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Thermodynamics of black and white holes in ensemble of Planckons

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes that black hole entropy is an integer counting correlated pairs of its $N$ Planck-mass constituents, $S_{BH}(N)=N(N-1)/2$, which reduces to Bekenstein-Hawking entropy for large $N$.

desk verdict A clean combinatorial toy for black hole entropy whose RN extension is postulated rather than derived, with a real algebraic slip in the equipartition claim; worth a serious referee but only after revision. read the letter →

arxiv 2506.13145 v5 pith:32KVPOGI submitted 2025-06-16 gr-qc hep-ph

G.E. Volovik This is my paper
classification gr-qchep-ph
keywords PlanckonsblackholeentropyTsallis-CirtostatisticsintegerReissner-Nordströmwhitenegativebackreactioncosmologicalconstantquantization
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 builds a toy model in which a black hole of mass $M=N m_P$ is an ensemble of $N$ Planckons, objects of reduced Planck mass $m_P=1/\sqrt{8\pi G}$. Each Planckon carries zero entropy, and all the entropy comes from correlations between pairs, so the entropy is the integer $S_{BH}(N)=N(N-1)/2$; at large $N$ this coincides with the Bekenstein-Hawking area law. Splitting a hole into parts of sizes $N_1$ and $N_2$ then has probability $w=e^{-N_1 N_2}$, which automatically includes the back-reaction correction to Hawking radiation. The same counting is extended to charged Reissner-Nordström holes, where the entropy stays $N(N-1)/2$ independent of charge, to white holes, where it becomes negative, and to the cosmological horizon, where it quantizes the cosmological constant. The point of the model is to give the thermodynamic formulas of black holes a discrete, combinatorial origin rather than a continuum-area origin.

What carries the argument

The central object is the Planckon ensemble: $N$ objects of mass $m_P=1/\sqrt{8\pi G}$ with zero individual entropy, whose only thermodynamic degrees of freedom are the $\binom{N}{2}=N(N-1)/2$ correlations between pairs. The composition rule $\sqrt{S(M_1+M_2)}=\sqrt{S(M_1)}+\sqrt{S(M_2)}$ is the $δ=2$ non-extensive Tsallis-Cirto statistics, and the integer pair count turns it into an exact discrete rule. Splitting probabilities are computed as $w=e^{-\Delta S}$, the entropy difference between final and initial configurations, which reproduces and extends the tunneling picture of Hawking radiation with back reaction. For charged holes the machinery is the split of horizon entropy into a positive outer-horizon part and a negative inner-horizon part, whose Tsallis-Cirto composition gives $S_{RN}=4\pi G M^2$; for the cosmological horizon the same counting gives quantized Hubble-volume entropy and a quantized cosmological constant.

What would settle it

For a Reissner-Nordström black hole of fixed mass $M$ and charge $Q$, compute the standard horizon-area entropy $S=A_+/4G=\pi r_+^2/G$. The paper's composition rule predicts $S=4\pi G M^2$ independent of $Q$; the area law predicts a smaller value that decreases with $Q$. A direct calculation of the thermodynamic entropy, for example from the tunneling rate between charged black-hole states or from a microstate count in a quantum-gravity model, would show which expression is the true entropy and would settle the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the Bekenstein-Hawking entropy, $S=A/(4G)=4\pi G M^2$, is the large-$N$ limit of an exact discrete entropy $S_{BH}(N)=N(N-1)/2$ for a hole built from $N$ Planckons with $M=N m_P$. The entropy is the number of unordered pairs of constituents, that is, the correlations among gravitationally attracted Planckons. This single formula determines the rates of all splitting and merging processes through $w=e^{-\Delta S}$; for example, splitting into parts $N_1+N_2$ has rate $e^{-N_1 N_2}$, and complete evaporation into $N$ free Planckons has rate $e^{-S_{BH}(N)}$. The same pair-counting entropy applies to a Reissner-Nordström black hole with charged Planckons, so charge does not enter the entropy until the extremal limit, where gravitational attraction and Coulomb repulsion cancel and the hole becomes unstable. White holes are the time-reversed objects and carry $S_{WH}(N)=-N(N-1)/2$, which makes their formation an exponentially rare fluctuation without violating the second law.

Load-bearing premise

The load-bearing premise is that the thermodynamic entropy of a charged black hole is the Tsallis-Cirto composition of a positive outer-horizon entropy and a negative inner-horizon entropy, $S=(\sqrt{S_+}+\sqrt{|S_-|})^2=4\pi G M^2$, rather than the usual outer-horizon area law that depends on charge.

