New Perspectives on the Irregular Singular Point of the Wave Equation for a Massive Scalar Field in Schwarzschild Space-Time

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors discussed the massive scalar field equation in Schwarzschild spacetime background. Although this is a calculation, I think it could be useful in several physical situation. Anyway I have some questions. 

The wave inside the horizon, even the scalar wave could fall into the origin if the Schwarzschild spacetime is black hole and the inverse could occur if the spacetime is white hole. It is difficult for me to find such behaviours in IIIA. 

Physically the monodromy could be important. When the radius r goes to infinity, there are two independent solutions corresponding to \epsilon = 1 and \epsilon = -1 as in (6.2). For both cases, what could be the behavior near the horizon? Physically we often choose \epsilon = -1. Then what could be the condition near the horizon to avoid \epsilon = 1 mode?

About the behavior near the horizon, say, in IIIB. Is the behaviour of the wave inside the horizon identical with that outside the horizon? Is there any difference? 

There might be answers in the manuscript but I have not found them. 
I would be happy if the authors could clarify my questions. 

Author Response

In this paper, the authors discussed the massive scalar field equation in Schwarzschild spacetime background. Although this is a calculation, I think it could be useful in several physical situation. Anyway I have some questions.

The wave inside the horizon, even the scalar wave could fall into the origin if the Schwarzschild spacetime is black hole and the inverse could occur if the spacetime is white hole. It is difficult for me to find such behaviours in IIIA.
Answer: as far as we can see, one cannot obtain such behaviours from our IIIA, now subsection 3.1. However, we have added
a comment: the behaviour at the origin r=0 agrees with the findings by Persides in the massless case. A bounded solution at
the origin exists in our massive model.

Physically the monodromy could be important. When the radius r goes to infinity, there are two independent solutions corresponding to \epsilon = 1 and \epsilon = -1 as in (6.2). For both cases, what could be the behavior near the horizon? Physically we often choose \epsilon = -1. Then what could be the condition near the horizon to avoid \epsilon = 1 mode?
Answer: we have added a footnote in Sec. VI. Our local solution therein only holds at large r, and hence cannot have
implications for the physics near the horizon at r=2M.

About the behavior near the horizon, say, in IIIB. Is the behaviour of the wave inside the horizon identical with that outside the horizon? Is there any difference?
Answer: it is an open problem how to relate behaviours of the wave in the left and right neighbourhoods of r=2M.
However, we have added a remark at the end of subsection 3.2: our limiting behaviour at r=2M agrees with the result by
Persides in the massless case.

There might be answers in the manuscript but I have not found them.
I would be happy if the authors could clarify my questions.

Reviewer 2 Report

Comments and Suggestions for Authors

Comments are attached.

Comments for author File: Comments.pdf

Author Response

-We have addressed your comments from 1 to 5 in items from (i) to (v) in
Section IX. Each of your comments might lead to a separate paper in order
to find a complete answer. Our discussion reflects the best that we
can do at present.
-We have improved the style and amended the misprints that you have discovered.
-The sign in the Klein-Gordon equation is consistent with our signature of the metric.
-We write "neighbourhood" because we use British English.
-We have cited also very recent papers, e.g., Refs. 20, 21, 25, but our original
results at large r are new and are more directly inspired by the Poincare' work
in Refs. 22, 23. We do not think that this is a drawback.

Reviewer 3 Report

Comments and Suggestions for Authors

- The language in the manuscript should be formal, without historical indicators (para 1 of introduction).

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Review Report on Ms. Symmetry-3663053

Title: New Perspectives on the Wave Equation for a Massive Scalar Field in Schwarzschild Spacetime

 

The authors worked out the solutions of spherical wave equation using Fourier modes.  They also presented the integral representation of massless scalar field and associated solutions.

The subject under discussion is traditional and has literature concentration. Unfortunately, the authors were not very successful to show the significance of this work due to several shortcomings and a lack of formal write up. Perhaps it is one of the initial manuscripts from the authors. The authors should cover the following lacks in their text:

- The title should be precise to the topic?

- The abstract should be revised keeping all the four portions in view including introduction, problem statement, methodology, and results.

- Though the authors worked on three dimensional spherical solutions of wavs, they have not mentioned it anywhere in the text.

- The language in the manuscript should be formal, without historical indicators (para 1 of introduction).

-Introduction should contain necessary introduction of the topic, must have the problem statement (not found anywhere), significance of the problem, and the aims and objectives.

- A massive literature review should be included. There are only nine related references which are insufficient for the topic under discussion.

- Several equations require references (examples are Eqs.(2.1), (2.3), and many more).

- The equations for massive source are not visible. How that massive source effect the spherical waves?

- What are C_{lm}, Y_{lm}, \beta_{l\omega\mu}, J_{l\omega\mu}, \alpha, l, m, \omega, and \mu etc. Terminology and mathematical expressions should be mentioned.

- Differentiate between \beta_{l\omega\mu} and \beta_0?

- What are Heun equations? Why are they important? Why the authors are discussing these equations?

- The authors did not explained any of the Figures.

- A portion containing the discussion of the results in necessary for the manuscript.

- Conclusion do not contains any of the results, discussion of the results, comparison of the waves without and with massive source.

