Black Hole and Wormhole Solutions in Einstein–Maxwell Scalar Theory

Round 1

Reviewer 1 Report

The authors consider the four-dimensional Einstein-Maxwell theory with massless canonical and phantom scalar fields. The authors obtain the static exact solutions with the spherical symmetry and show that some of these solutions represent the charged black hole geometries which are analogous to the Reissner-Nordström solution. The authors also find that other solutions represent the geometries with wormholes and naked singularities. I find this work interesting but I recommend the authors to consider the following points:

i) The authors claim that the metric (50) can be transformed to the metric (62) with the quantization condition (64) for the sake of the analytic extension across the horizon. However, such an extension would give an infinite class of solutions from a single solution (50). Moreover, while the solution (52) is similar to the solution (50), the metric (70) after the analytic extension of the metric (52) does not have such a quantization condition. Then the authors may explain the derivation of the metric (62) with the condition (64) in detail and discuss why the metric (50) has such rich structures compared to the metric (52).

ii) The authors argue that the black holes described by the metrics (62) and (70) have infinite horizon areas. Does such a divergence depend on the radial coordinate choice? If so, the authors might use another coordinate like the coordinate (32) to obtain finite horizon areas.

iii) The word $q_e$ below the equation (11) would be replaced with the word $Q$.

iv) The word “(12)” below the equation (15) would be replaced with the word “(13)”.

v) The words “I, II and III” below the equation (67) would be replaced with the words “III, II and I”.

vi) The authors may add the specific values of $a$ in the captions of the figures 1, 2 and 7.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

  I have examined  this manuscript, in which the authors  study blackholes (BH) and hypothetical wormholes (WH)  as non-trivial structures of the spacetime and intent to perform some classification.   They examine the specifically the case of the Einstein-Maxwell-Scalar theory.  They rely heavily on the previous work [7] in their list of references.  This work does not contain new results, but  as a review article it can be of some interest. However, there are some issues that in my opinion the authors should address before publication:
 
 - The authors should better clarify the conventions they use. For example, by looking at the form of the metric in different places of the work one would infer that its signature is  (+, -, -, -).  However, when they refer to the action (1) they say that t he parameter \epsilon can assume the values +1, which gives positive kinetic energy for the scalar field, and -1  to represent the “phantom” scalar field with negative kinetic energy.  This is strange because they already factored out a global minus sign in that equation. Therefore in the metric  (+, -, -, -)  the canonical situation with positive energy would correspond to \epsilon=-1, whereas the phantom situation  with negative kinetic energy should be \epsilon=+1.  In my opinion there is a mistake/typo here.    In order for the interpretation of the authors to be correct they should use the opposite metric (-, +, +, +).  But I cannot see it is the case e.g. in (6) and other places in the article.  For this reason it is imperative to clarify the sign conventions that are being used.  They should also indicate the conventions on the curvature tensor.
 
 - This clarification is important since as the authors point out scalar fields may favor the appearance of naked singularities instead of BH, unless it appears as a phantom field. So it is important to which sign of \epsilon a phantom field should be ascribed in this work.
 
 - In this presentation the authors perform the analysis directly in the Einstein frame (1).  This is different from what is done in Ref. [7] where the original presentation is in the Jordan frame and then perform a conformal transformation in the Einstein frame. I don't know if the authors are implying that these two presentations are equivalent.  In general the two frames are not equivalent because the geodesics followed by the particles are different and moreover in one frame there are interactions not existing in the other frame.
 
 - The notation on the right-hand-side of (10) does not correspond to a tensor relation compatible with the left-hand-side.  Why the authors use that notation?
 
 - Taking into account that the authors neglect the effect of any potential one should expect that the conclusions and even the classification of solutions is rather artificial since a simple change of the potential could change all the conclusions.  Is not that so?  In the case of the Brans-Dicke theory formulated in the Jordan frame to study the behaviour of a scalar field makes plenty more sense because there is a nonminimal coupling to gravity, which if viewed in the Einstein frame would generate a nontrivial potential. But starting from the Einstein frame without potential looks pretty artificial if not rather useless since the way back to the Jordan frame does not produce anything of interest, otherwise they should show what is obtained and why is interesting.  The authors should comment on that.  
 
 - If they did not want to introduce a scalar field potential with self-interactions, why they did not contemplate at least the possibility that the field is massive? Why to restrict themselves to the study of massless fields only?   Admitting a mass is the minimum the authors could have considered beyond a simple kinetic term and hence with a stiff equation of state.  In Brans-Dicke theory there is a sense not to introduce it because there is a nonminimal coupling to gravity. But here I cannot see the reason for such an oversimplification of the problem.  They should comment on that.
 
 - Why there is no interaction between the scalar field and the electromagnetic field? The authors speak of scalar charge in their article.  They refer to the charge of the BH but they do not clarify if the scalar field itself is charged.  The fact that there is no factor of 1/2 before the kinetic term of it somehow indicates that it is charged. If it is neutral, then a factor of 1/2 is missing. The authors should clarify this point and their normalizations. This adds up to one of my points above.
 
 - The authors should write "Reissner" correctly, as in some case it is written incorrectly.  They should also check for other typos.
 
 My conclusion is that the authors should provide some competent input to the above questions before the paper can be considered ready for publication in  Universe.
 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The paper is dedicated to classifying and studying charged black holes and wormholes solutions in the Einstein-Maxwell system in presence of a massless scalar field.

In my opinion, work has certain interesting results, however, the presentation and writing style makes them practically invisible now.

I strongly suggest the authors think to reconsider the writing style and mainly simplify it making it possible to follow and understand the novelty of the work. In this regard, I believe that it would be very useful to revise the writing about the goal of the paper. A similar suggestion will be about section 5. 

I suggest also reconsidering the reference list because in my opinion some interesting works have not been cited.

There are some other minor issues to be mentioned as well. One of them - no proper reference to Figure 6. 

It is highly recommended to check if the referring to certain equations and figures, in general, have been appeared properly.

I believe that after an intensive rewriting the paper will get a proper shape and can be recommended to be accepted.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I am satisfied with the corrections i), iii)-vi) except ii). If the black hole horizon area is infinite, the Bekenstein-Hawking entropy $S$ would become infinite. Moreover, since the Hawking temperature $T_H$, the electric charge $Q$ and the scalar charge $C$ would be finite, the black hole mass would become infinite according to the first law of black hole thermodynamics. However, it seems that the geometric masses $m$ in the present solutions would be finite. Is the first law of thermodynamics valid in the present system? Does the geometric mass coincide with the black hole mass defined by the ADM or the Komar formalisms? Then the authors may discuss the thermodynamic relation between the physical quantities $m, T_H, S, Q, C$ and its finiteness.

Author Response

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Author Response File: Author Response.pdf