Thermodynamics of Swirling Black Holes and Kerr-Taub-NUT

Diego Hidalgo; Ernesto Frodden; Francisco Manzano

REVIEW

Thermodynamics of Swirling Black Holes and Kerr-Taub-NUT

Ernesto Frodden, Diego Hidalgo, Francisco Manzano

High Energy Physics - Theory · Submitted 29 Oct 2026 · 43 references

Pith's verdict: A technically serious charge computation whose headline entropy result is a first-law inversion, not an independent derivation; the authors say so themselves.

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CitationDiego Hidalgo, Ernesto Frodden, Francisco Manzano. "Thermodynamics of Swirling Black Holes and Kerr-Taub-NUT." arXiv:2610.40529 (2026). https://pith.science/paper/LX2RXSMA

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Worth reading for the technology, not for the entropy claim as a measurement. The cone-improved surface charge method with K-terms is the real contribution: it produces r-independent charges for KS and KTN, and the interpretation of the Misner-string terms from Ballon Bordo et al. as K-term contributions at the horizon (5.5)-(5.9) is genuinely illuminating. The KS mass generator found by integrability, the exact AKS angular momentum, and the A^4 surface-charge check of the AKS mass are solid pieces of work. The Mathematica notebook helps.

The soft spot is the one the authors flag themselves. In Sections 3.2 and 5.1, S, T, and Omega are solved from delta M = T delta S + Omega delta J with the Kerr limit as boundary condition. The stress-test note is right: on the three-parameter phase space this determines S as the Kerr entropy function written in the new charges. Equation (3.24) is S_Kerr(M,J) in disguise, so S != A/4 is a consequence of the reconstruction, not an independent computation. The authors acknowledge this in 5.1 and the conclusion; the paper still presents it as the central result, and that overstates what has been shown. The honest statement is: within a first law built from conserved charges, the entropy-like function is not A/4. Whether that function is the physical entropy remains open.

Two secondary gaps, in proportion. The KS mass relies on exchanging an integral with an infinite series and extrapolating a closed form from four terms (3.8)-(3.11). It looks plausible and the K-term cancellation is neat, but that step is not rigorous. For AKS, the mass comes from applying the Christodoulou-Ruffini formula as a conjecture; the surface-charge check only runs to order A^4. The authors say both things explicitly. Neither sinks the paper, but both keep it conditional.

Who is this for? People working on exact solutions and black hole thermodynamics, especially on non-asymptotically flat or string-defected spacetimes. They will get a clear framework and concrete formulas. It deserves a serious referee; I would send it out, with the request that the conclusions distinguish what is computed from what is inferred. I would cite the charge computations; I would not cite the entropy mismatch as a result.

Referee report

Significance

The main assets of the paper, if the charge computations survive scrutiny, are the K-term prescription for obtaining surface-independent charges in the presence of conical and Misner defects, the exact angular momenta for KS and AKS (the total-derivative identity (4.13) makes the AKS computation rigorous and elegant), the structural observation that the first law fixes S, T, Ω uniquely once the Kerr limit is imposed, and the translation of the literature's 'Misner gravitational charges' into K-terms. The authors are commendably explicit about their two fragile inputs: the first-law inversion for KS and KTN and the conjectural Christodoulou-Ruffini mass for AKS. The supplied Mathematica notebook supports reproducibility. The counterweight is that the headline claims are conditional: the KS and KTN entropy is the solution of the first law, not an independently computed quantity, and the AKS mass is assumed and verified only to fourth order in the acceleration. The paper therefore establishes a coherent framework and a precise, testable proposal, but it does not yet establish the physical area-law contrast announced in the abstract.

