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Hamiltonian variational reconstruction of the 3D magnetic geometry of relativistic jets: accuracy across GRMHD models and the fundamental sign degeneracy

T0 review · 1 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Jet width and linear polarization recover 3D magnetic geometry but never the field's sense of direction.

desk verdict Clean free-field reconstructor plus a formal non-identifiability proof: unsigned jet geometry is recoverable from W and p at high fidelity; poloidal sense is not, and three recovery routes fail for named physical reasons. read the letter →

arxiv 2607.09569 v1 pith:JLLR7WNZ submitted 2026-07-10 astro-ph.HE

Buffoli Fabio This is my paper · ORCID
classification astro-ph.HE
keywords relativisticjetsmagneticfieldsGRMHDlinearpolarizationvariationalreconstructionsigndegeneracyFaradayrotationcircular
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Every resolved image of a relativistic jet encodes its magnetic field, yet synchrotron intensity and linear polarization are blind to which way the field points along the jet axis: both depend only on even combinations of the field components. This paper introduces H-MOG, a variational method that reconstructs the full three-dimensional magnetic field on a lattice from two projected observables—the jet width profile W(z) and the linear polarization degree p(z)—regularized by a Hamiltonian prior and optimized by automatic differentiation. Applied to ten GRMHD simulations spanning MAD and SANE accretion states and five black-hole spins, the method recovers the unsigned field orientation at alignment 0.95–0.98, far above chance. The sense of the poloidal field, however, is not recovered; the paper proves it cannot be, because both observables are invariant under global field reversal. Three attempted routes to break that degeneracy—an added rotation-measure term, full non-axisymmetric sampling, and a Blandford–Znajek spin prior—all fail for distinct physical reasons. The result draws a clean observational boundary: geometry is measurable with present VLBI data, while sense requires a parity-odd observable such as Faraday tomography or circular polarization.

What carries the argument

H-MOG, a Hamiltonian-regularized variational inversion that reconstructs amplitude and unit direction of the magnetic field on a Cartesian lattice from projected W(z) and p(z), optimized by automatic differentiation; its even prior and data terms make the sign degeneracy exact (Proposition 1).

What would settle it

Apply H-MOG to a GRMHD library that retains full non-axisymmetric structure and higher near-axis resolution, then check whether the unsigned alignment stays near 0.95–0.98 and whether any parity-even addition (or a parity-odd observable) can raise the signed alignment above chance.

Watch

Extended reading notes

Core claim

From jet width and linear polarization alone, H-MOG recovers the unsigned three-dimensional magnetic geometry of relativistic jets at ⟨|cos|⟩ ≃ 0.95–0.98 across ten GRMHD models; the poloidal sense cannot be recovered because those observables are invariant under B → −B, and three physically motivated attempts to break the degeneracy all fail for identifiable reasons.

Load-bearing premise

The truth fields used for scoring are axisymmetric mean fields after azimuthal averaging and resampling onto a uniform coarse Cartesian cube, so the high reported alignment measures recovery of large-scale mean geometry rather than the full native turbulent field.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper introduces H-MOG, a variational method that reconstructs the 3D magnetic field of a relativistic jet on a Cartesian lattice from two projected observables, the jet width W(z) and linear polarization degree p(z), regularized by a Hamiltonian prior and optimized with automatic differentiation. Applied to ten GRMHD models spanning MAD and SANE states and five black-hole spins, it recovers the unsigned field orientation at ⟨|cos|⟩ ≃ 0.95–0.98. Proposition 1 proves that W and p (and the retained even prior) are invariant under B → −B, so the poloidal sense is formally non-identifiable from these data. Three routes to break the degeneracy—an RM term, full 3D sampling, and a Blandford–Znajek spin prior—are tested and each fails for a distinct physical reason; the spin–sense relation in the simulated jets is shown to be non-monotonic and state-dependent. The authors conclude that recovering sense requires a parity-odd observable such as Faraday tomography or circular polarization.

