Magnetic seed generation by plasma heat flux in accretion disks

Felipe A. Asenjo; Nicol\'as Villarroel-Sep\'ulveda; Pablo S. Moya

REVIEW 3 major objections 5 minor 31 references

Magnetic seed generation by plasma heat flux in accretion disks

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper demonstrates that in an axisymmetric, radially stratified Keplerian disk around a Schwarzschild black hole, the heat-flux drive is the only non-vanishing magnetic seed source and it produces a seed magnetic field that grows…

desk verdict New heat-flux battery term in the relativistic vorticity equation is real and worth taking seriously, but the quantitative seed-field predictions rest on an uncontrolled linear-growth step. read the letter →

arxiv 2411.13222 v1 pith:FGVMO4K4 submitted 2024-11-20 astro-ph.HE

Nicolás Villarroel-Sepúlveda , Felipe A. Asenjo , Pablo S. Moya This is my paper · ORCID

classification astro-ph.HE

keywords accretiondisksmagneticseedgenerationheatfluxbatterygeneralizedvorticitySchwarzschildblackholefieldsplasmasBiermann

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

This paper aims to show that a non-ideal thermodynamic effect—the heat flux in the plasma—can generate a seed magnetic field from an initially zero field in a simple accretion disk model around a Schwarzschild black hole. In a disk in which every thermodynamic and hydrodynamic quantity depends only on the radial coordinate, the two standard seed mechanisms (the baroclinic Biermann battery and the relativistic drive) vanish identically, leaving the curl of the heat-flux vector as the only source. The paper demonstrates that this “heat flux battery” drives a magnetic seed field in the polar direction that grows linearly in time. If the mechanism operates in real disks, it would supply the initial magnetization that dynamo action later amplifies, addressing the long-standing question of where cosmic magnetic seeds come from.

What carries the argument

The load-bearing object is the heat-flux drive $\Xi_q=\nabla\times\Lambda$, where $\Lambda$ is the vector encoding the non-ideal contribution of the heat flux to the generalized vorticity equation, Eq. (11), which governs the sum of the magnetic field and the enthalpy-weighted fluid vorticity. In the radial-only model, $\Lambda$ has only an azimuthal component $\Lambda_\phi$ built from the Keplerian velocity, the Spitzer–Härm conductivity $\kappa\propto $T^{{5/2}}$$, the density $n(r)=n_0\Gamma(r)(r_S/r)^{3/2}$, and the temperature profile $T(r)=T_0(r_S/r)^\$\beta$(1-\sqrt{r_I/r})^\$\lambda$$. The curl of this component does not vanish, so it acts as a battery: it sources a polar generalized vorticity, and since the magnetic field is initially zero, the seed field is the part of $\$\Omega$$ that grows linearly in time. The radial structure of the result—the sign-change radii and the band where the battery dominates—is carried by the single function $h(r)$ in Eq. (17).

What would settle it

Compute the curl of the heat-flux drive $\Xi_q=\nabla\times\Lambda$ using a heat flux obtained from a general-relativistic radiative-transfer calculation rather than the assumed conduction law; if $\Xi_q$ vanishes or changes sign in the band $9.6<r/r_S<9.9$ for a realistic heat flux, the predicted seed field will not appear.

Watch

Extended reading notes

Core claim

The central claim is that the heat flux drive $\Xi_q=\nabla\times\Lambda$ is the only initial source for the time evolution of the magnetofluid’s generalized vorticity $\$\Omega$ = B+(m/q)\nabla\times(f\Gamma v)$ in an axisymmetric, radially dependent Keplerian disk around a Schwarzschild black hole. Because the baroclinic drive $\Xi_B$ and the relativistic drive $\Xi_R$ vanish when all gradients are radial, the seed field can only come from $\Xi_q$. With the assumed Tolman-law heat flux and Spitzer–Härm conductivity, $\Lambda$ has a single non-vanishing azimuthal component $\Lambda_\phi$, and its curl sources a polar seed field. For the adopted disk parameters ($\$\beta$=3/4$, $\$\lambda$=1/4$, $n_0=$10^{{10}}$\,\mathrm{cm}^{-3}$, $T_0=$10^{{6}}$\,\mathrm{K}$, $\kappa_0=$10^{{9}}$\,\mathrm{g\,$s^{{-3}}$\,$K^{{-1}}$}$), the field diverges at the innermost stable circular orbit (ISCO), changes sign at $r_{0B}/r_S\simeq 9.6$, and the heat-flux battery dominates the fluid-vorticity contribution in the bands $3<r/r_S<3.2$ and $9.6<r/r_S<9.9$.

