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A discussion on the symmetry of relativistic Vlasov gas and its accretion in Kerr-Newman black hole

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Pith's one-line read A collisionless charged gas around a rotating, charged black hole has a distribution function fixed by mass, energy, axial angular momentum, and total angular momentum; the resulting accretion drives the hole toward a neutral…

desk verdict Useful extension of Vlasov accretion to Kerr-Newman with charged particles, but the phase-space domains are computed for equatorial periastron and then applied at all angles, which likely skews the accretion rates and the Schwarzschild-evolution conclusion. read the letter →

arxiv 2506.08341 v4 pith:KNUYFVXY submitted 2025-06-10 gr-qc

Yongqiang Liu This is my paper
classification gr-qc MSC 83C5783C10 PACS 04.70.-s05.20.Dd
keywords VlasovgaskinetictheoryblackholeaccretionKerr-NewmanmetricCarterconstanthiddensymmetriesrateJüttnerdistribution
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 builds a kinetic description of a collisionless, charged Vlasov gas around a Kerr–Newman black hole. It argues that the spacetime's symmetries—stationarity, axisymmetry, and the hidden Carter symmetry—reduce the one-particle distribution function to a function of only the four constants of motion, $f=f(m,E,L_z,L)$, provided the gas is sourced from thermodynamic equilibrium at infinity. With that form, the paper analytically separates captured from scattered trajectories and computes number density, energy density, principal pressures, and mass, energy, and angular momentum accretion rates in the Locally Non-Rotating Frame. For a Jüttner-distributed neutral plasma, accretion of weakly charged gas removes both charge and spin from the hole, so a Kerr–Newman black hole evolves toward a Schwarzschild configuration.

What carries the argument

The load-bearing object is the reduced distribution function $f=f(m,E,L_z,L)$ of Eq. (30). It rests on three pieces: the Killing symmetries giving energy $E$ and axial angular momentum $L_z$; Carter's Killing tensor $C_{ab}=2\rho^2 l_{(a}k_{b)}+r^2 g_{ab}$, whose hidden symmetry supplies the Carter constant $L^2$ and completes integrability; and the action-angle coordinates $(P_0,P_1,P_2,P_3)=(\sqrt{-2H},E,L_z,L)$ with conjugate variables $Q^\mu$, in which the Vlasov equation forces $\partial f/\partial Q^0=\partial f/\partial Q^1=\partial f/\partial Q^2=0$, leaving at most linear dependence on $Q^3$. The equilibrium-at-infinity argument eliminates the $Q^3$ term. This reduced distribution, together with the periastron conditions $R=0$, $R'=0$ that define the absorption and scattering domains, carries all later computations of fluxes, densities, pressures, and accretion rates.

What would settle it

A concrete check is to start a kinetic simulation with a distribution function that has a non-zero linear $Q^3$ dependence and see whether the steady-state accretion rate onto a Kerr–Newman black hole differs from the rates computed here; a different result would show that Eq. (30) is not generic. Equivalently, a direct calculation of $\mathcal{L}_{Z_{L^2}} f$ for a physically motivated source at infinity that gives $\mathcal{L}_{Z_{L^2}} f \neq 0$ would falsify the reduction.

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Extended reading notes

Core claim

The paper's central claim is Eq. (30): in Kerr–Newman spacetime the distribution function of a collisionless Vlasov gas depends solely on the constants of motion, $f=f(m,E,L_z,L)$. The Killing vectors $\partial_t$ and $\partial_\varphi$ provide $E$ and $L_z$, while Carter's Killing tensor provides the hidden constant $L$; complete integrability then permits action-angle variables in which the Vlasov equation forces the distribution to be independent of three of the four angle coordinates. The remaining angle $Q^3$ is dropped by the steady-state/equilibrium-at-infinity argument, leaving the reduced form on which every later integral is built. From this foundation the paper derives the critical energy and angular momentum separating absorption from scattering, computes the densities and principal pressures in the LNRF with horizon-regularized quantities, and obtains analytic accretion rates. Numerical evaluation for a charge-symmetric Jüttner plasma shows the normalized mass and energy accretion rates are equal functions of $(a,Q)$ and are suppressed by both spin and charge, while the magnitude of the negative angular momentum accretion rate grows with $a$ and falls with $Q$; together these trends push the hole toward neutrality and zero spin.

