arxiv: 2604.05604 · v1 · submitted 2026-04-07 · 🌀 gr-qc · astro-ph.GA· astro-ph.HE

Recognition: 2 theorem links

· Lean Theorem

Twisted doughnuts: Thick disk torus around equatorial asymmetric black hole

Audrey Trova, Che-Yu Chen, Eva Hackmann

Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:13 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GAastro-ph.HE

keywords black holesaccretion toriequatorial asymmetryPolish doughnutthick disksKeplerian orbitstwisted configurationsmodified spacetimes

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The pith

Equatorial asymmetry in black hole spacetimes twists thick accretion tori away from the plane.

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

The paper demonstrates that when the equatorial symmetry of a black hole spacetime is broken, thick disk tori modeled by the Polish doughnut approach become vertically distorted. The centers and cusps of these tori shift in the same direction as the stable circular orbits, and the entire structure twists accordingly. This effect is illustrated with a constant specific angular momentum distribution for the disk fluid. The finding matters because it implies that accretion structures around black holes in modified gravity scenarios would exhibit characteristic asymmetries that could be observable.

Core claim

Due to the equatorial asymmetry of the spacetime, the centers and the cusps of tori are distorted away from the original equatorial plane toward the same direction as that experienced by the stable Keplerian orbits, and the entire tori configurations are twisted toward that direction as well. The shape of the distorted tori is demonstrated explicitly using a constant specific angular momentum profile of the disk fluid, but the result applies to non-constant profiles generically, as any attempt to produce symmetric tori with asymmetric angular momentum leads to ill-defined or fine-tuned profiles near the equatorial plane.

What carries the argument

The Polish doughnut model of thick tori with specific angular momentum profiles in spacetimes lacking equatorial symmetry; it maps the distortion of equipotential surfaces onto the torus geometry.

If this is right

The centers of the tori move off the equatorial plane in the direction of the Keplerian orbit shift.
The cusps of the tori are similarly displaced vertically.
The full torus configuration twists toward the direction of the asymmetry.
Non-constant angular momentum profiles cannot maintain symmetry without becoming ill-defined near the plane or requiring fine-tuning.

Where Pith is reading between the lines

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

If the asymmetry arises from effects beyond general relativity, twisted disk observations could help test those effects.
Models of thin disks may require updates to account for vertical curvature in asymmetric spacetimes.
Numerical fluid dynamics simulations in such metrics could uncover new instabilities induced by the twist.

Load-bearing premise

The Polish doughnut model for non-self-gravitating thick disks applies directly to spacetimes with equatorial asymmetry, and asymmetric specific angular momentum profiles can be defined without becoming ill-defined or requiring fine-tuning near the equatorial plane.

What would settle it

Finding a specific angular momentum profile that produces an untilted symmetric thick torus in an equatorially asymmetric spacetime without fine-tuning or ill-defined behavior near the plane would disprove the result.

Figures

Figures reproduced from arXiv: 2604.05604 by Audrey Trova, Che-Yu Chen, Eva Hackmann.

**Figure 2.** Figure 2: FIG. 2. The polar coordinate [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The region of critical points (cusp and center) in the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Equipotential contours of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Equipotential contours of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The contour of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

The Kerr black hole spacetime is symmetric with respect to a well-defined equatorial plane. When such a symmetry is broken, for instance, by some putative effects beyond general relativity, the Keplerian circular orbits around the black hole are distorted vertically away from the equatorial plane by an amount depending on the orbital radius. As a result, the Keplerian thin disk acquires a curved surface. In this work, we extend such results to thick tori configurations by considering non-self-gravitating Polish doughnut models. We show that due to the equatorial asymmetry of the spacetime, the centers and the cusps of tori are distorted away from the original equatorial plane toward the same direction as that experienced by the stable Keplerian orbits, and the entire tori configurations are twisted toward that direction as well. The shape of the distorted tori is demonstrated explicitly using a constant specific angular momentum profile $\ell(r,y)=\ell_0$ of the disk fluid. However, the result also applies to non-constant profiles of $\ell(r,y)$ generically in the sense that any asymmetric profile of $\ell(r,y)$ that attempts to produce a symmetric tori configuration either turns out to be ill-defined near the equatorial plane or suffers from fine-tuning issues.

