Pith. sign in

REVIEW 3 major objections 3 minor 32 references

The paper claims that for rotating regular black holes, the first law must be built in the full parameter space and only then constrained; doing it in the other order yields an unphysical 'effective temperature'.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 11:21 UTC pith:54CYKVRP

load-bearing objection The paper's central derivation is built on a metric mass function that contradicts the horizon equation it uses, so the full-phase-space result is unsupported as written; the order-of-operations idea is plausible and fixable. the 3 major comments →

arxiv 2601.06769 v2 pith:54CYKVRP submitted 2026-01-11 gr-qc

Constraints and Consistency of Rotating Regular Black Hole Thermodynamics

Qi-Quan Li , Yu Zhang , Hoernisa Iminniyaz This is my paper
classification gr-qc MSC 83C5780A10 PACS 04.70.Dy
keywords rotating regular black holesHawking temperaturethermodynamic phase space reductionfirst law of thermodynamicsregularity constraintBardeen black holeHayward black holesurface gravity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Regular black holes avoid singularities but their thermodynamics has been stuck with a known mismatch: the Hawking temperature from the first law disagrees with the geometric surface-gravity temperature. The paper claims to resolve this for a class of rotating regular black holes by treating the regularity condition M=q^3/alpha as a constraint that reduces the thermodynamic phase space. In the full unconstrained phase space {S,J,q,alpha}, the first law gives a temperature T^(I) that matches the geometric one; only after that should the constraint be applied. Applying the constraint first produces an 'effective temperature' T^(II) that obeys a different first law and has no physical meaning. The geometric temperature, by contrast, is insensitive to the order, so the paper identifies a procedural asymmetry as the origin of the old inconsistency.

Core claim

The central claim is that for rotating regular black holes, a self-consistent thermodynamics exists only if one works with the full phase space {S,J,q,alpha} first, derives the first law and all thermodynamic quantities there, and only then imposes the regularity condition M=q^3/alpha. In that full space, the first law dM=T dS + Omega dJ + Psi dq + A dalpha yields a temperature T^(I)_1 that equals the surface-gravity temperature T_1g. If the constraint is applied at the start instead, the reduced phase space yields a different temperature T^(II) which the first law no longer supports in the same form; the paper interprets T^(II) as an effective quantity with no physical meaning. The geometri

What carries the argument

The carrying mechanism is the order of operations between deriving thermodynamic quantities and imposing the regularity constraint. The squared-mass formula in the full phase space, Eq. (14), defines a first law with independent parameters S, J, q, alpha; its derivatives give the temperature T^(I)_1. The constraint M=q^3/alpha is then substituted in last. Applying the constraint first reduces the phase space to {S,q,alpha}, and the resulting temperature T^(II) is shown to relate to T^(I)_2 with a prefactor 1/(1+A q^3/M^2), marking it as an effective temperature rather than a true Hawking temperature.

Load-bearing premise

The whole derivation rests on an unproved form of the rotating metric's mass function: Eq. (13) assumes a different expression for m(r) than Eq. (12) states, and the paper never bridges that gap.

What would settle it

Compute the event-horizon equation and surface gravity directly from Eq. (11)-(12) with m(r)=M r^mu/(r^nu+q^nu)^{mu/nu}; if the resulting first-law temperature does not equal the geometric temperature, or if Eq. (13) is not reproduced, the claimed consistency in the full phase space fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The long-standing mismatch between first-law and geometric Hawking temperatures for this class of regular black holes is resolved by the 'full phase space first' rule.
  • For any rotating black hole model with a parameter constraint, thermodynamic quantities computed in the reduced phase space cannot be trusted; the constraint must be imposed after the first law is derived.
  • The geometric Hawking temperature is an intrinsic property of the metric, invariant under constraint ordering, so it can serve as the reference temperature for checking thermodynamic derivations.
  • The framework provides a basis for phase-transition and stability studies of rotating regular black holes, since all quantities are now mutually consistent.
  • The same reduction argument applies to static regular black holes and extends to axisymmetric spacetimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ordering principle is general, it suggests that any 'first law' written in a reduced parameter space for a constrained black hole family is generically a projection of the full first law, with a conformal factor like 1/(1+A q^3/M^2); testing that factor for other regular black hole families would be a direct extension.
  • The asymmetry between geometric and thermodynamic quantities under constraint application might have analogues in other constrained gravitational systems, for example black holes with scalar hair or dilatonic couplings where a parameter is fixed by a boundary condition.
  • A concrete testable extension: compute the thermodynamic temperature for the Bardeen limit (mu=3, nu=2) and Hayward limit (mu=nu=3) of Eq. (14) with the constraint applied last and check whether the match T^(I)=T_g survives in those special cases.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proposes a "full phase space first" procedure for the thermodynamics of a class of rotating regular black holes. Starting from a rotating "mother" spacetime with independent parameters {S,J,q,α}, the authors derive the first law and the Hawking temperature, then impose the regularity constraint M=q^3/α. They claim that this order of operations yields a thermodynamic temperature identical to the surface-gravity temperature, whereas imposing the constraint before deriving the first law gives a different, unphysical "effective temperature". The paper concludes that geometric temperatures are invariant under the order of constraint application, while thermodynamic temperatures are not, thereby resolving the long-standing mismatch for this class of black holes.

