Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

Pierre-Henri Chavanis

Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Maximizing entropy at fixed mass-energy and particle number yields the Fermi-Dirac distribution, the barotropic equation of state, and the Tolman-Oppenheimer-Volkoff and Tolman-Klein equations for a spherical relativistic Fermi gas.

desk verdict A careful, honest synthesis of known results—entropy maximization plus the static spherical metric yields TOV—but not a new derivation. read the letter →

arxiv 1908.10806 v1 pith:VAA6GRND submitted 2019-08-28 gr-qc

Pierre-Henri Chavanis This is my paper · ORCID

classification gr-qc MSC 83C5582B3085A15 PACS 04.40.Dg05.70.-a05.70.Fh95.30.Sf95.35.+d

keywords self-gravitatingFermigasgeneralrelativitymaximumentropyprincipleTolman-Oppenheimer-VolkoffequationsTolman-KleinrelationsFermi-Diracstatisticalensemblescaloriccurves

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

The paper constructs a statistical mechanics for a spherical self-gravitating Fermi gas in general relativity, claiming that a single variational principle—maximizing the entropy at fixed mass-energy and particle number—produces the full equilibrium structure: the Fermi-Dirac distribution, the barotropic equation of state, the Tolman-Oppenheimer-Volkoff hydrostatic equations, and the Tolman-Klein relations. The derivation is deliberately general: it holds for any convex entropy functional of the form $S = -k_B \int C(f)\, d^3r\, d^3p$, with Fermi-Dirac entropy serving as the illustrative example. If the claim is right, one does not need to impose hydrostatic equilibrium by hand; it falls out of thermodynamics, and the caloric curves of relativistic fermion stars follow from solving the differential equations the paper supplies. This matters because the same machinery then covers white dwarfs, neutron stars, and fermionic dark-matter halos within one formalism, and it clarifies which results in relativistic astrophysics genuinely require the Einstein equations versus which follow from entropy maximization alone.

What carries the argument

The load-bearing object is the entropy functional $S = \int s(r)[1-2GM(r)/(rc^2)]^{-1/2} 4\pi r^2 dr$ with local entropy density $s = -k_B \int C(f)\, dp$, where $C$ is convex, combined with $M(r) = (1/c^2)\int_0^r \varepsilon\, 4\pi r^2 dr$ and the particle number $N$ using the same proper volume factor. Varying $S$ at fixed $Mc^2$ and $N$, the convexity of $C$ makes the local extremum a global maximum and gives the Fermi-Dirac (or generalized) distribution; the global Lagrange multipliers then enforce the Tolman-Klein relations, and the derivative of the local thermodynamic relations converts them into the TOV equation. The two-step maximization—local thermodynamic equilibrium first, then global—is the machinery that separates microphysics (equation of state) from gravity (hydrostatic balance).

What would settle it

Take the identical Fermi-Dirac gas in a spherical box at fixed $N$ and $R$, compute equilibrium configurations by directly integrating the TOV equations with the Fermi equation of state, and compare the mass and $T_\infty$ versus binding-energy curves to those obtained by maximizing the entropy functional; any mismatch for one $(N,R)$ would falsify the claimed identity of the two routes.

Watch

Extended reading notes

Core claim

For a spherically symmetric system with metric $ds^2 = e^{\nu} c^2 dt^2 - r^2(d\theta^2 + \sin^2\theta\, d\varphi^2) - e^{\lambda} dr^2$, the paper takes the entropy and particle number as integrals over the proper volume element $e^{\lambda/2}4\pi r^2 dr$, with $e^{-\lambda}=1-2GM(r)/(rc^2)$, and maximizes $S$ at fixed mass-energy $Mc^2$ and particle number $N$. The first-order variation yields a constant $α = \mu(r)/k_B T(r)$ and $β_\infty = 1/k_B T_\infty$, from which the Tolman-Klein relations $T(r)=T_\infty e^{-\nu/2}$, $\mu(r)=\mu_\infty e^{-\nu/2}$, and the Oppenheimer-Volkoff equation $\frac{dP}{dr} = -\frac{\varepsilon+P}{c^2}\frac{GM(r)/r^2 + 4\pi G P r/c^2}{1 - 2GM(r)/(rc^2)}$ follow. The same calculation in the nonrelativistic limit recovers uniform temperature, the Gibbs law $\mu(r)+m\Phi(r)=\text{constant}$, and Newtonian hydrostatic equilibrium. The author stresses that this is not a derivation of the full Einstein equations, because the metric coefficient in the proper volume element is already taken from the Einstein equation.

