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On the Smarr formulas for electrovac spacetimes with line singularities

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Pith's one-line read For stationary axisymmetric electrovac spacetimes, the total Komar mass decomposes into a sum of Smarr terms for black-hole horizons and for Misner and Dirac strings, so line singularities carry mass, angular momentum, and charge.

desk verdict A careful and mostly convincing derivation of horizon-level Smarr formulas without NUT or magnetic terms, but the global string-level formula rests on a finite-part subtraction whose regulator independence is not established. read the letter →

arxiv 1908.10617 v1 pith:FOE4UKN5 submitted 2019-08-28 gr-qc hep-th

Gerard Clément , Dmitri Gal'tsov This is my paper · ORCID
classification gr-qchep-th MSC 83C5783C2283C40 PACS 04.70.Bw04.40.Nr
keywords SmarrformulaMisnerstringDiracNUTparameterKomarmassrodstructureKerr-NUTspacetimeEinstein-Maxwellequations
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

The paper extends Smarr's black-hole mass formula to spacetimes whose symmetry axis carries line singularities: Misner strings, Dirac strings, and struts. Using the rod-structure description in which horizons and defects are placed on the same footing, it derives a global formula in which the total Komar mass is a sum of Smarr terms over horizon rods and string rods. For Kerr-NUT and dyonic Kerr-Newman-NUT spacetimes, each Misner and Dirac string gets a definite mass, angular momentum, and electric charge, and the total mass is finite only after a symmetric gauge choice. The horizon Smarr formula is shown to contain neither a magnetic-charge term nor a NUT term, correcting earlier statements. If correct, the calculation gives a coherent physical bookkeeping for NUT-charged and magnetically charged black holes with string defects.

What carries the argument

The machinery is the rod structure of stationary axisymmetric spacetimes in Weyl coordinates, where horizons are timelike rods and line defects are spacelike rods on the polar axis, with constant Killing directions along each rod. The paper recasts the Komar-Tomimatsu surface integrals as boundary terms evaluated at the turning points between rods, using the gravitational and electromagnetic Ernst potentials, and adds the electromagnetic bulk terms that earlier treatments dropped. Divergent angular momenta of semi-infinite rods are regularized by choosing the symmetric Misner-string gauge and subtracting the divergent length term to define finite reduced string angular momenta $\tilde{J}_\pm$.

What would settle it

Thicken the Misner strings of Kerr-NUT with asymmetric tuning $s \neq 0$ into narrow tubes with a regular matter source, compute the total energy and angular momentum from the full stress-energy distribution, and check whether a finite Smarr relation holds without subtracting a divergent length term; if it does, the symmetric-gauge regularization would be shown to be not the only consistent choice.

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Extended reading notes

Core claim

The central result is the global Smarr decomposition $$M = \sum_H \left(2\Omega_H J_H + \frac{\kappa_H A_H}{4\pi} + \Phi_H Q_H\right) + \sum_S \left(2\Omega_S J_S + \frac{\kappa_S A_S}{4\pi} + \Phi_S Q_S\right),$$ in which each horizon rod and each string rod contributes its own angular-velocity term, area term, and electric-potential term to the total Komar mass. For Kerr-NUT with symmetrically tuned Misner strings this reduces to $M = 2T_H S_H + 2\Omega_H J_H + 2\Omega_+ \tilde{J}_+ + 2\Omega_- \tilde{J}_-$, where $\tilde{J}_\pm$ are the regularized finite angular momenta of the two semi-infinite strings. The horizon mass Smarr formula $M_H = 2\Omega_H J_H + 2T_H S_H + \Phi_H Q_H$ contains no NUT and no magnetic-charge term, so those charges enter the global mass only through the string contributions.

Load-bearing premise

The derivation assumes that the Komar-Tomimatsu surface integrals can be applied rod by rod to weak distributional line singularities—that the Ostrogradsky decomposition is valid when Dirac or Misner strings lie on the axis—and that the divergent infinite-rod angular momenta can be tamed by the symmetric gauge choice and by dropping the divergent length term.

