The Komar charge in presence of the Holst term and the gravitational Witten effect

Jos\'e Luis V. Cerdeira; Tom\'as Ort\'in

REVIEW 2 major objections 3 minor 48 references

The Komar charge in presence of the Holst term and the gravitational Witten effect

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Adding a parity-odd Holst term to first-order gravity shifts the conserved mass of a NUT-charged spacetime, so a solution with zero mass parameter carries nonzero mass.

desk verdict Solid Holst-modified Komar charge derivation with a new Taub-NUT computation; the ADM mass identification is the load-bearing assumption that needs defense. read the letter →

arxiv 2506.15904 v2 pith:HZFDY6GD submitted 2025-06-18 hep-th gr-qc

José Luis V. Cerdeira , Tomás Ortín This is my paper

classification hep-thgr-qc MSC 83C4083C5783D05

keywords KomarchargeHolsttermBarberoparameterTaub-NUTspacetimeNUTNoether-WaldgravitationalthetaeffectSmarrformula

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 asks what happens to the usual conserved charge of general relativity when the first-order action is augmented by the parity-odd Holst term, multiplied by a coupling $\alpha$. It shows that the generalized Komar charge becomes $K[k] = (1/16\pi G)(-\star d\hat{k} + \alpha d\hat{k})$: the standard charge plus a topologically closed, magnetic-dual term. For time translations in the Taub-NUT spacetime, the extra term integrates to the NUT charge, so the relation between the mass parameter $m$ and the conserved mass $M$ becomes $M = m - \alpha N$. A solution with $m = 0$ therefore carries mass $-\alpha N$, a gravitational analogue of the familiar $\theta$-angle effect in gauge theory, where a topological coupling shifts the electric charge of a magnetic monopole.

What carries the argument

The load-bearing object is the generalized Komar charge 2-form $K[k] = (1/16\pi G)(-\star d\hat{k} + \alpha d\hat{k})$, defined as the negative of the Noether-Wald charge of the combined first-order action. The first term is the standard Komar charge; the second is the contribution of the Holst term, proportional to the exterior derivative of the Killing one-form $\hat{k}$. This new term is closed by itself and, when integrated over a 2-sphere, contributes only when $d\hat{k}$ is globally closed but not exact, which is exactly the situation for the time-translation Killing vector of Taub-NUT. The mechanism mirrors electromagnetism, where a gauge potential with a string singularity produces a nonzero magnetic charge integral.

What would settle it

Compute the canonical mass at spatial infinity of the $m = 0$ Taub-NUT solution directly from the Hamiltonian boundary term built from the metric's falloff, without using the Komar prescription. If that surface integral vanishes, the paper's $M = -\alpha N$ is a charge-convention statement rather than a property of the spacetime's energy; alternatively, check whether the entropy function $S(M,N_\pm,\alpha)$ implied by the shifted first law is integrable, since a failure of integrability would undercut the thermodynamic interpretation.

Watch

Extended reading notes

Core claim

The central discovery is that the Holst term, although it drops out of the equations of motion, contributes to the Noether-Wald charge and hence to the conserved charges measured at infinity. In first-order variables, the combined action produces the generalized Komar charge $K[k] = (1/16\pi G)(-\star d\hat{k} + \alpha d\hat{k})$, whose first term is the standard Komar charge and whose second term is closed by itself. On the Taub-NUT solution, the integral of $d\hat{k}$ over the sphere at infinity defines the NUT charge $N$, and evaluating the generalized Komar integral gives $M = m - \alpha N$. The paper also derives the modified first law $\delta M = T\delta S + \psi_+\delta N_+ + \psi_-\delta N_- + \Phi_\alpha \delta\alpha$, with $\Phi_\alpha = T\, \partial S/\partial\alpha$, while the Smarr formula retains its standard form.

Load-bearing premise

Everything rests on identifying the generalized Komar integral at infinity with the ADM mass; if the physical mass is instead read off from the metric falloff (the parameter $m$), the Holst term only redefines which charge is called mass and no new physical effect remains.

Editorial extensions

If this is right

In first-order gravity with a Holst term, the conserved mass of a Taub-NUT spacetime is $M = m - \alpha N$, so a solution with $m = 0$ is not massless.
The Smarr formula $M/2 - ST - \psi_+N_+ - \psi_-N_- = 0$ keeps its form, but the Misner string strength charges $N_\pm$ acquire $\alpha$-dependent shifts.
The first law acquires a work term $\Phi_\alpha \delta\alpha$ with $\Phi_\alpha = T\, \partial S/\partial\alpha$, so the entropy must be treated as a function of $M$, $N_\pm$, and $\alpha$.
The NUT charge gains a charge-theoretic definition $N = -(1/8\pi)\int_{S^2_\infty} d\hat{k}$, reinforcing its interpretation as a magnetic dual of mass.