Significance. If the results hold, the paper cleanly separates what current VLBI/EHT linear-polarization and width data can determine (large-scale helical geometry) from what they cannot (poloidal sense). The formal invariance argument, the multi-model GRMHD campaign with reported snapshot scatter, the synthetic-jet validation across pitch and noise, and the explicit diagnosis of three failed recovery routes are all strengths. The finding that the spin–sense relation is not monotonic in these turbulent jets is an astrophysical result in its own right and cautions against silently imposing a BZ prior. The work is directly relevant to ongoing and next-generation EHT analyses of M87* and Sgr A*.

major comments (1)
  1. The high ⟨|cos|⟩ scores are measured against the same axisymmetrized, azimuthally averaged Cartesian cubes that the simplified forward model (ε ∝ |B|²) sees (Sect. 3.2). The paper states this limitation and does not claim recovery of native non-axisymmetric turbulence, so the central geometry claim remains sound for the large-scale mean field. A short, explicit statement in the abstract or conclusions that the reported accuracy is for the axisymmetric mean geometry would prevent over-reading by non-specialists.
minor comments (5)
  1. Table 1 and Table D.1 both report reconstruction statistics; a single consolidated table (or a clear pointer that D.1 is the complete version) would reduce redundancy.
  2. In Sect. 2.3 the plane-of-sky angle is written χ = arctan(Bz/Bx) + π/2; a brief note on the branch choice and how it is handled under automatic differentiation would aid reproducibility.
  3. Fig. 12 caption and text use “verso” for the mean poloidal sense; standard English “sense” or “sign” would be clearer for an international readership.
  4. Appendix C and Figs. D.1–D.2 usefully document convergence; a one-sentence statement of the final data residual level (already sub-percent) in the main text of Sect. 2.4 would help readers who skip the appendix.
  5. The frame-dragging term H_frame is correctly set to h = 0 for the main runs; stating the numerical value of h used in the failed Test 3 (Sect. 6.4) would make that experiment fully reproducible from the text alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: geometry skill is scored against independent GRMHD truth never seen by the optimizer; the sign non-identifiability is a proved invariance of the observables, not a fitted tautology.

full rationale

The paper’s load-bearing chain is self-contained and externally scored. H-MOG minimizes a cost built only from projected W(z) and p(z) plus an even Hamiltonian prior; the reconstructed field is then compared cell-by-cell to GRMHD truth cubes that the optimizer never receives. That is ordinary inverse-problem validation, not a prediction forced by a fitted input. Proposition 1 is a short, definitional invariance proof (ε∝|B|² and 2χ → 2χ+2π leave I, W, Q, U, p unchanged; retained prior terms are even in S), establishing non-identifiability of sense rather than claiming to derive sense from even data. Prior-weight sweeps (Sect. 4.6) show ⟨|cos|⟩ stays high and ⟨cos⟩ stays near zero, so the prior does not smuggle the reported geometry or flip the sense. The three degeneracy-breaking tests (RM term, full-3D sampling, BZ spin prior) are empirical failures with named physical reasons, not self-citations or uniqueness theorems imported from the same author. There is no self-citation load-bearing chain (single author; library and theory citations are external), no fitted constant renamed as a prediction, and no ansatz smuggled via prior work that itself only assumes the target. The axisymmetric resampling of the truth cubes is a modeling limitation the paper states explicitly; it does not make the skill score circular, because the same forward model generates both the inverted observables and the scoring field. Score 0 is therefore the correct finding.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The geometry-recovery claim rests on a standard synchrotron emissivity assumption, a deliberately even Hamiltonian prior whose weights are free parameters, and scoring against axisymmetrized GRMHD cubes. The non-identifiability claim rests only on the definitions of W and p plus the evenness of the retained prior terms; it needs no free parameters. No new physical entities are postulated.