Load-bearing premise

The reduction to $f=f(m,E,L_z,L)$ depends on the assumption that a gas sourced from thermodynamic equilibrium at infinity must have a phase-space distribution function independent of the angle variable $Q^3$; if that equilibrium condition fails, the distribution can carry a linear $Q^3$ term and every accretion rate computed from the reduced form would change.

Editorial extensions

If this is right

  • For a Jüttner-distributed neutral plasma the normalized mass and energy accretion rates of a Kerr–Newman black hole are identical functions of the spin parameter $a$ and charge $Q$, and both decrease as $a$ or $Q$ increases.
  • The magnitude of the angular momentum accretion rate, which is negative, increases with spin $a$ and decreases with charge $Q$, so rotating holes lose angular momentum through accretion.
  • A Kerr–Newman black hole accreting weakly charged plasma evolves toward smaller charge and smaller spin, i.e., toward a Schwarzschild black hole, across the parameter range including extremal cases.
  • Pressure anisotropy ($P_1\neq P_2\simeq P_3$) and radial-pressure suppression at the horizon are intrinsic features of collisionless dynamics: they persist in Schwarzschild spacetime and are not caused by rotation or charge.
  • The analytic absorption and scattering domains provide closed-form integration boundaries, so all reported observables can be reproduced without orbit-by-orbit ray tracing.

Reading between the lines

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

  • If the equilibrium-at-infinity assumption is relaxed, the distribution function can retain a linear $Q^3$ term; a natural test is to build a steady-state gas with a non-trivial $Q^3$ dependence and compute whether the accretion rates shift. This is an editorial extension, not a claim in the paper.
  • The charge-asymmetric density profiles (negative charges enhanced, positive charges suppressed) imply a net electric current into the hole; the paper fixes the background geometry, so the back-reaction of the induced electromagnetic field on the gas is left for future work.
  • The exact coincidence of the mass and energy accretion rate functions likely comes from the exponential $e^{-zE}$ form of the Jüttner distribution; other distribution families should break the coincidence and could provide a kinetic diagnostic of the phase-space distribution.
  • If the Schwarzschild attractor behavior holds dynamically, the coupled system $\dot M$, $\dot a$, $\dot Q$ could be integrated to give the neutralization and spin-down timescales, connecting kinetic accretion to the 'no-hair' relaxation of black holes.
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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

4 major / 5 minor

Summary. The paper studies collisionless Vlasov gas in Kerr-Newman spacetime. Section 2 claims that spacetime symmetries constrain the one-particle distribution function to the form f=f(m,E,L_z,L) (Eq. (30)), with the dependence on the angle variable Q^3 eliminated. Section 3 analyzes charged-particle trajectories, derives periastron conditions, and defines absorption and scattering domains in momentum space (Eqs. (49)-(50)). Section 4 computes particle flux, energy-momentum tensor, LNRF-frame observables, asymptotic quantities at infinity, and horizon-regularized accretion rates for a Jüttner-distributed two-species plasma. Section 5 presents numerical profiles and accretion rates for Kerr, Reissner-Nordström, and extremal Kerr-Newman black holes, and concludes that accretion of weakly charged plasma drives a Kerr-Newman black hole toward a Schwarzschild configuration. The central quantitative claims, including the evolution-to-Schwarzschild conclusion, rest on the phase-space domain construction and on the numerical evaluation of the accretion integrals.

