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REVIEW 3 major objections 4 minor 71 references

Doubly regular black holes

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Curvature-regular black holes usually fail thermodynamic regularity, and spin makes double regularity nearly unattainable.

desk verdict A useful new label and a clean static existence proof, but the rotating analysis needs a fixed-J derivative check before the restrictiveness claim can be trusted. read the letter →

arxiv 2507.23250 v2 pith:XTLSTOEL submitted 2025-07-31 gr-qc astro-ph.HE

Arthur G. Suvorov , Pedro Bargueño This is my paper
classification gr-qcastro-ph.HE MSC 83C5783C75 PACS 04.70.-s04.70.Dy
keywords blackholethermodynamicsDaviespointsregularholescurvatureregularityNewman-JanisalgorithmheatcapacityKerr-Newmanmetricspin-inducedsingularities
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

Black holes can fail in two independent ways: the curvature can blow up, and the heat capacity can blow up at so-called Davies points. This paper asks whether the popular 'regular' black hole models, which are smooth and curvature-bounded at the centre, are also free of thermodynamic blow-ups. It finds they generally are not: Hayward and Simpson-Visser holes are curvature-regular but develop Davies points in their phase space. The paper exhibits one static family, a hybrid Bardeen metric, that is 'doubly regular,' and then shows that spinning it up through the Newman-Janis algorithm typically brings the Davies point back. The claim matters because observed black hole spins already exceed the Kerr Davies value, so a doubly regular object would behave smoothly where the Kerr metric predicts a thermodynamic phase transition.

What carries the argument

The load-bearing objects are the Davies point, a point in charge-spin phase space where the heat capacity diverges because $(\partial T/\partial M)$ at fixed hairs vanishes (the paper's Eq. 13), and the two regularity notions: curvature regularity, meaning the Kretschmann invariant $K$ is bounded over physical space, and thermodynamic regularity, meaning the heat capacity is bounded over phase space. Because the temperature $T = \kappa_H/2\pi$ needs only the surface gravity, the classification runs without committing to a specific gravitational action. The Newman-Janis algorithm converts the static seeds into rotating metrics in a fixed form (Eq. 5), whose surface gravity is read off by anomaly cancellation (Eq. 10), letting the paper check double regularity numerically even when the horizon equation is transcendental.

What would settle it

Recompute the heat capacity $C_X = T(\partial S/\partial T)_X$ for the rotating hybrid Bardeen metric using the Wald entropy in an explicit theory while holding $J = aM$ fixed rather than $a$: if $(\partial T/\partial M)_J$ does not cross zero near $a \approx 0.75$, the claimed spin-induced Davies point is an artifact of holding $a$ fixed. Separately, evaluate the Wald entropy itself at $Q = M/\sqrt{e}$ for the Simpson-Visser family: if the entropy diverges there, condition (13) fails and the Davies point could be an entropy singularity rather than a heat-capacity one.

Watch

Extended reading notes

Core claim

Using only the metric potentials, the Hawking temperature can be defined theory-agnostically from the surface gravity, so a heat-capacity divergence, a Davies point, can be spotted wherever $(\partial T/\partial M)_X$ vanishes at fixed hairs. Surveying static curvature-regular spacetimes, the paper finds that Reissner-Nordström, extended Simpson-Visser, and weighted Hayward families all harbour Davies points (or lose curvature regularity), while a new hybrid Bardeen family (Eq. 26) with $k \geq 3$ and small $\zeta$ is both curvature-regular and thermodynamically regular. Rotating this seed via the Newman-Janis algorithm introduces a Davies point near $a \approx 0.75$ regardless of hair $Q$, so double regularity collapses under spin for this class. As a counterweight, the paper exhibits a 'CFL V' metric (Eq. 27) whose fixed dimensionful constants raise the static heat capacity enough that its Newman-Janis rotation remains doubly regular, proving that the set of stationary doubly regular objects is non-empty but 'especially restrictive.'