free parameters (7)
  • global prior weight α_prior = 0.5
    Scales the entire Hamiltonian relative to the data term; default 0.5, varied in sensitivity tests.
  • external-pressure weight α = 0.05
    Weight of the collimation term H_ext; default 0.05.
  • shear-layer weight β = 0.3
    Weight of the edge-pitch term H_shear; default 0.3.
  • anti-kink weight λ = 1
    Weight of the smoothness term H_kink; fixed at 1.
  • divergence penalty weight = 0.05
    Fixed weight on L_div; set to 0.05.
  • Adam learning rate and iteration count = 0.05 / 400
    Optimizer hyperparameters; lr=0.05, 400 iterations chosen for sub-percent residual convergence.
  • lattice size N and domain half-width L = 96 / 30 rg
    Discretization choices; N=96, L=30 rg throughout.
assumptions (4)
  • domain assumption Synchrotron emissivity is proportional to magnetic energy density, ε∝|B|².
    Stated in Sect. 2.3; standard simplification that makes W depend only on energy distribution.
  • ad hoc to paper With frame-dragging coefficient h=0, every retained prior term is even under S→−S.
    Explicitly enforced (Sect. 2.2, Appendix A) so that the sign degeneracy is exact rather than approximate.
  • domain assumption The axisymmetric mean field after azimuthal averaging faithfully represents the geometry that dominates the projected observables W(z) and p(z).
    Sect. 3.2; deliberate modeling choice that enables stable scoring but discards turbulent 3D structure.
  • standard math Jet width is the FWHM-equivalent second-moment width of the transverse brightness profile; polarization degree is formed from line-of-sight integrated Stokes Q and U.
    Eqs. (4)–(5); conventional definitions used as the data term.

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Cite this review

Pith. "Pith review of Hamiltonian variational reconstruction of the 3D magnetic geometry of relativistic jets: accuracy across GRMHD models and the fundamental sign degeneracy." pith.science (2026). https://pith.science/paper/JLLR7WNZ

@misc{pith2026260709569,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian variational reconstruction of the 3D magnetic geometry of relativistic jets: accuracy across GRMHD models and the fundamental sign degeneracy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLLR7WNZ}},
  note         = {Machine review of arXiv:2607.09569}
}
read the original abstract

Every resolved image of a relativistic jet encodes its magnetic field, yet no image records which way the field points along the axis. Synchrotron intensity and linear polarization are blind to this: both probe the field only through even combinations of its components. We introduce H-MOG, a variational method that reconstructs the 3D field on a lattice from two projected observables, the jet width W(z) and the linear polarization p(z), regularized by a Hamiltonian prior and optimized with automatic differentiation. We apply H-MOG to ten GRMHD simulations spanning MAD and SANE states and five black-hole spins (a* = -0.94, -0.5, 0, +0.5, +0.94), and test three routes to break the intrinsic sign degeneracy of the reconstruction: a Faraday rotation-measure term, full 3D sampling, and a Blandford-Znajek spin prior. The unsigned field orientation is recovered at <|cos|> ~ 0.95-0.98, far above the random expectation of 0.5. The sense of the poloidal field, however, is not recovered, and we prove it cannot be: W and p are invariant under B -> -B. All three routes to break this degeneracy fail for distinct physical reasons; the spin-sense relation in these turbulent jets is not monotonic, differing sharply between MAD and SANE. Recovering the sense requires a parity-odd observable: Faraday tomography or circular polarization.

Figures

Figures reproduced from arXiv: 2607.09569 by the authors.