-Massive oscillations in the graphs (Figs. 9 and 10) should be explained. Why these oscillations are very high?

-Section headings should be much formal.

-Open problems should be separated from the conclusions.

The addition of this information will make the manuscript more interesting for the symmetry readership.

Author Response

The authors worked out the solutions of spherical wave equation using Fourier modes.  They also presented the integral representation of massless scalar field and associated solutions.

The subject under discussion is traditional and has literature concentration. Unfortunately, the authors were not very successful to show the significance of this work due to several shortcomings and a lack of formal write up. Perhaps it is one of the initial manuscripts from the authors. The authors should cover the following lacks in their text:

- The title should be precise to the topic?
Answer: the title is now slightly longer and describes more accurately the original contribution of our paper.

- The abstract should be revised keeping all the four portions in view including introduction, problem statement, methodology, and results.
Answer: the abstract is now longer and describes the full content of our paper.

- Though the authors worked on three dimensional spherical solutions of wavs, they have not mentioned it anywhere in the text.
Answer: the ranges of our spherical coordinates and the expression of the metric are now clearer.

- The language in the manuscript should be formal, without historical indicators (para 1 of introduction).
Answer: we have omitted an explicit mention of Einstein, Persides and Wardell in the first paragraph of the introduction.

-Introduction should contain necessary introduction of the topic, must have the problem statement (not found anywhere), significance of the problem, and the aims and objectives.
Answer: the introduction contains a new paragraph with general considerations on partial differential equations in
physics and mathematics. The subsequent account of the contents is even more accurate.

- A massive literature review should be included. There are only nine related references which are insufficient for the topic under discussion.
Answer: we now cite an important paper by Cohen and Wald, our Ref. 15. However, since our paper is a regular paper,
we have focused on the papers truly used in our investigation. A review would have deserved instead hundreds of citations.

- Several equations require references (examples are Eqs.(2.1), (2.3), and many more).
Answer: we have added a reference before Eq. (3), but it would have been too hard to cite all authors
who work on the equations we need.

- The equations for massive source are not visible. How that massive source effect the spherical waves?
Answer: At the beginning of Sec. I we now stress that we are studying neither the coupled system
of Einstein and scalar-field equations nor the scalar self force model with a source.

- What are C_{lm}, Y_{lm}, \beta_{l\omega\mu}, J_{l\omega\mu}, \alpha, l, m, \omega, and \mu etc. Terminology and mathematical expressions should be mentioned.
Answer: we have described more clearly the meaning of our mathematical symbols.

- Differentiate between \beta_{l\omega\mu} and \beta_0?
Answer: Yes, \beta_{0} does not depend on l, \omega, \mu.

- What are Heun equations? Why are they important? Why the authors are discussing these equations?
Answer: we had already Appendix A devoted to Heun equations. Now we refer to it as soon as we talk
about Heun in Sec. IV.

- The authors did not explained any of the Figures.
Answer: Our Figure captions are detailed, but we have now summarized in the main text the
meaning of the varous Figures.

- A portion containing the discussion of the results in necessary for the manuscript.
Answer: our results are listed again in Sec. IX.

- Conclusion do not contains any of the results, discussion of the results, comparison of the waves without and with massive source.
Answer: see previous answer

-Massive oscillations in the graphs (Figs. 9 and 10) should be explained. Why these oscillations are very high?
Answer: the \varphi term in Eq. (54) depends on \omega, as shown by Eq. (51). Since it grows as \omega grows,
the phase of Eq. (54) leads to increasing oscillation in the high-frequency region of the plot.

-Section headings should be much formal.
Answer: as far as we can see, our section headings are formal

-Open problems should be separated from the conclusions.
Answer: our Sec. IX is so brief that it would have been inappropriate to split it
into two separate sections.

 

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript studies the radial wave equations in a Schwarzschild background with the presence of a massive scalar field.

The authors review the established formalism in the first few sections in a logical and clear manner, and present their new finding, an exact local solution in the neighbourhood of r → ∞. Notably, the formalism is extended in the black hole perturbation theory in a Schwarzschild spacetime, which is an important field in the black hole physics. Thus, this manuscript makes a novel contribution to the study of differential equations and black hole physics. I recommend the manuscript to be published in Symmetry.

Author Response

The authors review the established formalism in the first few sections in a logical and clear manner, and present their new finding, an exact local solution in the neighbourhood of r → ∞. Notably, the formalism is extended in the black hole perturbation theory in a Schwarzschild spacetime, which is an important field in the black hole physics. Thus, this manuscript makes a novel contribution to the study of differential equations and black hole physics. I recommend the manuscript to be published in Symmetry.
Answer: we are glad that our work merits this report.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In the reply by the authors, my questions have not been fully answered. But the reasons are fully given and I may agree to the authours. Then I may recommend this paper for publication. 

Reviewer 2 Report

Comments and Suggestions for Authors

The formula (2.1) has not yet been corrected (this might have been an issue during the PDF compilation process).

 

Reviewer 3 Report

Comments and Suggestions for Authors

Review Report on Ms. Symmetry-3663053: Revised Version

Title: New Perspectives on the Wave Equation for a Massive Scalar Field in Schwarzschild Spacetime

 

I recommend this manuscript for publication in its present state.