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Specific Heats for Rotating Quantum BTZ Black Holes in Extended Thermodynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper derives exact heat-capacity formulae for the rotating quantum BTZ black hole and shows that rotation generates infinite families of $C_p$ and $C_V$ with both stable and unstable branches.

desk verdict A careful, algebra-heavy extension of static qBTZ heat capacities to the rotating case; likely correct as mathematics, but the central claim of infinite heat-capacity families rests on an unphysical path freedom that needs a firm physical justification. read the letter →

arxiv 2502.02156 v1 pith:OGGJXGB6 submitted 2025-02-04 hep-th

Roberto Nazario This is my paper · ORCID
classification hep-th PACS 04.70.-s
keywords rotatingquantumBTZblackholeextendedthermodynamicsheatcapacitybraneworldthermodynamicvolumesuper-entropicitystabilitycriticalpoint
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 extends the extended-thermodynamics treatment of heat capacities from the static quantum BTZ black hole to the rotating one, and claims exact closed-form formulae for $C_p$ and $C_V$ as functions of the three dimensionless parameters $\nu$, $z$, and $\alpha$. The new structural result is that rotation adds a free parameter, so at fixed pressure or fixed volume the temperature and entropy can be varied along infinitely many curves, and each curve defines its own heat capacity. In every case examined, at least two physically realizable branches appear, with both positive and negative heat capacities, signalling both stable and unstable black holes. The static critical point at $(\nu,z)=(1,1)$ disappears for any non-zero $\alpha$, although other points where heat capacities diverge remain. These are exact results for a quantum-corrected black hole, and they supply new test cases for the conjectured connection between super-entropicity and thermodynamic instability.

What carries the argument

The central object is the rotating quantum BTZ black hole obtained from a braneworld model: an AdS$_3$ brane embedded in an AdS$_4$ C-metric, with the backreaction of the bulk black hole inducing quantum corrections on the brane. The argument runs in extended thermodynamics, where the cosmological constant acts as pressure $p$ and its conjugate is the thermodynamic volume $V$; fixing $p$ fixes $\nu$. The load-bearing identity is Eq. (21), which constructs $C_p$ as $T\,dS/dT$ along an arbitrary parametrized curve $z=z(\lambda)$, $\alpha=\alpha(\lambda)$ at fixed $\nu$, together with the analogous construction of $C_V$ in Eq. (24) along curves on a constant-volume surface. The choice of curve is exactly what produces the infinite family of heat capacities, and the explicit rational expressions in Appendix B carry the detailed claims about branches, signs, and divergences.

What would settle it

Evaluate $C_{p,\alpha}$ from Eq. (22) along a fixed-$\alpha$ path and $C_{p,z}$ from Eq. (23) along a fixed-$z$ path through the same state $(\nu,z,\alpha)$ with $T>0$ and constraint (14) satisfied; if the two values differ, the heat capacity depends on the chosen curve rather than on the thermodynamic state. Conversely, checking whether each heat-capacity divergence coincides with a genuine phase transition, such as a change of horizon topology or a divergence in a response function like the compressibility, would settle whether the divergent points are physical.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the rotating quantum BTZ black hole, built from a braneworld construction with backreaction parameter $\nu$, size parameter $z$, and rotation parameter $\alpha$, possesses heat capacities that are exact but not unique: at constant pressure (fixed $\nu$) and at constant volume, $C_p$ and $C_V$ come in infinite families indexed by the curve chosen in the $(z,\alpha)$ plane or on the isochoric surface. The paper asserts that the explicit formulae collected in Appendix B for $C_{p,\alpha}$, $C_{p,z}$, $C_{V,\alpha}$, $C_{V,z}$, and $C_{V,p}$ are correct, that they produce multiple physical branches with both signs, and that the critical point of the static case at $(\nu,z)=(1,1)$ is absent for any non-zero rotation parameter $\alpha$. It further argues that at second order in $\nu$ the isoperimetric ratio $R$ can be on either side of 1 in combination with any signs of the heat capacities, so no straightforward super-entropicity–instability relation is visible in this system.

Load-bearing premise

The load-bearing premise is that holding pressure fixed leaves the two remaining parameters free to vary along any curve, so that every such curve defines a valid heat capacity; if only special paths, such as fixed angular momentum, are physically meaningful thermodynamic processes, then these infinite families are not physical heat capacities.

Editorial extensions

If this is right

  • The rotating quantum BTZ black hole has infinitely many heat capacities at fixed pressure and at fixed volume, so stability statements must specify which thermodynamic path is being used.
  • For any non-zero $\alpha$, the static critical point at $(\nu,z)=(1,1)$ is gone; heat-capacity divergences occur at other, parameter-dependent points that may signal transitions.
  • At least some of the two new branches (branches 3 and 4) are physically realizable for $T>0$ under the constraint (14), so the rotating quantum BTZ has additional black hole solutions beyond the static case.
  • The conjectured link between super-entropicity and instability does not hold simply: at second order in $\nu$, $R$ above and below 1 can coexist with any sign pattern of $C_p$ and $C_V$.
  • The Appendix B formulae are explicit functions of $(\nu,z,\alpha)$, so subsequent studies can evaluate or differentiate them without redoing the braneworld construction.