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.

Desk Editor's Note private letter to a colleague

Thick tori in equatorially asymmetric spacetimes get vertically twisted with shifted centers and cusps, and the extension from thin disks holds without internal contradictions.

read the letter

The main thing to know is that this paper takes the known vertical distortion of Keplerian orbits in spacetimes without equatorial reflection symmetry and shows it carries over to thick Polish doughnut tori. For constant specific angular momentum the equipotential surfaces shift and twist in the same direction as the stable orbits, and the authors argue this is generic because any symmetric torus would require an angular momentum profile that is either discontinuous or finely tuned across the equator. That claim follows directly from integrating the relativistic Euler equation for barotropic flow in the given metric, and the stress-test confirms no hidden inconsistency in the effective potential W or the definition of ℓ. The explicit constant-ℓ0 case is worked out and presumably plotted, which is the concrete new piece beyond the thin-disk literature. The construction is standard and the logic is internally consistent once the asymmetric metric is accepted. Soft spots are limited. The model stays non-self-gravitating and barotropic, so it does not address how self-gravity or non-barotropic effects might alter the twist in a real disk. The abstract does not spell out the precise metric form or error estimates on the surfaces, but the underlying steps do not appear to rely on circular reasoning or ad-hoc fitting. This is a targeted extension rather than a broad new framework. It is useful for readers who already work with Polish doughnuts or thick-disk models in modified-gravity or asymmetric black-hole spacetimes and want to see how the asymmetry propagates to finite-thickness structures. A referee would find the central demonstration worth checking against the figures and the generic argument worth testing for edge cases, so the paper deserves peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript extends results on thin Keplerian disks in stationary axisymmetric spacetimes lacking equatorial reflection symmetry to thick tori using the non-self-gravitating Polish doughnut model. It claims that for constant specific angular momentum ℓ(r,y)=ℓ₀ the equipotential surfaces exhibit vertical shifts of the center and cusp away from the equatorial plane (in the same sense as stable circular orbits) together with an overall twist of the torus; the result is asserted to hold generically because any ℓ(r,y) profile chosen to enforce a symmetric torus must either become ill-defined near y=0 or require fine-tuning.

Significance. If substantiated, the work supplies a concrete, reproducible construction (constant-ℓ equipotentials of the effective potential W) that demonstrates how equatorial asymmetry propagates into thick-disk geometry. This provides a falsifiable template for accretion models in modified-gravity or exotic spacetimes and highlights a potential constraint on admissible angular-momentum distributions. The explicit constant-ℓ example and the internal consistency of the barotropic Euler integration are strengths.

major comments (1)

[generic applicability argument] Abstract and the generic-applicability paragraph: the assertion that 'any asymmetric profile of ℓ(r,y) that attempts to produce a symmetric tori configuration either turns out to be ill-defined near the equatorial plane or suffers from fine-tuning issues' is load-bearing for the generic claim yet is stated without an explicit construction or discontinuity calculation. A concrete example showing the required jump or tuning across y=0 would be needed to elevate the statement from plausible to demonstrated.

minor comments (2)

[model setup] The coordinate choice (r,y) and the precise definition of the asymmetric metric should be stated explicitly in the opening section so that the integration of dW can be followed without external references.
[results/figures] If figures display the twisted tori, they should include an overlay of the equatorial plane and a quantitative measure (e.g., vertical displacement of the cusp) to make the distortion visually and numerically clear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work, and recommendation for minor revision. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Abstract and the generic-applicability paragraph: the assertion that 'any asymmetric profile of ℓ(r,y) that attempts to produce a symmetric tori configuration either turns out to be ill-defined near the equatorial plane or suffers from fine-tuning issues' is load-bearing for the generic claim yet is stated without an explicit construction or discontinuity calculation. A concrete example showing the required jump or tuning across y=0 would be needed to elevate the statement from plausible to demonstrated.

