Pith. sign in

REVIEW 2 major objections 3 minor 2 cited by

Thermodynamic Instabilities of Generalized Exotic BTZ Black Holes

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Generalized exotic BTZ black holes with small α violate the reverse isoperimetric inequality yet can have positive C_V, refuting the conjecture that super-entropic holes always have C_V < 0.

desk verdict A clean, useful counterexample to Johnson's C_V conjecture, but its force depends on thermodynamic definitions imported from the authors' earlier paper. read the letter →

arxiv 1908.01254 v2 pith:XHLR7VXH submitted 2019-08-04 gr-qc

Wan Cong , Robert B. Mann This is my paper · ORCID
classification gr-qc MSC 83C57 PACS 04.70.Dy
keywords extendedblackholethermodynamicsreverseisoperimetricinequalitysuper-entropicholesgeneralizedexoticBTZspecificheatatconstantvolumethermodynamicinstabilitygravitationalChern-Simonsactioncosmologicalaspressure
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

This paper tests a recent proposal that any black hole violating the reverse isoperimetric inequality must have negative specific heat at constant volume, C_V < 0, as a sign of thermodynamic instability. Using the family of generalized exotic BTZ black holes—solutions of a convex combination of Einstein–Hilbert and gravitational Chern–Simons actions—the authors compute C_V, C_P, and the isoperimetric ratio R. They find that for sufficiently small α the black hole violates R ≥ 1 yet has positive C_V, a counterexample to the proposal. However, in every such case C_P < 0, so the black hole is still thermodynamically unstable; the paper therefore suggests a broader conjecture: all super-entropic black holes are thermodynamically unstable.

What carries the argument

The central object is the generalized exotic BTZ metric, obtained from the action $I = \alpha I_{\mathrm{EH}} + \gamma I_{\mathrm{GCS}}$ with $\gamma = 1-\alpha$, which makes the conserved mass $M = \alpha m + \gamma j/\ell$ and angular momentum $J = \alpha j + \gamma \ell m$. The argument is carried by the thermodynamic quantities derived for this family: entropy $S = \tfrac{1}{2}(\pi \alpha r_+ + \pi \gamma r_-)$, volume $V = \alpha \pi r_+^2 + \gamma \pi r_-^2\left(\tfrac{3r_+}{2r_-} - \tfrac{r_-}{2r_+}\right)$, and the ratio $R = \tfrac{1}{2}\sqrt{4\alpha - 2\gamma r_-^3/r_+^3 + 6\gamma r_-/r_+}$. For $\alpha < 1$ the ratio satisfies $R \leq 1$ with equality only in the extremal limit, so the holes are super-entropic. The sign of $C_V$ is found through the identity $C_V = C_P - T V \alpha_p^2/\kappa_T$, and its divergences occur exactly where $\partial V/\partial P|_{T,J} = 0$.

What would settle it

Evaluate the Euclidean action of the generalized exotic BTZ solution directly, without assuming the earlier volume and entropy formulas, and check whether a small-$\alpha$ black hole with $R < 1$ still has positive $C_V$; alternatively, scan the $(P, J, \alpha)$ space for any point with $R < 1$, $C_V > 0$, and $C_P \geq 0$, which would falsify the broader instability conjecture.

Watch

Extended reading notes

Core claim

The authors' central claim is that generalized exotic BTZ black holes with a sufficiently small α parameter are counterexamples to the conjecture of [1]: they violate the reverse isoperimetric inequality (R < 1, i.e., they are super-entropic, having more entropy than the isoperimetric bound would allow) but possess positive C_V. At the same time, whenever C_V > 0 the specific heat at constant pressure C_P is negative, so the black holes are unstable; the authors propose as a broader conjecture that every black hole violating the reverse isoperimetric inequality is thermodynamically unstable. The result sharpens the physical meaning of super-entropy: violation of the isoperimetric bound need not by itself produce C_V < 0, but it may still preclude thermodynamic stability.

Load-bearing premise

Everything rests on the inherited thermodynamic identifications from the authors' earlier work: the entropy, volume, and isoperimetric ratio formulas for generalized exotic BTZ black holes; if those definitions are not the correct conjugate variables, the reported signs of $C_V$ and the super-entropic violation would not follow.

Editorial extensions

If this is right

  • The conjecture of [1], that super-entropic black holes always have $C_V < 0$, is false as stated.
  • Generalized exotic BTZ black holes with $\alpha < 1/2$ and sufficiently large $r_+$ have $C_V > 0$ but $C_P < 0$, so they remain thermodynamically unstable.
  • More exotic holes ($\alpha < 1/2$) have negative $C_P$ except at extremality, while more standard holes ($\alpha > 1/2$) have non-negative $C_P$ and negative $C_V$.
  • $C_V$ diverges whenever $\partial V/\partial P|_{T,J} = 0$, marking a natural boundary between thermodynamic branches.
  • Standard BTZ black holes ($\alpha = 1$) saturate the isoperimetric inequality with $R = 1$ and have $C_V = 0$.

