Super-entropy bumblebee AdS black holes

B. Eslam Panah

REVIEW 2 major objections 3 minor 72 references

Super-entropy bumblebee AdS black holes

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For positive Lorentz-violating parameter $l$, bumblebee AdS black holes are super-entropy black holes, and the same parameter range makes their heat capacity at constant pressure negative, linking the property to thermodynamic instability.

desk verdict The new super-entropy classification is recoverable only with a corrected volume formula, and the CP stability argument is wrong, so the claimed confirmation of the Cong-Mann conjecture fails. read the letter →

arxiv 2501.09317 v1 pith:MUNPUSR3 submitted 2025-01-16 gr-qc hep-th

B. Eslam Panah This is my paper · ORCID

classification gr-qchep-th

keywords bumblebeegravityLorentzsymmetryviolationAdSblackholessuper-entropythermodynamicstabilityheatcapacityinverseisoperimetricratioextendedphasespace

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 tries to establish that bumblebee AdS black holes—solutions whose metric carries a background bumblebee field that breaks Lorentz symmetry—are super-entropy black holes and thermodynamically unstable in the extended phase space when the Lorentz-violating parameter is positive. It first computes the horizon, temperature, entropy, mass, and free energy, finding that large black holes can be locally and globally stable in the non-extended ensemble. The decisive step is the extended-phase-space analysis: the heat capacity at constant pressure cannot be positive for allowed values of the parameter, and the isoperimetric ratio $R=(1+l)^{-1/6}$ drops below one for $l>0$. If correct, this confirms the proposed connection between super-entropy and thermodynamic instability for Lorentz-violating AdS black holes.

What carries the argument

The load-bearing object is the inverse isoperimetric ratio $R=(3V/4\pi)^{1/3}(4\pi/A)^{1/2}$, whose value below one defines a super-entropy black hole. Into this ratio the paper feeds the thermodynamic volume $V=(4\pi/3)\sqrt{1+l}(S/\pi)^{3/2}$ and the horizon area $A=4S$ for the bumblebee AdS metric, obtaining $R=(1+l)^{-1/6}$. The Lorentz-violating parameter $l$ therefore controls whether the ratio falls below one. On the stability side the argument uses the heat capacity $C=2((1+l)\Lambda S/\pi-1)S/(1+(1+l)\Lambda S/\pi)$ in the non-extended ensemble and $C_P=2S(1+8PS)/(8PS-1)$ in the extended ensemble with pressure $P=-\Lambda(1+l)/(8\pi)$, together with the Helmholtz free energy for global stability.

What would settle it

Recompute $V$ by differentiating the printed mass $M(S,P)=(3+8PS)\sqrt{S}/(6\pi^{1/2}\sqrt{1+l})$ at fixed entropy: the derivative gives $V=(4\pi/3)(S/\pi)^{3/2}/\sqrt{1+l}$, and substituting this into Eq. (30) gives $R=(1+l)^{1/6}$, which exceeds one for $l>0$ and reverses the claimed super-entropy classification.

Watch

Extended reading notes

Core claim

The paper claims that four-dimensional bumblebee AdS black holes are super-entropy black holes whenever the Lorentz-violating parameter satisfies $l>0$. Substituting the thermodynamic volume and the horizon area into the inverse isoperimetric ratio gives $R=(1+l)^{-1/6}<1$, exactly the super-entropy condition. The same parameter range makes the heat capacity at constant pressure negative, so the black holes cannot be thermodynamically stable in the extended phase space, and the paper presents this pairing as confirmation that super-entropy black holes are thermodynamically unstable.

Load-bearing premise

A single factor of $\sqrt{1+l}$ in the thermodynamic volume decides the answer: if the printed volume formula is taken literally, the super-entropy condition flips, so the conclusion rests on the unstated volume formula behind Eq. (31).