Load-bearing premise

The central premise is that the disk’s heat flux is conduction-dominated and follows the assumed conductivity law with the adopted temperature profile, and that the magnetic seed grows only linearly in time; if either fails, the battery need not operate.

Editorial extensions

If this is right

In an axisymmetric, radially stratified Keplerian disk, the heat-flux battery is the only magnetic seed source; the baroclinic (Biermann) and relativistic drives vanish identically.
Starting from zero magnetic field, a seed field grows linearly in time, with its polarity set by the heat-flux profile and reversing at $r_{0B}/r_S\simeq 9.6$ for the adopted parameters.
The heat-flux battery dominates the fluid-vorticity contribution in the radial bands $3<r/r_S<3.2$ and $9.6<r/r_S<9.9$, identifying the regions where the mechanism is strongest.
Because the mechanism only requires the curl of $\Lambda$ to be non-zero, it should apply to other astrophysical plasmas with energy flux, whether from conduction, convection, radiation, or particle flux.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If this works, black hole accretion disks would have a built-in way to make the first magnetic field, so dynamo models would no longer need to assume an initial seed; the predicted radial sign flip at about $9.6\,r_S$ could be looked for in the magnetic polarity structure of real disks.
The battery should be added to general-relativistic magnetohydrodynamic simulations; whether the linearly growing seed survives the non-linear phase is a testable question that this paper leaves open.
For a rotating black hole, frame dragging will alter the connection terms in Eq. (11), so the heat-flux battery may shift its sign-change radius or dominate in different bands; this is a natural extension of the Schwarzschild calculation.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper identifies a genuinely new source term in the generalized vorticity equation — the heat flux drive Ξq = ∇×Λ — and shows that in a purely radial, axisymmetric Keplerian disk it is the only nonvanishing seed source. That is a real addition to the battery literature. The derivation from the stress-energy tensor through the 3+1 split is coherent, and the authors are appropriately careful about the regime of Spitzer conductivity. The comparison with the Cosmic Battery is useful.

The weak part is the step from the initial time derivative to a finite field. Eq. (20) gives ∂tB at t0; Eq. (21) multiplies by the dynamical time ς, using the approximation ∂tB ≈ B/ς imported from Asenjo et al. (2013). The paper does not establish that the right-hand side stays roughly constant over ς, nor that the ∇×(v×Ω) term and Lorentz feedback remain negligible. The source itself evolves on the heat-flux timescale, and the initial-state assumption ∂tT = ∂tp = ∂tn = 0 while ∂t f ≠ 0 is at least in tension, since f is constructed from n, T, and p. That tension is not addressed. So the existence of a nonzero initial heat-flux drive is robust, but the magnitude of the seed and the sign-change radius at 9.6 rS are not.

The quantitative predictions are also tied to the assumed temperature profile (β=3/4, λ=1/4), the conduction-dominated Tolman heat flux, and the chosen κ0, n0, T0. The zeros and domination windows shift with β and λ, as the authors state, but the paper does not map that sensitivity. And the plotted field strengths in Gauss cannot be reproduced without a black hole mass, since ς depends on M; the mass is never given. No code or data are provided.

Net: this is a solid conceptual contribution that deserves a serious referee, but it needs a major revision before the numbers can be used. The referee should push for (1) a controlled estimate or derivation of the linear-growth validity range, (2) a resolution of the initial-equilibrium inconsistency, and (3) a statement of the black hole mass and parameter sensitivity. I would not desk-reject it; the new drive is a legitimate addition to the seed-generation toolbox.