Significance. The question addressed is relevant: the inconsistency between thermodynamic and geometric Hawking temperatures for regular black holes is a known obstruction, and an extension to rotating cases is a natural step. The explicit mass formula and temperature expressions, if correct, would provide a concrete computational framework for a class of rotating regular black holes and a methodological principle applicable to other constrained parameter spaces. However, the central derivation as written is undermined by an inconsistency between the stated mass function and the horizon equation used throughout. The conceptual point about phase-space reduction is plausible, but the present manuscript does not yet establish it rigorously.

major comments (3)
  1. [Thermodynamics of rotating regular black holes, Eqs. (11)-(13)] The mass function stated in Eq. (12) is m(r)=M r^μ/(r^ν+q^ν)^{μ/ν}, which already satisfies the regularity constraint M=q^3/α (Eq. (9)). Substituting it into Δ in Eq. (11) yields Eq. (20), not the 'before constraint' horizon equation Eq. (13). Eq. (13) corresponds instead to m(r)=M−(q^3/α)[1−r^μ/(r^ν+q^ν)^{μ/ν}], i.e., Eq. (7). Thus the full-phase-space 'mother' line element is never defined. Since Eq. (14) and the temperature T^(I)_1 are derived from Eq. (13), the central derivation is disconnected from the stated spacetime. The authors must replace Eq. (12) with Eq. (7) for the full phase space (and clarify that Eq. (12) is the constrained form), or re-derive all thermodynamic quantities from the metric actually used.
  2. [Eqs. (13), (17), (18)] The claimed identity T^(I)_1=T_1g is not an independent consistency check. Both quantities are obtained from the same horizon equation (13): T^(I)_1 from the derivative of the mass formula derived from (13), and T_1g by inserting (13) into the surface-gravity formula (18). Unless the line element that yields (13) is given and the surface gravity is evaluated on that explicit metric, the 'geometric' temperature is not an independent benchmark. The authors should supply the explicit mother metric and verify Eq. (18) from it.
  3. [Eq. (23)] The derivation and meaning of Eq. (23) are unclear. The symbol A is used for the conjugate potential in Eq. (16), but in the text after Eq. (23) it appears as an undefined coefficient in T^(II)=T_2^(I) M^2/(M^2+A q^3). Also the phrase 'phase space {S,q,α}' omits J; the reduced phase space should be specified. This equation is used to conclude that T^(II) is an 'effective temperature' with no physical meaning, so it should be derived explicitly from the constraint dα=(3q^2/M)dq−(q^3/M^2)dM and the first law.
minor comments (3)
  1. [Eqs. (17), (19), (21), (22)] The symbol r is used without a subscript; earlier r_H denotes the horizon radius. Please use r_H consistently or state that r denotes the horizon radius in these formulas.
  2. [Eq. (14)] Eq. (14) is an implicit equation for M because a=J/M enters on the right-hand side. The text should state this explicitly and describe how the partial derivatives in Eq. (16) are computed from this implicit relation.
  3. [Eq. (12) vs Eq. (7)] The paper should clearly distinguish between the unconstrained mass function (7), used for the mother spacetime, and the constrained mass function (12), used only after imposing Eq. (9). As written, the reader cannot tell which mass function is being used in each subsequent equation.