Figure 1
Figure 1. The jet width is blind to the magnetic pitch. Left: W(z) for five values of the pitch µ; the curves overlap. Right: the maximum spread of W across µ is at the level of 10−15, i.e. machine precision. −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0 z 0.2 0.4 0.6 0.8 1.0 p ola riz atio n d e g r e e p(z) polarization encodes the pitch μ = 0.4 μ = 0.8 μ = 1.2 μ = 1.6 μ = 2.0 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The polarization degree p(z) for the same five values of the pitch. Unlike the width, p(z) responds strongly to the pitch, varying by more than a factor of two. barely budges: variations sit at 10−15, i.e. machine precision ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Reconstruction of a synthetic jet with known truth. Left: the true field, in the y = 0 plane. Centre: the H-MOG recon￾struction from W and p only. Right: the true and reconstructed pitch profiles µ(z). The direction field is recovered at ⟨cos⟩ = 0.97. 0.0 0.5 1.0 1.5 2.0 2.5 jet pitch μ = Bϕ/Bpol 0.0 0.2 0.4 0.6 0.8 1.0 r e c o n str u ctio n s kill ⟨c o s ⟩ robustness across pitch reconstruction null [PITH_FULL_IM… view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Robustness of the reconstructed geometry to observa￾tional noise. The unsigned alignment ⟨| cos |⟩ remains close to 0.95 up to 40% noise on W and p, and degrades only to 0.86 at 100%, never approaching the null value of 0.5 (dotted). better than MAD ones on average, bu…
Figure 6
Figure 6. Figure 6: Sensitivity of the reconstruction to the prior weights, for a synthetic jet (top) and a representative GRMHD snapshot (bot￾tom). Each panel varies one weight around its default (red line): the global prior weight αprior, the external-pressure weight α, and the shear we…
Figure 7
Figure 7. Figure 7: Geometric reconstruction accuracy ⟨| cos |⟩ across the GRMHD library, as a function of black-hole spin, for the MAD and SANE states. The alignment stays in the range 0.95–0.98 for all ten models, far above the random null of 0.5 (dotted). Error bars show the snapshot-t…
Figure 8
Figure 8. Figure 8: Three-dimensional magnetic field lines of a representative GRMHD jet. Left: the simulated field. Right: the H-MOG reconstruction from projected observables. The collimation and helical winding are recovered; the field orientation matches, while its sense is unconstrain…
Figure 9
Figure 9. Figure 9: The sign degeneracy and the observable that breaks it. Left, centre: the width W(z) and polarization p(z) computed from a field B and from −B are identical. Right: the transverse rotation-measure profile reverses sign between the two, because RM is linear in B. The RM …
Figure 10
Figure 10. Figure 10: Adding a rotation-measure term to the loss does not recover the sense. As the weight of the RM term is increased over four orders of magnitude, the signed alignment ⟨cos⟩ (squares) stays near zero, while the unsigned geometry ⟨| cos |⟩ (circles) remains high and in fa…
Figure 12
Figure 12. Figure 12: Mean poloidal field sense ⟨Bz sign(z)⟩ in the jet, mea￾sured directly in the simulated fields, against black-hole spin. The dotted line is the naive Blandford–Znajek expectation (sense = sign(a∗)). MAD models (circles) do not follow it and carry a coherent negative se…
Figure 13
Figure 13. Figure 13: Spatial distribution of the sign error in the slice y = 0, for a representative GRMHD model. Left: the sign of Bz in the true field. Centre: the sign in the H-MOG reconstruction, which settles into a single coherent sense across the jet. Right: where the two disagree,…

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Works this paper leans on

98 extracted references · 1 linked inside Pith

  1. [1]

    and Olivares, H

    Porth, O. and Olivares, H. and Mizuno, Y. and Younsi, Z. and Rezzolla, L. and others , title=. ApJS , year=

  2. [2]

    Etienne, Z. B. and Paschalidis, V. and Haas, R. and Moesta, P. and Shapiro, S. L. , title=. Phys. Rev. D , year=

  3. [3]

    McKinney, J. C. and Tchekhovskoy, A. and Blandford, R. D. , title=. MNRAS , year=

  4. [4]

    and Nakamura, M

    Asada, K. and Nakamura, M. , title=. ApJL , year=

  5. [5]

    and Nakamura, M

    Asada, K. and Nakamura, M. , title=. A&A , year=

  6. [6]

    and Asada, K

    Lu, R.-S. and Asada, K. and Krichbaum, T. P. and others , title=. Nature , year=

  7. [7]

    McKinney, J. C. and Tchekhovskoy, A. and Blandford, R. D. , title=. MNRAS Lett. , year=

  8. [8]

    and Porth, O

    Olivares, H. and Porth, O. and Mizuno, Y. and Rezzolla, L. , title=. Comput. Phys. Commun. , year=

Show all 98 references
  1. [9]

