REVIEW 2 major objections 6 minor 49 references
Coordinate Independence of the Schwarzschild Black Hole Accretion Vlasov Gas Model
T0 review · 2 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read Coordinate choice does not change the density, pressures, or accretion rates of a collisionless gas falling into a Schwarzschild black hole.
desk verdict The coordinate-invariance theorem is real and the expansions check out, but the abstract's 3/2 k_B entropy reduction is an artifact of mixing entropy-flux conventions; the paper needs revision, not rejection. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the most general stationary spherically symmetric metric ds² = −b²dt² − 2h dtdr + a²dr² + r²(dθ² + sin²θ dφ²), together with the identity a²b² + h² = 1 that holds for the Schwarzschild spacetime in all coordinate representations considered (orthogonal, Eddington-Finkelstein, and Painlevé-Gullstrand type). This identity makes the volume element √−g = r² sinθ and reduces the conservation equations to ∂_r(r²J^r) = 0 and ∂_r(r²T^r_t) = 0, so the accretion rates take the simple coordinate-invariant forms −4πr²J^r and −4πr²T^r_t. Action-angle variables are constructed to solve the Vlasov equation, and the absorption/scattering split is controlled by the critical angular momen
What would settle it
Recompute the entropy accretion rate with the standard Boltzmann entropy current S^μ = −k_B ∫ p^μ f(ln f − 1) dvol; if the specific entropy of accreted particles then differs from the global average by 1/2 k_B instead of 3/2 k_B, the paper's entropy claim is refuted while its coordinate-independence claim is untouched.
Extended reading notes
Core claim
For stationary, spherically symmetric Vlasov gas accretion onto a Schwarzschild black hole, the particle number density n = √(−g_{μν}J^μJ^ν), the eigenvalues of the mixed stress-energy tensor T^μ_ν (energy density ρ, radial pressure p_rad, tangential pressure p_tan), and the accretion rates ṅ and Ė are invariant under coordinate changes, even though the components J^μ and T^{μν} are coordinate-dependent. The simple flux formulas −4πr²J^r and −4πr²T^r_t hold in every coordinate system because of the identity a²b² + h² = 1 satisfied by the Schwarzschild metric in the general stationary spherically symmetric form. In the classical (Maxwell–Jüttner) limit, the mean energy of captured particles i
Load-bearing premise
The 3/2 k_B entropy reduction depends on an entropy-flux definition (Eq. 139) that omits the standard '+f' term; with the conventional Boltzmann entropy current used for the global average, the reduction changes to 1/2 k_B, though the coordinate-independence claims are unaffected.
Editorial extensions
If this is right
- The accretion theory becomes fully geometric: any coordinate system in the ansatz yields the same density, pressures, and accretion rates, so future work can choose coordinates for convenience (e.g., horizon-penetrating for numerics) without worrying about physical predictions.
- The particle and energy accretion rates take the simple, universal forms −4πr²J^r and −4πr²T^r_t at every radius, not just at the horizon, because the metric identity reduces the flux conservation equations to radial constancy.
- The angular-momentum filtering means the black hole preferentially absorbs lower-energy, lower-entropy particles; in the classical limit their mean energy is m₀+k_BT instead of m₀+3/2 k_BT, which changes how the black hole's mass and entropy budgets are computed in kinetic models.
- At low temperature the three statistics (Fermi-Dirac, Maxwell-Jüttner, Bose-Einstein) converge, so the coordinate-independence and energy-filtering results are robust across quantum statistics.
Reading between the lines
- Using the standard Boltzmann entropy current (with the −1 term) instead of the truncated flux in Eq. (139) would shift the specific-entropy reduction to 1/2 k_B; the qualitative conclusion that accreted particles carry lower entropy would survive, but the quantitative 3/2 k_B figure would change.
- The same angular-momentum filtering should operate in other spherically symmetric spacetimes: whenever an effective potential has a centrifugal barrier, captured particles will be biased toward lower energies, so the mean-energy deficit m₀+k_BT versus m₀+3/2 k_BT may be a generic feature of collisionless capture.
- The coordinate-invariant scalars n, ρ, p_rad, p_tan provide natural diagnostic quantities for numerical simulations of kinetic accretion, which often use horizon-penetrating coordinates; comparing these scalars across codes with different coordinate choices would directly test implementation consistency.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies stationary, spherically symmetric accretion of a collisionless Vlasov gas onto a Schwarzschild black hole, using the general stationary metric with off-diagonal g_tr = -h(r). It constructs action-angle variables, solves the Liouville/Vlasov equation, and computes the particle current and stress-energy tensor for Fermi-Dirac, Maxwell-Jüttner, and Bose-Einstein distributions. The central claim is that the particle number density n = sqrt(-g^{μν}J_μJ_ν), the eigenvalues of the mixed stress-energy tensor (energy density ρ, radial pressure p_rad, tangential pressure p_tan), and the accretion rates ˙n and ˙E are independent of the coordinate choice among Schwarzschild, Eddington-Finkelstein, and Painlevé-Gullstrand-type coordinates. Asymptotic expansions at infinity and near the horizon are presented, together with numerical results at finite radii. The paper also claims that in the classical limit captured particles have mean energy m_0 + k_B T and specific entropy lower than the global Maxwell-Jüttner average by 3/2 k_B.