Editorial extensions

If this is right

  • The horizon Smarr relation for a black hole with NUT or magnetic charge is the standard one, so putative NUT or magnetic terms in horizon mass formulas are artefacts of including string contributions.
  • For Kerr-NUT, the total mass equals the ADM mass $m$, the Kerr proportionality $J = aM$ holds globally including the strings, and the two Misner strings rotate in opposite directions with angular velocities $\Omega_\pm = \mp 1/(2n)$.
  • Finiteness of the total Komar angular momentum selects the symmetric tuning $s=0$ of the Misner strings as the physically meaningful gauge.
  • In the vanishing-NUT limit, the string terms of the dyonic Kerr-Newman-NUT formula reduce to an effective horizon magnetic-potential term, recovering the earlier dyonic Reissner-Nordstrom result.
  • For multi-black-hole spacetimes with struts, the global mass is generically a sum of separate Smarr terms for each horizon and each string, not a single simple relation.

Reading between the lines

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

  • Editorial inference: if the string area terms are someday given a thermodynamic role, the natural reading is a separate entropy-like contribution for each Misner string, independent of the horizon entropy; the authors explicitly refrain from this interpretation.
  • Editorial inference: the symmetric-gauge selection rule suggests a general principle for stationary NUT spacetimes: only configurations with balanced, oppositely rotating Misner strings admit finite conserved charges, which would constrain admissible multi-center NUT solutions.
  • Editorial inference: the same rod-by-rod decomposition could assign masses and angular momenta to the struts in binary black-hole systems, separating the energy carried by the binding defect from the horizon energies.
  • Editorial inference: the rod formalism extends to higher dimensions with $D-2$ commuting Killing vectors, so a Smarr decomposition of the same type should be derivable for line singularities in black-ring or higher-dimensional black-hole spacetimes and could be checked against known examples.
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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 / 5 minor

Summary. The paper reformulates Tomimatsu's Komar-integral approach in terms of the rod structure of stationary axisymmetric electrovac spacetimes. It derives rod-wise Smarr-type formulas for finite timelike rods (horizons) and finite or infinite spacelike rods (Misner and Dirac strings), culminating in the global decomposition Eq. (3.32), and applies the construction to Kerr-NUT and dyonic Kerr-Newman-NUT spacetimes. The paper claims that the horizon Smarr formula contains no independent NUT or magnetic-charge term, that finiteness of the total Komar angular momentum selects the symmetric Misner-string gauge s = 0, and that the physical Misner-string contributions to the global mass formula are captured by the reduced string angular momenta J̃± defined in Eq. (3.48).

Significance. If the central claim holds, the paper provides a systematic method for assigning Komar masses, angular momenta, and charges to line singularities, and it gives Smarr-type mass formulas for NUTty spacetimes without imposing the Misner periodicity condition. The explicit Kerr-NUT and dyonic KN-NUT computations are nontrivial and serve as independent checks of the rod-level algebra. The main strengths are that the Smarr relations are derived algebraically from the Komar-Tomimatsu identities rather than fitted, and that the horizon formula's independence from NUT and magnetic-charge terms is a clean, falsifiable statement. The main weakness is the treatment of divergent semi-infinite string angular momenta: the reduced charges (3.48) are introduced by a finite-part subtraction whose regulator independence is not established, so the physical identification of string charges is not yet fully rigorous. This concern is load-bearing for the paper's central claim and is the reason I recommend major revision rather than acceptance.