Significance. If the analysis were fully correct, the paper would provide a useful analytic framework for kinetic accretion in stationary axisymmetric charged spacetimes: explicit absorption/scattering boundaries, LNRF observables, and asymptotic formulas that recover the Schwarzschild Jüttner-gas limits. The paper does contain genuinely useful components: the periastron conditions (41)-(42), the first-order-in-κ expressions (44)-(46), the regularized horizon quantities (74), and the closed-form asymptotic results (70)-(73). The numerical exploration over a,Q∈[0,1] for multiple black-hole families is also a reasonable first survey. However, the load-bearing domain construction is restricted to equatorial-plane periastrons and is then applied at arbitrary θ, which directly affects the accretion rates and the main evolutionary conclusion. The symmetry derivation in Sec. 2.3 also overstates what is proven, since the elimination of all Q^3 dependence is an equilibrium assumption rather than a consequence of the Vlasov equation or Killing symmetries.

major comments (4)
  1. [Sec. 3, Eqs. (47)-(50)] The critical boundaries E_c and \bar L_c are derived only for trajectories whose periastron lies on the equatorial plane. The text states this explicitly: "It is crucial to emphasize that this analysis specifically addresses particle trajectories whose periastron lies on the equatorial plane (θ=π/2)" (Sec. 3, after Eq. (46)), and Fig. 1 is restricted to θ=π/2. Nevertheless, the domains D_abs and D_scat in Eqs. (49)-(50) are used in the momentum-space integrals (53)-(56) and in the horizon accretion integrals (75)-(77) at arbitrary θ with σ∈[0,2π]. For θ≠π/2, the general periastron conditions (41)-(42) show that the capture threshold depends on θ and σ, so the equatorial boundary misclassifies orbits. This biases the number/energy density profiles and, more importantly, the mass, energy, and angular-momentum accretion rates in Figs. 4-5, on which the Sec. 5.3 conclusion that accretion drives Kerr-Newman toward Schwarzschild is based. The authors must either restrict the entire analysis and conclusions to equatorial-plane periastrons or solve the general θ-dependent capture boundary and use it in all integrals.
  2. [Sec. 2.3, Eqs. (27)-(30)] The derivation of f=f(m,E,L_z,L) is not as conclusive as the abstract and Sec. 2.3 claim. The Vlasov equation and the Killing constraints (28) eliminate dependence on Q^0, Q^1, Q^2, but they do not eliminate dependence on Q^3. The linear term Q^3 \hat f in Eq. (29) is excluded only by the additional "natural physical conditions" that the distribution be a steady state depending solely on action variables; the text itself uses phrases like "we argue" and "natural choice". Single-valuedness on the invariant torus excludes the linear term, but it does not exclude a periodic function of Q^3, which would satisfy the stated equations and be a single-valued function on phase space. Thus the statement in the abstract that the distribution function is "shown" to depend solely on (m,E,L_z,L) overstates the result. Since the subsequent Jüttner model is assumed, the accretion calculations are not invalidated, but the symmetry section should be reframed as an assumption or supplemented with a rigorous argument (e.g., a steady-state/ergodic condition) that rules out all Q^3 dependence.
  3. [Sec. 4, Eqs. (49)-(56)] The notation for the capture threshold in the domain definitions is ambiguous in a way that can change all results. In Eqs. (49)-(50), D_abs and D_scat list the radial-momentum bound simply as \bar L_c, but Eq. (47) defines \bar L_c as a function of the critical energy E_c(r), not as a function of the integration variable E. In the integrals (53)-(56), the lower (or upper) limit must be the capture angular momentum \bar L_c(E) evaluated at the energy E being integrated, not a fixed number determined by E_c(r) at some radius. If the numerical implementation used a constant \bar L_c, then the domains are mis-specified. The authors should write \bar L_c(E) explicitly in (49)-(50) and clarify how the mapping E↦\bar L_c(E) is obtained from (41)-(42).
  4. [Sec. 5, Figs. 4-5] The numerical results underlying the main physical conclusion lack convergence tests and error estimates. The accretion integrals (75)-(77) contain the singular factor 1/√Δ, and the regularization \dot Q_Δ = lim_{r→r_H} √Δ \dot Q in Eq. (78) requires careful numerical extrapolation. No grid-resolution studies, tolerance values, or estimated relative errors are reported for the curves in Figs. 4-5, even though the claim of identical functional dependence of mass and energy accretion rates and the monotonic trends in a and Q depend on the accuracy of these integrals. The authors should provide a convergence check (e.g., doubling the radial/angular/momentum grid and reporting the change) and state the numerical precision of the plotted quantities.
minor comments (5)
  1. [Sec. 4.1, Eq. (56)] Eq. (56) contains a typographical error: the second integral is written as ∫_{E_c}^{+∞} d\bar L, but it should be ∫_{E_c}^{+∞} dE, consistent with Eq. (54) and with the following integration variable \bar L in the third integral.
  2. [Sec. 4.5, Eq. (74) and Fig. 3 caption] The regularization of the energy density is defined as \tilde ε ≡ e^{2ν} ε in Eq. (74), which is the correct finite combination near the horizon, but the caption of Fig. 3 writes \tilde ε ≡ e^{-2ν} ε. The caption should be corrected to match Eq. (74).
  3. [Sec. 3, Eq. (47)] In the line \bar L_c = g_{\bar L}(a,Q,θ,E_c,σ,q), the symbol E_c appears both as the critical energy and as the argument of the function g. Since Eq. (47) defines E_c as a function of r, the notation is confusing; the authors should write \bar L_c = g_{\bar L}(a,Q,θ,E_c(r),σ,q) and explain how this gives the boundary curve \bar L_c(E) used in the domains.
  4. [Sec. 4.1, Eqs. (54)-(56)] The factor 2 in the scattering integrals is said to account for "the symmetric treatment of pre- and post-scattering trajectories" but this is not explained. For a periastron-passing trajectory, the radial momentum changes sign at the turning point; the factor 2 is correct only if the two branches have equal weight, which should be stated explicitly.
  5. [Sec. 4.3, Eq. (67)] The quantity \dot M is called the mass accretion rate, but J^a defined in Eq. (51) is the particle-number current. Since the Jüttner model in Eq. (69) has f∝δ(P_0−m), the identification \dot M = m∫_S J^a \bar n_a holds only if the particle mass is set to unity. The text should state this convention explicitly to avoid a dimensionful inconsistency.

