BTZ Black-Bounce to Traversable Wormhole

Round 1

Reviewer 1 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

First of all, we would like to thank the referee for minutely reading our manuscript. The points raised helped to improve the paper. Below we provide the answers, point by point. 

Reviewer point 1)
In the line 56, $f (r) \neq 0$ is correct. 

Answer to point 1)
The referee is correct. The expression was corrected. 

Reviewer point 2) 
In the line 64, Ref. [] is missing. 

Answer to point 2)
The reference was added.

Reviewer point 3) 
Calling $\omega(r)$ in Eq. (11) as a parameter may not be a good idea as it depends on r. 

Answer to point 3)
We have replaced the term "parameter" by "function".

Reviewer point 4) 
In lines 84 and 86 there are errors. 

Answer to point 4)
The errors were corrected.

Reviewer point 5) 
 Coming to Eq. (19) which is the charged version. I believe that this is not the most appropriate form. The author could consider f (r) = −M + (r^2+a 2)/ l^2 − q^2 ln (r^2+a^2/l^2) which is more consistent with the entire notation. Furthermore, the sign of l^2 matters not l as it is in charged BTZ.

Answer to point 5)
The referee is right. We have corrected the ansatz in the f(r) function according to the prescription discussed in section II, which is $r\rightarrow\sqrt{r^2+a^2}$. The results for the charged section were recalculated, yielding slight modifications in comparison with the first version of the manuscript.  

Reviewer point 6) 
Adding the corresponding Penrose diagram will be very helpful to a reader unfamiliar with the model.

Answer to point 6)
The referee is correct about the necessity of discussing the Penrose diagram and causal structure.  However, as we discussed in detail in the paper, there is no need to draw a proper Penrose diagram, since they can be related to cases already studied in the literature. To clarify this, we added, at the end of section II

``Finally, some comments about the causal structure are necessary. First of all, for the black hole case ($a^2<Ml^2$), the Penrose diagram is identical to the one with $a=0$. The reason is that the only influence of $a$ is to change the radius of the horizon.  We also have that all the curvature tensors are the same at $r=0$.  Therefore, the causal structure is the same. The only case worth mentioning is the one-way wormhole ( $a^2=Ml^2$).  In this case, we have that $f=r^2/l^2$.  Curiously, this is exactly equal to the standard BTZ black hole with $M=0$. The Penrose diagram is given in \cite{Banados:1992gq}.  The diagram has a null throat at $r=0$. The only difference with the Schwarzschild case is that the infinity is now timelike and represented by a vertical line.''

After the above changes, we believe that our manuscript deserves publication in Universe.

Reviewer 2 Report

The comments are attached 

Comments for author File: Comments.pdf

Author Response

First of all, we would like to thank the referee for minutely reading our manuscript. The points raised helped to improve the paper. Below we provide the answers, point by point.
 

Reviewer  point 01)
``There are many typoes and grammar mistakes in the abstract and introduction, authors need to improve it.''

Answer to point 01)
We have improved the grammar. 

Reviewer point 02)
Keywords are missing. I think the author must add some keywords.

Answer to point 02)
Keywords were added. 

Reviewer  point 03)
A proper physical motivation is required for Eq. 1. In the current form, this equation has a meaningless role. Further, the Authors need to cite the proper reference for the regulator parameter.

Answer to point 03
The referee is correct and we have changed the beginning of section two. Now it reads:
``The usual solution for the charged BTZ black hole is given by \cite{BTZ}
\begin{eqnarray}\label{BTZ}
ds^2=f(r)dt^2-\frac{1}{f(r)}dr^2-r^2d\phi^2;\;   f(r)=-M+\frac{r^2}{l^2}-2 q^2 \ln \left(\frac{r}{l}\right).
\end{eqnarray}
The parameter $M$ is the black hole mass, $l$ is related to the cosmological constant so that $\Lambda=-\frac{1}{l^2}$ and $q$ is the charge. The above solution has, as in the Schwarzschild case, horizons and a singularity at $r=0$.  Recently, a black-bounce was proposed by Simpson and Visser. It is a way to obtain a solution that interpolates between regular black holes and a wormhole. This is controlled by a regulator parameter $a$, introduced by replacing $r\to \sqrt{r^2+a^2}$\cite{Simpson:2018tsi}. We will construct, here, a BTZ black-bounce in the same way. First, we consider the uncharged case, so that the black-bounce metric is obtained from (\ref{BTZ}) and given by''

Reviewer  point 04)
Traversable wormhole totally depends on the wormhole throat radius, is there a need for any horizon? Any difference and good justification are required in this regard. May authors include a justification before Eq.3.

Answer to point 04)
The referee is correct. The fact is that we can have horizons and this will depend on the value of the parameter $a$. The changes introduced in the answer to point 03 help to clarify this. We also have introduced further clarifications below Eq. (2).  Now it reads

 ``To analyze the horizons of the above solution, we note that the radial null curves are given by 
$$
\frac{dr}{dt}=\pm \left(-M+\frac{r^2+a^2}{l^2}\right)
$$
Notice that if $a^2>l^2 M$ we have $f(r)\neq0\,\,\forall\,r\,\in\,(-\infty, \infty)$ and that $dr/dt\neq 0$.  Thus we identify this region as a (2+1)-dimensions traversable wormhole\cite{Morris:1988cz}. In the case when $a^2=l^2 M$ we have $\lim_{r\rightarrow0}f(r)=0$ and that $\lim_{r\rightarrow0}dr/dt= 0$. This allows us to characterize this case as a one-way wormhole since we have an extremal null throat at $r=0$\cite{Simpson:2018tsi}. Finally, if $a^2<l^2M$ we have $dr/dt= 0$ at the location of the horizons, symmetrically placed at''

Reviewer  point 05)
 I am really confused about the modified mass, is it a total mass? a good explanation is required in the revised version.

