Universal thermodynamic topological classes of static black holes in Conformal Killing Gravity

Bekir Can L\"utf\"uo\u{g}lu; Di Wu; Hao Chen; Hassan Hassanabadi; Meng-Yao Zhang; Soroush Zare; Zheng-Wen Long

REVIEW 4 major objections 5 minor 97 references

Universal thermodynamic topological classes of static black holes in Conformal Killing Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For static black holes in Conformal Killing Gravity, the sign of the CKG parameter $\lambda$ alone chooses the universal thermodynamic topological class: charged solutions fall into $W^{0+}$ ($\lambda>0$) or $W^{1+}$ ($\lambda<0$), while…

desk verdict Novel sign-of-λ classification for CKG black holes is undermined by a false β(∞)=∞ assumption for λ>0; fixable but not publishable as is. read the letter →

arxiv 2506.03695 v2 pith:WMJKHHZJ submitted 2025-06-04 gr-qc

Hao Chen , Di Wu , Meng-Yao Zhang , Soroush Zare , Hassan Hassanabadi , Bekir Can Lütfüoğlu , Zheng-Wen Long This is my paper · ORCID

classification gr-qc MSC 83C57 PACS 04.70.-s

keywords blackholethermodynamicsthermodynamictopologyConformalKillingGravitywindingnumbertopologicalclassoff-shellfreeenergychargedAdSSchwarzschild

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

Conformal Killing Gravity (CKG) is an extension of general relativity whose metric carries an extra term $-\frac{\lambda}{5}r^4$, and this paper claims that the sign of $\lambda$ alone fixes the universal thermodynamic topological class of a static black hole. For the charged AdS and Reissner-Nordström solutions, $\lambda>0$ places the hole in the $W^{0+}$ class and $\lambda<0$ in the $W^{1+}$ class; for the Schwarzschild and Schwarzschild-AdS solutions, the classes are $W^{1-}$ and $W^{0-}$, respectively. The classification is read from winding numbers of a vector field built from the off-shell free energy, with $+1$ marking stable states and $-1$ unstable ones. The upshot is that the CKG parameter acts as a switch that changes which black hole states can exist at low and high temperature, and the paper also finds that pressure does not move a solution between classes.

What carries the argument

The machinery is the generalized off-shell free energy $F=M-S/\tau$ supplemented by an auxiliary angle $\Theta\in(0,\pi)$ through $\tilde{F}=F+1/\sin\Theta$. The two-component vector field $\phi=(\partial\tilde{F}/\partial r_h,\partial\tilde{F}/\partial\Theta)$ has a zero at each black hole state, and the winding number of that zero is $+1$ for a locally stable state (positive heat capacity) and $-1$ for an unstable one; the sum of all winding numbers is the topological class $W$. The CKG parameter $\lambda$ enters through the mass and temperature of the solution, and its sign determines whether the vector field points left or right at the boundaries $r_h=r_m$ and $r_h\to\infty$, which is what selects among the four universal classes $W^{1+}$, $W^{0-}$, $W^{0+}$, and $W^{1-}$.

What would settle it

Take the same charged AdS CKG black hole at a different parameter point, for instance $P r_0^2=1$, $q/r_0=0.5$, $\lambda/r_0=0.5$, and compute the full set of winding numbers. If the result is not $W^{0+}$ (or, with $\lambda<0$, not $W^{1+}$), the claimed sign-only universality fails; the same check can be run for the uncharged cases as $|\lambda|$ is varied or as $P\to 0$.

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Extended reading notes

Core claim

The central claim is that the $\lambda$-dependent term in the CKG metric function $f(r)=1-\frac{2M}{r}+\frac{q^2}{r^2}-\frac{\Lambda}{3}r^2-\frac{\lambda}{5}r^4$ changes the asymptotic behavior of the inverse-temperature function $\beta(r_h)$ and thereby the topological index $W$ of the black hole. With $\lambda>0$, the charged AdS black hole satisfies $\beta(r_m)=\infty$ and $\beta(\infty)=\infty$, which yields the class $W^{0+}$; with $\lambda<0$, $\beta(\infty)=0$ and the class becomes $W^{1+}$. The same two classes are assigned to the Reissner-Nordström black hole in CKG, while the Schwarzschild and Schwarzschild-AdS black holes are classified as $W^{1-}$ for $\lambda>0$ and $W^{0-}$ for $\lambda<0$. The classes are obtained by summing winding numbers of the vector field $\phi$ over the zero points in the $(r_h,\Theta)$ plane, and the paper reports that they do not depend on pressure.

