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The Komar charge in presence of the Holst term and the gravitational Witten effect
by José Luis V. Cerdeira and Tomás Ortín
Submission summary
| Authors (as registered SciPost users): | Tomás Ortín |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202506_00057v1 (pdf) |
| Date submitted: | June 30, 2025, 10:39 a.m. |
| Submitted by: | Tomás Ortín |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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 $-\alpha N$
Current status:
Reports on this Submission
Report
Unfortunately, I cannot recommend this paper for publication, as I do not believe the level of novelty rises to the level of a SciPost Physics Core paper. There are many papers on the same subject, the modification of the Komar charge from the Holst term, as mentioned by the authors, e.g., Refs. [23-25] and [26-28]. What is the novel insight from this work? To the knowledge of this reviewer, the investigation of Smarr formula from the Komar charge is relatively straightforward. For instance, the Black Hole mechanics of Taub-NUT-AdS solution was studied in 2205.10043 via precisely the same point of view that charges from the Holst term. I would like to see the calculation in the present manuscript applied to a novel problem, in a way that produces some new physical insight.
Recommendation
Reject
Dear Editor,
We would like answer to the second referee, starting with the first line of
his/her report: the goal f the paper is NOT ``to investigate the Komar charge
and the Smarr formula of the Taub–NUT solution from the effect of including
the Holst term.'' We think that the referee has overlooked the title and the
abstract, which refers to the gravitational Witten effect, which is the main
observation of our paper.
As the referee says, and we assume from the onset (actually, providing many
references), the Komar charge of GR with a Holst term has been derived before,
although we use a different formalism. This derivation is necessary to setup
the ensuing discussion and it is not the main result of our paper. It is not
our role here to review the differences with, for instance, the reference
2205.10043 mentioned by the referee, even though some of them are relevant. In
that reference the first law of black hole mechanics is studied in presence of
a Holst term. The referee claims that ``the investigation of the Smarr formula
from the Komar charge is relatively straightforward''. However, finding the
Komar charge in many situations is a complicated problem by itself (see e.g.
2506.14024) and the derivation of the Smarr formula for asymptotically
Taub-NUT spaces has puzzled many researchers for a long time, and it is far
from straightforward, as can be seen in, for instance, in 1903.08668,
1905.03785, 1908.10617 and 2505.15349.
In any case, we just want to address the main question the referee asks ``What
is the novel insight from this work?''
The main novel insight of our paper is that the Holst term gives rise to a
gravitational version of the Witten effect, modifying the very definition of
mass. If, in absence of Holst term, the mass and NUT charges are $M=m$ and
$N=n$, its presence leads to $M=m+\alpha n$ which means that $M$ can be
different from zero even when $m=0$ in presence of a non-vanishing NUT
charge. In 2205.10043, for instance, these two contributions are dealt with
separately and there is an ``electric'' and a ``magnetic'' first law
(eqs. 6.33, 6.24 and 6.35) which are then combined into a single complex
expression (eqs. 6.38 and 6.39) and the gravitational Witten effect we are
proposing here is never mentioned, as it is not mentioned in the referee's
report at all even though it is part of the title and it is explicitly
mentioned in the abstract as the main idea of the paper.
If the referee believes that this observation/insight (that the Holst term
gives rise to a gravitational version of the Witten effect) is not novel,
perhaps the he/she can indicate a reference in which it is
discussed. Otherwise, we believe the main observation made in this paper is
novel enough to be accepted in SciPost Physics Core.
Yours,
Tomas Ortin (as corresponding author)
Report
Dear Editor,
first of all my apologies to the authors for the delay. I have read their paper. It seems to me that the results reported are already known, and for this reason I cannot recommend publication of the paper. But I may be wrong, and thus if the authors would like to appeal, it would be very helpful if they could highlight in a precise way what they are proposing as new results.
Let me explain. After reviewing existing material in sections 2 and 3, the authors arrive at a generalized Komar charge (italic in their text), eq (33). But this charge is already known, e.g. [22-25] as they cite. So it seems that there is no new results so far. The final section 4 contains an example of application of the generalized Komar charge to the NUT solution, and to the study of the Smarr formula for the NUT solution. But also this seems to me a known result, see e.g.
- https://www.sciencedirect.com/science/article/pii/S0370269320300745
- https://arxiv.org/abs/0809.5218
and also comments/references in [11,28,35] cited by them.
On a more technical level, I have an issue also with the authors' description and use of Noether's current, equation (27). While the result (28-29) is correct, the Noether current should be defined without the two E' terms in (27), and should be closed only on-shell, and not off-shell. Otherwise it would be a trivial conservation law, with little to be learned from about the dynamics of the system. In physical terms, energy is conserved on solutions, not on arbitrary trajectories off-shell. In mathematical terms, I would say that (27) as defined is not closed off-shell, contrarily to what stated, because dE' is not identically zero. It is of course zero on-shell, and this is how one proves closure on-shell of the correctly defined Noether current. At least this is the way I understand it, and the way it is explained for instance in the Iyer-Wald '94 paper. If the authors disagree with that, they should point it out, and also provide explicit proofs of the closedness of (27) and of (28-29).
Recommendation
Reject
Anonymous on 2025-09-23 [id 5855]
Dear Editor,
We have written a long, detailed and explicit answer to the referee's objections in the attached file.
Yours,
Tomas Ortin (as corresponding author)
Attachment:
answertoreferee.pdf