Load-bearing premise

The calculation assumes spherical symmetry and uses the Einstein equation's metric coefficient $1 - 2GM(r)/(rc^2)$ as the volume factor in the entropy, so if that metric relation is not valid during the approach to equilibrium, the derived TOV and Tolman-Klein equations do not follow.

Editorial extensions

If this is right

The equilibrium structure of a relativistic Fermi gas—white dwarfs, neutron stars, fermion dark-matter halos—is fixed once the entropy functional and the conserved quantities $M$ and $N$ are chosen; no separate hydrostatic assumption is needed.
The caloric curves $T_\infty(E)$ and phase diagrams of self-gravitating fermions in general relativity can be constructed by integrating the provided TOV-type equations with Fermi integrals, determining $T_0$ and $\alpha$ from the particle-number constraint.
At equilibrium the temperature and chemical potential vary with the metric coefficient: $T(r) = T_\infty e^{-\nu/2}$ and $\mu(r) = \mu_\infty e^{-\nu/2}$, so equilibrium in general relativity means gradients of $T$ and $\mu$, not uniformity.
Microcanonical and canonical ensembles yield identical equilibrium configurations but different stability ranges; canonical stability implies microcanonical stability, not conversely, and stability changes occur at turning points of energy or temperature.
A thermodynamically stable state is dynamically stable; in general relativity, microcanonical stability is conjectured equivalent to dynamical stability with respect to the Vlasov-Einstein equations, so the late part of the equilibrium series is expected to be dynamically unstable—unlike Newtonian gravity, where all monotone isotropic equilibria are stable.

Reading between the lines

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

The derivation's dependence on a metric coefficient imported from the Einstein equations means the 'thermodynamic derivation of TOV' is conditional: in modified-gravity or non-spherical settings, the same entropy maximization would give different equilibrium equations, so entropy arguments alone cannot single out Einstein gravity.
Because the argument works for any convex $C(f)$, generalized power-law entropies become just another route to a barotropic star: the entropy only determines $P(\varepsilon)$, while gravity enters only through the volume factor, suggesting that all equilibrium self-gravitating fluids, whatever their statistics, obey the same TOV skeleton.
The Tolman temperature gradient in the post-Newtonian limit, $\nabla T/T = g/c^2$, is tiny but in principle observable; a precision temperature map of a hot optically thin gas in a strong gravitational field would provide a direct test of the relativistic equilibrium predicted here.
A numerical eigenvalue analysis of the second variations of $S$ and $F$ for specific $N$ and $R$—not given in this paper—would test the paper's conjecture that microcanonical stability coincides with dynamical stability, settling whether the branch after the first turning point is genuinely dynamically unstable.

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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

Worth knowing: this is a review-style derivation paper, not a new physics claim. Chavanis assembles the maximum-entropy route to the TOV equations and Tolman-Klein relations for an arbitrary convex entropy, with the Fermi-Dirac case worked out in detail. The derivations are careful and the paper is unusually explicit about its own limits: Sec. III.E.7 says in plain words that the metric coefficient (105) is imported from the Einstein equation and that the full Einstein equations are not derived. That honesty deserves credit.