Editorial extensions

If this is right

For $l>0$, every bumblebee AdS black hole satisfies $R=(1+l)^{-1/6}<1$, so the entire family is classified as super-entropy, independent of mass or horizon radius.
In the extended phase space, $C_P$ is negative for all allowed $l>-1$ in the AdS case, because $C_P>0$ would require $l<-1$; hence no bumblebee AdS black hole is thermodynamically stable in that ensemble.
The Lorentz-violating parameter shifts the horizon and the phase-transition critical point in opposite directions for $l>0$ and $-1<l<0$, so stable regions in the non-extended ensemble grow with $|l|$ for one sign and shrink for the other.
The result provides a new realization of the conjecture that super-entropy black holes are thermodynamically unstable, extending the pattern to Lorentz-violating AdS solutions.

Reading between the lines

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

Editorial inference: because $R$ depends only on $l$ and not on the entropy, a constraint on $l$ from any single observation would classify the whole family at once, with no dependence on which black hole is observed.
Editorial inference: applying the same ratio to rotating or charged bumblebee black holes would show whether $l>0$ remains sufficient for $R<1$ once the thermodynamic volume is no longer simply proportional to entropy.
Editorial inference: the analysis implies an ensemble mismatch—large black holes can pass the non-extended stability checks while failing the extended-phase-space check—so the physical stability verdict depends on whether the cosmological constant is treated as a fluctuating pressure.

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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

Short version: this is a routine thermodynamics calculation on a known bumblebee AdS metric. The only genuinely new item is the super-entropy claim R=(1+l)^(-1/6) for l>0. That result can be recovered from the paper's mass formula, but only if you replace the printed thermodynamic volume (26) with the one that actually follows from differentiating Eq. (25). With the printed volume the ratio comes out (1+l)^(1/6), which reverses the classification. So the central result as written is not supported.

The bigger problem is the CP analysis. From Eq. (28), CP=2S(1+8PS)/(8PS-1). For AdS, P=-Lambda(1+l)/(8 pi)>0, so the numerator is positive and CP>0 iff 8PS-1>0. Substituting P gives -Lambda(1+l)S/pi>1. Since Lambda<0, dividing through by Lambda flips the inequality: 1+l > -pi/(Lambda S), i.e. l > pi/(|Lambda|S)-1. The paper instead writes l0 is incompatible with CP>0. In fact, for any fixed l>0, sufficiently large S makes CP positive. So the claimed consistency with CP<0, and with it the confirmation of the Cong-Mann conjecture, collapses.

What is good: the non-extended phase space parts—temperature, heat capacity, free energy—are standard and consistent with the formulas. The paper engages the relevant bumblebee-gravity literature and the citation pattern looks fine. The writing is clear enough, and the errors are algebraic rather than conceptual. But the two algebraic errors sit at the load-bearing joints: the super-entropy classification depends on an unstated corrected volume, and the instability claim depends on an inequality solved with the wrong sign. Either one alone would require a major revision; together they break the abstract's central claim.

Who it is for: people interested in bumblebee black hole thermodynamics might want the corrected version on the arXiv. As it stands, I would not cite it in this form. I would still send it to a referee rather than desk reject—the subfield will want the corrected calculation checked, and the errors are specific enough to verify quickly—but it should not be accepted without fixing them.

Referee Report

2 major / 3 minor

Summary. The paper studies thermodynamic stability of four-dimensional bumblebee AdS black holes, using the metric and mass function from Maluf and Neves. In non-extended phase space it computes the Hawking temperature, heat capacity, and Helmholtz free energy, and concludes that large black holes are locally and globally stable. In extended phase space it defines the pressure P=-L(1+l)/(8pi), computes the heat capacity at constant pressure C_P, and claims that C_P cannot be positive for l>0. It then computes the isoperimetric ratio R and claims R=(1+l)^{-1/6}<1 for l>0, so the black holes are super-entropy black holes and thermodynamically unstable, confirming the Cong-Mann conjecture. The non-extended phase-space calculation is mostly standard, but the central extended-phase-space claims are undermined by an algebraic sign error and by an inconsistency between the printed thermodynamic volume and Eq. (31).