Load-bearing premise

The classification assumes that a heat-capacity divergence is exactly a zero of $(\partial T/\partial M)$ at fixed hairs, which requires the entropy to stay smooth in the theory at hand, and for the rotating plots it assumes the derivative is taken at fixed angular momentum $J$ rather than fixed spin $a$; if either fails, the claimed Davies points are not the physical heat capacity.

Editorial extensions

If this is right

  • If thermodynamic regularity is desired for astrophysical black holes, most proposed regular extensions of Kerr-Newman are ruled out, narrowing the viable pool.
  • The static hybrid Bardeen family (26) is a concrete doubly regular example for $k \geq 3$ and small $\zeta$, so double regularity is realizable in static, asymptotically flat spacetimes.
  • Spinning a doubly regular seed through the Newman-Janis algorithm tends to destroy thermodynamic regularity; for hybrid Bardeen, a Davies point appears near $a \approx 0.75$ for any $Q$.
  • Stationary doubly regular holes exist, namely the CFL V metric with $\zeta_1 \gtrsim M$ and $\zeta_2 \neq 0$, but the thermodynamic-regularity property depends on dimensionful constants fixed by the theory rather than on free hairs.
  • Astrophysical super-Davies spins (above $a \approx 0.681$ for Kerr) need not signal a thermodynamic phase transition if the true metric is doubly regular.

Reading between the lines

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

  • The paper's criterion (13) is entropy-blind: a theory whose Wald entropy itself diverges at the same phase-space point would break the equivalence between a zero of $\partial T/\partial M$ and a heat-capacity singularity, so testing the hybrid Bardeen family against an explicit $f(R)$ or higher-curvature action would show whether double regularity survives beyond the theory-agnostic level.
  • Recomputing the rotating figures with angular momentum $J = aM$ held fixed, rather than the spin parameter $a$, is a natural next check; if the sign change near $a \approx 0.75$ for the rotating hybrid Bardeen disappears under fixed-$J$ differentiation, rotation would be less hostile to double regularity than the paper concludes.
  • The paper's own discussion of late-time quasi-normal mode tails suggests a testable extension: computing those tails for the hybrid Bardeen metric (26) specifically could yield an observational signature that distinguishes doubly regular holes from Schwarzschild-like ones.
  • If rotation generically re-introduces Davies points for Newman-Janis-generated families with negative static heat capacity, then double regularity effectively pins the sign of the static heat capacity, a structural constraint that any ultraviolet completion of general relativity would have to reproduce.
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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 introduces the notion of 'doubly regular' black holes, defined as spacetimes that are both curvature-regular (CR, bounded Kretschmann invariant) and thermodynamically regular (TR, no Davies points where the heat capacity diverges). Using a theory-agnostic framework based on the Hawking temperature and the criterion that Davies points correspond to (∂T/∂M)_X → 0, the authors examine several static families (Reissner-Nordström, Simpson-Visser, weighted Hayward, and a new hybrid Bardeen-like metric) and their Newman-Janis rotating extensions. They find that most curvature-regular families are not thermodynamically regular, that the hybrid Bardeen metric (26) is claimed to be doubly regular for k≥3 and small ζ, and that rotation generally destroys this property, with a constructed stationary CFLV metric (27) provided as a counterexample. The central claim is that requiring both curvature and thermodynamic regularity strongly restricts the pool of viable regular black hole models.

