Astrophys.586, A38 (2016), arXiv:1510.07243 [astro- ph.HE]

    Pith/arXiv arXiv 2016

  61. [62]

    C. M. Fromm, A. Cruz-Osorio, Y. Mizuno, A. Nathanail, Z. Younsi, O. Porth, H. Olivares, J. Davelaar, H. Falcke, M. Kramer, and L. Rezzolla, Astron. Astrophys.660, A107 (2022), arXiv:2111.02518 [astro-ph.HE]

    Pith/arXiv arXiv 2022

  62. [63]

    Cruz-Osorio, C

    A. Cruz-Osorio, C. M. Fromm, Y. Mizuno, A. Nathanail, Z. Younsi, O. Porth, J. Davelaar, H. Falcke, M. Kramer, and L. Rezzolla, Nature Astronomy6, 103 (2022), arXiv:2111.02517 [astro-ph.HE]

    Pith/arXiv arXiv 2022

  63. [64]

    Cruz-Osorio, C

    A. Cruz-Osorio, C. Meringolo, C. M. Fromm, Y. Mizuno, S. Servidio, A. Nathanail, Z. Younsi, and L. Rezzolla, Astrophys. J.1001, 227 (2026), arXiv:2512.14835 [astro- ph.HE]

    arXiv 2026

  64. [65]

    Akiyama, R.-S

    K. Akiyama, R.-S. Lu, V. L. Fish, S. S. Doeleman, A. E. Broderick, J. Dexter, K. Hada, M. Kino, H. Na- gai, M. Honma, M. D. John son, J. C. Algaba, K. Asada, C. Brinkerink, R. Blundell, and et al., Astrophys. J.807, 150 (2015), arXiv:1505.03545 [astro-ph.HE]

    Pith/arXiv arXiv 2015

  65. [66]

    S. S. Doeleman, V. L. Fish, D. E. Schenck, C. Beaudoin, R. Blundell, G. C. Bower, A. E. Broderick, R. Cham- berlin, R. Freund, P. F riberg, M. A. Gurwell, P. T. P. Ho, M. Honma, M. Inoue, T. P. Krichbaum, and et al., Science338, 355 (2012), arXiv:1210.6132 [astro-ph.HE]

    Pith/arXiv arXiv 2012

  66. [67]

    R¨ oder, A

    J. R¨ oder, A. Cruz-Osorio, C. M. Fromm, Y. Mizuno, Z. Younsi, and L. Rezzolla, Astron. Astrophys.671, A143 (2023), arXiv:2301.09549 [astro-ph.HE]

    Pith/arXiv arXiv 2023

  67. [68]

    Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, IEEE Transactions on Image Processing13, 600 (2004)

    2004

  68. [69]

    Mizuno, Z

    Y. Mizuno, Z. Younsi, C. M. Fromm, O. Porth, M. De Laurentis, H. Olivares, H. Falcke, M. Kramer, and L. Rezzolla, Nature Astron.2, 585 (2018), arXiv:1804.05812 [astro-ph.GA]

    Pith/arXiv arXiv 2018

  69. [70]

    Wang and A

    Z. Wang and A. Bovik, IEEE Signal Processing Letters 9, 81 (2002)

    2002