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REVIEW 3 major objections 6 minor 62 references

RPST-Inspired Formalism for Black Holes in Flat Spacetime

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Pith's one-line read This paper argues that flat black holes with Rényi entropy can be given a restricted phase space thermodynamics in which the inverse Rényi parameter and a response potential act as a new variable pair, yielding van der Waals-like and…

desk verdict A self-consistent formal extension of RPST to flat Rényi black holes, but the physical content is limited by the fact that the new coordinate λ is imposed rather than derived. read the letter →

arxiv 2411.18256 v2 pith:75KAV5PI submitted 2024-11-27 hep-th gr-qc

Bidyut Hazarika , Prabwal Phukon This is my paper · ORCID
classification hep-thgr-qc MSC 83C5780A10
keywords restrictedphasespacethermodynamicsRényientropyflatblackholesReissner-NordströmholeKerrvanderWaalstransitionHawking-Pageresponsepotential
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 claims that asymptotically flat black holes, when described by Rényi entropy with deformation parameter $\lambda=1/\alpha$, can be given a restricted phase space thermodynamics once $\lambda$ is promoted to an independent thermodynamic coordinate paired with a response potential $\zeta$. For flat Reissner-Nordström and Kerr black holes the authors verify a first law of the form $d\tilde M = T\,dS + \tilde\phi\,d\tilde Q + \zeta\,d\lambda$ (with $\Omega\,dJ$ replacing the charge term for Kerr) together with the Euler relation, and they show that the $T$--$S$ and $F$--$T$ curves exhibit van der Waals-like first-order phase transitions below a critical charge or angular momentum. They also find Hawking-Page-like transitions in the $\zeta$--$\lambda$ plane. If correct, this would mean the phase structure usually attributed to a cosmological constant can emerge from the entropy statistics alone.

What carries the argument

The load-bearing object is the Rényi entropy rewritten as $S=\lambda\ln(1+S_0/\lambda)$ with $\lambda=1/\alpha$, combined with the rescaling $G\to\kappa^2/\lambda$ and $\tilde Q\to\kappa Q/\sqrt G$. This combination makes the rescaled black hole mass a homogeneous degree-one function of $S$, the charge or angular momentum, and $\lambda$, which is exactly what guarantees the first law and the Euler relation; the response potential $\zeta=\partial\tilde M/\partial\lambda$ is the conjugate force generated by that homogeneity. The phase structure then comes from the non-monotonic $T(S)$ produced by the exponentials $e^{S/\lambda}$ in the mass formulas.

What would settle it

Recompute the canonical ensemble for a flat RN black hole with fixed physical charge $Q$ and the standard Bekenstein-Hawking entropy $S_0=\pi r_+^2/G$, and check whether $F(T)$ has a swallowtail; the paper's transition appears only after promoting $\lambda$ to a fluctuating variable, so a calculation that fixes $\alpha$ and never differentiates with respect to it should show no van der Waals critical point. A direct numerical search for the predicted critical exponents in the heat capacity near $T_C$ for fixed $\tilde Q<\tilde Q_C$ would also settle whether the transition is genuine or an artifact of the coordinate choice.

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Extended reading notes

Core claim

The central claim is that flat black holes in Rényi statistics obey the same thermodynamic form as AdS restricted phase space thermodynamics, with the inverse Rényi parameter $\lambda=1/\alpha$ playing the role of the central charge and the response potential $\zeta=\partial\tilde M/\partial\lambda$ playing the role of the chemical potential. The proof mechanism is a rescaling $G\to\kappa^2/\lambda$ and $\tilde Q\to\kappa Q/\sqrt G$, under which the rescaled mass $\tilde M=M\kappa$ becomes a first-order homogeneous function of $S$, $\tilde Q$ (or $J$), and $\lambda$; homogeneity then forces the first law and the Euler relation $\tilde M = T S + \tilde\phi \tilde Q + \zeta\lambda$ (or $TS+\Omega J+\zeta\lambda$) to hold. From the explicit mass functions the authors compute critical points: for RN, $\tilde Q_C = \sqrt{(7-4\sqrt3)/\pi}\,\lambda$ and $T_C\approx0.256236$, below which the isocharge $T$--$S$ curve is non-monotonic and $F$--$T$ has a swallowtail; for Kerr, $S_C=0.483833\lambda$ and $J_C=0.0193724\lambda$ with the same swallowtail structure. In the $\zeta$--$\lambda$ plane they identify Hawking-Page-like transitions, and in the RN iso-voltage case they report a universal $\zeta$--$\lambda$ curve independent of $S$ and $\phi$.