Significance. The topic is relevant: it concerns whether Lorentz-violating AdS black holes can be super-entropy black holes and whether super-entropy implies thermodynamic instability. The paper makes its formulas explicit, which allows the calculations to be checked by hand, and the non-extended phase-space part (temperature, heat capacity, free energy) is largely consistent. If the central claim were correct, it would provide a concrete example supporting the Cong-Mann conjecture in a Lorentz-violating theory. However, the main conclusion is not supported by the paper's own equations: Eq. (28) actually allows C_P>0 for l>0 and sufficiently large entropy, and Eq. (31) does not follow from the printed Eq. (26). The manuscript therefore does not establish its advertised result, and a local correction cannot preserve the stated conclusion.

major comments (2)

[III.B, Eq. (28) and following text] The stability analysis contains a sign error. The condition C_P>0 from Eq. (28) is 8PS-1>0. Substituting P=-Lambda(1+l)/(8pi) with Lambda<0 gives 1+l > pi/(-Lambda S), i.e. l > pi/(-Lambda S)-1. The paper instead writes l < pi/(Lambda S)-1 and concludes that l<-1 is required. The division by the negative quantity Lambda was done without reversing the inequality. For any fixed l>0, all sufficiently large S satisfy 8PS-1>0, so C_P>0 is possible; indeed C_P has a divergence at S=1/(8P) and is positive beyond it. Thus the statement in Section III.C that 'C_P cannot be positive when l>0' is contradicted by the paper's own Eq. (28). This invalidates the claimed consistency between super-entropy and C_P<0 and the confirmation of the Cong-Mann conjecture.
[III.C, Eqs. (26), (30), (31)] The thermodynamic volume printed in Eq. (26) is inconsistent with Eq. (25). Differentiating Eq. (25) at fixed S gives V=(4pi/3)(S/pi)^{3/2}/sqrt(1+l), not the printed expression with the reciprocal factor sqrt(1+l). Substituting the printed Eq. (26) into Eq. (30) yields R=(1+l)^{1/6}, not the quoted (1+l)^{-1/6}. The claimed super-entropy classification R<1 for l>0 therefore relies on an unstated corrected volume expression rather than on the equation the paper cites. This is a load-bearing inconsistency for the central claim.

minor comments (3)

[Eq. (16)] The chain-rule expression for dM/dS is malformed; it reads '(\partial M/\partial S) = (\partial M/\partial r_+) (\partial S/\partial r = )' and should be written with the correct derivative factors.
[After Eq. (29)] The sentence 'T=(\partial M(S,P)/\partial S)|_V = 0' is not the correct definition of the Hawking temperature; the standard result C_V=0 for static black holes with V proportional to S is true, but the displayed identity is misleading and should be rewritten.
[Reference [24]] Reference [24] is incomplete: it reads 'Phys. Rev. Lett. Bluhm, 090801 (2002)' and needs the correct author list and page/article identifier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the thermodynamic derivation is a direct application of standard definitions to a given metric; the paper's algebraic and sign errors are correctness defects, not circular reductions.