Circularity Check

0 steps flagged

No significant circularity: the thermodynamic temperature is checked against the independent geometric surface gravity, and no fitted input or load-bearing self-citation is present.

full rationale

The paper's central derivation is not circular. The geometric Hawking temperature is defined independently from surface gravity (Eq. 18), while the thermodynamic temperature is obtained by differentiating the squared-mass formula (Eq. 14), which is generated by inverting the horizon condition (Eq. 13) in the phase space {S, J, q, alpha}. The equality T^(I)_1 = T_1g is then an external consistency check, not a definition or a fitted input. The later claim that T^(II) is an 'effective temperature' with no physical meaning is based on its failure to match the geometric result, again an independent benchmark. No parameter is fitted to data, and the load-bearing metric/algorithmic ingredients (Type-I rotating metric, Newman-Janis algorithm, Fan-Wang Lagrangian) come from external references rather than from the present authors' own prior results. Even if Eq. (12) and Eq. (13) appear mutually inconsistent—Eq. (12)'s mass function yields Delta = r^2 + a^2 - 2 M r^(mu+1)/(r^nu+q^nu)^(mu/nu), i.e. Eq. (20), while Eq. (13) uses the pre-constraint mass function Eq. (7)—that inconsistency is a correctness or consistency problem in the derivation chain, not a case of a result reducing by definition to its inputs. No self-citation is load-bearing, and no 'prediction' is forced by construction from fitted data.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The model parameters (\mu, \nu, q, \alpha) are inputs from the nonlinear electrodynamics Lagrangian, not fitted to data or tuned to produce the claimed consistency. The load-bearing assumptions are the area-law entropy, the validity of the NJA rotating metric, the standard surface-gravity temperature formula, and the transfer of the static regularity constraint to the rotating family. The metric mismatch in Eq. (12)/(13) makes the most basic axiom--the form of the rotating mother metric--unreliable.

axioms (5)
  • domain assumption S = A/4 (Bekenstein-Hawking area law) holds for rotating regular black holes.
    Used implicitly when setting S=\pi(r_H^2+a^2) before Eq. (14); imported from standard black hole thermodynamics without derivation.
  • domain assumption The Type-I rotating metric Eq. (11) is a valid rotating solution of Einstein gravity coupled to the NED Lagrangian Eq. (6), with mass function m(r).
    Adopted from Kar et al. [32] via the Newman-Janis algorithm; the field equations are not re-derived, and the mass function is internally inconsistent (Eq. (12) vs. Eq. (13)).
  • standard math The geometric Hawking temperature is given by T=\Delta'(r_H)/(4\pi(r_H^2+a^2)) for this Kerr-like metric.
    Standard Killing-horizon result used in Eqs. (18) and (22).
  • domain assumption The regularity condition lim_{r->0} m(r)=0 from the static seed extends to the rotating metric, giving M=q^3/\alpha.
    Eq. (9) is derived from the static mass function; the paper assumes it carries over to the rotating family without independent verification.
  • domain assumption The first law in the full phase space {S,J,q,\alpha}, dM=T dS+\Omega dJ+\Psi dq+A d\alpha, is the correct starting point for constrained reduction.
    This is the paper's core principle; it is asserted and verified only by matching the geometric temperature for the specific class considered.

pith-pipeline@v1.3.0-alltime-deepseek · 43 in / 23402 out tokens · 348595 ms · 2026-08-03T11:21:02.565630+00:00 · methodology

0 comments
read the original abstract

The thermodynamics of regular black holes has long suffered from a self-consistency problem: the Hawking temperature derived from the first law disagrees with the geometrically defined one, signaling that the thermodynamic quantities obtained in previous studies are unreliable. To resolve this, we construct a rotating ``mother'' black hole whose parameters are initially independent, forming an extended thermodynamic phase space. Thermodynamic quantities are first derived in this unconstrained phase space, and only then is the regularity condition imposed. We find a fundamental asymmetry between the geometric and thermodynamic derivations: the geometric temperature is invariant under the order of constraint application, whereas the thermodynamic temperature requires the correct order of operations to yield a physically meaningful result. This establishes a self-consistent thermodynamic framework for rotating regular black holes, in which all thermodynamic quantities are guaranteed to be physically correct, laying a rigorous foundation for investigating phase transitions and thermodynamic stability.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 2 canonical work pages