Referee Report

3 major / 5 minor

Summary. The paper studies the generation of a seed magnetic field by a heat-flux battery in the unified magnetofluid framework of Mahajan and Asenjo in Schwarzschild spacetime. For a radially dependent, axisymmetric Keplerian plasma disk, the baroclinic and relativistic drives vanish because all thermodynamic gradients are parallel, leaving the heat-flux drive Ξq = ∇×Λ as the only source in the generalized-vorticity equation. Assuming an initially vanishing magnetic field and a linear-in-time approximation, the authors derive Eq. (21) for the generated poloidal field, evaluate it using a Tolman-law heat flux with Spitzer-Härm conductivity and a Shakura-Sunyaev temperature profile, and report a sign-change radius at 9.6 rS and dominance windows where the heat-flux battery exceeds the time-evolving fluid-vorticity term.

Significance. If supported, the paper introduces a genuinely new seed-field mechanism that complements the Biermann and Cosmic batteries for weakly magnetized accretion disks. The identification that radial-only axisymmetric equilibria suppress the usual batteries and leave a non-vanishing heat-flux curl is clean, and the algebraic derivation from Eq. (3) to Eq. (20) is mostly coherent. The paper makes falsifiable predictions in the form of derived zero radii and dominance windows rather than fitting parameters to a target field, which is a clear strength. However, the central quantitative claims rest on an uncontrolled short-time approximation and an internally inconsistent initial equilibrium, so the specific field magnitudes and sign-change radii should be treated as provisional until those points are addressed.

major comments (3)

[Section 3, Eq. (21)] The step from Eq. (20) to Eq. (21) uses the approximation ∂tBi(t)|_{t0}^{ς} ≈ Bi(ς)/ς with ς = r/(|v|α), but this is a linear extrapolation that requires the right-hand side of Eq. (20) to remain approximately constant over the interval ς and requires the dropped ∇×(v×Ω) term in Eq. (11) to stay negligible. Neither condition is established. The heat-flux source itself is not time-independent: the energy equation is used to compute ∂t f, so f, T, and p change on a heat-flux timescale, and once B is nonzero the Lorentz force feeds back into v and Λ. The paper gives no estimate of the heat-flux or feedback timescales relative to ς, which is a dynamical (orbital) time. If the source decays or oscillates before ς, Eq. (21) overestimates the seed field, and the reported zero at r0B/rS = 9.6 and the dominance windows can shift or disappear. The authors should either present Eq. (21) as a formal short-time expansion B(t) = t∂tB(0) + O(t²) with a quantified validity range, or provide timescale estimates showing that the linear growth is a good approximation at t = ς.
[Section 3, paragraph on initial equilibrium] The initial state is assumed to have zero time derivatives for the fluid velocity, density, temperature, and pressure, but a nonzero heat flux is retained and the energy equation is used to obtain ∂t f ≠ 0 for the enthalpy-related quantity f = h/(mn), with h = ρ + p. Since f depends on the same thermodynamic variables whose time derivatives are set to zero, this combination is inconsistent: if n, T, and p are initially stationary, then f is initially stationary as well, and the second term in Eq. (20) should vanish. The paper needs to state a consistent ordering in which the background is stationary and the heat-flux-induced changes are first-order, carrying all time-derivative terms systematically, or allow the fluid variables to evolve from the outset. As written, the relative importance of the heat-flux battery versus the fluid-vorticity term in Eq. (21) is not uniquely defined because the ∂t f term is retained even though its thermodynamic inputs are held fixed.
[Section 4, heat-flux model dependence] The quantitative results depend on the specific choice of heat flux in Eq. (13), namely the Tolman law with Spitzer-Härm conductivity, and on the temperature profile in Eq. (14) with β = 3/4 and λ = 1/4. The paper correctly notes the strong dependence on β and λ, but it does not give corresponding qualifications for the conductivity model. If the disk heat flux is radiative or turbulent rather than collision-dominated conduction, the form of q_j changes and the sign-change radius, the dominance windows, and possibly even the sign of the generated field can change. To support the astrophysical conclusion that heat flux is the main seed driver in accretion disks, the authors should state the collisionality/optical-depth regime in which Eq. (13b) applies and, if possible, estimate the resulting uncertainty in r0B and in the dominance regions.

