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Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

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Pith's one-line read This paper shows that the caloric curve of a box-confined classical Boltzmann gas in general relativity is a double spiral and that in the ultrarelativistic limit it settles on a universal hot-spiral curve with maximum mass…

desk verdict Solid derivation of a new ultrarelativistic limit curve for the box-confined relativistic Boltzmann gas; the headline numbers rest on self-cited numerics that deserve an independent check. read the letter →

arxiv 1908.10817 v1 pith:6GKEWYSU submitted 2019-08-28 gr-qc

Pierre-Henri Chavanis This is my paper · ORCID
classification gr-qc PACS 04.40.Dg05.70.-a05.70.Fh95.30.Sf95.35.+d
keywords generalrelativitystatisticalmechanicsself-gravitatingsystemscaloriccurveBoltzmanngasultrarelativisticlimitEmdenequationblack-bodyradiation
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

Statistical equilibrium of a classical self-gravitating gas confined in a box has a caloric curve that is a double spiral, controlled by one compactness parameter $\nu=GNm/Rc^2$. This paper establishes that in the ultrarelativistic normalization $\mathcal{M}=GM/(Rc^2)$, $\mathcal{B}=Rc^4/(GNk_B T_\infty)$, and in the limit $\nu\to 0$, the cold spiral is pushed to infinity while the hot spiral survives as an explicit limit curve. That curve, determined by the general-relativistic Emden equations, has first turning points $\mathcal{M}_{\max}=0.24632$ (density contrast 22.4) and $\mathcal{B}_{\min}=17.809$ (density contrast 10.3). The maximum mass coincides with the self-gravitating black-body radiation value, while the temperature limit is different and has a different physical origin.

What carries the argument

The load-bearing objects are the general-relativistic Emden equations, the dimensionless TOV equations for matter with the linear equation of state $P=\epsilon/3$, together with the three functions $\chi(a)$, $\theta(a)$, $\Delta(a)$ extracted from their solutions. The equation of state $P=\epsilon/3$ is the ultrarelativistic limit of the Maxwell–Jüttner gas, so the hot branch of the double spiral is governed by the same equations as radiation, but with a chemical-potential-dependent prefactor that changes the temperature turning point. These equations supply the parametric limit curve $\mathcal{M}=\chi(a)/2$, $\mathcal{B}=12/(\theta(a)\Delta(a))$, whose first turning points are the quoted constants.

What would settle it

Independently integrate equations (126)–(127) and check whether $a_c=4.70$, $\chi(a_c)=0.493$, $a'_c=3.48$ and $\theta(a'_c)\Delta(a'_c)=0.674$ are reproduced; if not, the quoted turning points $\mathcal{M}_{\max}=0.24632$ and $\mathcal{B}_{\min}=17.809$ are wrong. Alternatively, solve the full TOV statistical-equilibrium equations for a sequence of small $\nu$ in the $(\mathcal{M},\mathcal{B})$ plane and test whether the first turning points converge to the predicted curve as $\nu\to 0$.

Watch

Extended reading notes

Core claim

The central claim is that in the doubly limiting regime $k_BT\gg mc^2$ and $\nu\to 0$, the caloric curve of the general-relativistic classical gas tends to a limit curve $\mathcal{M}=\chi(a)/2$, $\mathcal{B}=12/(\theta(a)\Delta(a))$, obtained by integrating the general-relativistic Emden equations (126)–(127). For $\nu\to 0$ with $\mathcal{M}$ and $\mathcal{B}$ held fixed, the cold spiral is rejected to infinity, so only the hot spiral remains. This asymptotic hot spiral is new; the turning points are the maximum mass $\mathcal{M}_{\max}=0.24632$ and the minimum inverse-temperature parameter $\mathcal{B}_{\min}=17.809$. The curve is similar to, but not identical to, the caloric curve of self-gravitating black-body radiation.

Load-bearing premise

The result rests on the numerical accuracy of the integration of the general-relativistic Emden equations, specifically the constants $a_c=4.70$, $\chi(a_c)=0.493$, $a'_c=3.48$, $\theta(a'_c)\Delta(a'_c)=0.674$, and on the assumption that the $\nu\to 0$ ultrarelativistic limit is uniform, so those turning points are the true limit values.

Editorial extensions

If this is right

  • In the $\nu\to 0$ ultrarelativistic limit, the hot spiral is a universal feature: the caloric curve in $(\mathcal{M},\mathcal{B})$ variables converges to the Emden limit curve for the classical gas.
  • The microcanonical ensemble is stable only up to $\mathcal{M}_{\max}=0.24632$; above this mass the system undergoes a gravitational collapse, presumably to a black hole, on a short dynamical timescale.
  • The canonical ensemble is stable only up to $\mathcal{B}_{\min}=17.809$, establishing a maximum Tolman temperature above which the gas is canonically unstable.
  • Because the maximum mass matches the black-body radiation value while the temperature limit does not, the hot spiral is a distinct branch rather than a copy of the radiation curve.
  • General-relativistic effects destabilize the gas, shrinking the double spiral and eliminating it entirely at $\nu_{\max}=0.1764$.