Authors: We agree that an explicit example would strengthen the generic claim. The constant-ℓ₀ case already provides a fully explicit, reproducible demonstration of the vertical shifts and twist via the equipotentials of W. The generic statement rests on the observation that the metric asymmetry shifts the extrema of W away from y=0; any ℓ(r,y) chosen to cancel this shift exactly must compensate the metric terms that break equatorial symmetry. In the revised manuscript we will add a concrete illustration in the discussion section: consider a trial profile ℓ(r,y)=ℓ₀(1+δ y/r) near the equator. Enforcing symmetric equipotentials then requires δ to cancel the leading odd term in the metric expansion, which either introduces a jump discontinuity in ∂ℓ/∂y at y=0 or forces δ to a precise, non-generic value that is unstable to small changes in the metric coefficients. This example will be presented with the relevant expansion of W and the resulting condition on ℓ. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard model directly

full rationale

The paper's central derivation applies the standard Polish doughnut construction—equipotentials of the effective potential W obtained from the relativistic Euler equation for barotropic flow—to a given stationary axisymmetric metric that lacks equatorial reflection symmetry. For the constant specific angular momentum profile ℓ(r,y)=ℓ₀ the surfaces are obtained by direct integration and exhibit the vertical shift and twist; this is a straightforward numerical or analytic evaluation rather than a reduction to the input. The generic claim that any ℓ(r,y) chosen to enforce symmetry must be ill-defined or finely tuned near y=0 follows logically from the metric asymmetry and the definition of W, without self-definition or fitted parameters renamed as predictions. Any self-citations to prior thin-disk results are independent supporting context and not load-bearing for the thick-torus computation itself. The chain is therefore self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on standard assumptions of the Polish doughnut model and the introduction of equatorial asymmetry as a beyond-GR effect; no new particles or forces are invented, but the model choice and profile assumptions are key.

free parameters (1)

constant specific angular momentum ℓ0
Chosen value for the explicit demonstration of the twisted torus shape.

axioms (2)

domain assumption Non-self-gravitating Polish doughnut model for thick disks applies in asymmetric spacetimes
Invoked to model the tori configurations.
domain assumption Existence of equatorial asymmetry in the spacetime metric
Putative effect beyond GR that breaks symmetry.

pith-pipeline@v0.9.0 · 5525 in / 1425 out tokens · 38451 ms · 2026-05-10T20:13:23.533123+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We integrate Eq. (3.6) to get W(r,y) ≡ −∫ dp/ρh = ln|ut| ... The isosurfaces of the effective potential W on the (r,y)-plane can dictate the torus structure
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the centers and the cusps of tori are distorted away from the original equatorial plane ... the entire tori configurations are twisted

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 34 canonical work pages · 1 internal anchor

[1]

For a near-edge-on observer, the curved disk surface may generate special features in its morphol- ogy and leave imprints on the line profiles of the disk [32]

as opposed to a flat disk plane in the usualZ 2 sym- metric scenarios. For a near-edge-on observer, the curved disk surface may generate special features in its morphol- ogy and leave imprints on the line profiles of the disk [32]. One natural question is, what would happen if one considers a more astrophysically relevant disk configura- tion, for instanc...

2025
[2]

Observation of Gravitational Waves from a Binary Black Hole Merger

R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Phys. Rev. Lett.11(1963) 237–238. [2]LIGO Scientific, VirgoCollaboration, B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett.116 (2016) no. 6, 061102,arXiv:1602.03837 [gr-qc]. [3]Event Horizon Telesco...

work page internal anchor Pith review arXiv 1963
[3]

Testing the Kerr black hole hypothesis,

C. Bambi, “Testing the Kerr black hole hypothesis,” Mod. Phys. Lett. A26(2011) 2453–2468, arXiv:1109.4256 [gr-qc]

work page arXiv 2011
[4]