Major objections (3)

[Section 3.1.2, Eqs. (3.8)-(3.11), (3.13)] The KS mass M = m/(1−η²), and with it every quantity in Section 3.2, rests on two analytic steps that are assumed rather than justified: the term-by-term exchange of the ȷ-series with the θ-integral in (3.8)-(3.9), and the extrapolation of a general term from the four computed orders. The closed forms in (3.11) are consistent with the displayed pattern (the first line follows from Σ_{n≥1}(6n+1)xⁿ = x(7−x)/(1−x)²), but the pattern is inferred, not proved, and the same extrapolation is used for the cone contribution (3.13). Moreover, the series in (3.10) converges only for |η| < 1, while the resulting charges are used on the full parameter space and M has a pole at η = 1 that is not discussed. Because no independent check of the KS mass is provided, in contrast with the AKS case which has the O(A⁴) surface-charge check, the section needs either a justification or a numerical/higher-order verification of the interchange and the extrapolation, or an independent derivation of M; a check at representative parameter values using the supplied Mathematica notebook would already settle the issue. The authors acknowledge this: “However, this is risky because it is unclear whether the limit commutes with integration.”
[Section 3.2 (Eqs. (3.23)-(3.31)) and Section 5.1 (Eqs. (5.1)-(5.3))] In the KS and KTN cases the first law is solved, not tested. The dimensional match between the three parameters and the three unknowns that the authors point out is correct and interesting, but it means that (3.23) is a system of differential equations for S, T, Ω rather than an identity to be verified. Where dM and dJ are independent and T is nonzero, dM = T dS + Ω dJ forces dS into the span of dM and dJ, so locally S = F(M,J); the Kerr slice (ȷ = 0, or n = 0 for KTN) is two-dimensional and fixes F to be the Kerr entropy function. Explicitly, (3.24) is the standard Kerr entropy evaluated at the charge pair (M,J) = (m/γ, am/γ), as is also visible from the rescalings (3.27)-(3.31), and (5.1) is the Kerr entropy at (M,J) = (m, (a−3Cn)m); T and Ω are then the corresponding Kerr thermodynamic derivatives. The first law therefore holds by construction once the charges are known, and the statement S ≠ Area/4 is a comparison between this chosen thermostatic function and the horizon area, not an independent derivation of an area-law violation. The authors acknowledge that they inferred rather than computed the temperature and entropy (Section 5.1; Conclusion), but the abstract still presents the mismatch as a finding. I recommend reformulating the central claim as a well-defined but conditional proposal and adding an independent check for at least one case; the surface gravity of the mass generator (3.21), the Euclidean action on a conically regularized geometry, or the quantum-field computation the authors themselves sketch would each serve. The authors acknowledge this: “An independent derivation, not relying on the first law itself, is therefore needed to confirm them.”
[Section 4.1.2-4.1.3, Eqs. (4.15)-(4.18), (4.21)-(4.23)] The AKS mass, and with it the AKS first law, is derived from the Christodoulou-Ruffini relation, which the paper states as a conjecture. Given M(S,J) of this form, the first law δM = TδS + ΩδJ follows by differentiation, so the thermodynamical content of the AKS section is carried by the exact angular momentum (4.11), the horizon area (4.15), and the conjecture itself. The independent evidence is the surface-charge check δM = /δQξ of Section 4.1.3, which is genuine and nontrivial but truncated at order A⁴ and constructed with a generator ξ inferred via the first law itself in (4.21)-(4.22). Since the AKS case provides the paper's clean example of S = Area/4, the strength of the advertised contrast with KS and KTN is proportional to the reliability of the conjecture and of the truncated check. I recommend extending the check to higher order in A or performing an exact computation of /δQ_{∂t}, whose structural similarity to the KS computation suggests feasibility, and, until then, presenting the AKS mass in the abstract and conclusions as conjectural with partial verification rather than as a derived result.