Reading between the lines

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

  • If the broader conjecture survives, super-entropy itself, rather than the sign of $C_V$, may be the robust marker of instability, and searches for thermodynamically stable ultra-spinning or charged black holes would be expected to fail.
  • The $C_V$ divergences where $\partial V/\partial P|_{T,J}=0$ suggest second-order phase transitions between thermodynamic branches that could be probed with equal-area constructions on the $P$–$V$ plane.
  • A direct Euclidean on-shell computation of the free energy for small $\alpha$ could verify whether positive $C_V$ corresponds to a local maximum of entropy, bypassing the inherited volume definitions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This manuscript tests the recent conjecture of Johnson [1] that black holes violating the reverse isoperimetric inequality necessarily have negative specific heat at constant volume C_V. The authors study the family of generalized exotic BTZ black holes, whose thermodynamic variables were obtained in Ref. [19], and compute C_P and C_V using the standard identity C_V = C_P - T V alpha_T^2 / kappa_T. For P=1 and J=0.1, they find that for sufficiently small alpha, C_V becomes positive at large r_+ while the isoperimetric ratio R is less than 1, which would constitute a counterexample to the conjecture. They also observe that C_P<0 whenever C_V>0, so these black holes remain thermodynamically unstable, and they propose a broader conjecture that all super-entropic black holes are thermodynamically unstable.

Significance. If the computation is correct, the paper provides a compact and useful counterexample to a recent conjecture, and it sharpens the discussion by distinguishing between stability with respect to C_V and stability with respect to C_P. The generalized exotic BTZ family is a natural test bed because the super-entropic property and the extended thermodynamic variables have already been established. The paper is clearly written, the figures are helpful, and the authors are appropriately careful in distinguishing the stronger and weaker versions of the conjecture. The main contribution is therefore a sharpened counterexample rather than a new computational method. However, the central claim is an application of existing results from Ref. [19], and the manuscript would benefit from being more self-contained about the key formulas that control the sign of C_V.

major comments (2)
  1. [Eqs. (10)-(12)] The central counterexample rests on the entropy S, thermodynamic volume V, and isoperimetric ratio R in Eqs. (10)-(12), all of which are quoted from Ref. [19] without derivation or independent check. The authors verify the first law and Smarr relation, but this is necessary rather than sufficient: different choices of S and V can satisfy the same first law, and the sign of C_V in Fig. 2 is controlled by the gamma-dependent term in V. Since the paper is explicitly a counterexample, I ask that the authors include a concise derivation of these formulas, or at least a derivation of V and R from the action I = alpha I_EH + gamma I_GCS, so that the sign of C_V can be certified rather than assumed.
  2. [Fig. 2 and the following text] The paper claims that generalized exotic BTZ black holes with a sufficiently small alpha parameter are counterexamples, but the positivity of C_V is only demonstrated numerically at P=1, J=0.1 and over a limited range of r_+. No closed-form expression for C_V or analytic condition for its sign is given, so the extent of the counterexample region is not established. I ask for an analytic criterion (for example, the location of the divergence where dV/dP|_{T,J}=0 and the threshold in alpha) or, alternatively, a clear statement that a single numerical example is intended to disprove the universal conjecture. The current wording is stronger than the evidence presented.
minor comments (3)
  1. [Abstract / opening paragraph] The first paragraph contains a typo: 'thermodynamic volues' should be 'thermodynamic volumes'.
  2. [Eq. (13) and surrounding text] The symbol r_m is used without definition; the horizon radii were defined as r_plus and r_minus, so r_m presumably denotes the inner horizon r_minus. Please define it and make the notation consistent.
  3. [Footnote 2] The phrase 'charged Schwarzschild-AdS' is imprecise; the charged non-rotating solution is usually called Reissner-Nordstrom-AdS rather than Schwarzschild-AdS.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity; the counterexample is a new computation from prior thermodynamic variables that are not fitted to the conjecture.

