REVIEW 2 major objections 4 minor 1 cited by
A note on the calculation of the Komar integral in the Lorentzian Taub-NUT spacetime
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Misner strings reappear on every time slice of stringless Taub-NUT
desk verdict A short but genuine clarification: Misner's stringless Taub-NUT still puts string singularities on every spacelike slice used for Komar integrals, so the string contributions to the Smarr formula cannot be avoided. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the Komar charge 2-form $K[k]$ associated to the horizon-generating Killing vector $k$, integrated over a spacelike hypersurface with boundaries at the bifurcation sphere $B_H$, the sphere at infinity $S^2_\infty$, and cones $T_\pm$ surrounding the Misner strings (the axis line singularities of the one-form $A=2N(\cos\theta+s)d\varphi$, analogous to Dirac strings). The mechanism is the identity $0=\int_{\Sigma_3} dK[k]$ together with Stokes' theorem, which turns the sum of boundary integrals into the Smarr formula; when a string is present, one excises a narrow cone around it and adds the string boundary integral, written as $\psi_\pm N_\pm$ with $\psi_\pm$ the string surface gravities and $N_\pm$ the conjugate charges. The paper's additional machinery is Misner's two-patch construction, in which overlapping northern and southern Taub-NUT patches are glued by $t_{(+)}=t_{(-)}+4N\varphi$ with $t_{(\pm)}$ periodic with period $8\pi N$, and the observation that the induced metric on any $t=\text{const}$ slice of this regular spacetime is exactly the metric of a singular patch, so the string reappears at the slice level.
What would settle it
Compute explicitly the pullback of the Misner-glued metric to the hypersurface $t_{(+)}=0$ and examine the induced 3-metric near the negative $z$-axis; if it turns out to be regular there, or if the integral of the Komar 2-form over a small cone around that axis vanishes in the zero-radius limit, then the string contributions are not needed and the Smarr formula $M=2TS$ would be the correct one for the stringless spacetime.
Extended reading notes
Core claim
The central claim is that Misner's string-removal construction does not remove the Misner strings from the spacelike hypersurfaces on which the Komar integrals are evaluated. Choosing $t_{(+)}=0$ forces $t_{(-)}=-4N\varphi$, and the induced metric is the same as that of the $t=0$ slice of the singular solution with $s=-1$, carrying a Misner-string singularity along the negative $z$-axis; the choices $t_{(-)}=0$ and $t_{(+)}=2N\varphi$ give strings on the positive axis and on both semiaxes. Therefore the identity $0=\int_{\Sigma_3} dK[k]$ must be evaluated on a hypersurface with conical excisions around those string lines, and the boundary integrals acquire string contributions $\psi_\pm N_\pm$ exactly as in the computation for spacetimes with explicit Misner strings. The conclusion is that the Smarr formula for the stringless Taub-NUT spacetime is the same general formula $M=2(TS+\psi_+N_++\psi_-N_-)$, with one of the pairs $(N_+,N_-)$ vanishing according to which semiaxis carries the string, and not the naive $M=2TS$. The paper also observes that these different string-carrying hypersurfaces are physically inequivalent, so observers on different slices experience different physics.
Load-bearing premise
The argument relies on the slice $t_{(+)}=0$ having a genuine Misner-string singularity in its induced metric along the negative $z$-axis; the paper states this inherited singularity rather than displaying the explicit pullback calculation that proves it.
Editorial extensions
If this is right
- The stringless Taub-NUT spacetime satisfies the same Smarr relation as the singular one, $M=2(TS+\psi_+N_++\psi_-N_-)$, so the NUT charge contributes to the mass through hypersurface string terms rather than through the sphere integrals.
- Any spacelike hypersurface used in the Komar computation carries at least one Misner string; there is no globally regular time slice for the connection $A$.
- Thermodynamic charges and potentials are hypersurface-dependent: the slicings $t_{(+)}=0$, $t_{(-)}=0$, and $t_{(+)}=2N\varphi$ yield different $N_\pm$ and $\psi_\pm$, and hence different physics.
