Thermodynamic topological classification of magnetically charged slowly rotating Kerr black holes in nonlinear electrodynamics with a cosmological constant

Peng Zhao; Yu-Die Wan; Zheng-Wen Long

REVIEW

Thermodynamic topological classification of magnetically charged slowly rotating Kerr black holes in nonlinear electrodynamics with a cosmological constant

Peng Zhao, Yu-Die Wan, Zheng-Wen Long

General Relativity and Quantum Cosmology · Submitted 15 Sep 2026

Pith's verdict: A clear but methodologically limited realization of the predicted W^{1-} subclass; the central claim stands within the reduced prescription the authors adopt, but the ensemble issue is load-bearing and should be stated more sharply.

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arXiv ID2609.16568 (version 2)

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CitationPeng Zhao, Yu-Die Wan, Zheng-Wen Long. "Thermodynamic topological classification of magnetically charged slowly rotating Kerr black holes in nonlinear electrodynamics with a cosmological constant." arXiv:2609.16568 (2026). https://pith.science/paper/AY6JGLG6

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Editor's note

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This paper gives the first concrete example of the W^{1-} thermodynamic topological subclass, which was predicted from symmetry arguments last year. The authors work through a slowly rotating Kerr black hole with magnetic charge in Kruglov's nonlinear electrodynamics, in both dS and AdS. The derivation is clean, the parameter scans are thorough, and the text is honest about where the approximations live. If the classification holds, it is a meaningful advance within the black hole thermodynamics topology subfield.

The main soft spot is the reduced grand-canonical ensemble. The authors fix Omega and Phi_m as intensive variables but then substitute on-shell expressions that depend on r_h, which is not consistent with the ensemble they claim. They flag this themselves after Eq. (6) and again in the conclusions, but the limitation is the central load-bearing concern. The stress-test note gets this right: if the Legendre transform is not integrable for fixed Omega and Phi_m, the defect curve and winding numbers could change. The paper does not claim the classification would survive a strict fixed-Omega, fixed-Phi_m ensemble, but the presentation could make this clearer. A minor revision should add a sentence that the result is only valid within the reduced prescription, not a proof for the full grand-canonical ensemble. The slow-rotation approximation and the formal large-radius limit in dS are acceptable for a first study, and the omission of static and Maxwell limits is noted.

Who is this for? Readers working on thermodynamic topology classifications of black holes, especially those interested in the W^{1-} subclass. It is a solid, honest contribution within its own framework. I would support publication after the authors sharpen the discussion of the ensemble limitation. The paper deserves a serious referee.

— about the subfield, but I would send it.

Referee report

Major objections (2)

[Section II, Eq. (6) and following text; Section IV] The reduced grand-canonical off-shell prescription used for the free energy substitutes on-shell expressions for Ω(J) and Φ_m (implicitly via the metric and mass function) while treating Ω and Φ_m as fixed intensive parameters. The authors themselves state that a strict fixed-Ω, fixed-Φ_m treatment requires verification of the complete first law, which is not provided. This is a load-bearing concern because the entire vector-field construction and the resulting W^{1-} classification depend on the functional form of F in Eq. (13). If the Legendre transform is not integrable, the defect curve and winding numbers could differ, potentially changing the classification. The authors should either (a) verify the first law and the integrability of the transform for this model, or (b) explicitly state that the classification holds only under the reduced prescription and discuss the conditions under which the result might be invalid. Currently the claim is presented as definite ("realizes"), while the caveat is only in Section IV; the abstract and introduction should also reflect the provisional nature.
[Section III A, Eq. (13) and prceeding paragraph] The derivation of Eq. (13) is not shown. The generalized free energy F = M - S/τ - Ω J - Φ_m q_m is stated, but the explicit expression for Φ_m q_m is never given, nor is it clear how the terms in Eq. (13) arise from the metric and thermodynamic quantities of the NLED model. The magentic potential Φ_m is a relevant intensive variable in the grand-canonical ensemble, yet it disappears from the final expression without justification. The authors should explain how the term -Φ_m q_m is evaluated (or why it is effectively absorbed into the other term) and provide intermediate steps leading from Eq. (6) to Eq. (13). Without this, the core calculation is opaque and cannot be independently verified.