minor comments (5)

[Section 2, Eq. (9)] The derivation of Eq. (9) from the stress-energy tensor is presented in a very compressed way; an appendix or an intermediate display equation showing how the three terms arise would improve reproducibility.
[Section 3, Eq. (14)] The displayed temperature profile contains an unclear typeset factor; it should read T(r) = T0 (rS/r)^β (1 − sqrt(rI/r))^λ, as stated in the text. Please correct the equation.
[Section 3, paragraph after Eq. (15)] The symbol for the seed-generation timescale is written as ζ in the caption of Fig. 2 but as ς elsewhere; the notation should be unified.
[Section 3, paragraph on Keplerian orbits] The term 'photosphere' should be 'photon sphere' when referring to rph = 3rS/2 as the innermost unstable circular orbit for massless particles; photosphere has a different astrophysical meaning.
[Section 3, numerical values] The chosen values n0 = 10^10 cm^-3, T0 = 10^6 K, and κ0 = 10^9 g·cm·K^-1·s^-3 are taken from different references and the text does not indicate whether they are mutually consistent for a single disk model; a brief comment on this would help the reader judge whether the field magnitudes in Gauss are representative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the heat-flux seed field is derived from the assumed disk profiles, not fitted to the predicted field.

full rationale

The derivation chain is self-contained in the relevant sense. The heat-flux drive Xi_q = curl(Lambda) is obtained by substituting the assumed Tolman-law heat flux, Spitzer conductivity, and radial temperature and density profiles into the 3+1 vorticity equation; no parameter is fitted to the resulting seed field, and the vanishing radii r0B, r0omega, and r0q follow from the same assumed profiles rather than being imposed. The linear-growth estimate used to turn the initial time derivative into Eq. (21) is a first-order Taylor approximation cited from Asenjo et al. (2013); although this is a self-citation, it is parameter-free, its stated assumption (initially zero magnetic field and first-order evolution) does not include the heat-flux result, and the heat-flux drive itself is derived in this paper, so the citation is not load-bearing in a circular sense. The paper explicitly acknowledges that the quantitative results depend on the chosen velocity, temperature, and heat-flux profiles and that Spitzer conductivity is only valid for an electron-proton plasma; these are model limitations, not circular reductions. No equation in the paper defines the predicted field in terms of itself or fits a parameter to the target observable.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The model's predictions depend on five explicit constants (n0, T0, kappa0, beta, lambda) and an unspecified black hole mass. These are not fitted to the target magnetic field, but they control the magnitude and sign-change radii. The physical framework imports several domain assumptions (unified magnetofluid theory, Tolman heat flux, Spitzer conductivity) from the cited literature.

free parameters (6)

n0 = 10^10 cm^-3
Normalization of plasma density from Bhattacharyya et al. (2000); sets overall B scaling.
T0 = 10^6 K
Normalization of disk temperature; sets overall B scaling.
kappa0 = 10^9 g cm s^-3 K^-1 (as written, unit missing cm)
Thermal conductivity normalization from Meyer-Hofmeister & Meyer (2006); sets heat flux magnitude.
beta = 3/4
Temperature index in Eq. (14); controls radii where B, omega, and Xi_q vanish.
lambda = 1/4
Temperature index in Eq. (14); controls radii where B, omega, and Xi_q vanish.
black_hole_mass_M = not specified
Absolute B magnitude depends on rS = 2GM/c^2; figures are in Gauss but M is never stated.