full rationale

The derivation chain is self-contained in the relevant sense. The bumblebee AdS metric (Eqs. 5 and 7) is taken from Maluf-Neves [1]; the mass, temperature, heat capacities, and super-entropy ratio follow by applying the standard AMD mass formula, area-law entropy, and the definition R=(3V/4π)^(1/3)(4π/A)^(1/2) of Refs. [68-70]. No free parameter is fitted to data, and no output quantity is defined in terms of the quantity it is supposed to predict. The only author self-citation (Ref. [67] for the phrase 'physical limitation point') is terminological and non-load-bearing. The central super-entropy claim R=(1+l)^(-1/6) is algebra from the volume and area; it is not circular, even though it is mathematically suspect because Eq. (26) as printed gives R=(1+l)^(1/6), so achieving the printed result requires an unstated corrected volume. Likewise, the conclusion that C_P<0 for all l>0 follows from a sign error when dividing 8PS-1>0 by Λ<0; for large S, C_P can be positive for l>0. These are correctness defects in the manuscript, not circular reductions. Accordingly, no circular step is identified.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on adopting the known bumblebee AdS metric and standard black hole thermodynamic definitions. No new particles, fields, forces, or conserved quantities are introduced. The only free parameter is the Lorentz-violating parameter l, inherited from the bumblebee gravity theory.

free parameters (1)

l (Lorentz-violating parameter) = not fitted
l is an input parameter from the bumblebee gravity model. The central claim, R<1 iff l>0, depends on l, but the paper does not determine or fit it. It is scanned over in figures and treated as a free theory parameter.

assumptions (4)

domain assumption The bumblebee AdS metric solution in Eqs. (5)-(7) is a valid solution of the bumblebee gravity action in Eq. (1).
The paper adopts the metric from Ref. [1] without re-deriving it. All thermodynamic quantities are computed from this metric.
domain assumption The area law entropy S=A/4 and the AMD mass formula apply to bumblebee black holes.
Section III sets S=πr+^2 and uses M=m0/√(1+l) (Eqs. 13-15) with reference to the Ashtekar-Magnon-Das approach. This is assumed, not derived in this paper.
domain assumption The thermodynamic pressure in extended phase space is P=-Λ(1+l)/(8π).
Eq. (24) defines pressure from the effective cosmological constant (1+l)Λ. This is a modeling choice for the extended phase space.
domain assumption The super-entropy condition R<1, defined by the isoperimetric ratio in Eq. (30), applies to this black hole.
The paper applies the Cvetic-Gibbons-Kubiznak-Pope definition of the isoperimetric ratio without justification beyond citing Ref. [70].

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Cite this review

Pith. "Pith review of Super-entropy bumblebee AdS black holes." pith.science (2026). https://pith.science/paper/MUNPUSR3

@misc{pith2026250109317,
  author       = {Pith},
  title        = {Pith review of: Super-entropy bumblebee AdS black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MUNPUSR3}},
  note         = {Machine review of arXiv:2501.09317}
}

abstract

Motivated by the effect of the bumblebee field on thermodynamic instability in (non)extended phase space, we study the thermodynamic instability for the bumblebee AdS black holes. For this purpose, first, we evaluate the effect of the bumblebee field (or Lorentz-violating parameter) on the event horizon for AdS black holes. Then, in non-extended phase space, we study the effect of the bumblebee field on the heat capacity and the Helmholtz free energy to investigate the local and global thermal stability areas, respectively. Next, we extend our study on the extended phase space by seeking on stable area by using the heat capacity at constant pressure ($C_{P}$). Finally, we evaluate the super-entropy black hole condition and indicate that the bumblebee AdS black holes are super-entropy black holes when $l>0$, which is consistent with the condition $C_{P}<0$.

Figures

Figures reproduced from arXiv: 2501.09317 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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To achieve this, we will analyze the heat capacity of these black holes

Local stability Our objective in this study is to examine the local stability of AdS black holes in bumblebee gravity. To achieve this, we will analyze the heat capacity of these black holes. The heat capacity, in the canonical ensemble, provides important information about the thermal structure of black holes . It indicates whether the system is thermall...

work page

[2] [2]

In other words, the negative of the Helmholtz free e nergy determines the global stability of a thermo- dynamic system

Global stability In the context of the canonical ensemble, the global stability of a t hermodynamic system can be studied by Helmholtz free energy. In other words, the negative of the Helmholtz free e nergy determines the global stability of a thermo- dynamic system. Therefore, by using the Helmholtz free energy, w e want to evaluate the global stability ...

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