Significance. If the coordinate-invariance result stands, it is a useful formal clarification of the Vlasov accretion formalism: the explicit h_0/H_0 cancellations in Secs. VIII and IX support the claim that the physical observables are well defined, and the low-temperature horizon results reproduce the known Rioseco-Sarbach values. The paper contains no parameter fitting, and the expansions are detailed enough to be checked. However, part of the invariance is by construction—n is a norm of a vector, the pressures are eigenvalues of a tensor, and the accretion rates are conserved Killing currents—so the formal invariance alone is not a new physical principle. The advertised entropy-reduction result is not supported as stated: the entropy flux in Eq. (139) is defined with a nonstandard f ln f current and a covariant momentum component, and using the standard Boltzmann entropy current changes the numerical claim from 3/2 k_B to 1/2 k_B. Since this result appears in the abstract and the concluding section, the paper requires revision before it can be accepted.
major comments (2)
- [Sec. X, Eq. (139)] The entropy flux is defined as 4πr² k_B ∫_{abs} f ln f p_r dvol. This is not the standard relativistic Boltzmann entropy current S^μ = -k_B ∫ p^μ f(ln f - 1)dvol. Two differences matter: (i) the covariant momentum component p_r is used instead of the contravariant p^r entering S^r, so the flux is not a contraction of the entropy current with the radial normal and is not manifestly coordinate-invariant; (ii) the +f term in ln f - 1 is omitted. When the standard entropy current is used consistently for both the absorbed subensemble and the asymptotic thermal gas, the absorbed specific entropy is k_B(-ln A + z + 2), not k_B(-ln A + z + 1). The global value quoted in the same section, k_B(-ln A + z + 5/2), uses the standard convention. The claimed reduction is therefore 1/2 k_B, not 3/2 k_B. The abstract and Sec. XII's thermodynamic conclusion do not follow from the calculation as written.
- [Sec. X, Eq. (140)] The substitution -4π∫ f p_r dvol → ˙n and -4π∫ E f p_r dvol → ˙E is not an identity for the metric (1). The accretion rates are defined through the contravariant component J^r = ∫ p^r f dvol, and p^r = g^{rμ}p_μ = b² p_r + h E when a²b²+h² = 1. Thus the integral over p_r differs from the integral defining ˙n by metric-dependent terms involving ∫ E f dvol. Even aside from the missing +f term, Eq. (139) does not define the physical entropy accretion rate unless this relation is established or the flux is redefined with p^r. The authors should either prove the required cancellation or replace Eq. (139) by the standard S^r = -k_B ∫ p^r f(ln f - 1)dvol.
minor comments (6)
- [Eq. (84)] The expression for n∞ is written with g_{tt}, g_{tr}, g_{rr} as if using covariant components with the metric itself; since the quantity is sqrt(-g^{μν}J_μJ_ν), the inverse metric should appear. Please correct the notation to avoid ambiguity.
- [Fig. 1 caption] The caption lists 'p^H_rad/n^H' twice; one instance is presumably meant to be a different quantity (e.g., p^H_tan/n^H). Please fix.
- [Sec. II, Eq. (3)] The identity a²b² + h² = 1 is stated for the Schwarzschild metric. Since it is used throughout as a restriction on the metric family, make explicit that it holds for the coordinate representations considered here and is not a property of a generic stationary spherically symmetric spacetime such as Reissner-Nordström.
- [Abstract and Sec. I] The phrase 'most general stationary spherically symmetric spacetime' is stronger than what is analyzed: the paper specializes to Schwarzschild with the identity (3) and particular expansions for h(r). Please qualify the scope.
- [Sec. X, after Eq. (141)] The global specific entropy σ∞ = k_B(-ln A + z + 5/2) is quoted without derivation. Give the standard Maxwell-Jüttner entropy density formula so the comparison with Eq. (141) is transparent and the convention is clear.
- [General] There are minor typographical issues, e.g., 'similer' before Eq. (122), and in Table I the entries for z=10 and z=30 are identical across ε values to many digits; this may be intended but should be commented on.
Circularity Check
Coordinate-invariance theorem is partly definitional for scalar observables, and the advertised 3/2 k_B entropy reduction follows from a nonstandard entropy-flux convention rather than from the accretion dynamics; the core asymptotic calculations are otherwise self-contained.
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self definitional
[Sec. VII, Eq. (64) and Sec. VII A, text after Eq. (62)]
"The particle number density is defined as n=\sqrt{-g_{\mu\nu}J^\mu J^\nu}. This quantity is a scalar and therefore coordinate-independent... These three scalars \rho, p_{\rm rad}, p_{\rm tan} are coordinate-independent, as they are eigenvalues of a tensor."