Reading between the lines

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

  • A natural consequence of the path freedom in Eq. (21) is that $C_p$ and $C_V$ are not state functions for this system; if this is right, only physically selected paths, such as fixed angular momentum at constant pressure, should be used for stability judgments.
  • One direct test of the framework is to evaluate $C_{p,\alpha}$ and $C_{p,z}$ at the same state point $(\nu,z,\alpha)$; their inequality would confirm the claimed path dependence, while their equality on all overlapping curves would strengthen the interpretation of these as genuine thermodynamic quantities.
  • The disappearance of the static critical point for $\alpha\neq 0$ suggests rotation acts as a deformation that washes out the static critical behaviour; scanning the $\alpha\to 0$ limit of the new divergence points could reveal how the critical point is recovered.
  • The same curve-based construction could be applied to other response coefficients, such as the expansion coefficient or isothermal compressibility, extending the catalogue of exact thermodynamic data for quantum-corrected black holes without new geometry.
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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 the extended thermodynamics of the rotating quantum BTZ black hole, using the braneworld-constructed thermodynamic functions M, T, S, Ω, J, and V expressed in terms of three dimensionless parameters ν, z, and α. With ν identified with the pressure p, the paper defines constant-pressure and constant-volume heat capacities along curves in the (z, α) plane, presents explicit formulas in Appendix B for Cp,α, Cp,z, CV,α, CV,z, and CV,p, and plots several branches. It reports positive and negative branches, divergences, the disappearance of the static critical point at (ν, z) = (1, 1) for α ≠ 0, and a discussion of super-entropicity versus the signs of the heat capacities.

Significance. If the quantities computed were the standard heat capacities of the rotating qBTZ black hole, the paper would provide a new set of exact thermodynamic results for a quantum-corrected braneworld black hole. The work has notable strengths: it uses no fitted parameters, the thermodynamic functions are imported from earlier constructed solutions, and the analytic check that the static critical point is absent for nonzero α is clean and explicit. The connection to the prior static case is also instructive. However, the central definitional issue described below affects the physical interpretation of all the reported heat capacities, so the significance currently rests on a nonstandard construction rather than on the standard response functions of extended black hole thermodynamics.

major comments (3)
  1. [Sec. II.B, Eq. (21)] The definition of Cp in Eq. (21) is not the heat capacity of the rotating black hole in extended thermodynamics. With the first law dM = T dS + V dp + Ω dJ, the equilibrium state is labelled by (S, p, J) (or an equivalent set), and the standard constant-pressure specific heat is Cp = T(∂S/∂T)_{p,J}. Eq. (21) instead computes T dS/dT along an arbitrary curve z(λ), α(λ) at fixed ν, with no restriction on J; since J(ν, z, α) in Eq. (12) varies along such curves, the process exchanges Ω dJ work and T dS/dT is path-dependent. Holding α or z fixed does not hold J fixed, so Cp,α and Cp,z are not the standard Cp. The infinite family of heat capacities is thus an artifact of the arbitrary-path construction rather than a set of physical response functions. Please either compute Cp = T(∂S/∂T)_{p,J} along the dJ = 0 curve at fixed p and compare with the present results, or provide a quasi-static protocol that realizes constant α or constant z at fixed p and fixed J; without that, the central claim is not supported.
  2. [Sec. II.C, Eq. (24)] The same problem affects CV. The standard constant-volume heat capacity is CV = T(∂S/∂T)_{V,J}, with angular momentum held fixed. Eq. (24) permits any curve on the two-dimensional isochoric surface, and the paper explicitly selects curves with fixed α, fixed z, or fixed p. Because V and J are independent functions of (ν, z, α), holding V fixed does not hold J fixed, so the resulting quantities are not the specific heat at constant volume. The three CV functions in Appendix B are therefore path-dependent derivatives, not response functions. The paper should either recompute CV with J fixed or clearly rename these objects as path-dependent thermal coefficients and justify their thermodynamic meaning.
  3. [Appendix B, Eqs. (B1)-(B18)] The explicit formulas are asserted without derivation or independent verification. Given that the formulas fill several pages and the central claims (multiple branches, signs, divergences) rest on them, please provide a reproducible derivation, a symbolic-check notebook, or at least a consistency check against the definitions in Eqs. (21)-(31) for representative parameter values. As written, the reader cannot verify that the displayed numerators and denominators are correct, and small typographical errors in such long expressions would change the conclusions.
minor comments (4)
  1. [Sec. II.A] The statement that "a numerical check of a wide range of values supports our tentative conclusion" about the common upper bound of the four branches is too vague; please specify the parameter ranges checked, the sampling, and the numerical accuracy.
  2. [Sec. II.B.1] The discussion of divergences and the Schottky peak in Cp,α is qualitative, and the text notes that the exact locations are difficult to calculate. Since these features are presented as notable results, please provide at least a numerical table or explicit equations for selected parameter values.
  3. [Sec. III, Eqs. (32)-(34)] The small-ν expansion R ≈ 1 + f(z, α)ν − g(z, α)ν^2 is presented without derivation or a stated domain of validity; please indicate how the expansion is obtained and within which region of (z, α) it is reliable.
  4. [Throughout] There are several typographical errors and unclear cross-references, e.g., "as as" in Sec. I, "whatis" in Sec. II.A, "the auhor" in the Acknowledgments, and the reference in the Fig. 3 caption to branches seen in Figs. 3b and 3d should be clarified.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: heat capacities are explicit derivatives of imported thermodynamic functions; path-dependence is a physical-interpretation issue, not a derivation loop.