full rationale

The paper's counterexample to Johnson's conjecture is obtained by substituting the generalized exotic BTZ thermodynamic variables (eqs. 7-11, quoted from [19]) into the standard identity C_V = C_P - T V alpha_T^2/kappa_T (eq. 14) and evaluating the resulting signs. Nothing is fitted to the conjecture: alpha and gamma are fixed by the action, and the sign of C_V is computed, not imposed. The R < 1 conclusion follows from eq. (12), also from [19], but the conjecture being tested is an external target, so a paper can legitimately use prior results to test it. The only self-citation concern is that [19] shares an author with the present paper (Mann), but [19] is a parameter-free derivation from the combined EH+GCS action with stated assumptions that do not include Johnson's conjecture or positive C_V. Whether [19]'s entropy and volume formulas are the unique correct thermodynamic potentials is a correctness risk, not a circularity: the paper checks the first law and Smarr relation, and the absence of a uniqueness proof does not make the computation equivalent to its input. Since no equation is defined in terms of the target result and no fitted parameter is relabeled as a prediction, the derivation is not circular.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. Its conclusions rest on the correctness of thermodynamic definitions from prior work [19] and standard extended-thermodynamics identities.

assumptions (3)
  • domain assumption The thermodynamic quantities (M, J, T, S, V) and the isoperimetric ratio R for generalized exotic BTZ black holes, as given in Eqs. (7) through (12), are correct.
    These expressions are taken from the prior paper Frassino, Mann, and Mureika, arXiv:1906.07190, co-authored by one of the present authors. The present paper does not rederive them; an error in these formulas would affect the sign of CV.
  • domain assumption The extended first law dM = T dS + V dP + Omega dJ and the interpretation of the cosmological constant as pressure P = -Lambda/8pi apply to these black holes.
    This is the standard framework of extended black hole thermodynamics, adopted without proof in this paper.
  • standard math The relation CV = CP - T V alpha_p^2 / kappa_T (Eq. 14) is valid for these black holes.
    This is a standard thermodynamic identity connecting heat capacities, cited from [1].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Thermodynamic Instabilities of Generalized Exotic BTZ Black Holes." pith.science (2026). https://pith.science/paper/XHLR7VXH

@misc{pith2026190801254,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Instabilities of Generalized Exotic BTZ Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHLR7VXH}},
  note         = {Machine review of arXiv:1908.01254}
}
abstract

We examine the conjecture that black holes violating the reverse isoperimetric inequality have negative specific heat at constant volume $C_V$. We test this conjecture on the family of generalized exotic Ba\~nados, Teitelbiom and Zanelli (BTZ) black holes and find that $C_V$ can be positive even when the reverse isoperimetric inequality is violated, providing a counter example to the conjecture. However in all cases where $C_V$ is positive, the specific heat at constant pressure $C_P$ is negative, indicating that generalized exotic black holes are thermodynamically unstable, suggesting that a broader version of the conjecture might hold.

Figures

Figures reproduced from arXiv: 1908.01254 by the authors.

Figure 1
Figure 1. FIG. 1. Plot of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Charged accelerating black hole in $f(R)$ gravity

    hep-th 2019-08 conditional novelty 6.0 of 10

    A charged, slowly accelerating AdS black hole in f(R) gravity is constructed and shown to satisfy a first law, with an eta-dependent thermodynamic volume and a reverse isoperimetric inequality that allows super-entrop...

  2. Specific Heats for Rotating Quantum BTZ Black Holes in Extended Thermodynamics

    hep-th 2025-02 conditional novelty 5.0 of 10

    For the rotating quantum BTZ black hole, the paper derives path-dependent heat capacities at constant pressure and volume with multiple positive and negative branches.

Reference graph

Works this paper leans on

30 extracted references · 12 canonical work pages · cited by 2 Pith papers

  1. [19]

    A. M. Frassino, R. B. Mann, and J. R. Mureika, (2019), arXiv:1906.07190 [gr-qc]

    work page Pith review arXiv 2019

  2. [1]

    C. V. Johnson, (2019), arXiv:1906.00993 [hep-th]

    arXiv 2019

  3. [2]

    Kubiznak, R

    D. Kubiznak, R. B. Mann, and M. Teo, Classical and Quantum Gravity 34, 063001 (2017)

    work page 2017

  4. [3]

    Cvetic, G

    M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, Phys. Rev. D84, 024037 (2011), arXiv:1012.2888 [hep-th]

    arXiv 2011

  5. [4]

    B. P. Dolan, Class. Quant. Grav. 28, 125020 (2011), arXiv:1008.5023 [gr-qc]

    arXiv 2011

  6. [5]

    Kastor, S

    D. Kastor, S. Ray, and J. Traschen, Classical and Quan- tum Gravity 26, 195011 (2009)

    work page 2009

  7. [6]

    small black hole

    M. M. Caldarelli, G. Cognola, and D. Klemm, Class. Quant. Grav. 17, 399 (2000), arXiv:hep-th/9908022 [hep- th]. 1 We thank Clifford Johnson for pointing this out. 2 For instance in the charged Schwarzschild-AdS solution, a sin- gle temperature can admit two black hole solutions at the same pressure and charge. These belong respectively to “small black hole...

