Vaidya and Generalized Vaidya Solutions by Gravitational Decoupling

Round 1

Reviewer 1 Report

The authors apply the gravitational coupling method in relation to black holes. This is applied to the generalized Vaidya spacetime and different forms of the Vaidya spacetime are considered. The nature of singularities are considered in this context. The results are interesting and as far as I am aware this is the first application of the gravitational coupling to null dust and null strings. It can therefore be considered for publication.

There are several grammatical corrections that need to be made by the authors before publication.

Also several interesting papers on the generalized Vaidya spacetime have been published recently. These should be included in the references and discussed in the Introduction. This will help the reader to understand the importance of the Vaidya geometry:

 Faraoni et al, Eur. Phys. J C, 81, 232 (2021)

 Dey and Joshi, Arab. J. Math. 8, 269 (2019)

Brassel et al, Eur. Phys. J. C 82, 359 (2022)

Maharaj and Brassel, Class. Quantum Grav. 38, 195006 (2021)

 Nikolaev and Maharaj, Eur. Phys. J. C 80, 648 (2020)

 

Author Response

First of all, we would like to say thanks to a reviewer for the helpful comments.

We have checked the text of the paper and corrected errors in English.

Also, we have added all references and new text, which we write below:

In the introduction, the following information was added:

It is possible to solve Einstein's field equations for a gravitational source, whose energy-momentum tensor is expressed as (1) by solving Einstein's field equations for each component $\tilde{T}_{ik}$ and $\Theta_{ik}$ separately. Then, by a straightforward superposition of the two solutions, we obtain the complete solution corresponding to the source $T_{ik}$. Since Einstein's field equations are non-linear, the MGD-decoupling represents a novel and useful method in the search and analysis of solutions, especially when we face scenarios beyond trivial cases, such as the interior of stellar systems with gravitational sources more realistic than the ideal perfect fluid, or even when we consider alternative theories, which usually introduce new features difficult to deal with.

The conformal symmetries and embedding properties of the generalized Vaidya metric were studied in [39,45]. Other properties of this spacetimes can be found in [42,44].

In the conclusion we have added the following part:

The Vaidya metric describes a dynamical spacetime instead of a static spacetime as the Schwarzschild or Reissner-Nordstrom metrics do. In the real world, astronomical bodies gain mass when they absorb radiation, and they lose mass when they emit radiation which means that the space-time around them is time-dependent. As we pointed out the Vaidya spacetime can be used as the simplest model of gravitational collapse. New solutions by the gravitational decoupling method allow us to investigate the question of how an additional matter field will affect the gravitational collapse process. When we consider the gravitational collapse of Vaidya spacetimes, one might expect the naked singularity to form. New solutions can tell us how $\Theta_{ik}$ will influence the result of the gravitational collapse. Vaidya spacetimes are widely used nowadays and the important question of the global structure of new solutions is the direction of future research. We have already told that $\Theta_{ik}$ can be thought as the energy-momentum tensor of a dark matter. So the obtained solution can tell us how the well-known properties of the Vaidya spacetimes change when an additional matter field present. These properties should be also studied in the future.

42.Faraoni et al, Vaidya geometries and scalar fields with null gradients Eur.  Phys.  J C, 81, 232 (2021)

43.Dey and Joshi, Gravitational collapse of baryonic and dark matter Arab. J. Math. 8, 269 (2019)

44.Brassel et al, Charged radiation collapse in Einstein–Gauss–Bonnet gravity Eur. Phys. J. C 82 359 (2022)

45.Nikolaev and Maharaj, Embedding with Vaidya geometry Eur. Phys. J. C 80, 648 (2020)

46.Maharaj and Brassel, Junction conditions for composite matter in higher dimensions Class Quantum Grav. 38, 195006 (2021)

 

Reviewer 2 Report

This paper generalizes the previously known eternal hairy black hole solutions to the dynamical case where the black hole mass M depends on the spacetime coordinates. This work is along the lines toward understanding the important questions of hairy black hole formations and possibly also the naked singularity formation.