Editorial extensions

If this is right

  • Splitting a black hole into two parts with $N=N_1+N_2$ has probability $w=e^{-N_1 N_2}$, so single-Planckon emission has rate $e^{-(N-1)}$ and includes the back-reaction correction $1-m_P/M$ to the Hawking exponent.
  • The probability that a black hole completely evaporates into free Planckons is $e^{-S_{BH}(N)}$, giving entropy a direct probabilistic meaning: the entropy is the negative logarithm of the annihilation probability.
  • The entropy of a Reissner-Nordström black hole is the same as that of an uncharged hole of the same mass, $S=N(N-1)/2$, for all non-extremal charges; adiabatic variation of the fine-structure constant connects the two without entropy change.
  • White holes have negative entropy $-N(N-1)/2$, and their formation from Planckons is a rare thermodynamic fluctuation, so the second law is preserved; an entropy bound holds between the black-hole and white-hole values.
  • In the de Sitter sector, the Hubble volume has quantized entropy $N(N-1)/2$ and the vacuum energy density takes quantized values $Λ=3/(8G^2 N)$, linking horizon thermodynamics to a discrete cosmological constant.

Reading between the lines

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

  • Extension: if black hole entropy literally counts pairwise correlations among $N$ constituents, then a quantum theory of the hole should have a Hilbert-space dimension growing like $\exp[N(N-1)/2]$, with pair states as the fundamental degrees of freedom rather than horizon-area cells.
  • Extension: the charge independence of $S_{RN}$ is a testable departure from the usual area law; near-extremal holes with charge close to the critical value should show no entropy suppression until the instability threshold, unlike standard Bekenstein-Hawking predictions.
  • Extension: the negative entropy of white holes suggests a general principle for horizon thermodynamics, that time-reversed objects carry opposite entropy, which would also assign negative entropy to inner horizons of other charged or rotating black holes.
  • Extension: the model's prediction of a quantized cosmological constant $Λ\propto 1/N$ could be compared with observed bounds on vacuum energy by treating $N$ as the number of Planckons in the current Hubble volume, giving a specific integer value to test.
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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

5 major / 5 minor

Summary. The paper proposes a toy model in which a black hole consists of N Planckons of reduced Planck mass m_P=1/√(8πG). It assigns the integer entropy S_BH(N)=N(N-1)/2, derives splitting probabilities w=e^{-N1N2} including back-reaction effects, assigns negative entropy to white holes, and extends the model to Reissner-Nordström (RN) black holes and to de Sitter space. The central claims are the combinatorial entropy formula, the resulting Planckon emission rate, the charge independence of RN entropy, and the quantization of the cosmological constant.

Significance. If the model is taken as a toy model, it provides a concrete combinatorial interpretation of black-hole entropy (correlated pairs of constituents), yields explicit and falsifiable predictions for splitting and emission probabilities, and naturally incorporates the Parikh-Wilczek back-reaction correction. The Schwarzschild part of the paper is internally consistent and arithmetically transparent. The significance is limited by the fact that the RN charge-independence claim rests on an unproven composition postulate, and the relation of the model to standard black-hole thermodynamics (area law, first law) is not fully clarified. With these limitations made explicit, the paper could be a useful contribution to the discussion of non-extensive statistics and black-hole microstructure.