Minor objections (6)

[Section 4.1.2, Eq. (4.18b)] The displayed formula for the temperature contains a corrupted radical ('/radicaltp/radicalvertex/radicalvertex'), making the expression unreadable as printed; the typesetting must be repaired.
[Sections 3.1 and 4.1] The symbol η is defined as 4maȷ in Section 3 and then redefined as mA∆ϕ below Eq. (4.11) in Section 4; given the paper's own notation warning, the redefinition should be stated explicitly.
[Section 3.2, footnote 5] The phrase 'For a larger, we have' appears to be a typo for 'For large r, we have'; please correct it.
[Section 5.1, Eq. (5.9)] The identity (5.9) and the statement that the remaining term in (5.5) is non-integrable are presented with a bare 'one can check' but no derivation; please include the verification in an appendix or state explicitly that it is contained in the Mathematica notebook.
[Section 4.1.2, Eqs. (4.16a)-(4.16d)] The algebraic equivalence of the four displayed forms of M² is not shown; a brief indication of the route, or a pointer to the relevant notebook section, would help the reader trust line (4.16d).
[Section 5, last paragraph] Reference [41] is a published 2006 paper by Kerner and Mann, yet it is cited for 'future investigations' of the present authors; either the citation should support the claim that defects may play a global role, or a different reference is intended.

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The paper

The paper claims

The paper establishes the first law of black hole thermodynamics for Kerr swirling, accelerated Kerr swirling, and Kerr–Taub–NUT, and finds the entropy departs from A/4 whenever string-like defects are present.

The reading

This paper establishes the first law of black hole thermodynamics, $\delta M = T\,\delta S + \Omega\,\delta J$, for three spacetimes that lack the usual flat infinity: the Kerr black hole in a swirling universe (KS), its accelerating cousin (AKS), and the Kerr–Taub–NUT spacetime (KTN). The authors compute mass and angular momentum quasi-locally with a Noether-based surface-charge method, adding K-terms from thin cones cut around the string-like defects along the axes so that the charges no longer depend on the chosen surface. From the first law, with the ordinary Kerr limit as boundary condition, they read off temperature, angular velocity, and entropy: for KS and KTN the entropy differs from one quarter of the horizon area, while for the string-free AKS spacetime the standard $S = A/4$ survives. If this is right, the area law cannot be applied naively to black holes with string-like defects, and those topologically different spacetimes need a new geometrical framework for their entropy.

What carries the argument

The engine of the paper is the surface-charge method for exact symmetries: each Killing vector $\xi$ gives an on-shell conservation law $dk_\xi\approx 0$, and the integrated two-form defines the variation of a charge; where string-like defects puncture the surface, two K-terms — limits of integrals over thin truncated cones around the north and south axes — are added, and these terms exactly cancel the spurious radius-dependence of the sphere integral. Integrability in phase space then selects the mass-generating Killing vector as a linear combination $\xi=\alpha(\partial_t+\beta\partial_\phi)$, with $\alpha$ and $\beta$ fixed uniquely by demanding the Kerr limit; the first law becomes a closed differential system because the three unknown functions $T$, $S$, $\Omega$ match the three parameter directions $(m,a,\jmath)$ or $(m,a,n)$. For AKS the direct time-charge computation is abandoned in favour of the Christodoulou–Ruffini mass formula combined with the area-as-Noether-charge identity on a bifurcation surface, which are then cross-checked against the surface-charge method in an expansion in the acceleration.

Extended reading notes

Core claim

The central claim is that imposing $\delta M = T\,\delta S + \Omega\,\delta J$ — with $M$ and $J$ fixed by the improved surface-charge method and the Kerr limit used as the boundary condition — fully determines $S$, $T$, and $\Omega$ for KS, AKS, and KTN, with no extra stringy terms in the first law. For the Kerr swirling black hole the resulting entropy is $S=\pi\left(a^2+\frac{(m+\sqrt{m^2-(1-\eta^2)^2a^2})^2}{(1-\eta^2)^2}\right)$ with $\eta=4ma\jmath$, and for Kerr–Taub–NUT it is $S=2m\pi\left(m+\sqrt{m^2-(a-3Cn)^2}\right)$, where $n$ is the NUT parameter and $C=\{-1,0,+1\}$ labels the string placement; neither equals $A/4$. For KTN the earlier over-determination of the first law disappears without restricting the parameter space, and the Misner-string extra terms of a previous approach are traced back to surface-artifact pieces of the conserved charges. In contrast, for the accelerated Kerr swirling black hole, where the regularity conditions remove the conical defects, the first law holds with the usual $S=A/4$, the mass being fixed by the Christodoulou–Ruffini relation $M^2=S/(4\pi)+\pi J^2/S$ and verified against the surface-charge method up to order $A^4$.