Load-bearing premise

Everything rests on promoting $\lambda=1/\alpha$ to an independent thermodynamic coordinate and on rescaling $G$ to $\kappa^2/\lambda$; Rényi statistics alone does not require the deformation parameter to fluctuate, so if $\lambda$ is a fixed model parameter the response potential is a formal derivative and the phase transitions are artifacts of the chosen variables.

Editorial extensions

If this is right

  • Flat Reissner-Nordström and Kerr black holes acquire a first law and Euler relation of the same form as AdS restricted phase space thermodynamics, with $\lambda$ and $\zeta$ replacing the central charge and chemical potential.
  • Isocharge $T$--$S$ and $F$--$T$ curves for flat RN black holes show a first-order van der Waals-like phase transition for $0<\tilde Q<\tilde Q_C$, turning second-order at $\tilde Q_C=\sqrt{(7-4\sqrt3)/\pi}\,\lambda$ with $T_C\approx0.256236$.
  • Flat Kerr black holes show the same structure: below $J_C=0.0193724\lambda$ the $T$--$S$ curves are non-monotonic with a swallowtail in $F$--$T$, and the transition becomes second-order at the critical point.
  • In the $\zeta$--$\lambda$ plane the formalism predicts Hawking-Page-like transitions, with $\zeta=0$ marking the Hawking-Page temperature in the iso-voltage RN case.
  • Because the framework is extensive despite starting from nonextensive Rényi entropy, all of this phase structure can be analyzed with standard thermodynamic machinery.

Reading between the lines

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

  • If the correspondence is physical rather than formal, $\lambda$ may encode the effective number of microscopic degrees of freedom of a flat-space black hole, and the sign of $\zeta$ would then serve as a macroscopic order parameter for whether those degrees of freedom interact attractively or repulsively.
  • The reported universality of the iso-voltage $\zeta$--$\lambda$ curve for RN is striking; repeating the construction with a third conserved charge or in higher dimensions would test whether this universality reflects a hidden scaling symmetry of the Rényi ensemble.
  • Because the small-$\alpha$ expansion of the Schwarzschild mass reproduces the Schwarzschild-AdS mass with $\alpha\propto 1/l^2$, one extension is to compute the critical exponents of the flat-space transitions and compare them with the AdS restricted phase space values; matching exponents would strengthen the case that $\lambda$ truly substitutes for a central charge.
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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

3 major / 6 minor

Summary. The paper proposes an RPST-inspired thermodynamic formalism for asymptotically flat black holes using Rényi entropy. It defines λ = 1/α as a deformation parameter analogous to the central charge C of AdS/CFT, introduces a conjugate 'response potential' ζ = ∂M/∂λ, and rescales Newton's constant as G → κ²/λ. For Reissner-Nordström and Kerr black holes, the authors derive explicit rescaled masses, temperatures, electric potential/angular frequency, response potentials, first laws, and Euler relations. They then study T–S, F–T, and ζ–λ processes, claiming van der Waals-like first-order phase transitions below a critical charge or angular momentum, and Hawking-Page-like transitions in the ζ–λ plane. The algebraic structure is self-consistent, but the physical status of λ as an independent fluctuating thermodynamic coordinate is not established, and the paper itself concedes that the analogies 'may be purely mathematical.'