Circularity Check

1 steps flagged · score 5.0 of 10

The only concrete circularity is the Q3-elimination step: Eq. (30) is assumed as the 'natural' action-only steady-state ansatz, then presented as derived; the accretion-rate integrals themselves are independent computations.

  1. self definitional [Sec. 2.3, Eqs. (29)-(30)]
    "Under natural physical conditions—specifically, for a stable accretion system with a particle source in thermodynamic equilibrium at infinity—we argue that f cannot depend on Q3. ... for an integrable Hamiltonian system, the natural choice for a stationary distribution function describing a system in a steady state is one that depends solely on the action variables (the constants of motion Pµ), and not on the angle variables (Qµ). ... Therefore, the general form of the distribution function in Kerr-Newman spacetime reduces to: f=f(P0,P1,P2,P3)=f(m,E,Lz,L)."

    Eq. (29) already allows the most general symmetry-compatible form, f=f0(P0,P1,P2,P3)+Q3 fhat(P0,P1,P2,P3), because the Vlasov and Killing constraints only force L_ZL2 f to be constant along trajectories, i.e. at most linear Q3 dependence. The step to Eq. (30) removes the Q3 term by invoking a 'natural choice' in which a stationary distribution depends solely on the action variables Pµ. But that is precisely the claimed conclusion f=f(m,E,Lz,L). The equilibrium-at-infinity statement is therefore a restatement of the action-only ansatz, not an independent consequence of the Vlasov equation or the spacetime symmetries. The paper presents this stipulated form as a derived 'general form,' and all later observables and accretion rates inherit it.