Reading between the lines

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

The stated mass shift is convention-dependent: if the ADM mass is defined by the metric's $1/r$ falloff (the parameter $m$), the Holst term changes no physical mass, and $M = m - \alpha N$ is a redefinition of the charge conjugate to time translations rather than a new observable. Which convention is called 'mass' then becomes a choice.
The electric-magnetic analogy suggests a dual statement: for a would-be dual time-translation Killing vector, the roles of $\star d\hat{k}$ and $d\hat{k}$ swap, so the same construction may produce a NUT-charge shift proportional to the ordinary mass. Computing the dual Komar integral of the Holst-modified action would test this.
If the extended first law with $\Phi_\alpha \delta\alpha$ is integrable, the $\alpha$-dependence of the entropy could be verified by a Euclidean path-integral calculation for Taub-NUT with a Holst term, whose on-shell action should reproduce $S(M,N_\pm,\alpha)$.

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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

Quick take: the Noether-Wald derivation of the Holst-modified Komar charge is standard and correct, and the Taub-NUT evaluation giving M = m − αN is genuinely new. The charge modification was already in Refs. [23-25], which the paper cites properly, so the novelty is the explicit evaluation, the shifted Misner string charges, and the Witten-effect interpretation. Those are real additions.

The calculation is clean. Eq. (37) for K follows from the action, the algebra of the α dk term is right, and the sphere/string integrals are handled carefully with the Wu-Yang/Dirak trick. The derivation is self-contained: the NUT charge is defined independently via Eq. (46), so there is no fitting to a desired answer.

Now the soft spots. First, the normalization. Eq. (45) defines M ≡ (1/8π)∫K[k] with K normalized as in Eq. (37). For α = 0, the integral of K gives M/2—as the paper itself says—so the prefactor in Eq. (45) is off by 16π. The final result M = m − αN uses the correct factor implicitly, but the written equation is wrong and needs fixing.

Second, the abstract in the version I was given says the induced mass is +αN, while Eq. (48) gives −αN. The full-text abstract says −αN, so this is likely a version mismatch, but as submitted it is an outright contradiction that must be reconciled.

Third, the load-bearing interpretation. The paper identifies the generalized Komar integral with the ADM mass. But the metric falloff already defines a mass m, independent of α; the Holst term does not change the metric, so the 1/r potential at infinity is still controlled by m. Calling M = m − αN the ADM mass is a charge redefinition, not an observable shift. The analogy to the Witten effect is incomplete: in Yang-Mills, the θ-term shifts the electric flux extracted from the field strength, whereas here the gravitational potential is read from g_tt and is α-independent. The stress-test note makes exactly this point, and I think it holds up. The authors could fix this by presenting M as a generalized or dual charge rather than the ADM mass, or by showing that the canonical Hamiltonian generator of time translations actually picks up the αN term.

Bottom line: this is a competent, serious paper worth engaging with. The Taub-NUT computation is a real contribution. It deserves peer review, but the referee should push for a careful defense of the mass definition and for the two concrete corrections. I would send it back with a request for major revision, not desk-reject.

Referee Report

2 major / 3 minor

Summary. The paper studies first-order (Palatini) gravity with the Einstein–Hilbert action supplemented by a Holst term with Barbero parameter α. Using Noether–Wald methods, it derives a generalized Komar charge K[k] = (1/16πG)(−⋆dk̂ + αdk̂), whose α-dependent piece is identified with a magnetic-type dual of the standard Komar charge. The charge is then evaluated for the Lorentzian Taub–NUT spacetime: with the NUT charge defined by N ≡ −(1/8π)∫_{S∞}dk̂ = n, the integral at infinity is claimed to give the ADM mass M = m − αN. This leads the authors to a gravitational analog of the Witten effect, in which a solution with m=0 has a non-vanishing mass M = −αN. The paper also computes the α-shifts of the Misner string charges and proposes an extended first law with a chemical potential for α.

Significance. If the identification of the generalized Komar integral with the ADM mass is accepted, the paper provides a clean and intriguing gravitational analog of the Witten effect, linking the Barbero–Immirzi parameter to the NUT charge. The Noether–Wald derivation in §3 is explicit and self-contained, the algebra leading to the αdk̂ term is correct, and the string contributions in §4 are carefully computed. The definition of the NUT charge in Eq. (46) is independent of the mass-shift claim, so there is no circularity. However, the central physical conclusion depends entirely on the asserted, but not derived, identification of the generalized Komar integral with the ADM mass; the paper also contains a normalization inconsistency between Eqs. (37) and (45). The significance of the result is therefore conditional on resolving these issues.

major comments (2)

