Kerr Black Holes within a Modified Theory of Gravity

Round 1

Reviewer 1 Report

The article deals with the Kerr black hole in the context of a modified gravity, the Pseudo-Complex General Relativity. The idea is very interesting but of restricted interest. Here are my comments:

1) The introduction should be improved. At least a bibliographic revision about pc-General Relativity should be added. Also the problem that should be solved with such an approach must be clear.

2) The condition (1) means that $F_{\mu\nu}=0$ immediately, this presentation should be clarified.

3) In equation (15) how the coordinates x and y are related?

4) I don’t understand why the anisotropic part of the energy-momentum tensor is related to dark energy. Maybe more details should be given.

5) It seems to me that pc-General Relativity is some kind of Finsler geometry, maybe the authors could make some comments about that.

6) Some minor mistakes, such as “an-isotropic”, require a general revision in the manuscript.

Author Response

Response to the first referee:

The page numbers and references refer all to the new version of the manuscript:

1) Concerning the first main point, we added in the introduction a complete revision of the bibliography of the pc-GR. We added a couple of  phrases at the end of the first paragraph and from the 3rd paragraph on, further 7 paragraphs have been added. The 5th paragraph of those  refer to the motivation of constructing pc-GR and the questions which
we would like to answer. Some discussion of the range of the main parameter $b_n$ is given in the second last paragraph added.

Further justifications and explanations can be found in section 2 (old and new .version)

2) On page 3, eq. (2) and (3), new version, we show that $F_{\mu\nu}$ is not zero, where real magnetic and electric fields (these are the physical ones) are taken.

3) On page 5 to 6, starting in the last paragraph on page 5, comments are given why we can relate the origin of the classical fluid to vacuum fluctuations and why this fluid has to be anisotropic.

5) The metric used is not Finsler. We use in both zero-divisor components a Riemannian metric. We added a corresponding comment in the paragraph, after Eq. (8)

6) We have eliminated several minor mistakes and changed the English in various places.

Additional changes:
a) References: We added new references: 10, 13, 15, 20-26, 45 and 49, i.e.  in total 12 new references. The main part of the references are due to the complete revision of the Introduction.
b) The figures were reordered.

P.s.: We answered separately to all three referees. There, additional information can be found on the changes made. We hope that we addressed al concerns by the referees and that with these changes the  manuscript is acceptable for publication.

Reviewer 2 Report

In this paper the authors study several properties of a Kerr-like black hole in Pseudo-Complex General Relativity. In particular the authors study the shadow for several values of the parameters.

Over all, I think the paper contains new results. Although the justification for this theory and solution is not completely clear to me, I think these results could have some interest for the community.

Author Response

Response to the second referee:

The page numbers and references refer all to the new version
of the manuscript:

We added a justification of the theory in the introduction, were 7 new paragraphs are added. Further justifications and explanations can be found in section 2 (old and new version).
Below Eq. (5), the paragraph was extended, in order to explain the relation of $y^\mu$ to the components of the 4-velocity.
Below eq. (15) the paragraph was substantially augmented with an explanation of the relation of $T_{\mu\nu}$ to the dark energy.

Additional changes:
1) References: We added new references: 10, 13, 15, 20-26, 45 and 49, i.e. in total 12 new references. The main part of the references are due to the complete revision of the Introduction.
b) The figures were reordered.

P.s.: We answered separately to all three referees. There, additional information can be found on the changes made. We hope that we addressed al concerns by the referees and that with these changes the manuscript is acceptable for publication.

Reviewer 3 Report

The paper studies a rotating black hole in the presence of vacuum fluctuations, which are modeled as a dark energy term. They find that the event horizon ceases to exist even for tiny vacuum fluctuations.

The paper would be much more interesting if :

it discussed the validity of the approximations made (preferably in a separate section) use some numbers to give a sense of masses and scales (e.g., what if we could compare results for M87 with that for our galaxy. If the black hole was bigger, are the effects easier to see? or are other parameters more important?) discuss more existing work to put your work into perspective (e.g., Front. Astron. Space Sci. vol 3, page 29, 2016 or arXiv:1501.04250 derive closed form solutions for the trajectory of a particle on the surface of a collapsing star. Would quantum effects matter in your case? could your conclusions have any impact on cosmology as described in arXiv:1201.1298, arXiv:1304.0594, and arXiv:1305.6838?
State clearly what equations were solved to produce each figure, and make the Mathematica notebook available online so that the work is reproducible. in the introduction, summarize clearly what was concluded from the EHT results and in the conclusions state at which point pc-GR could be distinguished from each other. How does your work help? what parameters are constrained?

minor comment - correct: "wht" in conclusions.