    Bellan, P. M. , title=. J. Plasma Phys. , year=

  2. [10]

    Blandford, R. D. and Znajek, R. L. , title=. MNRAS , year=

  3. [11]

    and Narayan, R

    Tchekhovskoy, A. and Narayan, R. and McKinney, J. C. , title=. MNRAS Lett. , year=

  4. [12]

    Boroson, T. A. , title=. ApJL , year=

  5. [13]

    Nikonov, A. S. and Kovalev, Y. Y. and Kravchenko, E. V. and others , title=. MNRAS , year=

  6. [14]

    and Kino, M

    Hada, K. and Kino, M. and Giovannini, G. and others , title=. arXiv e-prints , year=. 2403.18359 , archivePrefix=

    arXiv

  7. [15]

    and Zhao, G.-Y

    Cui, Y. and Zhao, G.-Y. and Li, Z. and others , title=. Nat. Astron. , year=

  8. [16]

    and Philippov, A

    Parfrey, K. and Philippov, A. A. and Cerutti, B. , title=. Phys. Rev. Lett. , year=

  9. [17]

    Blandford, R. D. and Payne, D. G. , title=. MNRAS , year=

  10. [18]

    Komissarov, S. S. , title=. MNRAS , year=

  11. [19]

    and Konigl, A

    Vlahakis, N. and Konigl, A. , title=. ApJ , year=

  12. [20]

    , title=

    Lyubarsky, Y. , title=. ApJ , year=

  13. [21]

    , title=

    Toth, G. , title=. J. Comput. Phys. , year=

  14. [22]

    and Kemm, F

    Dedner, A. and Kemm, F. and Kroner, D. and Munz, C.-D. and Schnitzer, T. and Wesenberg, M. , title=. J. Comput. Phys. , year=

  15. [23]

    and Bodo, G

    Mignone, A. and Bodo, G. and Massaglia, S. and others , title=. ApJS , year=

  16. [24]

    Gammie, C. F. and McKinney, J. C. and Toth, G. , title=. ApJ , year=

  17. [25]

    and Narayan, R

    Yuan, F. and Narayan, R. , title=. ARA&A , year=

  18. [26]

    Zavala, R. T. and Taylor, G. B. , title=. ApJ , year=

  19. [27]

    Lister, M. L. and. MOJAVE. XVII. Parsec-Scale AGN Jet Kinematics Analysis Based on 19 Years of VLBA Observations at 15 GHz , journal=. 2018 , volume=

    2018

  20. [28]

    Homan, D. C. and Lister, M. L. and Aller, M. F. and others , title=. ApJ , year=

  21. [29]

    First M87 Event Horizon Telescope Results. VII. Polarization of the Ring , journal=. 2021 , volume=

    2021

  22. [30]

    First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near the Event Horizon , journal=. 2021 , volume=

    2021

  23. [31]

    Broderick, A. E. and McKinney, J. C. , title=. ApJ , year=

  24. [32]

    and Igumenshchev, I

    Narayan, R. and Igumenshchev, I. V. and Abramowicz, M. A. , title=. PASJ , year=

  25. [33]

    and Asada, K

    Nakamura, M. and Asada, K. and Hada, K. and others , title=. ApJ , year=

  26. [34]

    Ressler, S. M. and Tchekhovskoy, A. and Quataert, E. and Gammie, C. F. , title=. MNRAS , year=

  27. [35]

    and Hawley, J

    Beckwith, K. and Hawley, J. F. and Krolik, J. H. , title=. ApJ , year=

  28. [36]

    and McKinney, J

    Tchekhovskoy, A. and McKinney, J. C. and Narayan, R. , title=. ApJ , year=

  29. [37]

    and Shibata, K

    Koide, S. and Shibata, K. and Kudoh, T. and Meier, D. L. , title=. Science , year=

  30. [38]

    and Goodman, J

    Li, L.-X. and Goodman, J. and Narayan, R. , title=. ApJ , year=

  31. [39]

    Spruit, H. C. , title=. Lect. Notes Phys. , year=

  32. [40]