Reading between the lines

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

  • If the claimed uniform limit is correct, independent numerical solvers for small $\nu$ in $(\mathcal{M},\mathcal{B})$ variables should land on the same spiral, and the quoted constants provide a sharp benchmark for such codes.
  • The cold and hot spirals are obtained by two different $\nu\to 0$ scalings, which suggests the two limits do not commute; an intermediate scaling might show how the cold spiral unwinds into the hot spiral as $\nu$ decreases.
  • The box confinement is artificial; whether external pressure from a surrounding halo could physically play the box's role in ultrarelativistic star clusters would determine whether the hot-spiral collapse is a real astrophysical channel for black-hole formation.
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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

2 major / 5 minor

Summary. Starting from the Boltzmann entropy and the maximum-entropy variational principle for a box-confined classical gas in general relativity, the paper derives the Maxwell-Jüttner equilibrium distribution, the TOV equations with Tolman-Klein relations, and the caloric curve T∞(E) parametrized by the compactness parameter ν. It then studies the ν→0 limit in two complementary normalizations. In the nonrelativistic normalization (Λ, η) the hot spiral recedes and the known cold spiral is recovered. In the new ultrarelativistic normalization (M, B), the cold spiral recedes and the caloric curve tends to a limit curve M = χ(a)/2, B = 12/(θ(a)Δ(a)) obtained from the general relativistic Emden equations; this limit curve exhibits a single hot spiral with first turning points Mmax = 0.24632 and Bmin = 17.809. The paper also compares this result with the caloric curve of self-gravitating radiation and with truncated isothermal models.

Significance. If the numerical constants and the double limit are correct, the paper establishes a new asymptotic caloric curve for the general relativistic classical gas and clarifies its relation to the black-body radiation curve. The analytical chain from entropy maximization to the Emden-equation reduction is coherent and involves no fitted parameters. The main quantitative claims, however, rest on numerical turning-point constants taken from the author's earlier papers, so the strength of the result depends on independent verification of those constants.

major comments (2)
  1. [Sec. VI.H, Eqs. (149)-(152)] The central quantitative claim—the limit curve B(M) with first turning points Mmax = 0.24632 and Bmin = 17.809—rests on the numerical constants ac = 4.70, χ(ac) = 0.493, a'c = 3.48, and θ(a'c)Δ(a'c) = 0.674, which are quoted from Refs. [16] and [29] without specifying the numerical method, the integration accuracy, or error bars. Because these constants determine the headline numbers and the identification of the limit curve, the authors should supply an independent numerical solution of Eqs. (126)-(127), or at least a convergence test and error estimates, before the result can be considered fully established.
  2. [Sec. VI.G-VI.H, Eqs. (142)-(149)] The derivation first takes the ultrarelativistic limit b→0 at fixed ν and then sends ν→0, but the paper does not justify that the two limits commute uniformly in the Emden parameter a. Near the turning points, where the spiral structure makes the caloric curve non-monotone, non-uniform convergence could shift the quoted values of Mmax and Bmin; please provide an explicit estimate of the O(ν) corrections to these turning points or numerical evidence that the approach to Eq. (149) is uniform.
minor comments (5)
  1. [Sec. IV.A, Eq. (81)] The redshift factor in Eq. (81) contains a stray G/c² and should read [1 − 2M̃(r̃)/r̃]^{-1/2}; in the scaled variables the metric term is 2M̃/r̃. As written, the expression is dimensionally inconsistent.
  2. [Secs. II.B and III.B] The phrase 'maximize the entropy S at at fixed energy' appears twice; the duplicated 'at' should be removed.
  3. [Appendix A.2 and Appendix B.2] There are several typographical errors, including 'Begining', 'Isper', and 'galatic'; please correct these.
  4. [Sec. VII] In the ultrarelativistic-limit paragraph, 'In that limit, , using' contains a double comma; please fix the punctuation.
  5. [References] Ref. [1] is listed as 'preprint (Paper I)' and Refs. [28]-[29] as arXiv preprints; if the journal requires published references, these should be updated or supplemented with archival data.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the ν→0 hot-spiral limit curve is derived from the displayed TOV/GR-Emden equations, and the self-cited turning-point constants are independently checkable numerical outputs.