Author Response

Response to the third referee:

The page numbers and references refer all to the new version of the manuscript:

First of all, we found it difficult to concentrate in only one section all approximations made. The Introduction was changes substantially (at its own a new section) and the approximations were partially already discussed in the old section 2, which is now enlarged quite a bit. If we would join all in one section, many explanations would be out of
context.

1) We added in the Introduction 5 paragraphs where we discuss the justification of the theory. On page 3, 3rd paragraph, one finds a justification for the pseudo-complex extension (already a part of the old version). Also, on page 6, 2nd paragraph, the jsutification for the ansatz of the dark energy is given.

2) Below Eq. (15) we added a discussion on the relation of $T_{\mu\nu}$ to the dark energy.

3) In the first paragraph on page 2, we added two new references  ([23] and [24]), suggested by the referee. This work was new to us and we found the relation to thermodynamics very interesting and we suggest further work on it and what to do: Repeat the calculations, shown in these references, but with a modified metric suggested by pc-GR. Because up to now we did not consider these type of models for the evolution of the universe, we cannot tell what will be the differences. We mentioned an attempt by us to develop a cosmological model (new refs. [21.22]).

We could not find the papers [23,24] in a regular journal. We would appreciate if the referee could provide us with this information.

4) In the last paragraph on page 7 we added a further reference, suggested by the referee, related to the path of a particle following the surface of a collapsing star. Up to now, we only considered circular orbits around a black hole, i.e., the star already collapsed. One could repeat these calculations with the modified metric, suggested by pc-GR.
Very interesting is also the semi-classical quantum mechanical description of particles. However, in our approach all is classical. Only the presence of a dark energy fluid (treated classically) permits perhaps finding a relation to a full quantum mechanical description. see also page 19, last paragraph in the Conclusions.

5) With respect to the changes induced by a larger mass, we mention at the end of the third paragraph in the introduction, that the structure itself does not depend on the mass of the black hole, absolute numbers yes, they scale with the mass. In the 4th paragraph on page 2 we discuss the range of $b_n$. This parameter can have any value, but for the reasons explained, we restrict $b_n$ from 0 to a maximal value, from which on no event-horizon exists. The value of $n$ can be measured by the position of the dark ring. Because due to the low resolution of the EHT, this structure is washed out, so that up to now we cannot measure $n$. The parameter $b_n$ is not constraint, however the
rotational parameter is, as usual from 0 to 1m_0.

On page 9, second paragraph, the $n$-dependence is discussed and it is mentioned on how one probably can measure it.

6) The theory is completely classical, thus, no real quantum effects are included. This is mentioned, for example, on page 2, 5th paragraph and Page 6, second paragraph. See also the last paragraph added in the Conclusions.

7) With respect to the EHT results, we mention the low resolution at the end of the first paragraph in the Introduction, which does not permit distinguish between GR and pc-GR. See also the discussion on page 19, end of first paragraph.
There, we also discuss what we plan to do in order to see eventual differences. See also the remark at the end of the 3rd paragraph in the Conclusions, concerning GR and the light ring.

8) If differences are found, pc-GR can help to give indications on how to modify GR and on the quantization of gravitation. See also page 6, second paragraph and page 7, third paragraph. See also last paragraph in the Conclusions,

9) In the figures we added at the end comments, related to which equations or programs we use to obtain the figures.

10) We created a page on github:

https://github.com/peterottohess/phase-transitions

where the MATHEMATICA programs can be retrieved and the modified
C++ routine, needed in GYOTO.

11) References: We added new references: 10, 13, 15, 20-26, 45 and 49, i.e. in total 12 new references. The main part of the references are due to the complete revision of the Introduction.

12) The figures were reordered.

13) We revised again the manuscript in order to eliminate misprints and improve the English.

P.s.: We answered separately to all three referees. There, additional information can be found on the changes made. We hope that we addressed al concerns by the referees and that with these changes the manuscript is acceptable for publication.

Round 2

Reviewer 1 Report

In my opinion the authors took care of of the main issues in this manuscript. Now this is in a much better shape and suitable for publication.

Reviewer 3 Report

The manuscript has been revised. I recommend acceptance.