Significance. The distinction between curvature regularity and thermodynamic regularity is a useful conceptual contribution, and the paper provides explicit examples, including a new static doubly regular family and a rotating proof-of-concept. The theory-agnostic approach is attractive because it avoids commitment to a specific gravitational action. The paper also honestly acknowledges limitations, such as the use of bounded Kretschmann scalar rather than full geometric regularity and the assumption that the entropy functional is well behaved. However, the rotating analysis is potentially undermined by the ambiguity in the thermodynamic derivative variable (fixed J vs fixed a), and the parametric claims rest on limited numerical evidence. If the rotating results are recomputed at fixed angular momentum, the conclusion that rotation makes double regularity especially restrictive may change.

major comments (3)
  1. [Sec. V A, Figs. 4 and 5] The paper states after Eq. (10) that thermodynamic derivatives should be taken holding J fixed rather than a, but the rotating figures plot dT/dM as a function of the dimensionless spin a without specifying which variable is held fixed in the numerical derivative. If the derivatives are evaluated at constant a, then the sign change in Fig. 4 and its absence in Fig. 5 do not correspond to the physical heat capacity at constant angular momentum: in the Kerr limit, (∂T/∂M)_a is negative for all spins, whereas the Davies point at a≈0.681 appears only in (∂T/∂M)_J. The authors must provide the explicit derivative formula or describe the numerical procedure, and recompute the figures accordingly. This is load-bearing for the central claim that rotation makes double regularity especially restrictive.
  2. [Sec. V A] The conclusion that rotation introduces Davies' points 'irrespective of the value of Q' and 'for any k' is supported by a single numerical example with k=3, Q=1, ζ=1/100. No parameter scan over Q, ζ, or k is presented, and no analytic argument is given to exclude the possibility that some other parameter choices remain doubly regular. The claim should be either supported by additional data or weakened to the specific parameter set considered.
  3. [Sec. IV D, Fig. 3] The static hybrid Bardeen family is claimed to be doubly regular for k≥3 and small ζ, but Fig. 3 demonstrates this only for k=3 and ζ=1/100, with no error bars or description of the numerical root-finding and differentiation accuracy. Since this family is one of the paper's main results, the numerical method should be described in enough detail to assess reliability, and ideally a scan over k and ζ should be provided to justify the parametric claim.
minor comments (4)
  1. [Sec. III A] The definition of CR as bounded Kretschmann scalar is acknowledged to be weaker than full geometric regularity (e.g., Zakhary-McIntosh invariants and geodesic completeness). Since the paper's abstract and title refer to 'regular black holes,' it would be helpful to remind the reader at key points that the results apply to this specific CR notion.
  2. [Sec. II C, Eqs. (12)-(13)] The chain-rule step from Eq. (12) to Eq. (13) requires that (∂S/∂M)_X be nonzero and finite, which is an assumption in a theory-agnostic framework. The paper acknowledges this, but it could be stated more prominently as a limitation.
  3. [Sec. IV D, Eq. (26)] The hybrid Bardeen potential is said to have an event horizon for any Q for suitable small ζ; this condition should be stated explicitly (e.g., a bound on ζ in terms of Q/M) or proven.
  4. [Figs. 4 and 5] The x-axis labels are 'a (dimensionless)' while the text refers to 'spin'; since the metric uses a with units of length, the figures should clarify whether a is normalized by M (and that M=1 in the plots).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper constructs explicit examples and checks the target conditions, with no fitted parameter renamed as a prediction.

full rationale

The paper does not claim to derive thermodynamic regularity from an independent first-principles theory; it defines TR as boundedness of the heat capacity (equivalently dT/dM != 0 under an explicitly stated smooth-entropy assumption), then computes dT/dM for a set of explicit metrics. The Bardeen-like and CFL V examples are presented as constructions, with parameters chosen to exhibit the desired property, so existence-by-construction is the honest logical status and is not a prediction. The few citations to the authors' prior work (metric form in [36], physical discussion in [43]) are for standard results or context and are not load-bearing. The Davies-point criterion (13) is a mathematical consequence of the chain rule given the stated assumptions, not a circular input. The skeptic's fixed-J versus fixed-a concern is a consistency and correctness risk in the rotating numerics, not a circular reduction, and therefore does not raise the circularity score.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claims rely on several domain assumptions and two ad hoc parameters. Most are stated in the text; none are machine-checked.