    arXiv 2000

  8. [7]

    Thermodynamic Volume of Cosmological Solitons

    S. Mbarek and R. B. Mann, Phys. Lett. B765, 352 (2017), arXiv:1611.01131 [hep-th]

    work page Pith review arXiv 2017

Show all 30 references
  1. [8]

    R. A. Hennigar, D. Kubiznak, and R. B. Mann, Phys. Rev. Lett. 115, 031101 (2015), arXiv:1411.4309 [hep-th]

    2015 arXiv

  2. [9]

    Klemm, Phys

    D. Klemm, Phys. Rev. D89, 084007 (2014), arXiv:1401.3107 [hep-th]

    2014 arXiv

  3. [10]

    R. A. Hennigar, D. Kubizˇ n´ ak, R. B. Mann, and N. Mu- soke, JHEP 06, 096 (2015), arXiv:1504.07529 [hep-th]

    2015 arXiv

  4. [11]

    W. G. Brenna, R. B. Mann, and M. Park, Phys. Rev. D92, 044015 (2015), arXiv:1505.06331 [hep-th]

    2015 arXiv

  5. [12]

    S. M. Noorbakhsh and M. Ghominejad, Phys. Rev. D95, 046002 (2017), arXiv:1611.02324 [hep-th]

    2017 arXiv

  6. [13]

    S. M. Noorbakhsh and M. Ghominejad, (2017), arXiv:1702.03448 [hep-th]

    2017 arXiv

  7. [14]

    Feng, H.-S

    X.-H. Feng, H.-S. Liu, W.-T. Lu, and H. Lu, Eur. Phys. J. C77, 790 (2017), arXiv:1705.08970 [hep-th]

    2017 arXiv

  8. [15]

    S. M. Noorbakhsh and M. H. Vahidinia, JHEP 01, 042 (2018), arXiv:1708.08654 [hep-th]

    2018 arXiv

  9. [16]

    R. B. Mann, Proceedings, 2nd Karl Schwarzschild Meet- ing on Gravitational Physics (KSM 2015): Frankfurt am Main, Germany, July 20-24, 2015 , Springer Proc. Phys. 208, 105 (2018)

    2018

  10. [17]

    Banados, C

    M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992), arXiv:hep-th/9204099

    1992 arXiv

  11. [18]

    C. V. Johnson, (2019), arXiv:1905.00539 [hep-th]. 4

    2019 arXiv

  12. [20]

    more exotic

    indicated that charged BTZ black holes were super- entropic. Generalized BTZ black holes are also super- entropic, but demonstration of this is somewhat more subtle. These objects are (2 + 1)-dimensional black holes whose metric is given by the usual rotating BTZ metric ds2 =−...

    1908 arXiv

  13. [21]

    Witten, Nucl

    E. Witten, Nucl. Phys. B311, 46 (1988)

    1988

  14. [22]

    A. M. Frassino, R. B. Mann, and J. R. Mureika, Phys. Rev. D92, 124069 (2015), arXiv:1509.05481 [gr-qc]

    2015 arXiv

  15. [23]

    P. K. Townsend and B. Zhang, Phys. Rev. Lett. 110, 241302 (2013), arXiv:1302.3874 [hep-th]

    2013 arXiv

  16. [24]

    Carlip, J

    S. Carlip, J. Gegenberg, and R. B. Mann, Phys. Rev. D51, 6854 (1995), arXiv:gr-qc/9410021 [gr-qc]

    1995 arXiv

  17. [25]

    Ba˜ nados, Phys

    M. Ba˜ nados, Phys. Rev. D57, 1068 (1998)

    1998

  18. [26]

    Carlip and J

    S. Carlip and J. Gegenberg, Phys. Rev. D44, 424 (1991)

    1991

  19. [27]

    S. N. Solodukhin, Phys. Rev. D74, 024015 (2006), arXiv:hep-th/0509148 [hep-th]

    2006 arXiv

  20. [28]

    Such distinctions do not occur in the gen- eralized exotic BTZ case [19]

    for more details). Such distinctions do not occur in the gen- eralized exotic BTZ case [19]. However the divergence in Cv can be argued to separate the class of exotic solutions into different branches

  21. [29]

    Ba˜ nados, A

    M. Ba˜ nados, A. Gomberoff, and C. Mart´ ınez, Classical and Quantum Gravity 15, 3575 (1998)

    1998

  22. [30]

    Kubiznak and R

    D. Kubiznak and R. B. Mann, JHEP 07, 033 (2012), arXiv:1205.0559 [hep-th]

    2012 arXiv

Pith tools

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