- The Euclidean approach, which never chooses a hypersurface, does not see these string contributions and its known difficulties for the stringless Taub-NUT spacetime persist.
Reading between the lines
- Beyond the paper: the same mechanism may explain other apparent Smarr-formula inconsistencies in NUT-charged or magnetic-type spacetimes written in globally regular coordinates, where a hidden line singularity on the Cauchy surface supplies the missing boundary term.
- A concrete extension would be to scan the one-parameter family of hypersurfaces $t_{(+)}=\alpha\varphi$ and trace how the string locations and charges $\psi_\pm,N_\pm$ vary with $\alpha$; the paper only discusses three values.
- One could test the physical-equivalence claim by computing the Hamiltonian energy on each slice; if the charges genuinely differ, the stringless Taub-NUT spacetime has no observer-independent thermodynamics in the Lorentzian formulation.
- The result sharpens the Dirac-monopole analogy: just as the Dirac string cannot be removed from the vector potential, the Misner string cannot be removed from the initial-data slice, only moved.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Lorentzian Taub-NUT spacetime in which Misner's procedure has been used to remove the Misner-string singularities of the 4D metric. It claims that, despite the 4D regularity, any spacelike hypersurface used in a Komar-integral derivation of the Smarr formula inherits a Misner-string singularity: for example, the hypersurface t^(+)=0 in the northern patch has an induced metric of the s=-1 form with a singularity on the negative z-axis. The authors conclude that the string contributions computed in Ref. [9] must be included even in the stringless spacetime, so the Smarr formula is again Eq. (19), and that different choices of hypersurface can lead to different physics.
Significance. If the central claim is correct, the paper resolves an apparent inconsistency in the Lorentzian thermodynamics of the stringless Taub-NUT solution: the naive result M=2ST is not the full Smarr formula because the spacelike hypersurfaces necessarily contain Misner-string singularities whose Komar flux must be added. The observation that different time slices contain different strings and yield different thermodynamic charges is a potentially important and non-obvious conclusion. The paper is concise and clearly written, and it explicitly identifies the three hypersurfaces corresponding to s=0,±1. However, the main technical step—the explicit pullback computation and the evaluation of the Komar flux on the excised cone—is not shown, so the conclusion is presently conditional on an unverified calculation.
major comments (2)
- [Section 3, Eq. (20)] The load-bearing claim is that the induced metric on t^(+)=0 has a Misner-string singularity along the negative z-axis. This is asserted without the explicit pullback computation. Writing the pullback of Eq. (11) to t^(+)=0 gives ds^2_Σ = λ(r)[2N(cosθ-1)dφ]^2 − λ^{-1}(r)dr^2 − (r^2+N^2)dΩ^2; the paper should show explicitly why this 3-metric is singular at θ=π (for example, by exhibiting the non-vanishing of g_φφ on the axis or by giving the local transverse metric and its conical character). It should then compute the Komar flux through a small tube surrounding that axis and show that it equals the string contribution ψ_− N_− of Ref. [9]. This is not a formality: the 4D Komar 2-form is smooth in the stringless spacetime, so a non-vanishing cone flux is not automatic but follows from the nontrivial winding of the section around the south pole. If the flux vanished or differed from Ref. [9], the Smarr formula would reduce to M=2ST and the central claim would fail.
- [Section 3, final paragraph] The conclusion that different hypersurfaces 'lead to different physics' is not substantiated. The paper shows that different time slices contain different Misner-string singularities and that the formal string charges take different values, but it does not specify an observable physical quantity that would distinguish these choices. If the hypersurfaces are related by a gauge transformation, the difference might be a gauge artifact rather than a physical difference. The authors should either give an explicit invariant (for instance, a conserved charge measured by a family of observers, or a holonomy of the S^1 fibration) or soften the claim to say that the Komar charges are slice-dependent, without asserting a difference in observable physics.
minor comments (4)
- [Abstract] The abstract contains a typo: 'Mister-string singularities' should be 'Misner-string singularities'.
