Hawking flux of 4D Schwarzschild blackhole with supertransition correction to second-order

Dear Editor and Referee,

Editor, thank you very much for your chief-in-editor work for my manuscript,
and Referee, thank you very much for your review for my manuscript. 

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The points I have modified are in red in the pdf file, "shingo02(red).pdf".
On the other hand, the pdf file, "shingo02.pdf", is the version in which 
all the red points are checked, so "shingo02.pdf" is the final version at
present. 

Here, I would like to note that I colored the points that I have modified
and deleted to red commonly. This is because in the SciPost.sty in TeX,
for some technical reason of TeX, \out is unavailable. Therefore, I was
not able to distinguish between texts I modified and the texts I deleted
in my coloring in red.

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> as a proceeding manuscript I would suggest for a major revision to improve
considerably the English.

As for my English writing, I am terribly sorry but I cannot find many points
in my English writing which should be improved. I have checked my Engslish
taking long time before I submit this manuscript at the first time, already.

However, I have shorten each text in my manuscript as much as possible.
Therefore some texts may look to be a little bit difficult to read (it
should be basically good that texts are short in any case). This is really
true that I have completed my manuscript believing one should be able
to read if only one wants to read. 

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Specifically, I have chiefly modified as follows:

1. blackhole → black hole 

I first dared to write as "blackhole" to shorten the manuscript as much
as possible. But I think papers adopting "blackhole" are very rare; as
the example I can show now is only this, https://arxiv.org/abs/hep-th/0605041.
On the other hand, I think there are many papers adopting "spacetime", 
for example: https://arxiv.org/abs/1602.05197. Therefore, I left as it is
regarding "spacetime".

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2. Deleted a comment to Brown-Henneaux in P.2 as thinking it is meaningless.

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3. Added description for further motivation in my work at the last of Sec.1,
which is "There would ... this study.".

By the way, you say as,
> He shows that within this framework the Hawking flux does not get any
> correction to second-order which is similar to the Hawking temperature
> with no surprise.
But I think one might not be always able to say that the result I obtained
is no surprise.

This is because, surely at the stage where the near-horizon metric is
obtained, as I write in Sec.B.2, the Hawking temperature can be obtained.
However, for the reasons I write in Sec.1, which is "Since ... Sec.5).",
and for the reason no works have analyzed Hawking temperature and flux
to the second-order yet as I write in the last of Sec.1, I think one might
not be always able to say that the result I obtained is trivial until my
analysis is finished.

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3. Improved Sec.3 entirely.

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4. added some new paragraph in Sec., which is "This study ... as above".

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5. Deleted "lastly, I 'd like to  ..." in P.8, because it's meaningless really
at all.

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I hope referee could choose accept for this manuscript.
It is true that I have written this manuscript sincerely.

Best regards,
Shingo Takeuchi