    Meier, D. L. and Koide, S. and Uchida, Y. , title=. Science , year=

  33. [41]

    White, C. J. and Stone, J. M. and Gammie, C. F. , title=. ApJS , year=

  34. [42]

    Fishbone, L. G. and Moncrief, V. , title=. ApJ , year=

  35. [43]

    and Sadowski, A

    Narayan, R. and Sadowski, A. and Penna, R. F. and Kulkarni, A. K. , title=. MNRAS , year=

  36. [44]

    and Falcke, H

    Moscibrodzka, M. and Falcke, H. and Shiokawa, H. , title=. A&A , year=

  37. [45]

    Walker, R. C. and Hardee, P. E. and Davies, F. B. and others , title=. ApJ , year=

  38. [46]

    and Krichbaum, T

    Kim, J.-Y. and Krichbaum, T. P. and Lu, R.-S. and others , title=. A&A , year=

  39. [47]

    and Doi, A

    Hada, K. and Doi, A. and Kino, M. and others , title=. Nature , year=

  40. [48]

    Doeleman, S. S. and Fish, V. L. and Schenck, D. E. and others , title=. Science , year=

  41. [49]

    Gralla, S. E. and Lupsasca, A. and Marrone, D. P. , title=. Phys. Rev. D , year=

  42. [50]

    Gralla, S. E. and Lupsasca, A. , title=. Class. Quantum Grav. , year=

  43. [51]

    First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring , journal=. 2019 , volume=

    2019

  44. [52]

    and Gomez, J

    Mizuno, Y. and Gomez, J. L. and Nishikawa, K.-I. and others , title=. ApJ , year=

  45. [53]

    and Lyubarsky, Y

    Mizuno, Y. and Lyubarsky, Y. and Beckwith, K. and Hardee, P. E. , title=. ApJ , year=

  46. [54]

    and Lobanov, A

    Mertens, F. and Lobanov, A. P. and Walker, R. C. and Hardee, P. E. , title=. A&A , year=

  47. [55]

    and Narayan, R

    Sadowski, A. and Narayan, R. and McKinney, J. C. and Tchekhovskoy, A. , title=. MNRAS , year=

  48. [56]

    Balbus, S. A. and Hawley, J. F. , title=. ApJ , year=

  49. [57]

    Komissarov, S. S. and Barkov, M. V. and Vlahakis, N. and Konigl, A. , title=. MNRAS , year=

  50. [58]

    McKinney, J. C. , title=. MNRAS , year=

  51. [59]

    and Hada, K

    Giroletti, M. and Hada, K. and Giovannini, G. and others , title=. A&A , year=

  52. [60]

    and Lu, R.-S

    Akiyama, K. and Lu, R.-S. and Fish, V. L. and others , title=. ApJ , year=

  53. [61]

    First M87 Event Horizon Telescope Results. IX. Detection of Near-horizon Circular Polarization , journal=. 2023 , volume=

    2023

  54. [62]

    First Sagittarius A* Event Horizon Telescope Results. VII. Polarization of the Ring , journal=. 2024 , volume=

    2024

  55. [63]

    A&A , year=

    Horizon-scale variability of. A&A , year=

  56. [64]

    and Camilloni, F

    Meringolo, C. and Camilloni, F. and Rezzolla, L. , title=. ApJL , year=. 2507.08942 , archivePrefix=

  57. [65]

    and Rodriguez, M

    Lupsasca, A. and Rodriguez, M. J. and Strominger, A. , title=. JHEP , year=

  58. [66]

    and others , title=

    Lupsasca, A. and others , title=. ApJ , year=

  59. [67]

    2026 , volume=

    Spatially resolved polarization swings in the binary black hole candidate OJ 287 with first EHT observations , journal=. 2026 , volume=

    2026

  60. [68]

    and Prather, B

    Dhruv, V. and Prather, B. and Wong, G. N. and Gammie, C. F. , title=. ApJS , year=

  61. [69]