Answer to point 05)
The referee is correct.  The expression ``modified mass'' is not correct. What we have is a change in the relation between the mass M and the horizon radius.  To correct this, the explanations below equation (3) were changed to:
``Therefore, the relation between the mass $M$ and the horizon radius is modified to   
\begin{eqnarray}
M=\frac{a^2+r_h^2}{l^2}.
\end{eqnarray}
The above expressions recover the usual BTZ relation when $a\rightarrow 0$.''

Reviewer  point 06)
I think a reference is missing in line 64 after the sentence non-trivial topology.

Answer to point 06)
The reference was added.

Reviewer  point 07)
 There are spelling mistakes in the sentence before Eq. 5.

Answer to point 07)
Spelling corrected. 

Reviewer  point 08)
Strong energy condition and trace energy condition are not calculated in the current analysis. I think the authors need to calculate and include its graphical discussion.

Answer to point 08)
We agree with the referee about the strong energy condition and a discussion was introduced. However,  to analyze four energy conditions seem enougth, as in the standard literature.  Beyond this, the plots with five energy conditions were very confusing. Therefore we did not add the trace energy condition. However, if the referee insists, we can add it to the next version. 

Reviewer  point 09)
 Why authors are using two different notations for tangential pressure components, In Eqs. 9-10, they are using Pl and in the Fig. 2 they are using Pφ. I think throughout the study notations should be the same.

Answer to point 09)
The referee is right and the notation was corrected.

Reviewer  point 10)
Plot legends are missing in Fig. 2 (b), may the author include the plot legends in all Figs.

Answer to point 10)
The referee is correct and the legends were added. 

Reviewer  point 11)
Why authors are using the simple definition of entropy is a good justification.

Answer to point 11)
We use the usual definition. This is used in the original paper of Simpson-Visser and all the literature follows then. We see no reason to do not to consider it. 

Reviewer  point 12)
Is there any physical reason or justification for one way wormhole in the literature?

Answer to point 12)

In the new version of the manuscript, we point to the discussion in the Simpson-Visse paper. 

Reviewer  point 13)
Authors have done orbits analysis, I am really confused about that, they mean by around wormhole throat radius or around wormhole neck, I think the authors need to explain and provide a link with the wormhole
throat radius.

Answer to point 13)
The referee is correct and using neck and throat is confusing. We have changed the text and replaced ``neck'' by ``throat' everywhere. 

Reviewer  point 14)
Is there any physical motivation for charge a wormhole?

Answer to point 14)
The referee is correct. The reference for the charged black-bounce was missing. To correct this we added at the beginning of section VI

``Soon after the Simpson-Visser black-bounce was proposed, the charged version was studied in Ref. \cite{Franzin:2021vnj}. The authors found a solution that interpolates between a regular charged black hole and a charged wormhole.''

Reviewer  point 15)
As authors are working with thermodynamics, how can authors ignore critical points of discussion?

Answer to point 15)
The referee is correct. In fact, by correcting a small error in the regularized metric, we found that phase transitions happen. The new discussions and graphs were added in the sections about thermodynamics. 

Reviewer  point 16)
The main issues with the missing important literature. May the authors cite recently published work on the wormholes and improve their introduction, for example, Fortschr. Phys. 2022, 2200053, Fortschr. Phys. 2021, 2100048, Eur. Phys. J. C (2021) 81:426, Eur. Phys. J. C 81, 426 (2021);

Answer to point 16)

The referee is right and all the references were added. 

After the above changes, we believe that our manuscript deserves publication in Universe.

Reviewer 3 Report

In hits paper the authors describe some properties of the charged and non charged BTZ black bounce

manifolds. The authors found that 

For the uncharged case, the temperature is insensitive to the

bounce parameter. They also found that exotic matter is always needed to support the wormhole solution.

The paper is nicely written and the arguments are well explained.

Before publication I recommend the authors modify the following:

* After Eq. 4 a reference is missing.

* In sec. 3 Energy conditions. I would recommend the authors to use the term “matter” instead of “fluid”. 

After this minor points, I would recommend the paper for publication. 

Author Response

First of all, we would like to thank the referee for reading our manuscript. The points raised helped to improve the paper. Below we provide the answers, point by point. 

 
Reviewer point 01)
* After Eq. 4 a reference is missing.

Answer to point 01)
The reference was added.

Referee point 02)
* In sec. 3 Energy conditions. I would recommend the authors to use the term “matter” instead of “fluid”. 
 

Answer to point 02)
The referee is right. The term fluid was replaced by matter.

After the above changes, we believe that our manuscript deserves publication in Universe.

Round 2

Reviewer 1 Report

The authors have considered my suggestions appropriately. I recommend the paper for publication.

Reviewer 2 Report

The author has addressed my all comments. Now the paper can be accepted for publication in its present form.