Significance. If the λ-coordinate construction is accepted, the paper provides a way to extend restricted phase space thermodynamics to flat black holes, replacing the cosmological-constant/central-charge pair with a Rényi-parameter/response-potential pair. The explicit first-law and Euler-relation derivations for RN and Kerr are a useful reference, and the T–S critical behavior at fixed λ is a genuine property of the Rényi mass function, consistent with earlier literature. The paper is also transparent in deriving the homogeneity that underlies the Euler relation. However, the physical significance is substantially weakened because no ensemble with fluctuating λ is constructed; the ζ dλ term and the associated phase structure are imposed by the chosen rescaling and coordinate promotion. The unresolved factor-of-4 mismatch in the α–Λ correspondence further undermines the motivational analogy. The paper's own closing caveat about the purely mathematical nature of the analogy correctly identifies the main uncertainty.

major comments (3)
  1. [Sec. I, Eqs. (9)–(11)] The motivation contains an unresolved factor-of-4 discrepancy. Comparing the small-α expansion of the Rényi Schwarzschild mass with the Schwarzschild-AdS mass gives α ≈ 4G/(πl²) in Eq. (9), while substituting Λ = −3/l² into the relation from Ref. [62] gives α ≈ G/(πl²) in Eq. (11). The manuscript notes both results but does not reconcile them. Since this relation is the primary motivation for replacing the AdS length with λ and for the rescaling in Eq. (22), the authors should either identify which derivation is correct, explain the discrepancy (e.g., an order-of-magnitude correspondence with the factor absorbed into κ), or explicitly state that the formalism does not depend on the precise coefficient.
  2. [Sec. I and Sec. II, Eqs. (22), (27), (29)] The central construction promotes λ = 1/α to an independent thermodynamic coordinate and rescales G to κ²/λ. The first-order homogeneity of Eq. (24) and the Euler relation (29) then follow by construction: under S → βS, Q̃ → βQ̃, λ → βλ, the exponential e^{S/λ} is invariant, so M̃ → βM̃. In Rényi statistics, α (hence λ) is a fixed parameter labeling the ensemble; the paper does not construct a statistical ensemble in which λ fluctuates. Consequently, ζ = ∂M̃/∂λ in Eq. (27) is a formal derivative, and the phase transitions in the ζ–λ plane (Fig. 3) are properties of the chosen coordinate system rather than demonstrated properties of the underlying Rényi thermodynamics. The concluding sentence that the analogies 'may be purely mathematical' concedes exactly this gap. The authors should either construct an ensemble with fluctuating λ and identify ζ as a physical response, or explicitly restrict the claims to a mathematical analogy and demonstrate which results (e.g., the T–S behavior at fixed λ) are independent of the coordinate promotion.
  3. [Sec. II.1 and Sec. III, Eqs. (31), (34), (44), (45)] Several load-bearing critical values are stated without derivation. For the RN case, Eq. (31) gives S_C = λ ln(2(√3 − 1)) and Q̃_C = √((7 − 4√3)/π) λ, and the temperature T_C = 0.256236 is quoted; the complicated free-energy expression in Eq. (34) is also presented without derivation. Similarly, the Kerr critical values in Eqs. (44) and (45) are asserted. These quantities underlie the phase-diagram claims, so the authors should provide the solution of the equations defining the critical point, or at least outline the calculation and state the assumptions, so that the results can be verified.
minor comments (6)
  1. [Sec. I, Eq. (6)] The expansion in Eq. (6) states the remainder is O(α^{3/2}), but the first correction is O(α²); the remainder should be O(α²).
  2. [Sec. I, text after Eq. (7)] The sentence 'Comparing eq.(8) with the first two terms in eq.(7)' appears to compare Eq. (6) with Eq. (8), not Eq. (7); please correct the cross-reference.
  3. [Sec. II, Eq. (18)] Eq. (18) is labeled as an expression for the event-horizon radius, but it actually gives the Rényi entropy S in terms of S₀ and λ; the surrounding text should be rephrased.
  4. [Sec. II.1, paragraph after Fig. 1] The phrase 'flat charged AdS black hole' should be 'flat charged black hole' or 'RN black hole'; the black hole is not AdS in this formalism.
  5. [Sec. II.1, Fig. 3(b)] The claimed universality of the iso-voltage ζ–λ plot is stated without proof; the text should explain that for fixed φ̃, ζ depends on S and λ only through the ratio S/λ, which makes the curves independent of S.
  6. [General] There are several typographical inconsistencies, including 'R´enyi' accents, 'isovoltage' versus 'iso-voltage', 'e-charge' versus 'Q̃', and a stray 'l' after 'respectively' in Sec. II; a careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 8.0 of 10