What is actually new is the systematic presentation: the two-step maximization (local then global), the integrated Gibbs-Duhem relation for general entropies (Appendix E), and the clean separation of what follows from thermodynamics versus what comes from the metric. The appendices are useful references for anyone building caloric curves for relativistic fermions. The nonrelativistic limit recovery is done carefully.

The soft spots are proportionate. Novelty is genuinely low: the entropy-maximization route to TOV and Tolman-Klein is already in Tolman (1930), Klein (1949), Ipser (1980), and Bilic-Viollier (1999), and the author cites these. What the paper adds is unification and pedagogical clarity, not a new result. The abstract says entropy maximization 'implies' the TOV equations without immediately flagging that the metric (105) is substituted in by hand. The stress-test note is correct on the logic: the implication is conditional on the static spherical Einstein metric. But the paper itself says exactly this in Sec. III.E.7, so it is a scope limitation, not a flaw.

Also, the paper is very long (318 references) and self-citation heavy, but since the key equations are re-derived in the text, that is not a reliability problem.

Who gets value: someone who wants a self-contained reference for statistical mechanics of relativistic self-gravitating fermions, or a student wanting to see the variational derivation written out. A referee should engage; the paper is well above the desk-reject line. It is not a breakthrough, but it is a competent, honest synthesis.

Referee Report

0 major / 4 minor

Summary. This paper develops a maximum-entropy formalism for the statistical equilibrium of spherically symmetric self-gravitating systems in general relativity. It works for a general convex entropy of the form S = -k_B ∫ C(f) drdp and applies it explicitly to the Fermi-Dirac entropy. The central construction is to write the total entropy and particle number with the proper volume element built from the metric coefficient e^{-λ} = 1 - 2GM(r)/(rc²) (Eq. 105), and then extremize S at fixed mass-energy and particle number. The first-order conditions yield a uniform α = μ/(k_B T), the Tolman temperature gradient, the Oppenheimer-Volkoff equation, and, using the Einstein-equation relation (106), the Tolman and Klein relations. The paper also derives the canonical-ensemble version, gives the full set of equations for constructing caloric curves in terms of T(r) or ϕ(r), recovers the nonrelativistic limit, and discusses ensemble inequivalence and dynamical stability. Appendices establish the Gibbs-Duhem relation and the hydrostatic condition for a general form of entropy, and the paper is careful to state which results come from entropy maximization and which are imported from the Einstein equations.

Significance. The paper is a systematic synthesis that unifies previous derivations (Tolman, Klein, Bilic-Viollier, Ipser and others) into one coherent variational framework. Its main value is as a reference derivation and a toolkit: it provides the equations needed to compute caloric curves and phase diagrams for relativistic self-gravitating fermions, and it makes the logical structure of the derivation transparent. The extension of the hydrostatic-equilibrium result to arbitrary convex entropy in Appendix C is a strength, as is the explicit separation of what is derived from entropy maximization from what is assumed about the metric. The derivation is conditional on the static spherical metric relations (105) and (106), but this is openly acknowledged in Sec. III.E.7; it is a scope limitation, not an internal inconsistency. The paper is not a new physical prediction but a careful and useful systematization that should serve as a reference.

minor comments (4)

[Abstract; Secs. III.E.4-E.5] The wording 'It also implies the Tolman-Oppenheimer-Volkoff equations ... and the Tolman-Klein relations' is stronger than what is shown: the derivation uses the Einstein-equation metric coefficient (105) in Eqs. (122)-(123) and the Einstein-equation relation (106) to obtain the Tolman equation (156). Since Sec. III.E.7 already states this caveat, please qualify the abstract and the 'implies' statements in Secs. III.E.4-E.5, for example by writing 'together with the static spherical Einstein metric' or 'given the metric relations (105) and (106)', so that the conditional nature is visible at the point of the claim and not only in the later caveat.
[Sec. III.E.2] There is a typo in the sentence 'The vanishing of Eq. (140) with repsect to variations on δn': 'repsect' should be 'respect'.
[Sec. IV.G] In the reduction of Eq. (233) to the Newtonian expression (60), the identity ∫ P dV = (2/3) Ekin is used without comment; stating this identity explicitly would help the reader follow the c → ∞ limit.
[Sec. III.H] The potential ϕ(r) is defined in Eqs. (181)-(183) using absolute values |μ_∞| and |α|; a sentence specifying the allowed signs of α and μ_∞ for the Fermi case (and how the T = 0 limit ϕ ≥ 0 arises in Sec. III.K.1) would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: entropy maximization yields TOV and Tolman-Klein relations conditional on the Einstein metric, with the limitation explicitly disclosed.