major comments (3)
  1. [§3.3, Eq. (3.48)] The 'reduced string angular momenta' J̃± ≡ J± + ω±L±/4 = ω±M±/2 are defined by subtracting the linearly divergent length term from J±, but no regulator is specified and no regulator-independence proof is given. Since J± ∼ ∓(n/2)R for s = 0 (Eq. (3.35)), the subtraction is a finite-part prescription, not an evaluation of a Komar integral over the string. Because Eq. (3.47) and the first law (3.49) treat J̃± as physical charges, a different cutoff or smoothing of the semi-infinite rods could shift the finite part and thereby change the coefficients in the Smarr formula while preserving its algebraic form. The authors do acknowledge that these are finite parts, but the paper needs a well-defined limiting procedure (e.g., a matched asymptotic expansion or background subtraction) showing uniqueness of the finite part; this is load-bearing for the paper's central identification of string angular momenta.
  2. [§3, Eqs. (3.3)–(3.8)] The passage from the Komar integrals at infinity to the rod decomposition applies the three-dimensional Ostrogradsky theorem in a spacetime whose polar axis carries distributional sources. For Misner and Dirac strings the fields are not smooth on the axis and, in the Kerr-NUT case, the per-rod angular momenta diverge linearly (Eq. (3.35)); the usual justification for weak line singularities (finite period integrals, as for cosmic strings) does not automatically extend. Please state the precise distributional assumptions under which the boundary is exhausted by the small cylinders Σ_n and no additional surface terms arise at the turning points or along the axis, and verify these assumptions for the Kerr-NUT example.
  3. [§3.3, Eq. (3.49)] The generalized first law dM = T_H dS_H + Ω_H dJ_H + Ω_+ dJ̃_+ + Ω_- dJ̃_- differentiates the subtracted charges J̃±. The physical status of this relation is not on the same footing as the horizon first law unless J̃± are defined by a canonical, regulator-independent charge construction. As written, the relation is an identity following from the algebraic Smarr formula, and it could be satisfied by any subtraction that preserves the form of (3.47). Please clarify the sense in which (3.49) is a first law and not merely a differential identity among rod parameters.
minor comments (5)
  1. [Footnote 1] The footnote about reference [32] is polemical and should be removed or replaced by a neutral citation note; it does not affect the technical content.
  2. [Eqs. (3.19)–(3.23)] The symbol n is used both as a rod label (e.g., in Eqs. (3.19)–(3.23)) and as the NUT parameter in the examples; this double use makes the notation confusing. Please use an index such as I for the rods.
  3. [§3.3, Eq. (3.34)] The cancellation of the leading divergence for s = 0 is shown, but the finiteness of the remaining angular-momentum integral is not demonstrated; a short estimate of the next-order term would help the reader verify finiteness.
  4. [Appendix] The statement that the case m = n = s reduces to the Taub-NUT instanton is not supported by an explicit coordinate transformation; please provide the transformation or a reference.
  5. [Abstract and Introduction] The phrase 'previously unknown or incorrect' is vague; since the paper corrects prior literature, please list in the introduction exactly which formulas are new and which are corrected, with equation numbers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Smarr relations are algebraic consequences of the Komar-Tomimatsu rod integrals, and cited prior work is re-derived rather than invoked as a black box.

full rationale

The central derivation starts from the Komar surface integrals (3.1), applies the Ostrogradsky decomposition to obtain rod integrals (3.7) and (3.8), rewrites them using Ernst potentials in (3.19), (3.20), and (3.22), and then algebraically obtains the rod angular-momentum formula (3.23). The horizon Smarr formula (3.27) and the string Smarr formula (3.29) follow by substituting the identities T_H S_H = (z2 - z1)/4 and its spacelike analogue (3.31); this is a derivation from the definitions of Komar charges, horizons, and rod data, not a fit or a renaming of an input. The global relation (3.32) is simply the sum of the rod Smarr identities. In the Kerr-NUT example, the reduced string angular momenta are defined explicitly in (3.48) as J~± = J± + ω±L±/4, and the global relation (3.47) is exactly the string contribution 2Ω±J± + L±/2 recast as 2Ω±J~±; again this is an identity, not an empirical prediction. The paper cites [27] for the corrected Tomimatsu scheme, but the essential formulas are re-derived in the present text, so the self-citation is not load-bearing. The stated equivalence to relation (10) of [25] is an explicit comparison of variables, not an appeal to authority. The genuine limitations flagged by the paper — the distributional validity of rod-by-rod Ostrogradsky reduction for Misner and Dirac strings, and the absence of a regulator-independence proof for the finite-part subtraction defining J~± — are correctness or regularization concerns, not circularity: the paper does not secretly use the target Smarr formula to define its charges. The paper is self-contained in its algebraic derivation and checks it against explicit Kerr-NUT and dyonic Kerr-Newman-NUT examples. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No data are fitted. The only parameters chosen by hand are the gauge parameter s, set to zero by the finiteness requirement, and an infinite-rod regulator R that cancels from final expressions. The main physical input is the rod-structure description of stationary axisymmetric solutions and the validity of the Tomimatsu-Komar integral decomposition for distributional line singularities; these are standard tools in this subfield but are not independently machine-checked here. No new fundamental entities are invented; the paper assigns conserved charges to existing Misner and Dirac strings.