free parameters (5)
  • zeta (weighted Hayward) = zeta = 1/2 critical; zeta > 1/2 used
    Dimensionless weight chosen to push the Davies point past extremality; for k = 1 this gives TR but CR fails.
  • zeta (hybrid Bardeen) = 0.01 in examples
    Small dimensionless constant chosen so dT/dM stays negative; no derivation from an action.
  • k (hybrid Bardeen) = 3 in examples
    Integer power chosen >= 3 to make the third term Bardeen-like and suppress the Davies point.
  • zeta_1 (CFLV) = 3/2 M in example
    Length parameter chosen large enough to make the static heat capacity positive before rotation is added.
  • zeta_2 (CFLV) = 1/sqrt(10) M in example
    Second length parameter chosen nonzero; the value is a free choice, not derived from a theory.
assumptions (7)
  • domain assumption Hawking temperature is identified with surface gravity via T = kappa/2pi for all theories considered
    Sec. II A, Eq. (4); the paper calls this theory-agnostic, but modified-gravity frameworks may have different temperatures.
  • domain assumption Entropy is well behaved, so heat-capacity divergence reduces to dT/dM = 0
    Sec. II C, Eqs. (11)-(13); the authors explicitly assume this to avoid computing Wald entropy for each theory.
  • domain assumption Surface gravity for rotating Newman-Janis metrics is given by kappa = Omega/(2a) Delta'(r_H)
    Sec. II B, Eq. (10), cited to anomaly-cancellation literature; required for all rotating results.
  • domain assumption Newman-Janis algorithm preserves curvature regularity
    Sec. V, cited to [34]; not proven for the specific seeds used here.
  • domain assumption Kretschmann boundedness (CR) is the relevant notion of geometric regularity
    Sec. III A; the authors acknowledge it is weaker than geodesic completeness.
  • ad hoc to paper Thermodynamic regularity is a desirable property for black holes
    Sec. II D argues this at length, but it is presented as an assumption, not a theorem.
  • ad hoc to paper zeta_1 and zeta_2 in the CFLV metric are fixed by an unknown gravitational action and are not hairs
    Sec. V B; no action is given, so this is a postulate introduced to keep the example TR.

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Cite this review

Pith. "Pith review of Doubly regular black holes." pith.science (2026). https://pith.science/paper/XTLSTOEL

@misc{pith2026250723250,
  author       = {Pith},
  title        = {Pith review of: Doubly regular black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XTLSTOEL}},
  note         = {Machine review of arXiv:2507.23250}
}
read the original abstract

In addition to curvature singularities, electrovacuum black holes in general relativity exhibit thermodynamic singularities. These so-called Davies' points occur at nonextremal values of charge and spin where the heat capacity diverges and may indicate a type of theoretical incompleteness. The thermodynamic regularity of several families of static, asymptotically flat spacetimes with bounded curvature invariants is examined using a theory-agnostic framework, showing that, while they may be regular in physical space, they are generally not in phase space. The inclusion of angular momentum, via the Newman-Janis algorithm, makes the set of such "doubly regular" objects especially restrictive. It is argued that, if thermodynamic regularity is to be considered a desirable property for an astrophysical black hole, these considerations could be used to narrow down the viable pool of regular extensions to the Kerr-Newman metric.

Figures

Figures reproduced from arXiv: 2507.23250 by the authors.

Figure 1
Figure 1. FIG. 1. Variation of the horizon temperature as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. , showing dT /dM as a function of Q for k = 1 up to the extremal limit (Q = M) for a few values of ζ. For the ζ = 2/5 case (blue), a Davies’ point is visible for QD ≈ 0.94 which, while larger than for pure Hayward as described above, still resides within the permissible phase space (Q ≤ M). For the critical value ζ = 1/2 (dashed), we see that this migration pushes QD to unity, indicating that geometric and phase ext… view at source ↗
Figure 3
Figure 3. FIG. 3. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.