Minor objections (7)

[General] The notation for the topological subclass uses \overline{W}^{1-} in the abstract and W^{1-} elsewher; please unify consistently.
[Eq. (16)] The expression for τ(r_h) in dS is given without deriviation; please show or cite the source for the intermedate algebra.
[Figs. 1-3, 5-7] The figure captions include the label 'parameter' above the legend keys; this is redundant and should be removed. Also, the insets are very small; consider enlarging or providing a separate zoom panel.
[Figs. 4 and 8] The capitons contain the note 'The numerical position of the zero must be rechecked when the source figure is regenerated.' This appears to be an authorial internal note; either recalculate the zero or remove the note.
[Section III C, Eq. (37)] The reduced Kerr-AdS defect curve τ = 4ℓ²π r_h/(ℓ²+3r_h²) is derived from Eq. (36) by setting ϕ_rh = 0. Howver, Eq. (36) contains a denominator 2G(a_KA² - ℓ²)τ; the cancellation leading to Eq. (37) should be clarified: it requires τ≠0 and a_KA²≠ℓ². State explictly.
[References [19], [20]] Refs. [19] and [20] are listed with only the month and year; they appear to be unpublished preprints. If they are publicly available, please add arXiv identifiers or provide the published reference. If they are internal notes, consider whether they are necessary.
[Section IV] The paragraph listing limitations is helpful. Consider also stating that the robustness statement is restricted to the sampled parameter ranges and that the non-rotating limit a→0 requires separate analysis (as noted).

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The paper

The paper claims

Slowly rotating, magnetically charged Kerr black holes with nonlinear electrodynamics realize the predicted single-branch topological subclass $\overline{W}^{1-}$, with global charge $W=-1$, in both de Sitter and anti-de Sitter…

The reading

Slowly rotating Kerr black holes carrying magnetic charge in nonlinear electrodynamics sit in a specific thermodynamic-topological subclass, in both de Sitter and anti-de Sitter spacetimes. The paper's central result is that these black holes realize the predicted $\overline{W}^{1-}$ subclass: a single unstable thermodynamic branch, an inverse-temperature curve that begins at a finite nonzero value at the smallest horizon radius and diverges at large radius, and a global topological charge $W=-1$. This matters because no explicit rotating black-hole realization of this subclass had previously been found, and because it shows that the nonlinear electrodynamic correction can remove the stable large-black-hole branch that the AdS background normally stabilizes. Across the sampled rotation, nonlinearity, and magnetic-charge parameters, the classification does not change.

What carries the argument

The load-bearing machinery is the off-shell thermodynamic-topology vector field built from the generalized free energy $F=M - S/\tau - \$\Omega$ J - \Phi_m q_m$. Zeros of this vector field in the $(r_h,\Theta)$ plane correspond to on-shell black-hole states, and their winding numbers encode local stability, while the sum gives the global charge $W$. The NLED mass function $M(r)=\frac{$q_m^{{3/2}}$}{2\$beta^{{1/4}}$}\arctan\bigl(r\sqrt{q_m}\,\$beta^{{1/4}}$\bigr)$ replaces the constant point mass; its finite total mass and smooth radial growth are what change the inverse-temperature endpoints from Kerr-AdS behavior $\tau(0)=\tau(\infty)=0$ to a finite nonzero $\tau(0)$ and divergence $\tau(\infty)=\infty$, producing the single-branch $\overline{W}^{1-}$ pattern.

Extended reading notes

Core claim

Within the reduced grand-canonical off-shell prescription, the paper constructs a generalized free energy $F=M - S/\tau - \$\Omega$ J - \Phi_m q_m$ for the slowly rotating NLED-Kerr black hole and derives the inverse-temperature curve $\tau(r_h)$ of its topological defect. This curve is monotonic on $r_h>0$, tends to a finite nonzero value as $r_h\to 0$, and grows like $$r_h^{4}$$ as $r_h\to\infty$. The vector field has exactly one zero in the sampled domain, with winding number $w=-1$, so the global topological charge is $W=-1$; this combination of endpoint behavior and winding number is what defines the predicted $\overline{W}^{1-}$ subclass. The same endpoint structure and charge are found in both dS and AdS sectors, and scans over $a$, $\$\beta$$, and $q_m$ leave the classification unchanged. Comparison with the reduced Kerr-AdS benchmark, whose curve has $\tau(0)=\tau(\infty)=0$ and two branches with $W=0$, leads the authors to attribute the change to the NLED effective mass function replacing the point mass.

Load-bearing premise

The classification rests on one assumption: the reduced grand-canonical free energy, built by holding angular velocity and magnetic potential fixed, really is the correct thermodynamic potential for this black hole. The paper itself says a strict fixed-angular-velocity, fixed-magnetic-potential treatment would require verifying the full first law, which it does not do; if that verification fails, the defect curve and the $W=-1$ classification could change.