    L., et al

    Akiyama, K., Lu, R.-S., Fish, V. L., et al. 2015, ApJ, 807, 150

    2015

  62. [70]

    & Nakamura, M

    Asada, K. & Nakamura, M. 2011, ApJL, 745, L28

    2011

  63. [71]

    Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883

    1982

  64. [72]

    Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433

    1977

  65. [73]

    Broderick, A. E. & McKinney, J. C. 2010, ApJ, 725, 750

    2010

  66. [74]

    N., & Gammie, C

    Dhruv, V., Prather, B., Wong, G. N., & Gammie, C. F. 2025, ApJS, 277, 16

    2025

  67. [75]

    S., Fish, V

    Doeleman, S. S., Fish, V. L., Schenck, D. E., et al. 2012, Science, 338, 355

    2012

  68. [76]

    2019, ApJL, 875, L1

    Event Horizon Telescope Collaboration . 2019, ApJL, 875, L1

    2019

  69. [77]

    2021 a , ApJL, 910, L12

    Event Horizon Telescope Collaboration . 2021 a , ApJL, 910, L12

    2021

  70. [78]

    2021 b , ApJL, 910, L13

    Event Horizon Telescope Collaboration . 2021 b , ApJL, 910, L13

    2021

  71. [79]

    2023, ApJL, 957, L20

    Event Horizon Telescope Collaboration . 2023, ApJL, 957, L20

    2023

  72. [80]

    2024, ApJL, 964, L25

    Event Horizon Telescope Collaboration . 2024, ApJL, 964, L25

    2024

  73. [81]

    F., McKinney, J

    Gammie, C. F., McKinney, J. C., & Toth, G. 2003, ApJ, 589, 444

    2003

  74. [82]

    2011, Nature, 477, 185

    Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185

    2011

  75. [83]

    C., Lister, M

    Homan, D. C., Lister, M. L., Aller, M. F., et al. 2009, ApJ, 706, 1253

    2009

  76. [84]

    P., Lu, R.-S., et al

    Kim, J.-Y., Krichbaum, T. P., Lu, R.-S., et al. 2018, A&A, 616, A188

    2018

  77. [85]

    McKinney, J. C. 2006, MNRAS, 368, 1561

    2006

  78. [86]

    C., Tchekhovskoy, A., & Blandford, R

    McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2012, MNRAS, 423, 3083

    2012

  79. [87]

    P., Walker, R

    Mertens, F., Lobanov, A. P., Walker, R. C., & Hardee, P. E. 2016, A&A, 595, A54

    2016

  80. [88]

    Mizuno, Y., Lyubarsky, Y., Beckwith, K., & Hardee, P. E. 2014, ApJ, 780, 87

    2014

  81. [89]

    2018, ApJ, 868, 146

    Nakamura, M., Asada, K., Hada, K., et al. 2018, ApJ, 868, 146

    2018

  82. [90]

    V., & Abramowicz, M

    Narayan, R., Igumenshchev, I. V., & Abramowicz, M. A. 2003, PASJ, 55, L69

    2003

  83. [91]

    F., & Kulkarni, A

    Narayan, R., Sadowski, A., Penna, R. F., & Kulkarni, A. K. 2012, MNRAS, 426, 3241

    2012

  84. [92]

    2019, ApJS, 243, 26

    Porth, O., Olivares, H., Mizuno, Y., et al. 2019, ApJS, 243, 26

    2019

  85. [93]

    C., & Tchekhovskoy, A

    Sadowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, MNRAS, 439, 503

    2014

  86. [94]

    Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2011, MNRAS Lett., 418, L79

    2011

  87. [95]

    Toth, G. 2000, J. Comput. Phys., 161, 605

    2000

  88. [96]

    & Konigl, A

    Vlahakis, N. & Konigl, A. 2004, ApJ, 605, 656

    2004

  89. [97]

    C., Hardee, P

    Walker, R. C., Hardee, P. E., Davies, F. B., et al. 2018, ApJ, 855, 128

    2018

  90. [98]

    Zavala, R. T. & Taylor, G. B. 2004, ApJ, 612, 749

    2004

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