full rationale

The derivation chain is self-contained in the relevant sense. The paper defines S and N with the proper-volume factor χ(r) = [1 - 2GM(r)/(rc²)]^{-1/2} (Eqs. 122-123), obtained from integrating the Einstein equation (101). The subsequent extremization of S at fixed mass-energy and particle number leads to Eq. (154) for the temperature gradient and, with the thermodynamic relation (147), to the Oppenheimer-Volkoff equation (155). The TOV equation is therefore an output of the variational calculation, not an input: the only gravitational input is the metric coefficient (105), and Sec. III.E.7 explicitly states that the full Einstein equations are not derived and that Eq. (106) 'cannot be derived from the present thermodynamical approach.' The Tolman-Klein redshift relations (158)-(159) are then obtained by combining the thermodynamically derived temperature gradient with the Einstein-derived relation (106), again an openly stated conditional step rather than a hidden circular one. There are numerous self-citations, but the load-bearing variational argument is carried by the displayed equations and by external references (Tolman 1930, Klein 1949, Bilic and Viollier 1999); no fitted parameter is renamed as a prediction and no uniqueness theorem from the author's prior work is invoked to force the result. The central claim is thus conditional on the static spherical Einstein metric, not circular.

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

No free parameters are fitted to data; Lagrange multipliers (alpha, beta_infinity, T0, phi0) are determined by constraints and boundary conditions. No new particles, forces, or conserved quantities are introduced. The central claim rests on two domain assumptions: spherical symmetry and the metric form (105). The paper is transparent that the Einstein equations are not derived.

assumptions (6)

domain assumption The equilibrium configuration is spherically symmetric.
Sec. II opening: 'we assume since the start that the system is spherically symmetric'; Sec. III.A.1 restricts to radial motions. This is an input, not a consequence of the maximum entropy principle in GR.
domain assumption The metric coefficient e^{-lambda(r)} is given by Eq. (105), i.e., the proper volume element in S and N uses the solution of the Einstein equation (101).
Eqs. (122)-(123) use chi = [1 - 2GM(r)/(r c^2)]^{-1/2}; Sec. III.E.7 notes this imports gravitational effects and that the full Einstein equations are not derived.
domain assumption The system is confined in a finite box of radius R or has a surface where density vanishes at T=0.
Sec. II.C and Sec. III.C state a statistical equilibrium state exists only if the system is confined within a box of radius R, otherwise it evaporates.
domain assumption The entropy is of the form S = -k_B ∫ C(f) drdp with C convex, and local thermodynamic equilibrium is described by maximizing entropy density at fixed local energy and particle density.
Appendix C assumes C''(f)>0; the Fermi-Dirac entropy is a special case. The local-equilibrium step is the statistical-mechanics input.
domain assumption The energy-momentum tensor is that of a perfect fluid, Eq. (98).
Eq. (98) in Sec. III.A.1 is used to write the TOV equations and the entropy variation.
standard math Standard thermodynamic identities (Euler theorem, Gibbs-Duhem) are valid locally; the paper derives the integrated Gibbs-Duhem relation for arbitrary entropy in Appendix E.
Appendix E and Sec. III.D use integration by parts and the Euler homogeneous-function theorem; these are standard mathematical results.

