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REVIEW 3 major objections 4 minor 53 references

Super-entropic black holes in gravity's rainbow and determining constraints on rainbow functions

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Thermodynamic stability in gravity's rainbow forces the rainbow function $g^2$ into a horizon-radius-dependent interval and, through the inverse-isoperimetric ratio $R = 1/f^{1/(d-1)}$, identifies $f>1$ rainbow black holes as…

desk verdict A correct inequality and a neat identity, but the model-exclusion claims contradict the paper's own stability criteria and need revision. read the letter →

arxiv 2505.13555 v1 pith:IAQNND7V submitted 2025-05-19 gr-qc hep-th

Behzad Eslam Panah , Narges Heidari , Mana Soleimani , Maryam Kaveh This is my paper · ORCID
classification gr-qchep-th
keywords gravity'srainbowfunctionsblackholethermodynamicssuper-entropicholesinverseisoperimetricinequalitythermodynamicstabilityAdSheatcapacity
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

In gravity's rainbow, spacetime is modified by rainbow functions $f(\varepsilon)$ and $g(\varepsilon)$ of the probing particle's energy. This paper asks which of the three standard rainbow models can describe AdS black holes that are thermodynamically stable, in both ordinary and extended phase space. Computing the Hawking temperature, total mass, heat capacity, Helmholtz free energy, and constant-pressure heat capacity, the authors find that stability requires $g^2(\varepsilon)$ to satisfy a two-sided bound set by the cosmological constant, the spacetime dimension, and the horizon radius $r_+$; hence $g^2$ cannot be a constant, ruling out model (i). Using the inverse isoperimetric inequality, they show $R = 1/f(\varepsilon)^{1/(d-1)}$, so the rainbow function $f$ decides whether the hole is super-entropic, and they combine this with the suggested super-entropic-instability link to exclude model (iii) with $\lambda>0$. The surviving model is (ii), with $f=1$ and $g=\sqrt{1-\eta\varepsilon^n}$, which satisfies all stability conditions and gives $R\ge1$.

What carries the argument

The load-bearing object is the inverse-isoperimetric ratio $R = ((d-1)V/\omega_{d-2})^{1/(d-1)}(\omega_{d-2}/A)^{1/(d-2)}$, which compares a black hole's thermodynamic volume with that of a round ball of equal area; $R<1$ is the super-entropic regime. In this rainbow spacetime the identity $R=1/f(\varepsilon)^{1/(d-1)}$ collapses the whole isoperimetric content into the rainbow function $f$. The second mechanism is the inequality chain from $T>0$, $M>0$, $C>0$, $F<0$, and $C_P>0$, which compresses into the two-sided bound on $g^2$ that forces $g$ to be horizon-dependent and eliminates model (i). Together these two pieces convert thermodynamic-stability conditions into concrete constraints on the rainbow functions $f$ and $g$.

What would settle it

Evaluate the heat capacity (16) for model (iii) with $\lambda>0$ and $f=g=1/(1-\lambda\varepsilon)$ in AdS. If any horizon radius $r_+$ gives $T>0$, $M>0$, and $C>0$ while $R<1$, then a super-entropic rainbow black hole is thermodynamically stable, contradicting the paper's instability claim. A simpler check is to exhibit any super-entropic black hole with positive specific heat in any theory of gravity; the instability lemma used in Section III would then be false.