Load-bearing premise

The load-bearing premise is that imposing the first law and using the Kerr limit as a boundary condition picks out physically meaningful values of $S$, $T$, and $\Omega$ even though none of them is derived independently — the paper says so explicitly in Sections 3.2 and 5.1 — and a second fragile input, also acknowledged, is the conjecture that regular black holes satisfy the Christodoulou–Ruffini mass formula used to fix the AKS mass.

Editorial extensions

If this is right

Standard thermodynamic formulas — $T=\kappa_H/(2\pi)$, $\Omega$ read off the horizon metric, and $S=A/4$ — cannot be used as-is for KS and KTN; keeping them forces extra first-law terms that are not conservation-law charges.
The earlier treatment of the Kerr–Taub–NUT first law restricted the parameter space; the new one holds on the full $(m,a,n)$ space, so the entropy of eq. (5.1) is the unique candidate that connects continuously to the Kerr limit.
For the regular AKS spacetime the first law keeps the standard area entropy, so the swirling background alone does not break the area rule — the string defects are what make the difference.
The Christodoulou–Ruffini mass–entropy–angular-momentum relation survives in all three families once the area is replaced by the first-law entropy wherever needed.

Reading between the lines

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

If the inferred entropies are correct, the lesson reaches beyond these exact solutions: any axisymmetric black hole with conical defects should be expected to violate the area law, with defect contributions entering through punctured-surface or holonomy-type invariants rather than the horizon area alone.
An independent Hawking-type calculation of the temperature on the KS or KTN background — the authors themselves point to Bogoliubov coefficients — would settle the matter: matching eqs. (3.25) or (5.2) would validate the programme, while recovering $\kappa_H/(2\pi)$ would show the inferred entropy is an artifact.
The same K-term technology should transfer to charged or electromagnetic swirling black holes, a natural next family where one could ask whether the electric potential joins the first law while the entropy shift persists.
The AKS result suggests it is the defect topology, not the mere absence of asymptotic flatness, that drives the entropy shift, since AKS keeps $S=A/4$ despite a non-flat swirling infinity.

Original abstract

We establish the first law of black hole thermodynamics for the recently discovered Kerr swirling (KS) and accelerated Kerr swirling (AKS) black hole solutions. We also revisit the first law for the Kerr-Taub-NUT (KTN) spacetime. Guided by Noether's original insight that global symmetries give rise to conservation laws, we employ and adapt the surface charge method for computing the associated charges (mass and angular momentum) in the presence of spacetime string-like defects (conical singularities) along the axes of both KS and KTN black holes. We obtain finite integrated charges in a quasi-local manner, without relying on the non-flat asymptotic regions characteristic of these spacetimes. We then establish a first law without introducing independent stringy terms unrelated to symmetries. For both KS and KTN, the quantities that play the role of the black hole entropy differ from the simple geometrical rule of one-fourth of the horizon area. This mismatch arises because we keep first-law charges associated with conservation laws. It indicates that standard formulas may not be naively used for black holes with string-like defects; they have a different topology and suggest the need for a new geometrical framework. In stark contrast, for the AKS spacetime, which can be regularized outside the singularity and lacks string-like defects, we find that the charges are consistent with a first law having the usual one-fourth of the area entropy. We first determine the mass using the Christodoulou-Ruffini formula and then check its agreement with the result obtained from the surface charge method.

