Resonance of Gravitational Axion-like Particles
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper contains basic errors and should not be published.
The first two sections of the manuscript merely seem sloppily put together. For example: eq. (4) is very obviously missing a factor of 1/2. There is another unrelated missing factor of 1/2 in the ALP kinetic term in eq. (8). Besides fixing these, the authors should carefully go through all their calcualtions to make sure that every constant is correct, since these factors could show up in many different places, and the authors need to take greater care that these are handled correctly. These factors could show up in many calculations, spoiling any kind of quantitative results the authors might aim to produce. Another example of poor preparation is that the mention decay of the ALP to two gravitons, is not associated with any of the key works on this topic, which were completed not that long after the discovery of the chiral anomaly in pion decay. Moreover, contrary to what is stated following eq. (11), only the inner coordinate singularity of the Kerr spacetime is an event horizon. It is possible for particles to dip into the intermediate region (the ergosphere) and still espcate to spatial infinity, so this is not an event horizon. Notation is sloppy also; the one-stroke φ character is used for differnt quantities in eq. (8) and in the line following eq. (16).
However, all these deficiencies, although significant, are minor compared to the basic errors that the manuscript makes in interpreting the Pontryagin density. Since the analysis of this density and its affects on ALP physics are basic to the manuscript, there does not appear to be anything salvageable for potential later publication. Starting from section II.A, the whole analysis falls apart. I have no idea what the first paragraph of section II.A means. Neither the claim that the Pontryagin density is incomplete, nor the claim that there should be turbulence (in a vacuum solution) makes sense to me. Yet those issues are not as nonsensical as arbitrarily changing the Pontryagin density to include a temporal δ-function, with no justification whatsoever (and contrary to the geometrical meaning of this quantity.) The use of this δ-function certainly simplifies the analysis in the later sections, but since it is wrong to begin with, the calculations in those later sections are of no scientific value.
Author Response
{\bf Answer to Referee 1}
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{\bf 1)} {\it The first two sections of the manuscript merely seem sloppily put together. For example: eq. (4) is
very obviously missing a factor of 1/2. There is another unrelated missing factor of 1/2 in the ALP
kinetic term in eq. (8). Besides fixing these, the authors should carefully go through all their
calcualtions to make sure that every constant is correct, since these factors could show up in many
different places, and the authors need to take greater care that these are handled correctly. These
factors could show up in many calculations, spoiling any kind of quantitative results the authors might
aim to produce.}
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{\bf A:} The referee is concerned about the lack of $1/2$ factors in equations (4) and (8). However, it is important to emphasize that these formulas primarily serve as motivation and do not impact our findings, as we rely on the correct equations of motion. While it is reasonable to suggest addressing this issue, asserting that these factors are crucial to the overall calculations is not accurate at all. Furthermore, our calculations' accuracy results from several months of intensive work, including developing both algebraic and numerical methodologies.
%These methodologies are available for an objective review by the referee, should it be deemed necessary.
\pagebreak
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{\bf 2)} {\it Another example of poor preparation is that the mention decay of the ALP to two
gravitons, is not associated with any of the key works on this topic, which were completed not that
long after the discovery of the chiral anomaly in pion decay.}
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{\bf A:} The referee miss the point because we say in our paper (verbatim):
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{\it{
At first sight, the system above retains many properties of the
conventional axion but also differs substantially because when
coupled to gravity, it becomes dynamically a very different system,
These gravitational axions will be denoted generically as ALP.
Additionally, the coupling is physically well-motivated
[9] by the gravitational anomaly and, in analogy with the
chiral anomaly where $\pi_0 \to 2 \gamma$, [10,11] we
might also expect the decay $\varphi \to 2 g$, where $g$ are
gravitons. }}
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The references we cite are primarily the standard ones, including the work of Steve Adler, who previously collaborated with us on a similar problem, as well as the paper by Bell and Jackiw and the article by L. Alvarez-Gaumé and E. Witten. This section of the paper serves as motivation, particularly concerning the gravitational anomaly. It is important to note that we are aware there is no axion-like particle (ALP) decay process involving gravitons, given that a fully developed quantum theory of gravitation does not yet exist.
%The references we cite are, of course, the standard ones (Steve Adler’s work, who was our collaborator in
%the past on a problem similar to the one discussed here), the paper by Bell and Jackiw, and, of course,
%the article by L. Alvarez-Gaumé and E. Witten. Again, this part of the paper is a motivation
%(related to the gravitational anomaly), and, of course, we know that there is no ALP decay process
%involving gravitons... because a quantum theory of gravitation does not yet exist!.