[§4, Eqs. (37) and (45)] The normalization of the mass integral is internally inconsistent. For α=0 and G_N=1, in the Schwarzschild limit of the metric (38) one has ∫_{S∞}(−⋆dk̂)=8πm, so Eq. (37) gives ∫_{S∞}K = m/2. Substituting into Eq. (45), M ≡ (1/8π)∫_{S∞}K, yields M=m/16π, contradicting the value M=m used in Eq. (44a). Since the derivation of Eq. (48) is linear in this prefactor, the claimed M=m−αN inherits the same factor error. To be consistent, Eq. (45) should be M=2∫_{S∞}K if K is normalized as in Eq. (37), or alternatively the charge form in Eq. (37) should carry the standard Komar prefactor and Eq. (45) should read M=∫_{S∞}K. Please correct the prefactors and state the convention explicitly.
[§4, Eq. (45) and the paragraph after Eq. (48)] The identification of the generalized Komar integral with the ADM mass is asserted rather than derived. The ADM mass is normally defined by the 1/r falloff of the metric; for the metric in Eq. (38) that parameter is m and is independent of α, because the Holst term does not alter the equations of motion or the metric. The additional term αdk̂ is closed and non-exact and measures the NUT charge of Eq. (46), so the integral in Eq. (45) is a different observable—a linear combination of m and N. To support the claim of a gravitational Witten effect, the authors should show that the canonical Hamiltonian generator of asymptotic time translations in the first-order theory with the Holst term (including boundary terms) equals the right-hand side of Eq. (45), or should explicitly state that they are defining a new charge and justify why this new charge, rather than the metric falloff parameter m, should be regarded as the physical mass. Without such a derivation or clarification, Eq. (48) is a charge-redefinition rather than a prediction about the physical spacetime.

minor comments (3)

[Abstract] The sign of the mass shift is inconsistent: the submitted abstract states that a non-vanishing NUT charge induces a mass αN, while the full-text abstract and Eq. (48) give −αN. Please harmonize these.
[Throughout] There are several typographical errors: footnote 2 has “Host term” instead of “Holst term,” footnote 12 has “teh” instead of “the,” §5 has “sourced bow” instead of “sourced by,” and footnote 16 has “Φαδα” instead of “Φ_α δα.”
[§4, after Eq. (38)] The text sets G_N=1 after Eq. (38), but Eq. (37) still displays G_N^{(4)}. Please state clearly that all subsequent formulas with explicit prefactors use G_N=1 and re-check the displayed factors accordingly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the generalized Komar charge is derived from the action, the NUT charge integral is independent, and the mass shift is a computation rather than a fitted prediction.

full rationale

The derivation chain is self-contained: Eq. (13) defines the action, and the Noether-Wald construction in Sec. 3 yields K[k] in Eq. (37) directly from the presymplectic potential and the Lorentz parameter P^k. The NUT charge is defined independently by Eq. (46) as the integral of dk, and for the Taub-NUT metric that integral equals n (Eq. (47)); the alpha term in K then contributes alpha times that integral, giving Eq. (48). No parameter is fitted to the predicted quantity, and no result is imported from the authors' prior work to force the conclusion. The main caveat is physical rather than circular: Eq. (45) identifies the ADM mass with the generalized Komar integral, whereas the metric falloff parameter m gives the conventional ADM mass; if the Hamiltonian charge is used instead, the Holst term need not shift the mass. This is an assumption about charge identification, not a circular derivation. Self-citations (Refs. [16], [21], [36]) are used only for standard techniques or discussion and are not load-bearing; moreover, the necessary steps are shown in the paper itself. The normalization of Eq. (45) relative to Eq. (37) also appears inconsistent at alpha = 0, but that is a correctness issue, not circularity.

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

The derivation uses no fitted parameters: α is a coupling constant of the modified action, and n is an integration constant of the Taub-NUT solution. The main assumptions are the validity of the Noether-Wald charge formalism for first-order gravity and the definition of mass via the generalized Komar integral rather than metric falloff.

assumptions (4)

standard math Noether-Wald charge formalism with the presymplectic potential Θ of Eq. (17) provides the correct conserved charge for first-order gravity.
Used to derive Eq. (29) and Eq. (37); standard technique, see Refs. [16,21].
domain assumption The Komar integral at infinity defines the ADM mass, Eq. (45).
This is the load-bearing interpretive step; 'ADM mass' here is the generalized Komar charge, not the metric-falloff mass m.
standard math The Killing vector k=∂t leaves the torsion invariant, making δ_k ω^ab=0 on-shell.
Required for the on-shell closure of the generalized Komar charge and the Killing symmetry of all fields (Appendix).
domain assumption The Holst term does not affect the equations of motion but contributes to the symplectic structure.
Established in the first-order formalism and used in Section 3 to justify that only charges change.

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Pith. "Pith review of The Komar charge in presence of the Holst term and the gravitational Witten effect." pith.science (2026). https://pith.science/paper/HZFDY6GD

@misc{pith2026250615904,
  author       = {Pith},
  title        = {Pith review of: The Komar charge in presence of the Holst term and the gravitational Witten effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZFDY6GD}},
  note         = {Machine review of arXiv:2506.15904}
}

abstract

In the first-order formalism, the Einstein--Hilbert action can be modified by the addition of a Holst term multiplied by the Barbero parameter $\alpha$. This modification breaks parity although it does not affect the equations of motion. We show that the standard Komar charge is also modified by the addition of a topological term multiplied by the Barbero parameter $\alpha$. For the Killing vector that generates time translations, the value of the Komar integral at infinity is modified by the addition of a term proportional to the NUT charge $N$ and the parity-breaking Barbero parameter. Thus, as in the standard Witten effect, a non-vanishing NUT charge $N$ induces a non-vanishing mass $\alpha N$.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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