Editorial extensions

If this is right

The predicted $\overline{W}^{1-}$ subclass now has an explicit rotating black-hole realization, in both signs of the cosmological constant.
Over the sampled ranges of rotation $a$, nonlinearity $\beta$, and magnetic charge $q_m$, the inverse-temperature curves are only rescaled; the $W=-1$ classification does not change.
The stable large-black-hole branch of the reduced Kerr-AdS benchmark is absent here, so at high temperature there is no on-shell black-hole state at all.
In the dS sector, the large-radius divergence is a formal off-shell continuation beyond the cosmological-horizon bound, and the classification should be read with that caveat.

Reading between the lines

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

If the full first law with independent $\Omega$ and $\Phi_m$ is verified, the same endpoint mechanism would likely make the $\overline{W}^{1-}$ classification robust over a much wider parameter region; the present paper only samples finite parameter ranges, so untested regions remain open.
The formulas are singular at $a=0$, $q_m\to 0$, and $\beta\to 0$, so the topological class may change discontinuously at the static and Maxwell limits; checking those limits directly would test whether the boundary is genuine.
A natural extension is to test other NLED mass profiles with the same bounding behavior: if the finite-nonzero $\tau(0)$ endpoint is generic to regularized effective masses, other regular rotating black holes should fall in the same $\overline{W}^{1-}$ subclass.
Because the AdS sector has no stable large branch, the usual large-black-hole/thermal-radiation phase transition appears to be absent in this model; whether that absence survives a strict fixed-$\Omega,\Phi_m$ ensemble is a direct open question.

Original abstract

We provide an explicit rotating black-hole realization of the predicted $\overline{W}^{1-}$ thermodynamic topological subclass by studying a slowly rotating, magnetically charged Kerr black hole in nonlinear electrodynamics (NLED). Using a reduced fixed-parameter off-shell potential, we derive the inverse-temperature curve analytically rather than relying solely on numerical plots. In AdS spacetime, this curve begins at a finite nonzero value, increases monotonically, and diverges at large horizon radius; it therefore contains exactly one defect with $w=W=-1$ and has no stable large-black-hole branch. Positivity of the analytic coefficient establishes this result throughout the admissible slow-rotation parameter domain. Under the same reduced prescription, the Kerr-AdS benchmark instead has two branches and belongs to the $W^{0-}$ class, establishing a clear topological distinction between the NLED-Kerr and classical rotating cases. In dS spacetime, the formally extended off-shell curve remains monotonic and any isolated zero has $w=-1$; because the parameter-dependent horizon-admissible range is not fixed in the present reduced analysis, no physical dS asymptotic subclass is assigned.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,833 input tokens.

What the abstract itself states, as probabilities: claims first 0.15 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.31 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this work, we investigate the thermodynamic topological classification of slowly rotating Kerr black holes with magnetic charge in nonlinear electrodynamics (NLED) using a reduced grand-canonical off-shell prescription. By constructing the generalized off-shell free energy and the associated topological vector field, we determine the local winding numbers and the global topological charge of the NLED-Kerr black hole in both de Sitter (dS) and anti-de Sitter (AdS) spacetimes. The inverse-temperature curve starts from a finite nonzero value at the lower radial endpoint and diverges at large horizon radius. Together with the single zero of winding number $w=-1$, this behavior identifies the black hole as a realization of the predicted $\overline{W}^{1-}$ subclass, with global topological number $W=-1$ and a single unstable branch. The classification remains unchanged over the sampled rotation, NLED, and magnetic-charge parameters. Within the same prescription, comparison with the two-branch $W^{0-}$ Kerr-AdS result associates the NLED effective mass function with the changed endpoint behavior and the absence of a stable large-black-hole branch. Thus, the physical novelty is an explicit rotating NLED realization of the predicted $\overline{W}^{1-}$ subclass for both signs of the cosmological constant.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work, we investigate the thermodynamic topological classification of slowly rotating Kerr black holes with magnetic charge in nonlinear electrodynamics (NLED) using a reduced grand-canonical off-shell prescription.	contribution 0.99	0.93	0.05	asserted 0.87
s1	By constructing the generalized off-shell free energy and the associated topological vector field, we determine the local winding numbers and the global topological charge of the NLED-Kerr black hole in both de Sitter (dS) and anti-de Sitter (AdS) spacetimes.	method 0.91	0.95	0.07	asserted 1.00
s2	The inverse-temperature curve starts from a finite nonzero value at the lower radial endpoint and diverges at large horizon radius.	result 1.00	0.84	0.04	asserted 1.00
s3	Together with the single zero of winding number $w=-1$, this behavior identifies the black hole as a realization of the predicted $\overline{W}^{1-}$ subclass, with global topological number $W=-1$ and a single unstable branch.	interpretation 0.64	0.87	0.41	asserted 1.00
s4	The classification remains unchanged over the sampled rotation, NLED, and magnetic-charge parameters.	result 0.99	0.90	0.12	asserted 1.00
s5	Within the same prescription, comparison with the two-branch $W^{0-}$ Kerr-AdS result associates the NLED effective mass function with the changed endpoint behavior and the absence of a stable large-black-hole branch.	interpretation 0.98	0.76	0.09	asserted 0.75
s6	Thus, the physical novelty is an explicit rotating NLED realization of the predicted $\overline{W}^{1-}$ subclass for both signs of the cosmological constant.	interpretation 0.59	0.87	0.82	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.94.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** displays the rh − τ curves for the NLED-Kerr black hole in dS spacetime. For every sampled value of a, the curve is smooth and monotonic, in agreement with dτ /drh > 0. The inset resolves the lower-radius region and shows that τ approaches a finite positive value rather than zero. The two limits in Eq. (18) are therefore consistent with the plotted curves. The stability assignment is not based on the visual curvature of τ (rh) alone; it follows from the winding number of the corresponding vector-field zero discussed below