full rationale

The paper contains one concrete circular step: the central structural claim f=f(m,E,Lz,L) is obtained by assuming, as a 'natural choice,' that steady-state distribution functions depend only on action variables, which is exactly the result being derived. This is not a hidden fit and it is not a self-citation chain, but it does mean the headline symmetry statement is an input dressed as a derivation. The accretion-rate results themselves are not circular: they are computed directly from specified Jüttner distributions and phase-space integrals, reproduce Schwarzschild limits, and are not fitted to any data. Self-citation is not load-bearing here: Ref. [34] is by the author but appears only in the introduction as background on Kerr accretion models, while the domain analysis relies on Ref. [32], which is not a self-citation. A separate non-circular concern is the paper's own limitation, 'It is crucial to emphasize that this analysis specifically addresses particle trajectories whose periastron lies on the equatorial plane (θ=π/2),' followed by use of the resulting domains at all σ and θ in Eqs. (53)-(56) and (75)-(77); this is an internal-validity issue rather than a circular-reduction issue, so it is not counted in the score. Overall, the derivation chain is mostly self-contained, but the Q3-elimination step makes the central structural claim partially circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The main input is the equilibrium-at-infinity assumption that removes Q3, plus standard integrability results inherited from Carter and the Vlasov literature. The parameters a, Q, kappa, z, and alpha0 are model inputs, not fitting parameters.

free parameters (5)
  • black hole rotation a = varied over [0,1]
    Input Kerr-Newman spin parameter controlling geometry; accretion rates are studied as functions of it, not fitted to data.
  • black hole charge Q = varied over [0,1]
    Input electromagnetic charge parameter; accretion rates are studied as functions of it, not fitted to data.
  • dimensionless charge coupling kappa = 0, +-0.3 in numerics
    Chosen values for the weak electromagnetic interaction regime; first-order expansion in kappa is used in Sec. 3.
  • coldness z = z=1 in numerics
    Thermodynamic parameter of the Juttner distribution; fixed by hand for the numerical profiles.
  • Juttner normalization alpha0 = unspecified, cancels in normalized rates
    Amplitude of the equilibrium distribution function; arbitrary scale that drops out of normalized accretion rates.
assumptions (5)
  • standard math Kerr-Newman spacetime admits a Killing tensor C^ab and is completely integrable for charged particle motion.
    Inherited from Carter (Ref. [39]); used in Secs. 2.2-2.3 to define the fourth constant L and the action-angle-like coordinates.
  • domain assumption The distribution function is sourced from a stable thermodynamic equilibrium at infinity, so f_hat=0 in Eq. (29).
    Invoked in Sec. 2.3 to eliminate Q3 dependence; this is the key input behind the form f=f(m,E,Lz,L).
  • domain assumption Electromagnetic interaction is weak, kappa approx 0, and only first-order corrections are kept.
    Used in Sec. 3 to truncate the expansion in Eq. (44) and to simplify the critical energy and angular momentum expressions.
  • domain assumption The plasma consists of two equal-mass species with charges +q and -q, not a mass-disparate astrophysical plasma.
    Stated in the footnote in Sec. 4.4; limits the applicability of the neutrality evolution conclusions.
  • domain assumption The background Kerr-Newman geometry is fixed and backreaction is neglected.
    Used throughout the test-field calculation; Sec. 5.3 nonetheless draws conclusions about black hole evolution.

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Pith. "Pith review of A discussion on the symmetry of relativistic Vlasov gas and its accretion in Kerr-Newman black hole." pith.science (2026). https://pith.science/paper/KNUYFVXY

@misc{pith2026250608341,
  author       = {Pith},
  title        = {Pith review of: A discussion on the symmetry of relativistic Vlasov gas and its accretion in Kerr-Newman black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNUYFVXY}},
  note         = {Machine review of arXiv:2506.08341}
}
abstract