The paper is overall clear in terms of its presentation except that there are many minor grammar mistakes that the authors might want to fix. Considering that the technical results obtained in this paper might be useful for future studies of black holes, I think there is merit for this paper to be published in the journal of Universe if the following minor remarks can be addressed:

1. The authors discussed the locations of the singularities around the equations (29), (37) and (47). In what sense that the singularity is at r=0? Is it because the Kretschmann scalar has a 1/r^6 pole? But this is not immediately obvious as the numerator has complicated r-dependence. Moreover, is there a case where one has a naked singularity? I think it is important that the authors clarify these.

2. In the conclusions, the authors commented on future investigations of "the naked singularity formation, gravitational collapse, and global structure of these spacetimes." All these sound very interesting, but their direct connections to the work in this paper seem to be slim. I think the physical significance of this paper can be improved if the authors are able to comment further, at least conceptually, how the results in this paper can provide new insights for these topics. 

Author Response

First of all, we would like to say thanks to a reviewer for the helpful comments.

We have checked the paper for English mistakes and corrected them.

1. The structure of the singularity is the question of future research. In our work, we wanted to show that a new singularity, except for $r=0$, doesn't appear in these solutions. Also, we have added a reference for the recently published paper in Universe 'Near horizon thermodynamics of hairy black holes from gravitational decoupling' [47]. In this paper the structure of the Schwarzschild spacetime in these coordinates has been discussed but for $M=const$ which is valid in our case only if we consider the usual Vaidya spacetime which doesn't depend upon $r$ coordinate.

47.R. T. Cavalcanti, K. dos S. Alves, J. M. Hoff da Silva Near horizon thermodynamics of hairy black holes from gravitational decoupling Universe 2022, 8(7), 363

2. We have added some information in the conclusion. I am blind and can't mark it in pdf, so I write here what we have changed:

First of all, we have added extra references and the following information:

In the introduction, the following information was added:

It is possible to solve Einstein's field equations for a gravitational source, whose energy-momentum tensor is expressed as (1) by solving Einstein's field equations for each component $\tilde{T}_{ik}$ and $\Theta_{ik}$ separately. Then, by a straightforward superposition of the two solutions, we obtain the complete solution corresponding to the source $T_{ik}$. Since Einstein's field equations are non-linear, the MGD-decoupling represents a novel and useful method in the search and analysis of solutions, especially when we face scenarios beyond trivial cases, such as the interior of stellar systems with gravitational sources more realistic than the ideal perfect fluid, or even when we consider alternative theories, which usually introduce new features difficult to deal with.

The conformal symmetries and embedding properties of the generalized Vaidya metric were studied in [39,45]. Other properties of this spacetimes can be found in [42,44].

In the conclusion we have added the following part:

The Vaidya metric describes a dynamical spacetime instead of a static spacetime as the Schwarzschild or Reissner-Nordstrom metrics do. In the real world, astronomical bodies gain mass when they absorb radiation, and they lose mass when they emit radiation which means that the space-time around them is time-dependent. As we pointed out the Vaidya spacetime can be used as the simplest model of gravitational collapse. New solutions by the gravitational decoupling method allow us to investigate the question of how an additional matter field will affect the gravitational collapse process. When we consider the gravitational collapse of Vaidya spacetimes, one might expect the naked singularity to form. New solutions can tell us how $\Theta_{ik}$ will influence the result of the gravitational collapse. Vaidya spacetimes are widely used nowadays and the important question of the global structure of new solutions is the direction of future research. We have already told that $\Theta_{ik}$ can be thought as the energy-momentum tensor of a dark matter. So the obtained solution can tell us how the well-known properties of the Vaidya spacetimes change when an additional matter field present. These properties should be also studied in the future.