Load-bearing premise

Everything rests on the assumption that changing the strength of the CKG correction, the pressure, or the charge never changes which of the four classes a black hole falls into, only the sign of $\lambda$; the paper shows examples but does not prove this.

Editorial extensions

If this is right

For a charged AdS black hole in CKG with $\lambda>0$, the class is $W^{0+}$: a stable small black hole coexists with an unstable large one at low temperature, and no black hole state exists at high temperature.
For $\lambda<0$, the same black hole is $W^{1+}$: stable small and large black holes flank an unstable intermediate one, so a stable large black hole appears in the high-temperature limit.
For the Schwarzschild and Schwarzschild-AdS black holes, $\lambda>0$ gives $W^{1-}$ with both small and large states unstable, and $\lambda<0$ gives $W^{0-}$ with no low-temperature state and an unstable small plus stable large black hole at high temperature.
The pressure does not change the classification: the charged AdS result transfers to Reissner-Nordström, and the Schwarzschild-AdS result transfers to Schwarzschild.
At $\lambda=0$ the results reduce to the known general-relativity classes, so CKG with $\lambda>0$ actively shifts the charged AdS black hole from $W^{1+}$ to $W^{0+}$ and the Schwarzschild-AdS black hole from $W^{0-}$ to $W^{1-}$.

Reading between the lines

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

The sign-only rule is inferred from examples at one parameter point ($P r_0^2=0.1$, $q/r_0=0.1$, $|\lambda|/r_0=0.1$); a natural next step is to scan a grid of $P$, $q$, and $|\lambda|$ and check whether any configuration leaves its assigned class.
If the pressure-independence claim survives the $P\to 0$ limit, the flat-space Reissner-Nordström and Schwarzschild CKG black holes give direct checks of the same classes without a cosmological constant.
The same off-shell free-energy vector field could be applied to rotating or higher-dimensional CKG black holes; the construction is not tied to spherical symmetry, although the explicit metric and thermodynamic quantities would differ.
A direct analytic proof that $W$ depends only on $\operatorname{sgn}(\lambda)$ would turn the example-based classification into a theorem.

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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

The paper does something genuinely new: it runs the Wei-Liu-Mann off-shell free-energy topology machinery on Conformal Killing Gravity black holes and finds that the sign of the CKG parameter λ flips the universal thermodynamic class. The charged-AdS case is worked out in enough detail to follow the winding-number counts without external code, and the extension is legitimate, though not deep. The ground in the existing topological literature is fine; there is no circularity in the computation itself.

The problem is that the λ>0 classification is built on a false boundary condition. The paper states β(r_m)=∞ and β(∞)=∞ for λ>0 (Eq. 20), but its own Eq. (19) gives β ≈ -4π/(λ r_h³) → 0⁻ as r_h→∞ for λ>0. The curve does not run to +∞ at large r_h; it becomes negative and approaches zero from below. The W0+ class they assign depends on there being no black hole states at β→0+. If β(∞) is actually 0⁻, the positive-β branch ends at finite r_h, and the topology of the zero-point set changes. This is not cosmetic: the winding number at the boundary I1 is different, and the universality claim for λ>0 collapses.

There is also the overconstrained condition in Eq. (22): generation/annihilation points need only ∂β/∂r_h=0, not ∂²β/∂r_h²=0. The listed β_c values may come from the correct extremum condition, but as written the equation is wrong. The RN and Schwarzschild classifications in Section III.C and Table II are asserted with no shown derivation; the 'universal' claim is supported only by a few examples at fixed P, q, and |λ|.

The λ<0 side looks more defensible: there β(∞)=0⁺ is consistent with Eq. (24), and the W1+ classification for the charged-AdS case may survive correction. But the paper as written is not self-consistent. I would send it to a referee rather than desk-reject, because the error is specific, probably correctable, and the method is well established. The referee should demand a corrected treatment of the λ>0 asymptotics and an explicit derivation for the uncharged cases before acceptance.