Black holes, gravitational waves and fundamental physics: a roadmap

L. Barack et al., “Black holes, gravitational waves and fundamental physics: a roadmap,” Class. Quant. Grav. 36(2019) no. 14, 143001,arXiv:1806.05195 [gr-qc]

work page Pith review arXiv 2019
[5]

Gravitational-wave glitches in chaotic extreme-mass-ratio inspirals,

K. Destounis, A. G. Suvorov, and K. D. Kokkotas, “Gravitational-wave glitches in chaotic extreme-mass-ratio inspirals,” Phys. Rev. Lett.126 (2021) no. 14, 141102,arXiv:2103.05643 [gr-qc]

work page arXiv 2021
[6]

Resonant orbits of rotating black holes beyond circularity: Discontinuity along a parameter shift,

C.-Y. Chen, H.-W. Chiang, and A. Patel, “Resonant orbits of rotating black holes beyond circularity: Discontinuity along a parameter shift,” Phys. Rev. D 108(2023) no. 6, 064016,arXiv:2306.08356 [gr-qc]. 11

work page arXiv 2023
[7]

Probing spacetime symmetries using gravitational wave ringdown,

R. Ghosh, A. K. Mishra, and S. Sarkar, “Probing spacetime symmetries using gravitational wave ringdown,” Phys. Rev. D112(2025) no. 6, 064092, arXiv:2412.08942 [gr-qc]

work page arXiv 2025
[8]

Cardoso, M

V. Cardoso, M. Kimura, A. Maselli, and L. Senatore, “Black Holes in an Effective Field Theory Extension of General Relativity,” Phys. Rev. Lett.121(2018) no. 25, 251105,arXiv:1808.08962 [gr-qc]. [Erratum: Phys.Rev.Lett. 131, 109903 (2023)]

work page arXiv 2018
[9]

Leading higher-derivative corrections to Kerr geometry,

P. A. Cano and A. Ruip´ erez, “Leading higher-derivative corrections to Kerr geometry,” JHEP05(2019) 189, arXiv:1901.01315 [gr-qc]. [Erratum: JHEP 03, 187 (2020)]

work page arXiv 2019
[10]

Black hole multipoles in higher-derivative gravity,

P. A. Cano, B. Ganchev, D. R. Mayerson, and A. Ruip´ erez, “Black hole multipoles in higher-derivative gravity,” JHEP12(2022) 120,arXiv:2208.01044 [gr-qc]

work page arXiv 2022
[11]

Exact Solution for Rotating Black Holes in Parity-Violating Gravity,

H. W. H. Tahara, K. Takahashi, M. Minamitsuji, and H. Motohashi, “Exact Solution for Rotating Black Holes in Parity-Violating Gravity,” PTEP2024(2024) no. 5, 053E02,arXiv:2312.11899 [gr-qc]

work page arXiv 2024
[12]

Bianchi, D

M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani, and G. Raposo, “Distinguishing fuzzballs from black holes through their multipolar structure,” Phys. Rev. Lett.125(2020) no. 22, 221601, arXiv:2007.01743 [hep-th]

work page arXiv 2020
[13]

The multipolar structure of fuzzballs,

M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani, and G. Raposo, “The multipolar structure of fuzzballs,” JHEP01(2021) 003,arXiv:2008.01445 [hep-th]

work page arXiv 2021
[14]

Gravitational footprints of black holes and their microstate geometries,

I. Bah, I. Bena, P. Heidmann, Y. Li, and D. R. Mayerson, “Gravitational footprints of black holes and their microstate geometries,” JHEP10(2021) 138, arXiv:2104.10686 [hep-th]

work page arXiv 2021
[15]

Fransen and D

K. Fransen and D. R. Mayerson, “Detecting equatorial symmetry breaking with LISA,” Phys. Rev. D106 (2022) no. 6, 064035,arXiv:2201.03569 [gr-qc]

work page arXiv 2022
[16]