Reviewer 3 Report

Dear Authors,

 

I have carefully read your paper entitled  “Vaidya and generalized Vaidya solutions by gravitational decoupling”. The paper aims at determining different black hole solutions involving the Vaidya spacetime. The work is potentially interesting, however several important explanations are missing, like: in the Introduction the arguments are not clearly presented (e.g., gravitational decoupling method, Vaidya spacetime, hairy Schwarzschild black hole) and are also not clear which are the motivations behind your work; in the central part of the paper, where the black hole solutions are determined, there are steps and calculations that are not justified; there are also missing discussions on the obtained solutions, possible comparisons via plots or other estimations; the conclusions also does not clearly argument the possible implications and future perspective of the article. For these reasons, I would suggest to the authors to reorganize the paper and make it more consistent, because in the present status is complicate to be read and has also low scientific impact, in my opinion.

Author Response

First of all, we would like to say thanks to a reviewer for the helpful comments.

We have checked the text of the paper and corrected English mistakes.

In the introduction, we have added new information regarding the gravitational decoupling method. The main goal of our article was to obtain new solutions by the gravitational decoupling method that describes the dynamical hairy black holes. All hairy black hole solutions, which have been obtained so far, describe the static and stationary black holes and we are interested in dynamical solutions in order to apply them to the gravitational collapse problem in future investigations.

In the central and conclusion parts, we have proved that we can obtain the dynamical solutions by the gravitationally decoupling method. We show only two things regarding black hole solutions i.e. the location of the apparent horizon and that there are no any extra singularities except for $r=0$. The structure of the obtained solutions is the question of future research because we must realize the global structure of obtained solutions and how a new charge affects the famous properties in usual Vaidya spacetimes.  Some extra information has been added in the conclusion regarding this question. 

We have added new information. Unfortunately, I am blind and can't mark all changes in pdf, so I write about them here:

We have added 6 extra references and the following information:

In the introduction, the following information was added:

It is possible to solve Einstein's field equations for a gravitational source, whose energy-momentum tensor is expressed as (1) by solving Einstein's field equations for each component $\tilde{T}_{ik}$ and $\Theta_{ik}$ separately. Then, by a straightforward superposition of the two solutions, we obtain the complete solution corresponding to the source $T_{ik}$. Since Einstein's field equations are non-linear, the MGD-decoupling represents a novel and useful method in the search and analysis of solutions, especially when we face scenarios beyond trivial cases, such as the interior of stellar systems with gravitational sources more realistic than the ideal perfect fluid, or even when we consider alternative theories, which usually introduce new features difficult to deal with.

The conformal symmetries and embedding properties of the generalized Vaidya metric were studied in [39,45]. Other properties of this spacetimes can be found in [42,44].

In the conclusion we have added the following part:

The Vaidya metric describes a dynamical spacetime instead of a static spacetime as the Schwarzschild or Reissner-Nordstrom metrics do. In the real world, astronomical bodies gain mass when they absorb radiation, and they lose mass when they emit radiation which means that the space-time around them is time-dependent. As we pointed out the Vaidya spacetime can be used as the simplest model of gravitational collapse. New solutions by the gravitational decoupling method allow us to investigate the question of how an additional matter field will affect the gravitational collapse process. When we consider the gravitational collapse of Vaidya spacetimes, one might expect the naked singularity to form. New solutions can tell us how $\Theta_{ik}$ will influence the result of the gravitational collapse. Vaidya spacetimes are widely used nowadays and the important question of the global structure of new solutions is the direction of future research. We have already told that $\Theta_{ik}$ can be thought as the energy-momentum tensor of a dark matter. So the obtained solution can tell us how the well-known properties of the Vaidya spacetimes change when an additional matter field present. These properties should be also studied in the future.

 

 

Reviewer 4 Report

The author presents some generalizations of the Vaydia metric which are intended to describe a hairy black hole. for doing that they resort to the gravitational decoupling method, 

I believe that  the motivation of the paper is well grounded, the calculations seem to be correct and   the presented results may be of interest for researchers working in relativistic astrophysics.