The Euler relation, first law, and phase transitions are consequences of declaring λ=1/α an extensive coordinate and rescaling G→κ²/λ, not of Rényi statistics itself.

  1. self definitional [Sec. I, Eqs. (14), (22); Sec. II, Eqs. (24), (28), (29)]
    "If S, ˜Q, and λ are rescaled as S → βS, Q → βQ, and λ → βλ, then Eq. (24) implies ˜M → β ˜M which proves the first order homogeneity of ˜M. Using this mass in eq.(24), the first law of black hole thermodynamics can be written as d ˜M = T dS + ˜ϕ d˜Q + ζ dλ (28)... It is also very easy to prove the Euler relation E = T S + ˜ϕ ˜Q + ζ λ (29)."

    The homogeneity that makes the Euler relation and first law valid is manufactured by the preceding rescaling (22): replacing G by κ²/λ and Q by κQ/√G is what makes M̃(βS, βQ, βλ) = βM̃. In Rényi statistics, α (hence λ) is a fixed ensemble label; before this rescaling, M̃(βS, βQ, λ) is not βM̃(S, Q, λ). The first law is then just the total differential of the constructed M̃ with ζ ≡ ∂M̃/∂λ, so its 'derivation' is an identity, not an independent thermodynamic law. All later VdW and Hawking-Page-like transitions are properties of this chosen mass function rather than consequences of Rényi thermodynamics alone.

  2. self definitional [Sec. II.1, ζ−λ and ζ−T discussion after Eq. (27)]
    "ζ = 0 leads to the vanishing of the Gibbs free energy, G = ζλ. For ζ >0, the microscopic degrees of freedom of the RN black hole exhibit repulsive behavior, whereas for ζ < 0, they exhibit attractive behavior. The temperature at which ζ becomes zero is identified as the Hawking-Page temperature, THP."

    The Hawking-Page-like transition is read off from the zero of ζ, but ζ is defined in Eq. (27) as ∂M̃/∂λ, and the relation G = ζλ is the Euler relation already enforced by the rescaling in Eq. (22). Thus the transition criterion is definitionally tied to the same constructed coordinate and extensivity assumption; it is not an independently derived thermodynamic instability. The physical interpretation of ζ as a response potential is assigned after the fact, with no statistical ensemble in which λ fluctuates.

full rationale

The paper's central result — a first law with ζdλ, a valid Euler relation, and VdW/Hawking-Page-like phase transitions for flat Rényi black holes — is obtained by promoting the fixed Rényi parameter α to a variable λ = 1/α and then rescaling G → κ²/λ so that the mass becomes degree-one homogeneous in (S, Q, λ). Once that coordinate choice and normalization are made, the Euler relation follows from Euler's homogeneous-function theorem, the first law is the total differential of the constructed M̃, and the phase structures are algebraic properties of that M̃. The paper is transparent about the construction, even conceding in the conclusion that the analogies 'may be purely mathematical'; that honesty does not remove the definitional circularity in presenting these as derived results. No load-bearing self-citation chain is present: the cited prior work on Rényi flat black holes and RPST is background support, not the mechanism that forces the Euler relation. The formalism may be a useful mathematical model, but as a derivation from first principles it reduces to its own construction, giving a circularity score of 8.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The construction rests on the Rényi entropy ansatz and three ad hoc modeling choices: λ as a thermodynamic coordinate, the rescaling G=κ²/λ, and the analogy between λ and the CFT central charge. These choices do the work that produces the phase transitions; no independent evidence is supplied for any of them.