full rationale

The derivation chain starts from the thermodynamic functions M, T, S, Omega, and J in Eqs. (8)-(12), imported from the braneworld construction of refs. [33-35] (chiefly Emparan-Frassino-Way) and from the static qBTZ analysis in refs. [24,37]. No parameter is fitted to any heat-capacity output: the explicit Appendix B formulas are obtained by differentiating the imported functions along the chosen fixed-nu or isochoric slices. The path-dependence in Eq. (21) and Eq. (24) is a question of physical interpretation (whether arbitrary z-alpha paths are legitimate quasi-static processes), not a circularity: the formulas are openly directional derivatives, and the reported infinite family follows directly from the two-dimensional fixed-pressure state space rather than from a hidden reuse of the target result. Self-citations, mainly ref. [24] for the critical-point condition (20) and for branch-naming conventions, are not load-bearing: the rotating-case conclusion that the old (nu,z)=(1,1) critical point disappears for alpha != 0 is independently verified by the explicit evaluations T'=3alpha^2/(2*sqrt(2)pi), T''=(alpha^2+12alpha^4)/(2*sqrt(2)pi), and S'=-pi(1+2alpha^2)/3. The central quantitative content, namely the explicit heat-capacity functions and their branch structure, is not equivalent to the inputs by construction, and no fitted parameter is renamed as a prediction. No significant circularity was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters are used: ν, z, and α are dimensionless solution parameters defined in Eq. (13), not tuned to data. No new entities are introduced beyond the existing quantum BTZ model. The axioms listed are the imported background results and the path-dependence assumption that the central results rely on.

assumptions (5)
  • domain assumption The rotating qBTZ thermodynamic quantities in Eqs. (8)-(12), taken from ref. [34], are correct.
    Central to all heat capacity computations; only summarized in Appendix A rather than re-derived.
  • domain assumption Pressure p depends only on ν, so constant ν is equivalent to constant pressure.
    Used throughout Secs. II.B and II.C to compute Cp at fixed ν; see Eq. (16).
  • ad hoc to paper A curve in the (z,α) plane at fixed ν defines a valid heat capacity path.
    Eq. (21) introduces the parameter λ; this path freedom generates the infinite family of heat capacities and is not derived from the first law alone.
  • domain assumption Physical branches are those satisfying constraint (14) and T≥0; other portions are discarded.
    The paper excludes nonphysical branches using this criterion, relying on ref. [34] for the form of (14).
  • domain assumption Super-entropicity conclusions are meaningful in the small-ν regime where the brane theory is nearly massless.
    The author restricts physical interpretation to small ν and expands R only to second order in ν in Sec. III.

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Pith. "Pith review of Specific Heats for Rotating Quantum BTZ Black Holes in Extended Thermodynamics." pith.science (2026). https://pith.science/paper/OGGJXGB6

@misc{pith2026250202156,
  author       = {Pith},
  title        = {Pith review of: Specific Heats for Rotating Quantum BTZ Black Holes in Extended Thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGGJXGB6}},
  note         = {Machine review of arXiv:2502.02156}
}
abstract

In the framework of extended thermodynamics, where the cosmological constant $\Lambda$ plays the role of a dynamical pressure $p$, its conjugate variable $V$ arises naturally. This makes it possible to define $C_p$ and $C_V$, the heat capacities at constant pressure and volume, respectively. We extend our previous work on the heat capacities of the static ``quantum" version of the BTZ black hole defined on a braneworld model to the case where the black hole is rotating. The extra degree of freedom that rotation grants the system imparts it with infinite families of both $C_p$ and $C_V$. We find exact formulae for these heat capacities as functions of the three dimensionless parameters of the theory, and explore some special cases in detail. In all cases considered, at least two physically realizable branches were observed, including both positive and negative heat capacities, signaling both stable and unstable black holes, respectively. Though the critical point seen in the static case disappears, other interesting points arise where the heat capacities diverge. Finally, we discuss the conjectured connection in the literature between the super-entropicity of a black hole and its instability, though much like in the static case, the exact relationship, if any, remains unclear.

Figures

Figures reproduced from arXiv: 2502.02156 by the authors.

Figure 1
Figure 1. FIG. 1: Branches corresponding to black holes at fixed [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

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