assumptions (7)

domain assumption Plasma obeys unified magnetofluid equation with M_mu_nu = F_mu_nu + (m/q)S_mu_nu.
Inherited from Mahajan (2003); the central evolution equation (3) is not derived here.
ad hoc to paper Only non-ideal term in stress-energy tensor is heat flux; viscosity and resistivity are neglected.
States in Section 1: 'the heat flux term is the only considered non-ideal component'.
domain assumption Heat flux obeys Tolman law q_j = -(kappa/alpha) partial_j(alpha T) with Spitzer-Härm conductivity.
Eqs. (13a)-(13b); cited to Misner et al. (2017), Spitzer & Härm (1953).
ad hoc to paper Temperature profile T(r) = T0 (rS/r)^beta (1 - sqrt(rI/r))^lambda.
Eq. (14); approximation to Page-Thorne thin disk, with beta and lambda chosen from Shakura-Sunyaev.
ad hoc to paper Density profile n(r) = n0 Gamma(r) (rS/r)^(3/2).
Eq. (15); derived from continuity in the spiral-orbit limit zeta -> 0, which sets vr = 0.
ad hoc to paper Linear growth approximation partial_t B_i approximately B_i(ς)/ς.
Following Asenjo et al. (2013); converts time derivative to a field value without specifying validity range.
ad hoc to paper Initial equilibrium with zero time derivatives for fluid variables is consistent with the presence of a dissipative heat flux.
Section 3 states initial equilibrium, but later says heat flux injects energy and makes the system unstable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Magnetic seed generation by plasma heat flux in accretion disks." pith.science (2026). https://pith.science/paper/FGVMO4K4

@misc{pith2026241113222,
  author       = {Pith},
  title        = {Pith review of: Magnetic seed generation by plasma heat flux in accretion disks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGVMO4K4}},
  note         = {Machine review of arXiv:2411.13222}
}

read the original abstract

Context. Magnetic batteries are potential sources that may drive the generation of a seed magnetic field, even if this field is initially zero. These batteries can be the result of non-aligned thermodynamic gradients in a plasma, as well as of special and general relativistic effects. So far, magnetic batteries have only been studied in ideal magnetized fluids. Aims. We study the non-ideal fluid effects introduced by the energy flux in the vortical dynamics of a magnetized plasma in curved spacetime. We propose a novel mechanism for generating a heat flux-driven magnetic seed within a simple accretion disk model around a Schwarzschild black hole. Methods. We use the 3+1 formalism for the splitting of the space-time metric into space-like and time-like components. We study the vortical dynamics of a magnetized fluid with a heat flux in the Schwarzschild geometry in which thermodynamic and hydrodynamic quantities are only dependent on the radial coordinate. Assuming that the magnetic field is initially zero, we estimate linear time evolution of the magnetic field due to the inclusion of non-ideal fluid effects. Results. When the thermodynamic and hydrodynamic quantities vary only radially, the effect of the coupling between the heat flux, spacetime curvature and fluid velocity acts as the primary driver for an initial linearly time growing magnetic field. The plasma heat flux completely dominates the magnetic field generation at an specific distance from the black hole, where the fluid vorticity vanishes. This distance depends on the thermodynamical properties of the Keplerian plasma accretion disk. These properties control the strength of the non-ideal effects in the generation of seed magnetic fields.

Figures

Figures reproduced from arXiv: 2411.13222 by the authors.

**Figure 2.** (a) Magnitude of the linear-time evolution of the magnetic field |B|, heat flux battery ζ|Ξq| and fluid vorticity ζ(m/q)|∂ω/∂t| as functions of r/rS . (b) Ratio of the strengths of the heat flux battery and time evolution of the fluid vorticity as a function of r/rS . Narzilloev & Ahmedov 2022) and κ0 = 109g · s −3 · K −1 , following Ref. Meyer-Hofmeister & Meyer (2006). The charge and mass are taken to be those o… view at source ↗

**Figure 1.** Magnitude of the magnetic field produced in the equatorial plane in a time interval ς. The shaded area represents the inside of the horizon, the dashed line indicates the photosphere, the solid black line is the ISCO, and the solid white line is the radius at which the generated magnetic field passes through zero and inverts its sign. Equation (21) is solved numerically for β = 3/4, λ = 1/4, which indicates a black… view at source ↗

**Figure 3.** Generalized vorticity generated by the heat flux battery in a time interval ζ as a function of r/rS [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 15 canonical work pages

[1]

, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....
[3]