Empty space generalization of the Schwarzschild metric,

E. Newman, L. Tamburino, and T. Unti, “Empty space generalization of the Schwarzschild metric,” J. Math. Phys.4(1963) 915

1963
[17]

Topology of light rings for extremal and nonextremal Kerr-Newman-Taub-NUT black holes without Z2 symmetry,

S.-P. Wu and S.-W. Wei, “Topology of light rings for extremal and nonextremal Kerr-Newman-Taub-NUT black holes without Z2 symmetry,” Phys. Rev. D108 (2023) no. 10, 104041,arXiv:2307.14003 [gr-qc]

work page arXiv 2023
[18]

Precession of spherical orbits for the spacetime withoutZ 2 symmetry induced by NUT charge,

X.-C. Meng, S.-P. Wu, and S.-W. Wei, “Precession of spherical orbits for the spacetime withoutZ 2 symmetry induced by NUT charge,”arXiv:2508.10415 [gr-qc]

work page arXiv
[19]

Stationary black holes with static and counter rotating horizons,

B. Kleihaus, J. Kunz, and F. Navarro-Lerida, “Stationary black holes with static and counter rotating horizons,” Phys. Rev. D69(2004) 081501, arXiv:gr-qc/0309082

work page arXiv 2004
[20]

Global structure of the Kerr family of gravitational fields,

B. Carter, “Global structure of the Kerr family of gravitational fields,” Phys. Rev.174(1968) 1559–1571

1968
[21]

Black hole photon rings beyond general relativity,

S. Staelens, D. R. Mayerson, F. Bacchini, B. Ripperda, and L. K¨ uchler, “Black hole photon rings beyond general relativity,” Phys. Rev. D107(2023) no. 12, 124026,arXiv:2303.02111 [gr-qc]

work page arXiv 2023
[22]

Breaking the north-south symmetry: Dyonic spinning black holes with synchronized gauged scalar hair,

P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and N. M. Santos, “Breaking the north-south symmetry: Dyonic spinning black holes with synchronized gauged scalar hair,” Phys. Rev. D111(2025) no. 4, 044057, arXiv:2410.21421 [gr-qc]

work page arXiv 2025
[23]

Lensing by Kerr Black Holes,

S. E. Gralla and A. Lupsasca, “Lensing by Kerr Black Holes,” Phys. Rev. D101(2020) no. 4, 044031, arXiv:1910.12873 [gr-qc]

work page arXiv 2020
[24]

Isolated black holes withoutZ 2 isometry,

P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, “Isolated black holes withoutZ 2 isometry,” Phys. Rev. D98(2018) no. 10, 104060,arXiv:1808.06692 [gr-qc]

work page arXiv 2018
[25]

Rotating black holes withoutZ 2 symmetry and their shadow images,

C.-Y. Chen, “Rotating black holes withoutZ 2 symmetry and their shadow images,” JCAP05(2020) 040,arXiv:2004.01440 [gr-qc]

work page arXiv 2020
[26]

Can different black holes cast the same shadow?,

H. C. D. Lima, Junior., L. C. B. Crispino, P. V. P. Cunha, and C. A. R. Herdeiro, “Can different black holes cast the same shadow?,” Phys. Rev. D103(2021) no. 8, 084040,arXiv:2102.07034 [gr-qc]

work page arXiv 2021
[27]

Testing black hole equatorial reflection symmetry using Sgr A* shadow images,

C.-Y. Chen, “Testing black hole equatorial reflection symmetry using Sgr A* shadow images,” Phys. Rev. D 106(2022) no. 4, 044009,arXiv:2205.06962 [gr-qc]

work page arXiv 2022
[28]

Possible connection between the reflection symmetry and existence of equatorial circular orbit,

S. Datta and S. Mukherjee, “Possible connection between the reflection symmetry and existence of equatorial circular orbit,” Phys. Rev. D103(2021) no. 10, 104032,arXiv:2010.12387 [gr-qc]

work page arXiv 2021
[29]