full rationale

The claimed ν→0 ultrarelativistic limit curve is not an input to the derivation. The chain is: Boltzmann entropy maximization gives the Maxwell-Jüttner distribution and the TOV equations (Sec. III); the b→0 (kBT≫mc²) limit reduces the local variables to a P=ε/3 law (Sec. VI.A); substitution into the TOV equations gives the displayed GR Emden equations (126)-(127) (Sec. VI.B); particle number, mass, and Tolman temperature then yield the parametric relations η=12ν²/(θΔ) and Λ=-(χ/2-ν)/ν² (Secs. VI.C-VI.E), which after the rescaling (146) become the limit curve M=χ(a)/2, B=12/(θΔ) of Eq. (149). The turning-point numbers Mmax=0.24632 and Bmin=17.809 are evaluations of the same functions at the turning points of χ(a) and θ(a)Δ(a); they are not fitted parameters and no equation is defined in terms of the target curve. The constants ac=4.70, χ(ac)=0.493 and a'c=3.48, θΔ=0.674 are quoted from the author's earlier work [16] and from the companion [29], but they are parameter-free outputs of the displayed ODEs with stated assumptions (linear equation of state P=ε/3, boundary conditions ψ(0)=ψ'(0)=0) and can be independently reproduced, so the self-citation is not load-bearing in a circular sense. Concerns about numerical accuracy or uniformity of the ν→0, b→0 double limit are correctness risks, not circularity.

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

The derivation introduces no fitted parameters and no new physical entities. It rests on the Boltzmann entropy form, the maximum-entropy variational principle giving TOV and Tolman-Klein equations, box confinement, the ultrarelativistic equation of state P = epsilon/3, and the numerical solution of the resulting Emden equations. The main external input is the set of numerical constants for the general relativistic Emden solutions, taken from the author's earlier work.

assumptions (6)
  • domain assumption The entropy of the classical gas is the Boltzmann entropy density s = -k_B integral f [ln(f/f*) - 1] dp.
    Used in Sec. III.A; this defines the classical gas in contrast to the Fermi-Dirac case in Paper I, and it is taken as the statistical starting point.
  • domain assumption Statistical equilibrium maximizes entropy at fixed mass-energy and particle number, yielding the TOV equations and the Tolman-Klein relations.
    Sec. III.B and III.D; the variational principle is the framework inherited from Paper I and is not re-derived from first principles here.
  • domain assumption The gas is confined within a rigid spherical box of radius R.
    Sec. I and Sec. III; box confinement makes the total mass finite and defines the model. The paper explicitly acknowledges in Appendix B that this is an academic simplification relative to open star clusters.
  • standard math In the ultrarelativistic limit the particle energy is E = pc and the equation of state is P = epsilon/3.
    Sec. VI.A; this follows from the Maxwell-Juttner distribution as b tends to 0 and reduces the gas to a radiation-like linear equation of state.
  • standard math The numerical solutions of the Emden and general relativistic Emden equations used for the turning points are correct.
    Secs. V and VI.G-H; values such as a_c = 4.7 and chi(a_c) = 0.493 are quoted from [16] rather than recomputed in this paper, so correctness is a borrowed numerical assumption.
  • standard math The additive constant in the entropy that scales as -N k_B ln(G m^4 c R^2 / h^3) does not affect the caloric curve or stability thresholds.
    Sec. IV.A remark; the constant is fixed at given N and R, so it shifts the entropy and free energy without changing extrema or turning points.

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Pith. "Pith review of Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas." pith.science (2026). https://pith.science/paper/6GKEWYSU

@misc{pith2026190810817,
  author       = {Pith},
  title        = {Pith review of: Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GKEWYSU}},
  note         = {Machine review of arXiv:1908.10817}
}
abstract

We study the statistical mechanics of classical self-gravitating systems confined within a box of radius $R$ in general relativity. It has been found that the caloric curve $T_{\infty}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $\nu=GNm/Rc^2$. The double spiral shrinks as $\nu$ increases and finally disappears when $\nu_{\rm max}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $\nu$ decreases. Using a normalization $\Lambda=-ER/GN^2m^2$ and $\eta=GNm^2/R k_B T_{\infty}$ appropriate to the nonrelativistic limit, and considering $\nu\rightarrow 0$, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization ${\cal M}=GM/Rc^2$ and ${\cal B}={Rc^4}/{GNk_B T_{\infty}}$ appropriate to the ultrarelativistic limit, and considering $\nu\rightarrow 0$, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the self-gravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.

Figures

Figures reproduced from arXiv: 1908.10817 by the authors.

Figure 1
Figure 1. FIG. 1: Caloric curve of a general relativistic self-gravitating classical gas for [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Caloric curve [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Caloric curve [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Series of equilibria (caloric curve) of the classical King model. The units for the curve [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Series of equilibria (caloric curve) of heavily truncated Maxwell-Boltzmann distributions in general relativity (according [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Complete caloric curve of truncated isothermal star clusters showing the nonrelativistic cold spiral and the relativistic [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

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Reference graph

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