On the Radial Solutions of the Dirac Equation in the Kerr-Newman Black Hole Surrounded by a Cloud of Strings

Round 1

Reviewer 1 Report

The paper discusses the Kerr solution for black holes in the presence of a cloud of strings and the solution of the Dirac equation for a fermonic field around such a black hole. The manuscript is lacking discussions on the physics at hand, both on the motivations and consequences, in particular on the relevance and observability of what’s studied. Moreover, the end of section 3 announces two missing extra sections, which clearly indicate that the draft is unfinished. For those two reasons, I can not recommend the paper for publication and I urge the authors to appropriately revise their manuscript.

In more details:
* Introduction: “The information about the states of a black hole can be completely obtained from the three parameters” Please be more precise and rigorous. The no-hair theorem has assumptions, especially about stationarity and matter content (mostly abelian EM field).

* Introduction: Although pleasant to read, it is not clear what the history lesson on BHs bring as insight. The authors should spend more time justifying the physical relevance of considering a cloud of strings. Is there any hint of such clouds of strings in cosmology and/or astrophysics? Why bosonic strings, which are anomalous, except in 26 space-time dimensions? What’s the physical motivation for the present study?

* Section 2: eqn(3) is Nambu-Goto action for bosonic string worldsheet. M is the inverse string tension.  Using M is a bad notation, conflicting with BH mass. In Eqn (9) on the energy-momentum, are the authors assuming that these are non-interacting strings? Is such an assumption physically reasonable or motivated? There’s a missing step between eqn (10) and eqn (11).

* Section 2.4: What ensures that the Janis-Newman procedure produces here a solution to Einstein equations? Please explain. Is the cloud of strings also rotating with the BH? The authors should provide more physical explanations beside the lists of equations: what are the main measurable physical effects of the presence of a cloud of strings around the BH?

* Section 3: The authors should definitely interpret more their analytical results. Comments (and plots) on the radial solutions should be discussed. Is the effect of the cloud of strings simply a rescaling of the size of the horizon or is there more subtle effects? Are those effects on Spin-1/2 fields possibly observable?

* End of section 3: the paper is clearly unfinished, since the authors announce two more sections to the paper:
“In order to obtain solutions with a better visualization, we shall consider in the next two sections the solutions  of the Dirac equation describing two specific scenarios: the Dirac particle at the vicinity of a slowly rotating charged black hole surrounded by a cloud of a strings; and the case of a  massless Dirac field.”

Author Response

Dear Editor

 

Please find in what follows our answer to the questions and comments raised by the first reviewer, to whom we thank for their careful work and contributions to turn this paper better..

We expect that all questions were answered appropriately.

 

 

Question 1:

 

* Introduction: “The information about the states of a black hole can be completely obtained from the three parameters” Please be more precise and rigorous. The no-hair theorem has assumptions, especially about stationarity and matter content (mostly abelian EM field).

 

Answer 1:

 

We dicided to delete this paragraph.

 

Question 2:

 

* Introduction: Although pleasant to read, it is not clear what the history lesson on BHs bring as insight. The authors should spend more time justifying the physical relevance of considering a cloud of strings. Is there any hint of such clouds of strings in cosmology and/or astrophysics? Why bosonic strings, which are anomalous, except in 26 space-time dimensions? What’s the physical motivation for the present study?

 

Answer 2:

 

We wrote some paragraphs, starting on line 101, page 3 of the old version, explaining the reason to choose the background spacetime under consideration. We cited five interesting papers published recently, taking into account scenarios with cloud of strings.

 

 

Question 3:

 

* Section 2: eqn(3) is Nambu-Goto action for bosonic string worldsheet. M is the inverse string tension.  Using M is a bad notation, conflicting with BH mass. In Eqn (9) on the energy-momentum, are the authors assuming that these are non-interacting strings? Is such an assumption physically reasonable or motivated? There’s a missing step between eqn (10) and eqn (11).

 

Answer 3:

 

We changed the notation in Eqs. (3) to (8) in order to avoid confusion between the string parameter and the black hole mass. We assumed a cloud of non-interacting strings, following what Letelier (Letelier, P.S. Clouds of strings in general relativity. Physical Review D 1979, 20, 1294). We made it clear in the text.

 

Question 4:

 

* Section 2.4: What ensures that the Janis-Newman procedure produces here a solution to Einstein equations? Please explain. Is the cloud of strings also rotating with the BH? The authors should provide more physical explanations beside the lists of equations: what are the main measurable physical effects of the presence of a cloud of strings around the BH?

 

Answer 4:

 

The seed metric used in the method, ccorresponds exactly to the Reissner-Nordström one, with a solid déficit angle. This can be seen by appropriate redefiniton of coordinates as well as of mass and charge, in terms of the parameter which codifies the presence of the cloud of strings( See arXiv:1510.08526[gr-qc]). Thus, we obtained in the new coordinates, the Reissner-Nordström black hole metric, with a solid déficit angle, and we can use the method adopted by Newman et al, J. Math. Phys. 6, 918(1965).