  1. [1]

    Hawking and R

    S.W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A 314(1970) 529-548

    1970

  2. [2]

    Bardeen, GR5, 87 (1968)

    J. Bardeen, GR5, 87 (1968)

    1968

  3. [3]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80(1998), 5056-5059 doi:10.1103/PhysRevLett.80.5056 [arXiv:gr-qc/9911046 [gr-qc]]

    Pith/arXiv arXiv 1998

  4. [4]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B493(2000), 149-152 doi:10.1016/S0370-2693(00)01125-4 [arXiv:gr- qc/0009077 [gr-qc]]

    arXiv 2000

  5. [5]

    S. A. Hayward, Phys. Rev. Lett.96(2006), 031103 doi:10.1103/PhysRevLett.96.031103 [arXiv:gr- qc/0506126 [gr-qc]]

    arXiv 2006

  6. [6]

    Berej, J

    W. Berej, J. Matyjasek, D. Tryniecki and M. Woronowicz, Gen. Rel. Grav.38(2006), 885-906 doi:10.1007/s10714-006-0270-9 [arXiv:hep-th/0606185 [hep-th]]

    Pith/arXiv arXiv 2006

  7. [7]

    Burinskii and S

    A. Burinskii and S. R. Hildebrandt, Phys. Rev. D65(2002), 104017 doi:10.1103/PhysRevD.65.104017 [arXiv:hep-th/0202066 [hep-th]]

    Pith/arXiv arXiv 2002

  8. [8]

    Balart and E

    L. Balart and E. C. Vagenas, Phys. Rev. D90 (2014) no.12, 124045 doi:10.1103/PhysRevD.90.124045 [arXiv:1408.0306 [gr-qc]]

    Pith/arXiv arXiv 2014

  9. [9]

    M. S. Ma, Annals Phys.362(2015), 529-537 doi:10.1016/j.aop.2015.08.028 [arXiv:1509.05580 [gr-qc]]

    Pith/arXiv arXiv 2015

  10. [10]

    C. H. Nam, Gen. Rel. Grav.50(2018) no.6, 57 doi:10.1007/s10714-018-2380-6

    work page doi:10.1007/s10714-018-2380-6 2018

  11. [11]

    Z. Y. Fan and X. Wang, Phys. Rev. D94 (2016) no.12, 124027 doi:10.1103/PhysRevD.94.124027 [arXiv:1610.02636 [gr-qc]]

    Pith/arXiv arXiv 2016

  12. [12]

    M. S. Ma and R. Zhao, Class. Quant. Grav. 31(2014), 245014 doi:10.1088/0264-9381/31/24/245014 [arXiv:1411.0833 [gr-qc]]

    Pith/arXiv arXiv 2014

  13. [13]

    M. S. Ma, H. F. Li and J. H. Shi, Sci. China Phys. Mech. Astron.69(2026) no.1, 210411 doi:10.1007/s11433-025- 2753-6 [arXiv:2507.09551 [gr-qc]]

    arXiv 2026

  14. [14]

    Mukhopadhyay, Astrophys

    B. Mukhopadhyay, Astrophys. Space Sci. Proc.53 (2018), 3-15 doi:10.1007/978-3-319-94607-8 1

    work page doi:10.1007/978-3-319-94607-8 2018

  15. [15]

    C, Bambi, Astrophysical Black Holes: A Review, arXiv.1906.03871 [gr-qc]

    Pith/arXiv arXiv 1906

  16. [16]

    Kumar, A

    R. Kumar, A. Kumar and S. G. Ghosh, Astrophys. J.896(2020) no.1, 89 doi:10.3847/1538-4357/ab8c4a [arXiv:2006.09869 [gr-qc]]. 5

    Pith/arXiv arXiv 2020

  17. [17]

    Afrin, R

    M. Afrin, R. Kumar and S. G. Ghosh, Mon. Not. Roy. Astron. Soc.504(2021) no.4, 5927-5940 doi:10.1093/mnras/stab1260 [arXiv:2103.11417 [gr-qc]]