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Pith. "Pith review of Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas." pith.science (2026). https://pith.science/paper/VAA6GRND

@misc{pith2026190810806,
  author       = {Pith},
  title        = {Pith review of: Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VAA6GRND}},
  note         = {Machine review of arXiv:1908.10806}
}

abstract

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of self-gravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit $c\rightarrow +\infty$ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

(A1) In that case, the kinetic energy density and the pressure are give n by ǫkin = ∫ f p2 2m dp and P = 1 3 ∫ f p2 m dp

Nonrelativistic limit In the nonrelativistic limit, the kinetic energy of a particle is Ekin = p2 2m . (A1) In that case, the kinetic energy density and the pressure are give n by ǫkin = ∫ f p2 2m dp and P = 1 3 ∫ f p2 m dp. (A2) We have the general relations P = 2 3 ǫkin, E kin = 3 2 ∫ P d r. (A3)

[2] [2]

generalized thermodynamics

Ultrarelativistic limit In the ultrarelativistic limit, the energy of a particle is E = Ekin = pc. (A4) In that case, the energy density and the pressure are given by ǫ = ǫkin = ∫ f pc dp and P = 1 3 ∫ f pc dp. (A5) We have the general relations P = 1 3 ǫ = 1 3 ǫkin, E = Ekin = 3 ∫ P d r. (A6) Appendix B: Virial theorem for Newtonian systems In this Appen...

[3] [3]

II and III

Two-steps derivation To maximize the entropy S at ﬁxed energy E and particle number N , we proceed in two steps as in Secs. II and III. We ﬁrst maximize the entropy density s(r) at ﬁxed energy density ǫ(r) and particle number density n(r) with respect to variations on f (r, p) following the steps of Secs. II D and III D. The variational principle (126) fo...

[4] [4]

We ﬁrst consider the Newtonian gravity case

One-step derivation We now present a one-step derivation of the preceding results. We ﬁrst consider the Newtonian gravity case. The generalized entropy is S = −kB ∫ C(f ) drdp. (C12) The particle number and the mass are given by M = N m = m ∫ f drdp = ∫ mn dr = ∫ ρ dr. (C13) 39 The energy is given by E = Ekin + W = ∫ f Ekin(p) drdp + 1 2 ∫ ρΦ dr, (C14) wh...

[5] [5]

Newtonian gravity We ﬁrst consider the Newtonian gravity case but, for the sake of g enerality, we allow the particles to be relativistic in the sense of special relativity. The extremization of the entropy S at ﬁxed particle number N and energy E leads to a distribution function of the form (see Appendix C) f (r, p) = F [β (Ekin(p) + mΦ( r)) − α0] , (D1)...

[6] [6]

General relativity We now consider the general relativity case. The extremization of t he entropy S at ﬁxed mass-energy E and particle number N leads to a distribution function of the form (see Appendix C) f (r, p) = F [ E(p) kBT (r) − α ] , (D8) where F is deﬁned by Eq. (C4) and where α is constant. According to Eqs. (120) and (D8) the pressure is give n...

[7] [7]

(E1) An extensive variable (energy, entropy,...) is proportional to the ab solute size of the system

Standard derivation The ﬁrst law of thermodynamics can be written as dE = −P dV + T dS + µdN. (E1) An extensive variable (energy, entropy,...) is proportional to the ab solute size of the system. In other words, if one doubles all extensive variables, all other extensive quantities also b ecome twice as large. For example, E(αS, αV, αN ) = αE(S, V, N ), (...

[8] [8]

(C3) with Eq

Direct derivation of the integrated Gibbs-Duhem relatio n for a general form of entropy The local condition of thermodynamical equilibrium, obtained by maxim izing the local entropy at ﬁxed energy density and particle number density, is given by Eq. (C3) with Eq. (C4 ). Substituting Eq. (C3) into Eqs. (113), (114) and (120) we ﬁnd that the particle number...