It appears that the referee overlooked the statement, ``{\it These gravitational axions will be denoted generically as ALP} '' { which was explicitly stated in the paper}.
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{\bf 3)} {\it Moreover, contrary to what is stated
following eq. (11), only the inner coordinate singularity of the Kerr spacetime is an event horizon. It
is possible for particles to dip into the intermediate region (the ergosphere) and still espcate to spatial
infinity, so this is not an event horizon.}
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{\bf A:} This is a typo on our part; instead of saying that $r_{\pm}$ is a horizon, we should have written $r_+$. We have, of course, corrected this.
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{\bf 4)} {\it However, all these deficiencies, although significant, are minor compared to the basic errors that the
manuscript makes in interpreting the Pontryagin density. Since the analysis of this density and its
affects on ALP physics are basic to the manuscript, there does not appear to be anything salvageable
for potential later publication. Starting from section II.A, the whole analysis falls apart. I have no idea
what the first paragraph of section II.A means.Neither the claim that the Pontryagin density is
incomplete, nor the claim that there should be turbulence (in a vacuum solution) makes sense to
me. Yet those issues are not as nonsensical as arbitrarily changing the Pontryagin density to include
a temporal delta-function, with no justification whatsoever (and contrary to the geometrical meaning of
this quantity.) The use of this delta-function certainly simplifies the analysis in the later sections, but
since it is wrong to begin with, the calculations in those later sections are of no scientific value.}
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{\bf A:} If the manifold is orientable and the integration is well-defined, the modification is entirely consistent (this substitution is valid if the Dirac delta does not introduce singularities, which is indeed the case). In conclusion, the referee’s objection is incorrect.
The comment about turbulence is frankly ill-intentioned (or the referee didn’t read the article). The statement is
\red{Although we will not address the turbulence problem, we would like to
point out that the analog of quantized circulation is
\bb
\int d^4 x ~ \sqrt{-g} ~R^{\mu \rho \sigma} {\tilde R}_{\mu \nu \rho \sigma} = n,
\ee
where $n=0,\pm 1,\pm 2, \cdots$ what is is a standard theorem in geometry}.
We think it’s not even worth attempting to respond to this last point
Reviewer 2 Report
Comments and Suggestions for AuthorsThe work appears interesting in several parts. However, I've reservations that the authors might address before proceeding with resubmission.
1) The authors started with a given Lagrangian for axions. However, more details are needful concerning this Lagrangian, i.e., the authors have to motivate why the electromagnetic field is coupled in the way they did and which connections occur in the formalism of magnetogenesis, here fully-ignored.
2) The symmetry adopted seems to be cylindrical, why the spectral properties that are found seem computed with spherical symmetry? Explanations are required.
3) The results appear quite in line with previous literature, to be honest. The authors have to stress which novelties are found from their approach with respect to current literature.
4) According to point no. 1 and 3, in fact, the work appears lack of references.
5) The manuscript can be presented better, i.e., its form is not nice enough.
After these changes, I need to resee the manuscript again.
Comments on the Quality of English LanguageEnglish and style might be refined.
Author Response
{\bf 1)} {\it The authors started with a given Lagrangian for axions. However, more details are needful
concerning this Lagrangian, i.e., the authors have to motivate why the electromagnetic field is coupled
in the way they did and which connections occur in the formalism of magnetogenesis, here fullyignored.}
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{\bf A:} We do not fully understand the question posed by the referee, and we believe he is referring to equations (1) and (2) in the manuscript, which were introduced \textit{mutatis mutandis} without explicitly motivating the idea. The paper aims to identify the analog of \( F^{\mu \nu} \) with \( R^{\mu \nu \rho \sigma} \), as well as with the dual tensors. This identification is indeed more elaborate, and there are other terms that we have not included (this pertains to an effective Lagrangian). In the case of magnetogenesis, additional issues are present, know (and now include references in this version) that there are several interesting works in this direction, for example in V.~Domcke, B.~von Harling, E.~Morgante, and K.~Mukaida, \textit{Baryogenesis from Axion Inflation}, JCAP \textbf{10} (2019), 032. This reference and others have been added to the text. We hope this addresses the referee's question.