    Pith/arXiv arXiv 2021

  18. [18]

    R. V. Maluf and J. C. S. Neves, Phys. Rev. D97 (2018) no.10, 104015 doi:10.1103/PhysRevD.97.104015 [arXiv:1801.02661 [gr-qc]]

    Pith/arXiv arXiv 2018

  19. [19]

    E. T. Newman and A. I. Janis, J. Math. Phys.6(1965), 915-917 doi:10.1063/1.1704350

    doi:10.1063/1.1704350 1965

  20. [20]

    H. C. Kim, B. H. Lee, W. Lee and Y. Lee, Phys. Rev. D101(2020) no.6, 064067 doi:10.1103/PhysRevD.101.064067 [arXiv:1912.09709 [gr-qc]]

    Pith/arXiv arXiv 2020

  21. [21]

    C. A. Benavides-Gallego, A. A. Abdujabbarov and C. Bambi, Phys. Rev. D101(2020) no.4, 044038 doi:10.1103/PhysRevD.101.044038 [arXiv:1811.01562 [gr-qc]]

    Pith/arXiv arXiv 2020

  22. [22]

    C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-A ¨ ınou and A. Wang, Phys. Rev. D101 (2020) no.8, 084001 [erratum: Phys. Rev. D103 (2021) no.8, 089902] doi:10.1103/PhysRevD.101.084001 [arXiv:2003.00477 [gr-qc]]

    Pith/arXiv arXiv 2020

  23. [23]

    Z. Xu, X. Gong and S. N. Zhang, Phys. Rev. D101 (2020) no.2, 024029 doi:10.1103/PhysRevD.101.024029

    doi:10.1103/physrevd.101.024029 2020

  24. [24]

    Kumar and S

    R. Kumar and S. G. Ghosh, Eur. Phys. J. C78 (2018) no.9, 750 doi:10.1140/epjc/s10052-018-6206-1 [arXiv:1711.08256 [gr-qc]]

    Pith/arXiv arXiv 2018

  25. [25]

    Xu and J

    Z. Xu and J. Wang, Phys. Rev. D95(2017) no.6, 064015 doi:10.1103/PhysRevD.95.064015 [arXiv:1609.02045 [gr- qc]]

    Pith/arXiv arXiv 2017

  26. [26]

    Mazza, E

    J. Mazza, E. Franzin and S. Liberati, JCAP 04(2021), 082 doi:10.1088/1475-7516/2021/04/082 [arXiv:2102.01105 [gr-qc]]

    Pith/arXiv arXiv 2021

  27. [27]

    Bambi and L

    C. Bambi and L. Modesto, Phys. Lett. B721(2013), 329- 334 doi:10.1016/j.physletb.2013.03.025 [arXiv:1302.6075 [gr-qc]]

    Pith/arXiv arXiv 2013

  28. [28]

    Beltracchi and P

    P. Beltracchi and P. Gondolo, Phys. Rev. D104 (2021) no.12, 124066 doi:10.1103/PhysRevD.104.124066 [arXiv:2104.02255 [gr-qc]]

    Pith/arXiv arXiv 2021

  29. [29]

    Toshmatov, Z

    B. Toshmatov, Z. Stuchl ´ ık and B. Ahme- dov, Phys. Rev. D95(2017) no.8, 084037 doi:10.1103/PhysRevD.95.084037 [arXiv:1704.07300 [gr-qc]]

    Pith/arXiv arXiv 2017

  30. [30]

    Tsukamoto, Phys

    N. Tsukamoto, Phys. Rev. D97(2018) no.6, 064021 doi:10.1103/PhysRevD.97.064021 [arXiv:1708.07427 [gr- qc]]

    Pith/arXiv arXiv 2018

  31. [31]

    H. X. Zhang, Y. Chen, T. C. Ma, P. Z. He and J. B. Deng, Chin. Phys. C45(2021) no.5, 055103 doi:10.1088/1674- 1137/abe84c [arXiv:2007.09408 [gr-qc]]

    Pith/arXiv arXiv 2021

  32. [32]

    A. Kar, A. Dey and S. Kar, [arXiv:2510.11364 [gr-qc]]

    arXiv