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Extended reading notes

Core claim

The central claim is a selection rule for rainbow functions. For $d$-dimensional AdS black holes in gravity's rainbow, positivity of the temperature, total mass, heat capacity, negative Helmholtz free energy, and constant-pressure heat capacity together imply $$\frac{2\Lambda(\varepsilon) r_+^2}{(d-1)(d-2)} < $g^{2}$(\varepsilon) < -\frac{2\Lambda(\varepsilon) r_+^2}{(d-1)(d-2)},$$ which, for $\Lambda<0$, leaves the operative bound $g^2(\varepsilon) < -2\Lambda(\varepsilon) r_+^2/((d-1)(d-2))$ and makes $g^2$ depend on the horizon radius; no constant $g$ can satisfy this at every $r_+$, so model (i) is excluded. Separately, the inverse-isoperimetric ratio for these spacetimes is exactly $R=1/f(\varepsilon)^{1/(d-1)}$, so $f(\varepsilon)>1$ means the black hole violates the isoperimetric bound and is super-entropic. Taking the suggested correspondence between super-entropic black holes and thermodynamic instability as given, the paper concludes that $f>1$ holes are thermodynamically unstable, ruling out model (iii) with $\lambda>0$; model (ii), with $f=1$, obeys $R\ge1$ and passes every stability condition, making it the only admissible model among the three.

Load-bearing premise

The load-bearing premise is that the tentative link between entropy-exceeding (super-entropic) black holes and thermodynamic instability is actually true, together with the implicit requirement that stability hold for every horizon radius; if either assumption fails, the exclusions of model (iii) with positive λ and of model (i) no longer follow.

Editorial extensions

If this is right

  • Among the three standard rainbow models, only model (ii), with $f=1$ and $g=\sqrt{1-\eta\varepsilon^n}$, passes every thermodynamic stability condition and satisfies $R\ge1$.
  • A viable rainbow function must have $g^2(\varepsilon) < -2\Lambda(\varepsilon) r_+^2/((d-1)(d-2))$ for AdS black holes, so $g^2$ has to depend on the horizon radius; constant-$g$ rainbow deformations of Schwarzschild-AdS cannot be stable at every radius.
  • If the super-entropic instability correspondence holds, $f(\varepsilon)>1$ is a direct instability marker: rainbow black holes with $f>1$ are super-entropic and thermodynamically unstable, which forbids model (iii) for $\lambda>0$ and any analogous $f>1$ rainbow model.
  • In the extended phase space the same conditions become a lower bound on the thermodynamic pressure, $P>(d-2)(d-3)(\omega_{d-2}/4S)^{2/(d-2)}/16\pi$, tying the allowed rainbow parameters to an admissible pressure range.
  • The first law $dM=T\,dS+V\,dP$ and the Smarr relation $M=2(TS-PV)$ hold for all three models, so the selection among rainbow functions takes place inside a thermodynamically consistent framework.

Reading between the lines

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

  • A natural next test is to run the same inequality chain for charged or rotating rainbow black holes; if the $g^2$ window shifts, the excluded models could be revived in those sectors.
  • The identity $R=1/f^{1/(d-1)}$ is independent of $g$ and of the details of the metric function, so if the instability correspondence is ever proven, the stability classification of rainbow black holes becomes a property of $f$ alone, extending the paper's model-by-model verdict to any rainbow theory.
  • The paper's 'no constant $g$' conclusion is the all-radii version. If stability is required only over a bounded range of horizon radii, a constant-$g$ model such as model (i) could still describe stable black holes in that window, a weaker reading the paper does not address.
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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 / 4 minor

Summary. The paper studies d-dimensional Schwarzschild-AdS black holes in gravity's rainbow, with energy-dependent metric functions f(ε) and g(ε), and uses thermodynamic stability to constrain the rainbow functions. The authors compute the Hawking temperature, entropy, total mass, heat capacity, and Helmholtz free energy, and combine the positivity conditions for temperature, mass, heat capacity, and negative free energy into the window (26) for g²(ε). In extended phase space they write M(S,P), the thermodynamic volume, and a Smarr relation. They then use the inverse isoperimetric inequality to compute R = 1/f^{1/(d-1)}, concluding that f>1 makes the black holes super-entropic and thermodynamically unstable. On this basis they select among the three rainbow-function models: model (ii) is allowed, while model (i) and model (iii) with λ>0 are excluded. The central algebraic derivation of (26) is internally consistent, but the model-selection conclusions are not supported by the paper's own stability criteria.