Arnowitt, R., Deser, S., & Misner, C. W. 2008, General Relativity and Gravitation, 40, 1997–2027, 10.1007/s10714-008-0661-1

doi:10.1007/s10714-008-0661-1 2008
[4]

A., Mahajan, S

Asenjo, F. A., Mahajan, S. M., & Qadir, A. 2013, Physics of Plasmas, 20, 022901, 10.1063/1.4792257

work page doi:10.1063/1.4792257 2013
[5]

2015, Physical Review D, 91, 10.1103/physrevd.91.123005

Bhattacharjee, C., Das, R., & Mahajan, S. 2015, Physical Review D, 91, 10.1103/physrevd.91.123005

work page doi:10.1103/physrevd.91.123005 2015
[6]

V., Misra, R., & Datta, B

Bhattacharyya, S., Thampan, A. V., Misra, R., & Datta, B. 2000, The Astrophysical Journal, 542, 473–483, 10.1086/309502

work page doi:10.1086/309502 2000
[7]

1950, Zeitschrift für Naturforschung A, 5, 65–71, 10.1515/zna-1950-0201

Biermann, L. 1950, Zeitschrift für Naturforschung A, 5, 65–71, 10.1515/zna-1950-0201

work page doi:10.1515/zna-1950-0201 1950
[8]

F., Foucart, F., & Quataert, E

Chandra, M., Gammie, C. F., Foucart, F., & Quataert, E. 2015, The Astrophysical Journal, 810, 162, 10.1088/0004-637X/810/2/162

doi:10.1088/0004-637x/810/2/162 2015

Show all 31 references

[9]

1998, The Astrophysical Journal, 508, 859, 10.1086/306426

Contopoulos, I., & Kazanas, D. 1998, The Astrophysical Journal, 508, 859, 10.1086/306426

1998 doi
[10]

2015, The Astrophysical Journal, 805, 105, 10.1088/0004-637X/805/2/105

Contopoulos, I., Nathanail, A., & Katsanikas, M. 2015, The Astrophysical Journal, 805, 105, 10.1088/0004-637X/805/2/105

2015 doi
[11]

2018, Monthly Notices of the Royal Astronomical Society, 473, 721, 10.1093/mnras/stx2249

Contopoulos, I., Nathanail, A., Sądowski, A., Kazanas, D., & Narayan, R. 2018, Monthly Notices of the Royal Astronomical Society, 473, 721, 10.1093/mnras/stx2249

2018 doi
[12]

2012, Lecture Notes in Physics, Vol

Gourgoulhon, E. 2012, Lecture Notes in Physics, Vol. 846, 3+1 Formalism in General Relativity, 1st edn. (Springer Berlin), https://doi.org/10.1007/978-3-642-24525-1

2012 doi
[14]

1998 b , Monthly Notices of the Royal Astronomical Society, 295, L6, 10.1046/j.1365-8711.1998.29511447.x

---. 1998 b , Monthly Notices of the Royal Astronomical Society, 295, L6, 10.1046/j.1365-8711.1998.29511447.x

1998
[15]

E., & Contopoulos, I

Koutsantoniou, L. E., & Contopoulos, I. 2014, The Astrophysical Journal, 794, 27, 10.1088/0004-637X/794/1/27

2014 doi
[16]

Kulsrud, R. M. 2005, Plasma Physics for Astrophysics (Princeton University Press). http://www.jstor.org/stable/j.ctvzsmf0w

2005
[17]

D., Lifshitz, E

Landau, L. D., Lifshitz, E. M., Sykes, J. B., & Reid, W. H. 1959, Fluid Mechanics (Pergamon Press Oxford, England)

1959
[18]

Mahajan, S. M. 2003, Physical Review Letters, 90, 10.1103/physrevlett.90.035001

2003 doi
[19]

M., & Yoshida, Z

Mahajan, S. M., & Yoshida, Z. 2010, Physical Review Letters, 105, 10.1103/physrevlett.105.095005

2010 doi
[20]

2006, Astronomy & Astrophysics, 449, 443–447, 10.1051/0004-6361:20053997

Meyer-Hofmeister, E., & Meyer, F. 2006, Astronomy & Astrophysics, 449, 443–447, 10.1051/0004-6361:20053997