 

 

Question 5:

* Section 3: The authors should definitely interpret more their analytical results. Comments (and plots) on the radial solutions should be discussed. Is the effect of the cloud of strings simply a rescaling of the size of the horizon or is there more subtle effects? Are those effects on Spin-1/2 fields possibly observable?

 

Answer 5:

 

The last paragraph refers to this point.

 

 

Question 6:

 

* End of section 3: the paper is clearly unfinished, since the authors announce two more sections to the paper:

“In order to obtain solutions with a better visualization, we shall consider in the next two sections the solutions  of the Dirac equation describing two specific scenarios: the Dirac particle at the vicinity of a slowly rotating charged black hole surrounded by a cloud of a strings; and the case of a  massless Dirac field.”

 

Answer 6:

 

We apologize by this mistake. It is a wrong reference to a point which was not considered. We delete this paragraph.

 

 

Sincerely yours,

 

The authors

Author Response File: Author Response.pdf

Reviewer 2 Report

Happy new year!

 

Why do you use complex coordinates in Eqs.(58-59)?

 

What is the definition of \bar{rho}* in Eq.(77)?

 

The last paragraph above of the Conclusion is not clear for me, "we shall consider in the next two sections the solutions of the Dirac equation describing two specific scenarios:.." Where are these two sections?

Author Response

 

Dear Editor

 

Please find in what follows our answer to the questions and comments raised by the second reviewer, to whom we thank for their careful work and contributions to turn this paper better.

We expect that all questions were answered appropriately.

 

 

Question 1:

 

Why do you use complex coordinates in Eqs. (58-59)?

 

Answer 1:

 

 The complexification of coordinates is a step of the Newman-Janis algorithm, which is used to obtain a rotating black hole solution from its static counterpart. We made this clear in the text right above Eq. (59).

 

Question2:

 

What is the definition of \bar{rho}* in Eq.(77)?

 

Answer 2:

 

The definition of bar{rho}, which is given by \bar{\rho} = r + i a cos \theta, was included after Eq. (77).

 

Question 3:

The last paragraph above of the Conclusion is not clear for me, "we shall consider in the next two sections the solutions of the Dirac equation describing two specific scenarios:.." Where are these two sections?

 

Answer 3:

 

We apologize for the mistake. The paper has 4(four) sections.

We delete all references to these supposed extra sections.

 

 

Sincerely yours,

 

The authors

Author Response File: Author Response.pdf

Reviewer 3 Report

This paper is trying to investigate the behavior of fermions in non-trivial spacetime backgrounds by solving the Dirac equation under a charged and rotating black hole with a cloud of strings but the constitution of the article is incomplete and it lacks important contents.

Although this paper seems to have 6(or5?) sections according to LL120-125 on P3, the manuscript has only 4 sections. Actually “next two sections” in L400 on P19 cannot be found.

As the authors emphasize the existence of black holes in the Introduction, it is physically significant to evaluate the detection possibility of strings by some kind of observations such as gravitational waves and black hole shadows, with which potential readers would be more concerned than the detailed mathematical calculation process.

Moreover, the reviewer strongly recommends the authors to check the manuscript thoroughly before submission. For example, “sen” in Eq.(32) on P6 must be an apparent typo.

The conclusion is that this paper is inappropriate for publication in the present state.

Author Response

 

Dear Editor

 

Please find in what follows our answer to the questions and comments raised by the third reviewer, to whom we thank for their careful work and contributions to turn this paper better.

We expect that all questions were answered appropriately.

 

 

Question 1:

 

Although this paper seems to have 6(or5?) sections according to LL120-125 on P3, the manuscript has only 4 sections. Actually “next two sections” in L400 on P19 cannot be found.

 

Answer 1:

 

We apologize for the mistake. The paper has 4(four) sections.

We delete all references to these supposed extra sections.

 

Question 2:

 

As the authors emphasize the existence of black holes in the Introduction, it is physically significant to evaluate the detection possibility of strings by some kind of observations such as gravitational waves and black hole shadows, with which potential readers would be more concerned than the detailed mathematical calculation process.

 

Answer 2:

 

The possibility of detecting some effects produced by a cloud of strings, in astrophysics observations, was emphasized in the paragraph of lines 97 to 105. We highlighted the recent studies that analyzed the shadows, accretion and Other phenomena, in backgorunds of black holes with a cloud of strings.

 

Question 3

Moreover, the reviewer strongly recommends the authors to check the manuscript thoroughly before submission. For example, “sen” in Eq.(32) on P6 must be an apparent typo.

 

Answer 3:

 

This typo was corrected. We carefully corrected the English used in the text also.

 

 

Sincerely yours,

 

The authors

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors satisfyingly improved their manuscript according to the reports. The work is sound and I can recommend it for publication.

Reviewer 3 Report

Although the format may not be the reviewer’s concern, functions(ln, sin, …) should be written in Roman type.