**Figure 2.** displays the corresponding curves for fixed L = 1, qm = 1, and a = 0.02, with β varied over the values parameter qm qm = 1 qm = 2 qm = 3 qm = 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.05 0.10 0.15 0.20 τ rh 0.000 0.001 0.002 0.003 0.004 0.005 0.00 0.01 0.02 0.03 0.04 0.05 τ rh FIG. 3. Inverse-temperature curves τ (rh) for the NLED-Kerr black hole in dS spacetime, with L = 1, β = 0.1, a = 0.02, and qm = 1, 2, 3, 4. given in the caption. Since τ ∝ β 1/4 , increasing β shifts the curve upward, while leaving its monotonic form and endpoint types unchanged. The scan therefore changes the…

**Figure 3.** displays the magnetic-charge scan at fixed L = 1, β = 0.1, and a = 0.02. Because τ ∝ q −3/2 m , increasing qm shifts the curves downward. Their monotonicity, finite nonzero lower-endpoint value, and large-radius divergence are unchanged, so the sampled solutions remain in the W 1− subclass. 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 3.0 r+/r0 Θ FIG. 4. Normalized vector field n A in the (rh/r0, Θ) plane for the NLED-Kerr black hole in dS spacetime, with L = 1, qm = 1, β = 1, a = 0.05, and τ /r0 = 60. The black dot denotes the isolated zero

**Figure 6.** shows the NLED-parameter scan for fixed L = 1, qm = 1, and a = 0.02. The analytic scaling τ ∝ β 1/4 implies an upward shift as β increases. The monotonic shape and the two endpoint types remain unchanged, so this scan does not change the W 1− classification. This behavior is qualitatively consistent with the dS scan in

**Figure 7.** displays the magnetic-charge scan for fixed L = 1, β = 0.1, and a = 0.02. Increasing qm shifts the curve downward according to τ ∝ q −3/2 m , but does not change

**Figure 8.** displays the normalized vector field n A for L = 1, qm = 1, β = 1, a = 0.05, and τ /r0 = 60. One isolated zero is marked by the black dot. The negative Jacobian gives w = −1, identifying an unstable branch. No additional zero is visible in the displayed domain, and hence W = −1 for this configuration. Together with the dS result, this supports the same W 1− classification in both cosmological backgrounds. The numerical position of the zero must be rechecked when the source figure is regenerated. For the off-shell prescription and sampled parameter ranges considered here, the dS a

**Figure 9.** displays the reduced Kerr-AdS rh − τ curve for ℓ = 10 and aKA = 0.05. It has a single maximum at rmax = ℓ/√ 3, τmax = 2πℓ/√ 3, and vanishes as rh → 0 and rh → ∞. For 0 < τ < τmax, the curve contains an unstable small-black-hole branch (w = −1) and a stable large-black-hole branch (w = +1), which merge at the maximum. Within this reduced prescription, the two branches give W = 0 and the W0− classification. By contrast, the NLED-Kerr curve is monotonic, has one unstable branch, and gives W = −1, thereby realizing W 1− . Thus, within the stated reduced benchmark, the two endpoint

Showing 7 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.16568 by the authors.

References

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

Novel thermodynamic topological subclass from inverse-temperature asymptotics
gr-qc 2026-10 accept novelty 6.0 of 10

The charged McGuigan-Nappi-Yost black hole realizes a new thermodynamic topological subclass ˘W^{1+}, with winding number +1 and finite nonzero inverse temperature at large horizon.
Novel topological subclasses of a magnetic black hole in causal nonlinear electrodynamics
gr-qc 2026-10 accept novelty 6.0 of 10

A new thermodynamic topological subclass W^{0+↔1-} is found in a magnetic black hole, characterized by boundary-induced change of the global topological number from 0 to -1.

Discussion

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