major comments (5)
  1. [II.B, Eq. (9)] Equation (9) states M = 2N T_H with T_H = m_P/N. Substituting M = N m_P gives N m_P = 2m_P, which holds only for N = 2 and contradicts the claim that the relation holds for large N. The correct large-N relation from the model's thermodynamics is M ≃ N^2 T_H (with T = dM/dS ≃ m_P/N), so the equipartition argument as written does not support the pair-degree-of-freedom interpretation. Please correct the equation and the surrounding argument.
  2. [Introduction and II.B, Eq. (1)] The composition rule √S(M1+M2)=√S(M1)+√S(M2) is exact for the standard entropy S=4πGM², but the proposed entropy S(N)=N(N-1)/2 satisfies it only up to corrections of relative order 1/N. Since this rule is presented as the defining property of Tsallis-Cirto δ=2 statistics, the paper should state explicitly that the model matches that statistics only asymptotically, and clarify whether the splitting probabilities w=e^{-N1N2} in Eq. (11) are exact finite-N results or large-N approximations.
  3. [IV.A, Eq. (22)] The central claim that the RN entropy is independent of Q is based on the composition S_RN=(√S_+ + √|S_-|)², which is introduced as a postulate ("can be considered as"). This rule is not derived from the area law or from the first law; the standard area-law entropy S=A/4=πr_+²/G depends on Q through r_+. The adiabatic argument of §IV.B assumes S_RN is a function of M only (since M is held fixed), so it cannot serve as independent evidence for charge independence. Please state explicitly that Eq. (22) is an additional model axiom, and discuss the relation to, and possible falsification by, the standard first law dM = T_+ dS + Φ dQ.
  4. [IV.B, Eq. (26)] Equation (26) writes S_± with prefactor ±N(N-1)/8, whereas the standard horizon entropies from Eqs. (23)-(24) with M=Nm_P give prefactor N²/8. The replacement of N² by N(N-1) is necessary to obtain the exact integer S_RN=N(N-1)/2 from Eq. (22), but it is not justified as a geometrical result. Note that with the N(N-1)/8 prefactors the composition in Eq. (22) does cancel the αq² dependence exactly; however, the N(N-1) prefactor is an additional quantization assumption that should be labeled as such, otherwise the equations are inconsistent with the preceding area formulas.
  5. [IV.C, detailed-balance calculation] The derivation of 1/T = 4π(r_+ + r_-) = 8πM for the RN case uses the emission probability P_emission = exp(ω/|T_-|) exp(-ω/T_+), which contains an enhancement factor from the inner horizon. This is not the standard Hawking emission rate from the outer horizon, and the resulting T is neither T_+ nor the temperature appearing in the first law. Because this expression is used to conclude that RN thermodynamics coincides with Schwarzschild thermodynamics, the author should justify it or identify it explicitly as a definition within the model.
minor comments (5)
  1. [II.B, Eq. (7)] In Eq. (7) and the surrounding text, the symbol N is used both for the number of Planckons and for the binomial coefficient; please write \binom{N}{2} explicitly to avoid confusion.
  2. [II.C, Eq. (15)] Equation (15) and the following sentence contain the phrase "N = N(N-1)/2 exponents"; the middle N is a typographical confusion between the number of Planckons and the entropy value, and the sentence should read "N(N-1)/2 exponents".
  3. [III.A, Eq. (18)] Equation (18) and its text include "S_WH(N) = -N(N-1)/2 = -N = -S_BH(N)", where the middle term "-N" is erroneous and should read "-N(N-1)/2".
  4. [IV.B, Eq. (28)] The statement that for the current fine structure constant the critical charge is "about two electron charges" should specify the unit convention for q in Eq. (28), since numerically q_c = 1/√(8πα) ≈ 2.33 in units of the elementary charge only if q is measured in those units.
  5. [General] The paper would benefit from an explicit sentence distinguishing exact results within the model (such as w=e^{-N1N2}) from asymptotic results (such as S≈4πGM² for large N), since the current text often blurs this distinction.

Circularity Check

2 steps flagged · score 8.0 of 10

The RN charge-independence claim is Eq. (22) restated as Eq. (25); the adiabatic 'support' is the same definition.

  1. self definitional [Section IV.A, Eqs. (22) and (25)]
    "In the extended Thallis-Cirto statistics in Eq.(21), the entropy of Reissner-Nordström (RN) black hole can be considered as the composition of the entropies of the outer and inner horizons: SRN = (√SRN(r+) + √|SRN(r−)|)^2 = 4πGM^2 . ... Then Eq.(22) demonstrates that the entropy of the RN ensemble does not depend on the Planckon charge. ... SRN = N = N(N − 1)/2 ."

    Eq. (22) is itself the charge-independence result: because √S_+ ∝ (M + √(M²−αQ²)) and √|S_-| ∝ (M − √(M²−αQ²)), their sum squared cancels Q by construction, giving 4πGM². With M = N m_P and m_P² = 1/(8πG), this is exactly Eq. (25). So 'the statistical ensemble remains the same' is an input (a chosen composition rule for horizon entropies), not a derived consequence of the Planckon model. The non-extensive composition rule and the negative inner-horizon entropy are imported from the author's preceding Tsallis-Cirto papers and Ref. [17], not independently derived here.

  2. self definitional [Section IV.A, paragraph after Eq. (25)]
    "The composition rule for the Reissner-Nordström black hole shows that the total entropy of an RN black hole is completely determined by its mass M. This is consistent with the adiabatic transformation of the RN black hole into a Schwarzschild black hole, where the parameter α (the fine structure constant) adiabatically transforms to zero for fixed M and Q. During this adiabatic transformation, the entropy does not change."

    This paragraph presents the adiabatic invariance as confirmation, but it is a tautology: after Eq. (22) has defined S_RN to depend only on M, varying α at fixed M and Q cannot change S_RN. The standard area entropy S_RN(r+) = π r_+²/G in Eq. (23) does depend on Q, so the conclusion is not supported by ordinary black-hole thermodynamics. The adiabatic 'support' therefore adds no evidence beyond the definition already contained in Eq. (22).