Significance. If fully supported, the paper would offer a concrete thermodynamic selection among rainbow-function models and connect the super-entropic condition to gravity's rainbow. The derivation of Eq. (26) is transparent and largely self-contained once the metric function and AMD mass formula are imported from refs. [50-52]; the computation of R in Eq. (29) follows correctly from the paper's own volume and area formulas, and there is no parameter fitting. However, the central model-selection claims rely on an unproven conjecture and on an unjustified 'for all r_+' reading of Eq. (26), so the significance of the paper as a constraint on rainbow functions is not yet established.

major comments (3)
  1. [Section III (text after Eq. (29))] The exclusion of model (iii) for λ>0 is inconsistent with the paper's own stability criterion. For model (iii), f(ε)=g(ε)=1/(1-λε), and for λ>0 with 0<λε<1 one has g²>1. For AdS (Λ<0), the lower bound in Eq. (26) is negative while the upper bound grows like -2Λr_+²/((d-1)(d-2)), so for every fixed λ>0 there exist sufficiently large horizon radii r_+ for which Eqs. (11), (15), (17), and (19), and hence Eq. (26), are all satisfied. By the Section II criteria these are thermodynamically stable black holes. The later assertion that f>1 gives R<1 and therefore thermodynamic instability depends entirely on the 'suggested' conjecture of refs. [46-49], which the manuscript itself describes as unconfirmed; the assertion is used as a blanket theorem and directly contradicts the Section II stability analysis for the same solutions. This contradiction is load-bearing because it is the basis for rejecting model (iii) with λ>0; the paper must either prove the super-entropic instability link or reconcile the two criteria.
  2. [Section II (text after Eq. (26))] The conclusion that g² cannot be constant does not follow from Eq. (26) as stated. Equation (26) is an r_+-dependent interval; for a fixed constant value of g², and fixed Λ<0 and d, all four underlying conditions are satisfied for an interval of r_+ (in particular, for sufficiently large r_+). To infer that a constant g² is forbidden, one must additionally assume that thermodynamic stability is required for every allowed horizon radius r_+, an assumption that is neither stated nor justified. Without that assumption, model (i) with g(ε)=1 is not excluded by the thermodynamic analysis.
  3. [Section II (Eq. (28))] Eq. (28) states M=2(TS-PV) for d-dimensional black holes, but the scaling relation for the mass function (21) gives (d-3)M=(d-2)TS-2PV; equivalently, M=((d-2)/(d-3))TS-(2/(d-3))PV. Equation (28) is therefore valid only in d=4. Since the paper presents this as a d-dimensional result, it should be corrected or explicitly restricted to d=4.
minor comments (4)
  1. [Section II, below Eq. (17)] There is a typo in the text: 'two follwoing constrains' should be 'two following constraints'.
  2. [Section II, before Eq. (26)] The phrase 'evaluating the three conditions in Eqs. (11), (15), (17), and (19)' is inconsistent, since four constraints are listed and Eq. (17) itself contains two inequalities.
  3. [Section II, paragraph containing Eq. (22)] The text refers to 'the thermodynamic volume (V) of bumblebee AdS black holes'; this paper is not about bumblebee gravity, so 'bumblebee' should be removed.
  4. [Section II, paragraph containing Eq. (22)] The parenthetical 'i.e., V∝S' is inaccurate: from Eqs. (22) and (12), V ∝ S^{(d-1)/(d-2)}, not V ∝ S. The conclusion that C_V=0 only requires V to be a function of S independent of P.

Circularity Check

1 steps flagged · score 4.0 of 10

Super-entropic instability link is imported as a self-cited, unconfirmed ansatz; the core thermodynamic inequalities are not circular.

  1. ansatz smuggled in via citation [Section III, after Eq. (29)]
    "In Refs. [46–49], a link between super-entropic black holes and thermodynamic instability was suggested. Confirming this connection would be of great importance. ... Based on the relationship between super-entropic black holes and thermodynamic instability,d-dimensional AdS black holes in gravity’s rainbow exhibit thermodynamic instability whenf(ε)>1."