free parameters (2)
  • lambda (deformation parameter, inverse of Rényi parameter α) = not fixed; scales all critical quantities
    Introduced in eqs. (18)-(19) as the inverse Rényi parameter and promoted to a thermodynamic variable. All critical quantities, e.g. S_C=λ ln(2(√3-1)) and Q_C∝λ, scale with it, so it is not fixed by any data.
  • kappa (rescaling constant) = cancels in rescaled mass
    Introduced in eq. (22) via G→κ²/λ and Qtilde→κ Q/√G. It drops out of Mtilde but is an arbitrary rescaling freedom used to make the construction extensive.
assumptions (5)
  • domain assumption Rényi entropy S=(1/α)ln(1+αS0) is the correct entropy functional for black holes.
    The whole construction starts from eq. (1) and assumes Rényi statistics apply to flat black hole horizons; no derivation from microstates is given.
  • ad hoc to paper λ=1/α can be treated as an independent thermodynamic coordinate on the same footing as S, Q, J.
    Used to write the first law eq. (14) and to prove the Euler relation (29). The paper gives no physical mechanism by which λ fluctuates.
  • ad hoc to paper The rescaling G→κ²/λ (eq. 22) preserves the physics while making the mass homogeneous.
    This rescaling is chosen specifically so that Mtilde is homogeneous of degree one in S, Qtilde, λ; no independent justification is offered.
  • ad hoc to paper The analogy between λ and the CFT central charge C is meaningful.
    Throughout, λ is claimed to parallel C (Sec. I) based on scaling behavior, but no dual CFT for flat black holes is identified.
  • domain assumption The relation α≈G/(πl²) from ref. [62] is reliable.
    Used in Sec. I to motivate λ as a replacement for Λ/C. It disagrees by a factor 4 with the paper's own coefficient comparison eq. (9), and the discrepancy is not discussed.
invented entities (2)
  • λ deformation parameter as an independent thermodynamic variable
    purpose: Replaces the central charge C of AdS RPST, intended to encode degrees of freedom and deviation from Boltzmann-Gibbs statistics.
    No falsifiable handle outside the paper; it is the inverse of the Rényi parameter and is simply promoted to a thermodynamic coordinate.
  • ζ response potential
    purpose: Conjugate variable to λ in the first law, eqs. (14), (27), (41), used to define ζ-λ and ζ-T phase diagrams.
    Defined as ∂Mtilde/∂λ, so it carries no independent information beyond the chosen mass function.

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Pith. "Pith review of RPST-Inspired Formalism for Black Holes in Flat Spacetime." pith.science (2026). https://pith.science/paper/75KAV5PI

@misc{pith2026241118256,
  author       = {Pith},
  title        = {Pith review of: RPST-Inspired Formalism for Black Holes in Flat Spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75KAV5PI}},
  note         = {Machine review of arXiv:2411.18256}
}
abstract

In this work, we propose a novel formalism for the thermodynamics of flat black holes, inspired by the Restricted Phase Space Thermodynamics (RPST) framework. Our construction is motivated by the observed similarities in the thermodynamic behavior of flat black holes within the R\'enyi entropy framework and that of AdS black holes described by the Bekenstein entropy regime. The RPST framework is, by construction, exclusive to AdS black holes because it depends on the cosmological constant $\Lambda$, which is linked to the central charge $C$ of the dual conformal field theory (CFT). However, for non-AdS black holes, where $\Lambda$ is absent, we introduce a deformation parameter $\lambda$ to replace the central charge $C$. This RPST-inspired formalism incorporates $\lambda$ and its conjugate variable, the response potential $\zeta$, as a new pair of thermodynamic variables, analogous to the central charge $C$ and chemical potential $\mu$ in the AdS case. To illustrate the applicability of this formalism, we analyze two examples: the Reissner-Nordstr\"om (RN) flat black hole and the Kerr black hole.

Figures

Figures reproduced from arXiv: 2411.18256 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Thermodynamic behavior of the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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