- [Section 1, paragraph 1] There is a duplicated article in 'Notice that the the 1-form A'; it should read 'the 1-form A'.
- [Footnote 1] The parenthetical sentence 'all the coordinates in this piece of the should bear a (+) label' is missing the word 'solution'; it should read 'piece of the solution'.
- [Footnote 5] The footnote explaining the notation '.=', '=.' and '≑' is garbled and contains a typo ('noly' for 'only'). The definitions of the three types of identities are not clearly distinguished, and the sentence is not grammatical. Please rewrite it so that the on-shell, reducibility, and combined conditions are each associated with the correct symbol.
Circularity Check
No significant circularity; the paper's central claim is a geometric observation that does not reduce to its inputs.
full rationale
The paper's central claim is that in the stringless (Misner) Taub-NUT spacetime, the spacelike hypersurfaces used for Komar integrals necessarily inherit Misner-string singularities, so the string contributions of Ref. [9] must be included. This is an independent geometric statement: the hypersurface t(+)=0 is shown, via the gluing relation t(+)=t(-)+4Nφ, to have an induced metric of the s=-1 form, so the singularity analysis of Eqs. (9)-(10) applies directly. The Smarr formula (19) is quoted from external Ref. [9], not rederived from a fitted parameter, and no quantity is defined in terms of the result it is supposed to explain. The only self-citation is Ref. [18], a peripheral 'see also' for generalized Komar charges, which is not load-bearing for the argument. The note's conclusion that different hypersurfaces correspond to different s values is a direct consequence of the coordinate transformation (13), not a circular redefinition. Any concerns about whether the hypersurface t(+)=0 is actually spacelike or whether the induced-metric singularity is exactly a Misner string are correctness questions, not circularity, and fall outside the scope of this pass.
Assumptions & free parameters
assumptions (5)
- standard math The Komar integral identity ∫_Σ dK = ∫_∂Σ K holds for the Taub-NUT Killing field, and the contributions from the boundaries (horizon, infinity, strings) give the Smarr formula.
- domain assumption The two charts of the Misner construction are glued by t(+) = t(−) + 4Nφ, with φ identified mod 2π, making the time coordinate periodic with period 8πN.
- domain assumption The hypersurface t(+) = 0 (with t(−) = -4Nφ) is a valid spacelike hypersurface covering the entire space outside the horizon, and its induced metric is the same as the metric with a Misner-string singularity along the negative z-axis.
- domain assumption The string contributions in the stringless case are computed exactly as in Ref [9], using cone excision around the strings.
- domain assumption There is no globally regular gauge choice for the connection form A, so every hypersurface necessarily carries at least one Misner-string singularity.
Cite this review
Pith. "Pith review of A note on the calculation of the Komar integral in the Lorentzian Taub-NUT spacetime." pith.science (2026). https://pith.science/paper/3IWFTLPT
@misc{pith2026250515349,
author = {Pith},
title = {Pith review of: A note on the calculation of the Komar integral in the Lorentzian Taub-NUT spacetime},
year = {2026},
howpublished = {\url{https://pith.science/paper/3IWFTLPT}},
note = {Machine review of arXiv:2505.15349}
}
read the original abstract
It has recently been shown that one can derive consistent thermodynamical expressions in the Lorentzian Taub--NUT spacetime keeping the Misner-string singularities and taking into account their contributions in the Komar integrals. We show how the same results are obtained when the Mister-string singularities are removed by using Misner's procedure because, even though the complete spacetime has no such singularities anymore, they are unavoidable in all spacelike hypersurfaces which are used in the Komar integrals. Different choices of hypersurfaces may contain different strings and lead to different physics, though.
Forward citations
Cited by 1 Pith paper
-
The Komar charge in presence of the Holst term and the gravitational Witten effect
The Holst term shifts the Komar mass of Taub-NUT space by -αN, giving a gravitational analog of the Witten effect.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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