I have only two minor points:

1) The coordinate r in (9, 30, 38) is NOT spacelike (g_{rr}=0), it is a null coordinate. This coordinate r is different from the one appearing in (3), which is a spacelike coordinate (g_{rr} is not 0).

2) The last name: Misner is misspelled is several parts of the text.

 

Author Response

First of all, we would like to say thanks to a reviewer for the helpful comments.

We have checked the text and corrected English mistakes.

1. This coordinate transformation doesn't change $r$ coordinate, only $t\rightarrow v$ coordinate and this new time coordinate $v$ is null. This coordinate transformation for static hairy black holes by gravitational decoupling has been done in [47]. The reference has been added to the paper.
2. We have checked and corrected mistakes in English, thank you for this comment.

Also, we have added some new information to the introduction and conclusion. I am sorry, but I am blind and can't mark new information in pdf, so I inform about all changes here:

We have added 6 extra references and the following information:

It is possible to solve Einstein's field equations for a gravitational source, whose energy-momentum tensor is expressed as (1) by solving Einstein's field equations for each component $\tilde{T}_{ik}$ and $\Theta_{ik}$ separately. Then, by a straightforward superposition of the two solutions, we obtain the complete solution corresponding to the source $T_{ik}$. Since Einstein's field equations are non-linear, the MGD-decoupling represents a novel and useful method in the search and analysis of solutions, especially when we face scenarios beyond trivial cases, such as the interior of stellar systems with gravitational sources more realistic than the ideal perfect fluid, or even when we consider alternative theories, which usually introduce new features difficult to deal with.

The conformal symmetries and embedding properties of the generalized Vaidya metric were studied in [39,45]. Other properties of this spacetimes can be found in [42,44].

In the conclusion we have added the following part:

The Vaidya metric describes a dynamical spacetime instead of a static spacetime as the Schwarzschild or Reissner-Nordstrom metrics do. In the real world, astronomical bodies gain mass when they absorb radiation, and they lose mass when they emit radiation which means that the space-time around them is time-dependent. As we pointed out the Vaidya spacetime can be used as the simplest model of gravitational collapse. New solutions by the gravitational decoupling method allow us to investigate the question of how an additional matter field will affect the gravitational collapse process. When we consider the gravitational collapse of Vaidya spacetimes, one might expect the naked singularity to form. New solutions can tell us how $\Theta_{ik}$ will influence the result of the gravitational collapse. Vaidya spacetimes are widely used nowadays and the important question of the global structure of new solutions is the direction of future research. We have already told that $\Theta_{ik}$ can be thought as the energy-momentum tensor of a dark matter. So the obtained solution can tell us how the well-known properties of the Vaidya spacetimes change when an additional matter field present. These properties should be also studied in the future.

 

Round 2

Reviewer 1 Report

The revisions are acceptable. The paper may be published.

Author Response

Thank you for your comments and advice.

Reviewer 3 Report

Dear Authors,

 

I have revised the new version of the paper. Your improvements have been applied slightly applied only to Introduction and Conclusions. I am still of the opinion that the paper must be strongly improved together with the English style. My major comments are in the presentation of the results, where the Vaidya spacetime has been explained in the conclusions and not in the Introduction, and the main body of the text are calculations not commented. I would strongly suggest you to revise the structure of the paper making it more self consistent and easy to be followed by a reader not experts of the topics you have proposed. 

 

Author Response

We would like to thank the reviewer for the helpful comments, which we hope help us to improve our article.
Sections II, III and Iv are alike. We have done a gravitational decoupling for three different cases, but the method we use is the same for all three cases. For this reason, we have changed only the section 'The perfect fluid case'. We haven't added new formulas. However, we have extended the text between them. We have added more explanatory comments after formula 5 up to the end of the section.
At the conclusion, we describe the possible future investigation based on our results. We think that Vaidya spacetime and its application are described in the introduction at length.