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{\bf 2)}{ \it The symmetry adopted seems to be cylindrical, why the spectral properties that are found seem
computed with spherical symmetry? Explanations are required.}
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{\bf A:} The problem is written in spherical coordinates (or Boyer-Lindquist coordinates), which are more useful for addressing the dynamics of Kerr black holes. Naturally, cylindrical coordinates can also be used (and sometimes are), but this is uncommon, and the community in this area generally considers Boyer-Lindquist coordinates and spherical symmetry to be the most appropriate. In summary, it is a technical matter of convenience.
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{\bf 3)}{ \it The results appear quite in line with previous literature, to be honest. The authors have to stress
which novelties are found from their approach with respect to current literature.}
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{\bf A:} The Klein-Gordon equation in a Kerr background was studied in the early 1970s, primarily by the school of Kip Thorne, but developments in this area stalled for many years (up until just before the discovery of gravitational waves). Subsequently, the topic has started to develop again, but mainly in the case of the free equation—except for the important works by Detweiler, where the resonance in the theory was never fully discussed (even in Detweiler’s paper, this aspect is not entirely clear). Furthermore, when Pontryagin-type sources are present (and the Teukolsky equation applies), resonance and gravitational radiation, in our opinion, have not been previously discussed, and our paper is a contribution in this direction.
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{\bf 4)}{\it According to point no. 1 and 3, in fact, the work appears lack of references.}
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{\bf A:} We have corrected this issue and added many new references.
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{\bf 5)} {\it The manuscript can be presented better, i.e., its form is not nice enough.}
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{\bf A:} We have revised and improved the manuscript following this recommendation.
Reviewer 3 Report
Comments and Suggestions for AuthorsAuthors study asymptotics of the massive scalar field equations on the Kerr background assuming, in addition, the non minimal interaction with gravity through a Pontryagin (R^2) term. The work is written on the subject of current interest, and can be published in the present form. Authors should verify/improve their equations in the final version. For instance, the variational problem (1) do not implies the equations (2); in the variational problem (8) I do not see an interaction of scalar field with the Pontryagin term, and so on. I should also note that in the particle limit of scalar field theory, the resulting axion particle could exceed the speed of light if it interacts with EM field according to (2), see Sect. VI of the work: arXiv:1511.00645.
Author Response
{\bf 1)}{\it Authors study asymptotics of the massive scalar field equations on the Kerr background assuming,
in addition, the non minimal interaction with gravity through a Pontryagin ($R^2$) term. The work is
written on the subject of current interest, and can be published in the present form. Authors should
verify/improve their equations in the final version. For instance, the variational problem (1) do not
implies the equations (2); in the variational problem (8) I do not see an interaction of scalar field with
the Pontryagin term, and so on. I should also note that in the particle limit of scalar field theory, the
resulting axion particle could exceed the speed of light if it interacts with EM field according to (2),
see Sect. VI of the work:T arXiv:1511.00645.}
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{\bf A: }Thank you very much for the comment; we have explained this more clearly in the new version of the paper, and we have added the reference suggested by him.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe authors have made only minimal changes to the manuscript, not even deigning to correct an admitted error in eq. (4). I recommended rejection of the paper before, because its entire methodology is based on a nonstandard (and frankly, nonsensical) modification of the Pontryagin density. The quantity used in the paper differs from the actual Pontryagin density so fundamentally that there is nothing useful or physically meaningful that can be learned from the manuscript's calculations. Since the paper's results are incorrect, I cannot recommend anything except rejection.
Author Response
We have modified the manuscript to the recommendations of the first referee. The subsection discussing the Pontryagin density has been deleted and replaced by a discussion about the proposed source term.
A factor of 1/2 has been corrected where necessary.
We have also added a $\varphi$ field missing in the original manuscript in equation (8).
We insist that the referee's argument concerning the impact of the numerical factors on the final results is unjustified. The equation (16), the equation of motion analyzed in our work, depends on the factor $\kappa$, which contains all numerical factors. Moreover, this factor $\kappa$ can also be absorbed in a redefinition of the field, which has been done in (25). Finally, our results deal with ratios of the field, which renders the results independent of $\kappa$.
Reviewer 2 Report
Comments and Suggestions for AuthorsI am happy with the changes. I do accept the manuscript.
Comments on the Quality of English LanguageEnglish can be improved.
Author Response
Thanks for your feedback.