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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, 2026-10-30: 1 request, 7,664 input tokens.

What the abstract itself states, as probabilities: claims first 0.76 · states limitation 0.67 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.91 · 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.

We establish the first law of black hole thermodynamics for the recently discovered Kerr swirling (KS) and accelerated Kerr swirling (AKS) black hole solutions. We also revisit the first law for the Kerr-Taub-NUT (KTN) spacetime. Guided by Noether's original insight that global symmetries give rise to conservation laws, we employ and adapt the surface charge method for computing the associated charges (mass and angular momentum) in the presence of spacetime string-like defects (conical singularities) along the axes of both KS and KTN black holes. We obtain finite integrated charges in a quasi-local manner, without relying on the non-flat asymptotic regions characteristic of these spacetimes. We then establish a first law without introducing independent stringy terms unrelated to symmetries. For both KS and KTN, the quantities that play the role of the black hole entropy differ from the simple geometrical rule of one-fourth of the horizon area. This mismatch arises because we keep first-law charges associated with conservation laws. It indicates that standard formulas may not be naively used for black holes with string-like defects; they have a different topology and suggest the need for a new geometrical framework. In stark contrast, for the AKS spacetime, which can be regularized outside the singularity and lacks string-like defects, we find that the charges are consistent with a first law having the usual one-fourth of the area entropy. We first determine the mass using the Christodoulou-Ruffini formula and then check its agreement with the result obtained from the surface charge method.

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	We establish the first law of black hole thermodynamics for the recently discovered Kerr swirling (KS) and accelerated Kerr swirling (AKS) black hole solutions.	contribution 0.86	0.95	0.41	asserted 1.00
s1	We also revisit the first law for the Kerr-Taub-NUT (KTN) spacetime.	contribution 0.98	0.88	0.05	asserted 0.86
s2	Guided by Noether's original insight that global symmetries give rise to conservation laws, we employ and adapt the surface charge method for computing the associated charges (mass and angular momentum) in the presence of spacetime string-like defects (conical singularities) along the axes of both KS and KTN black holes.	method 0.98	0.91	0.06	asserted 0.99
s3	We obtain finite integrated charges in a quasi-local manner, without relying on the non-flat asymptotic regions characteristic of these spacetimes.	method 0.45	0.96	0.09	asserted 1.00
s4	We then establish a first law without introducing independent stringy terms unrelated to symmetries.	result 0.83	0.95	0.22	asserted 1.00
s5	For both KS and KTN, the quantities that play the role of the black hole entropy differ from the simple geometrical rule of one-fourth of the horizon area.	result 1.00	0.85	0.28	asserted 1.00
s6	This mismatch arises because we keep first-law charges associated with conservation laws.	interpretation 1.00	0.80	0.12	asserted 1.00
s7	It indicates that standard formulas may not be naively used for black holes with string-like defects; they have a different topology and suggest the need for a new geometrical framework.	interpretation 1.00	0.59	0.23	hedged possibility 0.97
s8	In stark contrast, for the AKS spacetime, which can be regularized outside the singularity and lacks string-like defects, we find that the charges are consistent with a first law having the usual one-fourth of the area entropy.	result 1.00	0.96	0.22	asserted 0.79
s9	We first determine the mass using the Christodoulou-Ruffini formula and then check its agreement with the result obtained from the surface charge method.	method 0.99	0.92	0.05	asserted 0.81

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

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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

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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.** Integration region for the surface charge in the presence of string-like defects. The spacelike volume Σϵ lies between the spheres S1 and S2 (r1 < r2) and outside the thin tubes TN and TS enclosing the north and south strings (red), taken as the truncated cones θ = ϵ and θ = π − ϵ. Z TS kξ = Kξ

Figures reproduced from arXiv:2610.40529 by the authors.

References

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Discussion

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View on Pith Pith review arXiv 2025