We investigate the kinetic properties of collisionless Vlasov gas in Kerr-Newman spacetime, analyzing how spacetime symmetries constrain the distribution functions. The distribution function is shown to depend on the constants of motion ($m, E, L_z, L$) and a configuration variable $Q^{3}$. Furthermore, we discuss in detail the different scenarios of charged particles being captured or scattered by the black hole in Kerr-Newman spacetime, and analytically derive the boundaries of the absorption and scattering domains in phase space. Within the Locally Non-Rotating Frame, we compute particle number density, energy density, principal pressures, and construct a set of diagnostic parameters to quantitatively measure the anisotropy information of the Vlasov gas (relative to a perfect fluid), and provide the analytic results of these physical quantities at spatial infinity. In addition, we discuss the accretion rates of the Kerr-Newman black hole and obtain their analytic expressions. Numerical results for the relative (normalized) mass and energy accretion rates reveal an identical parametric dependence: both are suppressed as the black hole's rotation $a$ and charge $Q$ increase. Conversely, the absolute value of the normalized angular momentum accretion rate (which is negative) increases with $a$, and $Q$ also moderately influences the angular momentum accretion rate through its effect on spacetime geometry. Accretion of weakly charged plasma drives charged black holes toward electrical neutrality while reducing angular momentum, ultimately favoring evolution toward Schwarzschild configurations. These findings provide new insights into kinetic accretion processes in spacetime geometries.

Figures

Figures reproduced from arXiv: 2506.08341 by the authors.

Figure 1
Figure 1. Critical parameters for a Kerr-Newman black hole ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. (a) Normalized particle number density nJ /nJ∞ in the comoving frame and (b) normalized particle number density n/n∞ in the LNRF, both evaluated on the equatorial plane. Dashed lines in (b) display the rescaled quantity ˜n regularized for horizon divergence. Fig. 2b displays the LNRF particle number density n, which similarly decreases with r. The tetrad factor e −ν induces divergence near the horizon rH; regulariza… view at source ↗
Figure 3
Figure 3. (a) Energy density ε and rescaled energy density ˜ε ≡ e −2ν ε. (b) Principal pressures Pi and rescaled radial pressure P˜ 1 ≡ e −2µ1P1, illustrating the anisotropic nature of Vlasov gas. Fig. 3a demonstrates the monotonic radial decrease of energy density ε, mirroring the behavior observed for particle number density. This correlation stems from the constant energy of individual particles along their worldlines, com… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Normalized (a) mass and (b) energy accretion rate parameters for characteristic black [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Angular momentum accretion rate parameters for characteristic black holes, normalized [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Coordinate Independence of the Schwarzschild Black Hole Accretion Vlasov Gas Model

    gr-qc 2026-07 conditional novelty 4.0 of 10

    Vlasov gas accretion onto a Schwarzschild black hole is shown explicitly to have coordinate-invariant density, pressures, and accretion rates, with captured particles carrying mean energy m0+kBT and lower specific entropy.

Reference graph

Works this paper leans on

41 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hoyle F and Lyttleton R A 1939Math. Proc. Camb. Phil. Soc.35405–415

    work page

  2. [2]

    Bondi H and Hoyle F 1944Mon. Not. R. Astron. Soc.104273–282

    work page

  3. [3]

    Bondi H 1952Mon. Not. R. Astron. Soc.112195–204 13 IOP PublishingJournalvv(yyyy) aaaaaa Authoret al

    work page

  4. [4]

    Space Sci.15153–160

    Michel F C 1972Astrophys. Space Sci.15153–160

    work page

  5. [5]

    Phys.34856–882

    J¨ uttner F 1911Ann. Phys.34856–882

    work page

  6. [6]

    Synge J L 1934Trans. R. Soc. Can.28127–171

    work page

  7. [7]

    Rev.1221342–1365

    Tauber G E and Weinberg J W 1961Phys. Rev.1221342–1365

    work page

  8. [8]

    Israel W 1963J. Math. Phys.41163–1181

    work page

Show all 41 references
  1. [9]

    Ehlers J 1971 General relativity and kinetic theoryGeneral Relativity and Cosmologyed Sachs R K (New York: Academic Press) pp 1–70