free parameters (2)
  • Misner string gauge parameter s = 0 (symmetric tuning)
    In the Kerr-NUT and KN-NUT metrics (2.9), s fixes the relative strength of the two semi-infinite Misner strings. Total Komar angular momentum is finite only when s = 0 (Eqs. (3.34)-(3.35)); the explicit Smarr calculations assume this symmetric choice.
  • Infinite-rod regularization length R = R tends to infinity (cancels)
    Semi-infinite Misner string angular momenta J_+- diverge as L_+-/4 with L_+- = R - sigma; the combinations J~_+- are finite. R is a regulator introduced in Eq. (3.44) and drops out of the final formulas.
assumptions (5)
  • domain assumption Stationary axisymmetric Einstein-Maxwell spacetimes with two commuting Killing vectors admit Weyl-Papapetrou coordinates and rod structure with constant rod direction vectors.
    Used in Section 2 to parameterize the metric (2.1) and define rods; constancy is cited to Harmark [36].
  • domain assumption The line singularities are weak distributional singularities with dim ker G(0,z) = 1, so charges can be defined by cylinder integrals.
    Section 2 states stronger singularities would create curvature singularities; the surface-integral treatment requires this weak-singularity condition.
  • domain assumption The Tomimatsu-Komar rod-charge formulas (3.19)-(3.20) from [27] are valid for spacelike rods as well as horizons.
    The paper extends formulas derived in [27] to string rods; this is asserted in Section 3.1 rather than re-derived from scratch.
  • domain assumption Ostrogradsky and Gauss theorems apply to the bulk with boundary cylinders around all rods and a sphere at infinity; boundary terms at string endpoints vanish or cancel.
    This enables the mass and angular momentum decomposition (3.3)-(3.8); it relies on integrability of the distributional sources.
  • standard math Kerr-NUT and dyonic Kerr-Newman-NUT metrics satisfy the Einstein-Maxwell equations with the stated Ernst potentials and vector potential (3.53).
    The explicit examples in Sections 3.3-3.4 use these known solutions and their complexified Ernst potentials as the testing ground.

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Pith. "Pith review of On the Smarr formulas for electrovac spacetimes with line singularities." pith.science (2026). https://pith.science/paper/FOE4UKN5

@misc{pith2026190810617,
  author       = {Pith},
  title        = {Pith review of: On the Smarr formulas for electrovac spacetimes with line singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOE4UKN5}},
  note         = {Machine review of arXiv:1908.10617}
}
read the original abstract

Using the revised Komar-Tomimatsu approach, we derive Smarr mass formulas for stationary axisymmetric solutions of the Einstein-Maxwell equations containing line singularities (defects) on the polar axis. In terms of the rod structure associated with Weyl representation of the metric, the horizons and the defects are formally similar up to differences due to their timelike/spacelike character. We derive (previously unknown or incorrect) horizon and global Smarr formulas in presence of a Newman-Unti-Tamburino (NUT) parameter. To avoid the divergence of the Komar angular momentum of semi-infinite Dirac and Misner strings, it is necessary to use a symmetric tuning. We also note that the horizon mass Smarr formula does not include either magnetic charge, or NUT parameter, correcting some statements in the literature. The contribution of each Misner string to the total mass consists in an angular momentum term, an electric charge term, and a length term, which can also be presented as the product of the spacelike analogue of surface gravity and the area of the string.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Taub-NUT as gravitational dyon with torsion

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    Taub-NUT's Misner strings are torsion defects; with this torsion, the spacetime is a soliton sourced by two spin-fluid beams with a cosmic-string-like equation of state.

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