2006 doi
[21]

W., Thorne, K

Misner, C. W., Thorne, K. S., & Wheeler, J. A. 2017, Gravitation (Princeton University Press)

2017
[22]

2022, Symmetry, 14, 10.3390/sym14091765

Narzilloev, B., & Ahmedov, B. 2022, Symmetry, 14, 10.3390/sym14091765

2022 doi
[23]

N., & Thorne, K

Page, D. N., & Thorne, K. S. 1974, The Astrophysical Journal, 191, 499, 10.1086/152990

1974 doi
[24]

J., Adams, M

Pilgram, J. J., Adams, M. B. P., Constantin, C. G., et al. 2022, High Power Laser Science and Engineering, 10, e13, 10.1017/hpl.2022.2

2022 doi
[25]

N., Peeters, A

Schobert, B. N., Peeters, A. G., & Rath, F. 2019, The Astrophysical Journal, 881, 56, 10.3847/1538-4357/ab2df6

2019 doi
[26]

I., & Sunyaev , R

Shakura , N. I., & Sunyaev , R. A. 1973, Astronomy & Astrophysics, 24, 337

1973
[27]

1953, Physical Review, 89, 977–981, 10.1103/physrev.89.977

Spitzer, L., & Härm, R. 1953, Physical Review, 89, 977–981, 10.1103/physrev.89.977

1953 doi
[28]

2017, Monthly Notices of the Royal Astronomical Society, 466, 705, 10.1093/mnras/stw3116

Sądowski, A., Wielgus, M., Narayan, R., et al. 2017, Monthly Notices of the Royal Astronomical Society, 466, 705, 10.1093/mnras/stw3116

2017 doi
[29]

2006, The Astrophysical Journal, 649, 345, 10.1086/506442

Tanaka, T., & Menou, K. 2006, The Astrophysical Journal, 649, 345, 10.1086/506442

2006 doi
[30]

Thorne, K. S. 1974, The Astrophysical Journal, 191, 507, 10.1086/152991

1974 doi
[31]

2008, Foundations of high-energy astrophysics (University of Chicago Press)

Vietri, M. 2008, Foundations of high-energy astrophysics (University of Chicago Press)

2008
[32]

2020, Phys

Zhang, C., Hua, J., Wu, Y., et al. 2020, Phys. Rev. Lett., 125, 255001, 10.1103/PhysRevLett.125.255001

2020 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

[3] [3]

Arnowitt, R., Deser, S., & Misner, C. W. 2008, General Relativity and Gravitation, 40, 1997–2027, 10.1007/s10714-008-0661-1

doi:10.1007/s10714-008-0661-1 2008

[4] [4]

A., Mahajan, S

Asenjo, F. A., Mahajan, S. M., & Qadir, A. 2013, Physics of Plasmas, 20, 022901, 10.1063/1.4792257

work page doi:10.1063/1.4792257 2013

[5] [5]

2015, Physical Review D, 91, 10.1103/physrevd.91.123005

Bhattacharjee, C., Das, R., & Mahajan, S. 2015, Physical Review D, 91, 10.1103/physrevd.91.123005

work page doi:10.1103/physrevd.91.123005 2015

[6] [6]

V., Misra, R., & Datta, B

Bhattacharyya, S., Thampan, A. V., Misra, R., & Datta, B. 2000, The Astrophysical Journal, 542, 473–483, 10.1086/309502

work page doi:10.1086/309502 2000

[7] [7]

1950, Zeitschrift für Naturforschung A, 5, 65–71, 10.1515/zna-1950-0201

Biermann, L. 1950, Zeitschrift für Naturforschung A, 5, 65–71, 10.1515/zna-1950-0201

work page doi:10.1515/zna-1950-0201 1950

[8] [8]

F., Foucart, F., & Quataert, E

Chandra, M., Gammie, C. F., Foucart, F., & Quataert, E. 2015, The Astrophysical Journal, 810, 162, 10.1088/0004-637X/810/2/162

doi:10.1088/0004-637x/810/2/162 2015