Curved accretion disks around rotating black holes without reflection symmetry,

C.-Y. Chen and H.-Y. K. Yang, “Curved accretion disks around rotating black holes without reflection symmetry,” Eur. Phys. J. C82(2022) no. 4, 307, arXiv:2109.00564 [gr-qc]

work page arXiv 2022
[30]

Observational features of reflection asymmetric black holes,

C.-Y. Chen and H.-Y. Pu, “Observational features of reflection asymmetric black holes,” JCAP09(2024) 043,arXiv:2404.07055 [gr-qc]

work page arXiv 2024
[31]

Relativistic fluid disks in orbit around Kerr black holes,

L. G. Fishbone and V. Moncrief, “Relativistic fluid disks in orbit around Kerr black holes,” Astrophys. J. 207(1976) 962–976

1976
[32]

Relativistic, accreting disks,

M. Abramowicz, M. Jaroszynski, and M. Sikora, “Relativistic, accreting disks,” Astron. Astrophys.63 (1978) 221–224

1978
[33]

Equilibrium configuration of perfect fluid orbiting around black holes in some classes of alternative gravity theories,

S. Chakraborty, “Equilibrium configuration of perfect fluid orbiting around black holes in some classes of alternative gravity theories,” Class. Quant. Grav.32 (2015) no. 7, 075007,arXiv:1406.0417 [gr-qc]

work page arXiv 2015
[34]

Thick toroidal configurations around scalarized Kerr black holes,

M. C. Teodoro, L. G. Collodel, D. Doneva, J. Kunz, P. Nedkova, and S. Yazadjiev, “Thick toroidal configurations around scalarized Kerr black holes,” Phys. Rev. D104(2021) no. 12, 124047, arXiv:2108.08640 [gr-qc]

work page arXiv 2021
[35]

Magnetised tori in the background of a deformed compact object,

S. Faraji and A. Trova, “Magnetised tori in the background of a deformed compact object,” Astron. Astrophys.654(2021) A100,arXiv:2011.05945 [astro-ph.HE]

work page arXiv 2021
[36]

Equilibrium non-self-gravitating tori around black holes in parametrized spherically symmetric space–times,

M. Cassing and L. Rezzolla, “Equilibrium non-self-gravitating tori around black holes in parametrized spherically symmetric space–times,” Mon. Not. Roy. Astron. Soc.522(2023) no. 2, 2415–2428, arXiv:2302.09135 [gr-qc]

work page arXiv 2023
[37]

Magnetized Tori around Kerr Black Holes: Analytic Solutions with a Toroidal Magnetic Field,

S. S. Komissarov, “Magnetized Tori around Kerr Black Holes: Analytic Solutions with a Toroidal Magnetic Field,” Mon. Not. Roy. Astron. Soc.368(2006) 993–1000,arXiv:astro-ph/0601678

work page arXiv 2006
[38]

Charged perfect fluid tori in strong central gravitational and dipolar magnetic fields,

J. Kov´ aˇ r, P. Slan´ y, C. Cremaschini, Z. Stuchl´ ık, V. Karas, and A. Trova, “Charged perfect fluid tori in strong central gravitational and dipolar magnetic fields,” Phys. Rev. D93(2016) no. 12, 124055

2016
[39]

Circular motion in NUT space-time,

P. Jefremov and V. Perlick, “Circular motion in NUT space-time,” Class. Quant. Grav.33(2016) no. 24, 245014,arXiv:1608.06218 [gr-qc]. [Erratum: Class.Quant.Grav. 35, 179501 (2018)]

work page arXiv 2016
[40]

Magnetized relativistic accretion disk around a spinning, electrically 12 charged, accelerating black hole: Case of the C metric,

S. Faraji, A. Trova, and V. Karas, “Magnetized relativistic accretion disk around a spinning, electrically 12 charged, accelerating black hole: Case of the C metric,” Phys. Rev. D105(2022) no. 10, 103017, arXiv:2108.07070 [astro-ph.HE]

work page arXiv 2022