From Scalar Clouds to Rotating Hairy Black Holes

Round 1

Reviewer 1 Report

The authors examines an exciting problem that discusses solutions for complex-valued scalar fields for the Klein-Gordon in two folds. In the first part, they discuss the solution of the Klein-Gordon equation with complex-valued scalar field with and without electric charge. Moreover, they investigate the effective potential in the presence of scalar clouds.  In the second part, they work in a stationary, axisymmetric, circular  and flat space time.  They used numerical methods and presented virtual results for Kerr and Kerr-Newman black holes. 

The paper is written clearly, but still there are few things can be corrected. 

1) For example in some equations, i.e. (11), (14)... the punctuation needed to be corrected. 

2) Also in several graphs authors can use different patters to avoid lapping in black-white printing. 

3) Another point is author calculated some values up to 8 or 9 meaningful digits, however, the assigned values are not given in this precision. 

4) A description of Komar mass and angular momentum could be useful for new readers. 

After these corrections, the manuscript can be accepted.

Author Response

Please see the attached .pdf file

Author Response File: Author Response.pdf

Reviewer 2 Report

The article “From scalar clouds to rotating hairy black holes” has two main goals:

(i) to analyze bound state configurations (scalar clouds) associated with a massive, electrically charged, test scalar field around a Kerr-Newman black hole and

(ii) to analyze solutions of the Einstein-Klein-Gordon equations that correspond to hairy black holes (i.e. the system scalar cloud + black hole when backreaction is taken into account).

The possibility of rotating hairy black holes (and the study of scalar clouds around rotating black holes) has been extensively studied since the seminal works of Hod (linear analysis, ref. [17]) and Herdeiro, Radu (non-linear analysis, ref. [16]). These 10 year-old articles initiated a new line of investigation in General Relativity and Astrophysics that has produced several important results and, still today, motivates original research.

The article under review is well written and technically sound. However, I am not convinced that the results presented are sufficiently novel to merit publication. More precisely, the existence and the properties of scalar clouds of Kerr-Newman black holes were previously investigated in detail in several references (which are not referenced in the article under review):

[A] https://doi.org/10.1103/PhysRevD.90.024051 [linear analysis, test field]

[B] https://doi.org/10.1103/PhysRevD.90.104024 [linear analysis, test field]

[C] https://doi.org/10.1016/j.physletb.2016.08.032 [non-linear analysis, Einstein-Maxwell-Klein Gordon]

The linear analysis in the paper is presented in Secs. II and III. It is not clear to me, among the results presented in these sections, what is really novel. In particular, it seems that the results for the charged field in the Kerr-Newman spacetime basically reproduce what has been found and shown in detail in refs. [A,B]. The condition (41) has also been found previously, as the authors acknowledge, in ref.[17]. The framework employed do not seem new either, having appeared previously in refs. [19,25,26] by some of the authors. On the other hand, the nonlinear analysis of the article is found in Sec IV, where solutions of the Einstein-Klein-Gordon system that correspond to hairy black hole configurations are presented. It is not clear to me, in this section, what is novel in comparison to the results presented in refs.[16,C] and by the authors in ref. [20].

Recognizing that the topic under investigation is important and interesting to several researchers in the field, and that I may have misunderstood the novelties in the article under review, I would like the authors to explain in detail what are the essential differences in their article in comparison to previous literature [specially the articles mentioned by me in the previous paragraph]. If the authors can successfully argue that their work is sufficiently new, I believe the article should be published in this special edition of Particles.

Besides this main point, I have other (minor) comments for the consideration of the authors:

I) In the case of an extreme Kerr black hole, the Klein-Gordon equation for scalar cloud configurations is a confluent hypergeometric equation (as shown in ref. [17]). However, this is not considered by the authors. What is the advantage of using the approach indicated in Appendix A in such a case of extremal Kerr black holes?

II) In eq. (22) the authors provide the first and second order derivatives of the radial function at the event horizon as boundary conditions. Since the differential equation is second order, isn’t it possible to write the boundary condition in terms of the function itself at the horizon? In other words, if one substitutes eq. (22) into eq.(20), wouldn’t one find R(rH)? Is there a reason for using the second derivative as a boundary condition instead of the function itself? [A similar question applies to eq. (30).]

III) To increase the interest of readers which are relatively new to the topic of scalar clouds, I suggest that the authors mention in the article that:

- Scalar clouds have been discussed in analogue models of gravity (they can, in principle, be observed in the laboratory):

https://doi.org/10.1103/PhysRevD.91.104038

https://doi.org/10.1016/j.physletb.2018.10.030

https://doi.org/10.1103/PhysRevD.103.045004

- Scalar clouds have been analyzed in theories beyond (3+1)-General Relativity. More precisely, scalar clouds were investigated for other rotating black hole metrics besides Kerr and Kerr-Newman. A few examples are:

https://doi.org/10.1088/1361-6382/aa7964

https://doi.org/10.1016/j.physletb.2017.08.017

https://doi.org/10.1140/epjc/s10052-020-8062-z

https://doi.org/10.1103/PhysRevD.106.024046

Author Response

Please see the attached .pdf file

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Dear editors and authors

First of all I would like to mention that I was aware of the general guidelines of the Special Issue (which are published in the website), including the fact that review papers are within the scope of the special issue. Nevertheless, I apologize to the authors and editors if I misunderstood the scope of the special issue. I was not aware of the additional guidelines that were sent to the authors (if these guidelines were public, I never saw them until now). In any case, in a review of the field of expertise of the author(s), I would expect to see a more general and detailed discussion of the field (as opposed to a paper that focuses on recent results obtained by the authors).

Having said all that, I recognize that the new version of the manuscript is much clear in distinguishing what are the novel results and what is review material of previous results published by the authors elsewhere. The explanations provided by the authors about the differences between this work and previous work have convinced me that there is a reasonable amount of interesting new results in the article. The authors' reply concerning the technical questions I had raised are also clear and satisfactory. Therefore I recommend the revised article for publication.