    The inference that f>1 implies thermodynamic instability is not derived from the paper's thermodynamic analysis. Eq. (29) only proves R=1/f^{1/(d-1)}, i.e., R<1 iff f>1, which is the definition of the label 'super-entropic'. The step from 'super-entropic' to 'thermodynamically unstable' is imported from refs [46-49], one of which (ref [48]) is by the same author, and the paper itself describes that link as merely 'suggested' and unconfirmed. This imported ansatz is then used to exclude model (iii) with λ>0, even though the paper's own stability inequalities (Eq. (26)) are satisfied by that model for sufficiently large r_+. The model-selection conclusion is thus carried by a self-cited conjecture rather than by the paper's derivation.

full rationale

The core thermodynamic derivation is self-contained: Eqs. (6)-(26) compute T, S, M, C, F, and CP from the imported metric and AMD mass formulas and combine positivity conditions; no parameter is fitted and the inequalities are not reverse-engineered. The ratio R=1/f^{1/(d-1)} follows algebraically from the paper's own volume and area formulas, so the super-entropic classification (R<1 iff f>1) is legitimate. The circular-adjacent step is in Section III: the exclusion of model (iii) with λ>0 rests on upgrading the 'suggested link' between super-entropic black holes and thermodynamic instability (refs [46-49], including same-author ref [48]) into a theorem, despite the paper's admission that confirming it 'would be of great importance.' This is a load-bearing self-citation/ansatz rather than a derivation, and it also conflicts with Eq. (26), which model (iii) with λ>0 can satisfy for sufficiently large r_+. Separately, the claim that g^2 cannot be constant follows only if Eq. (26) is imposed for every r_+, which the paper never states; that is a logical gap, not circularity. Overall, the main derivation is not circular, but the model-selection conclusion is partly carried by an unproved, partly self-cited assumption, so the paper does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the rainbow function parameters β, η, λ are inputs of the models, not determined by this paper. The central claim rests on imported formulas and on two assumptions flagged above.

assumptions (4)
  • domain assumption The metric function ψ(r,ε) (Eq. 6) is a valid solution of gravity's rainbow field equations with cosmological constant.
    Taken from ref [50]; not re-derived in this paper.
  • domain assumption The AMD approach (refs [51,52]) gives the total mass via Eq. (13).
    Standard method in AdS gravity; accepted without proof.
  • domain assumption Super-entropic black holes are thermodynamically unstable (from refs [46-49]).
    The paper calls this a suggested link, then uses it as a fact in Section III.
  • ad hoc to paper Thermodynamic stability must hold for all horizon radii r_+.
    Needed to conclude g^2 cannot be constant and to exclude model (i); never explicitly stated.

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Pith. "Pith review of Super-entropic black holes in gravity's rainbow and determining constraints on rainbow functions." pith.science (2026). https://pith.science/paper/IAQNND7V

@misc{pith2026250513555,
  author       = {Pith},
  title        = {Pith review of: Super-entropic black holes in gravity's rainbow and determining constraints on rainbow functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAQNND7V}},
  note         = {Machine review of arXiv:2505.13555}
}
abstract

This paper is motivated by the application of the inverse isoperimetric inequality to establish constraints on the parameters of gravity's rainbow. We investigate the thermodynamic (in)stability conditions for $d-$dimensional energy-dependent black holes, which are recognized as $d-$ dimensional black holes within the framework of gravity's rainbow. To achieve this, we calculate thermodynamic quantities such as Hawking temperature, entropy, total mass, and heat capacity in both extended and non-extended phase spaces for these black holes. We assess the physical and stable regions by utilizing these thermodynamic quantities alongside the inverse isoperimetric inequality, aiming to determine constraints on the rainbow functions. Finally, we show that by considering a constraint on the rainbow function, these black holes satisfy the super-entropic condition.

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