full rationale

The Schwarzschild/white-hole portion is an openly declared toy model, not circular in the objectionable sense: S_BH(N)=N(N−1)/2 is chosen to match the Tsallis-Cirto composition rule and the Bekenstein-Hawking entropy, and the splitting probabilities follow from the standard fluctuation relation w ∝ exp(−ΔS). That is internal consistency of a model, not a hidden reduction. The serious circularity is concentrated in the Reissner-Nordström extension. Eq. (22) defines the total RN entropy as (√S_+ + √|S_-|)^2, which algebraically cancels the charge and yields 4πGM²; substituting the already chosen M=N m_P immediately gives Eq. (25), S_RN(N)=N(N−1)/2. The claim that charge does not affect the Planckon ensemble is therefore the composition rule itself, restated. The adiabatic-transformation argument is not independent support, because Eq. (22) already makes S a function of M only, so varying α at fixed M,Q cannot change it by construction. The standard horizon-area entropy S_+=πr_+²/G does depend on Q, so the paper's charge independence is not a consequence of the usual area law. The remaining physical content—the interaction potential U(r), the critical charge, and the loss of stability at extremality—escapes this circularity, but the central RN claim reduces by definition to Eq. (22), forcing a high circularity score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central derivation of S=N(N-1)/2 is a postulate fitted to the known BH entropy; the RN result is forced by the assumed composition rule; Lambda quantization is asserted without derivation.

free parameters (3)
  • Planckon mass m_P = 1/sqrt(8*pi*G)
    Chosen so that S=N(N-1)/2 matches Bekenstein-Hawking entropy at large N; a fit to the known entropy.
  • Entropy assignment = N(N-1)/2
    The integer entropy formula is postulated, not derived; it is selected to match BH entropy and composition law asymptotically.
  • Planckon charge q = Q/N
    Charge is distributed among Planckons; the critical charge q_c=1/sqrt(8*pi*alpha) is derived from interaction balance.
assumptions (6)
  • ad hoc to paper A black hole is an ensemble of N Planckons of reduced Planck mass, each with zero individual entropy.
    Introduced in Section II.B as the toy model; no independent evidence.
  • ad hoc to paper Entropy comes from correlated pairs of Planckons, S=N(N-1)/2.
    Postulated in Eqs (6)-(7); the counting of pairs is the definition of the model.
  • domain assumption Extended Tsallis-Cirto delta=2 statistics with negative entropy for white holes (Eq 21).
    Taken from the author's prior work (Refs 1,2) and applied to white holes; allows negative entropy for horizon systems.
  • ad hoc to paper The total entropy of an RN black hole is the composition S_RN=(sqrt(S_+)+sqrt(|S_-|))^2.
    Eq (22); this is the premise from which charge-independence of entropy follows, not a derived result.
  • ad hoc to paper The fine structure constant can be adiabatically varied at fixed M and Q.
    Used in Section IV.A to argue adiabaticity of RN to Schwarzschild transition; a thought experiment.
  • domain assumption Holographic bulk-surface correspondence for de Sitter entropy in the Hubble volume.
    Section V.A; standard assumption in the author's framework, used to relate local entropy to horizon entropy.
invented entities (1)
  • Planckon
    purpose: Constituent of black holes, white holes, and de Sitter vacuum; carries reduced Planck mass and possibly charge.
    No falsifiable handle outside the model; the paper itself distinguishes Planckons from physical Planck-scale black holes (zero individual entropy).

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Cite this review

Pith. "Pith review of Thermodynamics of black and white holes in ensemble of Planckons." pith.science (2026). https://pith.science/paper/32KVPOGI

@misc{pith2026250613145,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of black and white holes in ensemble of Planckons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32KVPOGI}},
  note         = {Machine review of arXiv:2506.13145}
}
abstract

The Tsallis-Cirto non-extensive statistics with $\delta=2$ describes the processes of splitting and merging of black holes and their thermodynamics. Here we consider a toy model, which matches this generalized statistics and extends it by providing the integer valued entropy of the black hole, $S_{\rm BH}(N)=N(N-1)/2$. In this model the black hole consists of $N$ the so-called Planckons -- objects with reduced Planck mass $m_{\rm P}=1/\sqrt{8\pi G}$ -- so that its mass is quantized, $M=Nm_{\rm P}$. The entropy of each Planckon is zero, but the entropy of black hole with $N$ Planckons is provided by the $N(N-1)/2$ degrees of freedom -- the correlations between the gravitationally attracted Planckons. This toy model can be extended to a charged Reissner-Nordstr\"om (RN) black hole, which consists of charged Planckons. Despite the charge, the statistical ensemble of Planckons remains the same, and the RN black hole with $N$ Planckons has the same entropy as the electrically neutral hole, $S_{\rm RNBH}(N)=N(N-1)/2$. This is supported by the adiabatic process of transformation from the RN to Schwarzschild black hole by varying the fine structure constant. The adiabaticity is violated in the extreme limit, when the gravitational interaction between two Planckons is compensated by the repulsion between their electric charges, and the RN black hole loses stability. The entropy of a white hole formed by the same $N$ Planckons has negative entropy, $S_{\rm WH}(N)=-N(N-1)/2$.

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Reviewed August 7, 2026 · model on record in the stance chip above.