    1971

  2. [10]

    Reidel) pp 1–125

    Ehlers J 1973 Survey of general relativity theoryRelativity, Astrophysics and Cosmologyed Israel W (Dordrecht: D. Reidel) pp 1–125

    1973

  3. [11]

    Cercignani C and Kremer G M 2002The Relativistic Boltzmann Equation: Theory and Applications(Basel: Birkh¨ auser)

  4. [12]

    Narayan R, Yi I and Mahadevan R 1995Nature374623–625

  5. [13]

    Akiyama K, Alberdi A, Alef Wet al.2019Astrophys. J. Lett.875L5

  6. [14]

    Akiyama K, Alberdi A, Alef Wet al.2022Astrophys. J. Lett.930L16

  7. [15]

    Andr´ easson H, Kunze M and Rein G 2014Commun. Math. Phys.329787–808

  8. [16]

    Hadˇ zi´ c M, Lin Z and Rein G 2021Arch. Ration. Mech. Anal.2411–89

  9. [17]

    Proc.1548134–155

    Sarbach O and Zannias T 2013AIP Conf. Proc.1548134–155

  10. [18]

    Quantum Grav.31085013

    Sarbach O and Zannias T 2014Class. Quantum Grav.31085013

  11. [19]

    Proc.1577192–207

    Sarbach O and Zannias T 2014AIP Conf. Proc.1577192–207

  12. [20]

    Acuna-Cardenas R O, Gabarrete C and Sarbach O 2022Gen. Relativ. Gravit.547

  13. [21]

    Quantum Grav.34095007

    Rioseco P and Sarbach O 2017Class. Quantum Grav.34095007

  14. [22]

    Rioseco P and Sarbach O 2017J. Phys. Conf. Ser.831012009

  15. [23]

    Gamboa A, Gabarrete C, Dom ´ ınguez-Fern´ andez Pet al.2021Phys. Rev. D104083001

  16. [24]

    Space Sci.367109

    Liao J W and Liu D J 2022Astrophys. Space Sci.367109

  17. [25]

    Mach P and Odrzywo lek A 2021Phys. Rev. Lett.126101104

  18. [26]

    Mach P and Odrzywo lek A 2021Phys. Rev. D103024044

  19. [27]

    Mach P and Odrzywolek A 2022arXiv:2202.02173

    arXiv

  20. [28]

    Mach P, Cie´ slik A and Odrzywo lek A 2023Phys. Rev. D108124057

  21. [29]

    Cie´ slik A, Mach P and Odrzywo lek A 2024Phys. Rev. D110084014

  22. [30]

    Cie´ slik A and Mach P 2020Phys. Rev. D102024032

  23. [31]

    Li P, Yang J H and Xu S W 2025Phys. Lett. B866139555

  24. [32]

    Cie´ slik A, Mach P and Odrzywo lek A 2022Phys. Rev. D106104056

  25. [33]

    Mach P and Odrzywo lek A 2023arXiv:2306.02279

    arXiv

  26. [34]

    Li P, Liu Y Q and Zhai X H 2023Phys. Rev. D108124022

  27. [35]

    Rioseco P and Sarbach O 2023arXiv:2302.12849

    arXiv

  28. [36]

    Quantum Grav.40193001

    Rein G 2023Class. Quantum Grav.40193001

  29. [37]

    Quantum Grav.33155008

    Ames E, Andr´ easson H and Logg A 2016Class. Quantum Grav.33155008

  30. [38]

    Dark Universe42101292

    Cai Z Q and Yang R J 2023Phys. Dark Universe42101292

  31. [39]

    Carter B 2009Gen. Relativ. Gravit.412873–2938

  32. [40]

    Straumann N 2013General Relativity: With Applications to Astrophysics(Springer, Berlin)

  33. [41]

    J.178347–370 14

    Bardeen J M, Press W H and Teukolsky S A 1972Astrophys. J.178347–370 14

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