Referee Report

4 major / 5 minor

Summary. The paper applies the Wei-Liu-Mann generalized off-shell free energy formalism to static black holes in Conformal Killing Gravity (CKG). Using the CKG-charged AdS metric, it computes the inverse-temperature function β(r_h), identifies generation/annihilation points, and assigns universal thermodynamic topological classes. The central claim is that the sign of the CKG parameter λ alone determines the class: W^{0+} for λ>0 and W^{1+} for λ<0 for charged AdS/Reissner-Nordström black holes, and W^{1-} (λ>0) and W^{0-} (λ<0) for Schwarzschild/Schwarzschild-AdS black holes. The paper also tabulates stability properties and the number of generation/annihilation points for each case.

Significance. If correct, the result would be a compact and striking statement: a single sign choice in CKG flips the universal thermodynamic topology of static black holes. The manuscript uses an established topological method, provides explicit expressions for the free energy and the vector field, and contains no parameter fitting; the computations are direct. However, the derivation as written has internal inconsistencies in the asymptotic boundary conditions and in the critical-point condition, and the claimed universality is supported only by two numerical examples. The strengths are the transparent setup and the explicit formulas; the weaknesses are the unsupported universality and the incorrect statements that affect the classification logic.

major comments (4)

[Section III.A, Eq. (20); Section III.C, Eq. (25)] The boundary condition β(∞)=∞ for λ>0 is contradicted by the paper's own Eq. (19). Since β(r_h)=4πr_h³/(r_h²-q²+8πP r_h⁴-λr_h⁶), the denominator is dominated by -λr_h⁶ for λ>0, so β≈-4π/(λr_h³)→0⁻ as r_h→∞; for λ<0 it goes to 0⁺. Thus both signs give β(∞)=0 (with opposite signs), not the (∞,∞) or (∞,0) combinations stated in Eqs. (20) and (25). Because the text explicitly uses these boundary values to assign W^{0+} and W^{1-} classes, the λ>0 classification as written is not supported by the model's own temperature function.
[Section III.A, Eq. (22)] The condition ∂β/∂r_h = ∂²β/∂r_h² = 0 for generation/annihilation points is incorrect. A generic extremum of β(r_h) requires only ∂β/∂r_h=0; requiring the second derivative also to vanish selects stationary inflection points, which are not the points where pairs of black hole states are created or annihilated. The quoted β_c values (2.4171, 2.9273, 4.0467) correspond to ordinary extrema of Eq. (19), so the stated condition is inconsistent with the actual computation and would alter the GP/AP counts in Table II if applied literally.
[Section III.C and Table II] The universal-class claims for Reissner-Nordström (P=0) and Schwarzschild black holes are asserted without derivation. The text says only that 'a similar analytical methodology' is applied, but no β(r_h) analysis, vector-field plots, or winding-number calculations are presented for these cases. Moreover, the universality claim—that the class depends only on sign(λ) and not on P, q, or |λ|—is not proven; all explicit examples fix P r₀²=0.1, q/r₀=0.1, and λ/r₀=±0.1. The title and abstract promise a universal result, but the evidence is anecdotal.
[Section III.A, Eq. (23)] Equation (23) is introduced under the assumption q=0 ('If we ignore the influence of charge'), but it is then used together with q/r₀=0.1 to obtain the three β_c values. The manuscript does not explain how a q=0 expression applies to the q=0.1 case. The quoted numbers appear to follow from the full Eq. (19), making Eq. (23) and the surrounding text misleading and preventing reproduction of the critical values.

minor comments (5)

[Abstract and Section I] The phrase 'the smallest inner and the largest outer black hole states' is vague; these states should be defined in terms of the horizon-radius branches.
[Figures 2 and 6 captions] The phrase 'the arrow signifies that' is incomplete; the captions should state that the arrows represent the unit vector field ϕ/|ϕ| at the zero points.
[Table II] The abbreviations GP, AP, Un, and S are used without definition in the table caption; they should be defined explicitly.
[Section III.A, text after Eq. (20)] The sentence 'constraining the inversion temperature parameter to λ>0, β(r_m)=∞ and β(∞)=∞' is grammatically awkward and should be rewritten; also, the physical meaning of β(r_m)=∞ for the charged black hole should be clarified.
[References] A few references have inconsistent formatting, e.g., Ref. [20] lists only an arXiv number without a title, and Ref. [94] duplicates Ref. [91]; these should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the topological classes are computed directly from the off-shell free energy, and the same-author citations supply only an external classification framework.