The advertised coordinate independence of n, \rho, p_{\rm rad}, and p_{\rm tan} is introduced by construction: n is defined as the norm of the vector J^\mu, and \rho, p_{\rm rad}, p_{\rm tan} are defined as eigenvalues of the mixed stress-energy tensor. Both are scalars by the definitions, so their invariance under coordinate transformations is immediate and does not need a separate proof. The later asymptotic expansions show the expected cancellations of h_0 and H_0, but those expansions verify the definitions rather than independently deriving the invariance.
-
other
[Sec. X, Eqs. (139)-(141) and the following paragraph]
"The entropy flux through a spherical surface of arbitrary radius can be defined as \dot S=4\pi r^2 k_B \int_{\rm abs} f(x,p)\ln f(x,p) p_r d{\rm vol}_x(p). ... \sigma_{\rm abs}=-k_B\ln A+\bar E/T=k_B(-\ln A+z+1). At infinity... specific entropy is given by \sigma=k_B(-\ln A+z+5/2). This indicates that the entropy per particle of the accreted particles is lower than the global average by 3/2 k_B."
The 3/2 k_B reduction is fixed by the chosen entropy-flux convention, not derived from the dynamics. Eq. (139) uses p_r and omits the +f term of the standard Boltzmann entropy current S^\mu=-k_B\int p^\mu f(\ln f-1)d{\rm vol}. With that choice Eq. (141) gives \sigma_{\rm abs}=k_B(-\ln A+z+1). Comparing to the standard global value \sigma_\infty=k_B(-\ln A+z+5/2) yields exactly (z+5/2)-(z+1)=3/2 by construction. Using the standard current consistently for both subensembles gives \sigma_{\rm abs}=k_B(-\ln A+z+2), i.e. only 1/2 k_B below the global average. The advertised reduction is therefore a convention artifact.
full rationale
The main derivation chain is explicit and self-contained: metric form (1), equations of motion, action-angle variables, the distribution (33), phase-space integrals (44)-(49), integration limits (50)-(57), near-infinity and near-horizon expansions, and the accretion rates (112)-(113) are direct calculations with no fitted parameters and no predicted quantity that reduces to a fit. The critical angular momentum l_c(E) in Eq. (51) is derived in-text from R=0 and \partial_r R=0; Ref. [47] (same authors) is cited only as a 'well-known expression' and is not load-bearing. No uniqueness theorem is imported from the authors' prior work. However, the coordinate-invariance claims for n and the pressures are partly definitional: n is a vector norm and the pressures are tensor eigenvalues, so their invariance is built into the definitions. More importantly, the abstract-level claim that the specific entropy of accreted particles is lower by 3/2 k_B is convention-dependent. Eq. (139) defines the entropy flux without the standard +f term and with p_r; Eq. (141) then gives \sigma_{\rm abs}=k_B(-\ln A+z+1), and comparing to the standard global value k_B(-\ln A+z+5/2) makes the 3/2 k_B difference a direct consequence of the chosen definitions. Using the standard Boltzmann current consistently changes the reduction to 1/2 k_B. Thus the coordinate-independence portion has substantial independent content, but the advertised entropy-reduction result is partially circular by construction.
Assumptions & free parameters
assumptions (4)
- domain assumption Schwarzschild background and the identity a2b2+h2=1 hold for all coordinate representations considered
- domain assumption The steady-state distribution is f = delta(P0-m0) A/(exp[(E-mu)/kBT]+epsilon), with T and mu interpreted only as labels away from infinity
- standard math Absorption/scattering split is determined by the critical angular momentum l_c(E) of Eq. (51)
- ad hoc to paper Entropy flux definition Eq. (139) uses f ln f without the standard -f term of the Boltzmann entropy current
Cite this review
Pith. "Pith review of Coordinate Independence of the Schwarzschild Black Hole Accretion Vlasov Gas Model." pith.science (2026). https://pith.science/paper/DOA2DXEP
@misc{pith2026260719815,
author = {Pith},
title = {Pith review of: Coordinate Independence of the Schwarzschild Black Hole Accretion Vlasov Gas Model},
year = {2026},
howpublished = {\url{https://pith.science/paper/DOA2DXEP}},
note = {Machine review of arXiv:2607.19815}
}
abstract
This paper presents a detailed study of the coordinate dependence of Vlasov gas accretion onto a Schwarzschild black hole. Asymptotic results at infinity and near horizon are obtained via Taylor expansions for three different statistical distributions within the framework of the most general stationary spherically symmetric spacetime. Our findings demonstrate that the particle number density, energy density, radial and tangential pressures, and accretion rates are independent of the coordinate choice, even though individual components such as the particle current density and the stress-energy tensor explicitly depend on the coordinate system. Consequently, the accretion theory can be formulated without reference to any particular coordinate system. We also show that the mean energy of the accreted particles is $m_0+k_BT$, lower than the mean energy $m_0+\frac{3}{2}k_BT$ of the Maxwell-Boltzmann system in the classical limit. And the specific entropy of the accreted particles is lower than the global average by $\frac{3}{2}k_B$. This is because particles of lower energy are more easily accreted, while particles of higher energy are more readily scattered. We also present numerical results at finite radii for the relevant physical quantities.
Figures
Figures from the paper (2 more)
Reference graph
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