full rationale

This paper's derivation chain is self-contained with respect to the topology computation: the off-shell free energy F (Eq. 15), the vector field (Eqs. 17-18), the on-shell inverse temperature β (Eq. 19), and the contour integrals around the (r_h, Θ) rectangle determine the winding numbers and hence the W class. No parameter is fitted to the target classification, and the W^{0+}, W^{1+}, W^{0-}, W^{1-} labels are external categories imported from Ref. [67] and refined in Ref. [69]; the same-author citations (Refs. [69-71]) provide the classification framework and prior examples, but the CKG results are computed rather than read off from those citations. The uncharged Reissner-Nordström and Schwarzschild cases are asserted in Sec. III.C and Table II without displayed calculation, and the λ>0 boundary condition β(∞)=∞ in Eq. (20) appears inconsistent with the model's own Eq. (19), which gives β ~ -4π/(λ r_h^3) → 0^- for λ>0; these are completeness and correctness concerns, not cases where an output reduces to an input by construction. Accordingly no specific circular step is identified; the minor self-citations import a benchmark method rather than force the central claim, so the circularity score is 2.

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

No parameters are fitted; P, q and lambda are theory inputs. The central claim rests on the cited CKG solution, the cited topological method, and the unproved assertion that the class is universal in the sign of lambda. No new entities are invented.

assumptions (4)

domain assumption The CKG metric function (Eq. 11) and thermodynamic quantities (Eqs. 12-14) are correct.
Taken from Refs. [87,95,96]; the paper does not derive or independently verify them.
domain assumption The off-shell free-energy vector-field method of Refs. [30,67,69] applies to CKG black holes.
Used without proof that the method's assumptions (cavity, winding-number classification) survive the CKG modification.
ad hoc to paper The universal class is determined by the sign of lambda and not by P, q, or |lambda|.
Supported only by examples with P r0^2=0.1, q/r0=0.1, lambda/r0=+/-0.1; no general proof is given.
standard math The asymptotic behavior of beta(r_h) at r_h -> r_m and r_h -> infinity determines the winding number and class.
This is the classification scheme from Ref. [67] and earlier work; the paper relies on it.

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Cite this review

Pith. "Pith review of Universal thermodynamic topological classes of static black holes in Conformal Killing Gravity." pith.science (2026). https://pith.science/paper/WMJKHHZJ

@misc{pith2026250603695,
  author       = {Pith},
  title        = {Pith review of: Universal thermodynamic topological classes of static black holes in Conformal Killing Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMJKHHZJ}},
  note         = {Machine review of arXiv:2506.03695}
}

abstract

In this study, we develop universal thermodynamic topological classes for the static black holes in the context of the Conformal Killing Gravity. Our findings indicate that the Conformal Killing Gravity significantly reconstructs the thermodynamic properties of both the smallest inner and the largest outer black hole states. Additionally, it considerably alters the thermodynamic stability of black holes across both high-temperature and low-temperature regimes. This analysis shows that different CKG parameter settings will lead to $W^{0+}$ $(\lambda>0)$ and $W^{1+}$ $(\lambda<0)$ categories for the charged AdS black hole, the Reissner-Nordstr$\ddot{o}$m black hole in Conformal Killing Gravity is classified into the $W^{0+}$ and $W^{1+}$ categories. Furthermore, we examine the specific scenario where charge is neglected. The study reveals that within the framework of Conformal Killing Gravity, the Schwarzschild black hole similar to the Schwarzschild-AdS black hole, can be classified into the $W^{1-}$ and $W^{0-}$ categories. This work provides key insights into the fundamental nature of quantum gravity theory.

Figures

Figures reproduced from arXiv: 2506.03695 by the authors.

**Figure 3.** FIG. 3: The asymptotic behavior of vector field [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4: The contours [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 7.** Figure 7: FIG. 7: The asymptotic behavior of vector field [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: FIG. 8: The contours [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 11.** Figure 11: FIG. 11: The arrow signifies that, for the unit vector field of [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]

**Figure 14.** Figure 14: FIG. 14: The contours [PITH_FULL_IMAGE:figures/full_fig_p007_14.png]

**Figure 13.** Figure 13: FIG. 13: The contours [PITH_FULL_IMAGE:figures/full_fig_p007_13.png]

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