On the Traversable Yukawa–Casimir Wormholes

Round 1

Reviewer 1 Report

Dear Authors,

 

I have read with great interest you paper entitled “On traversable Yukawa-Casimir wormholes”. In this work, you consider wormholes in General Relativity, where the source term is modeled by the exotic stress-energy tensor based on the Casimir effect. In addition, you exploit a Yukawa correction in the shape function to model different wormhole solutions.

 

The work is interesting, but there are several points not well written and not clear. The English style must be strongly improved. I have some major concerns and I also highlight several points in the text, that needs to be improved. If the authors reply in a satisfactory and clear way to my questions and modify the  paper according to my suggestions, I will recommend the paper for publications, otherwise I am forced to reject it.

 

Major Issues

1. The physical background on the Casimir wormhole has been not presented at all. Normally, the Casimir wormhole should be a micro-astrophysical object, since it involves small quantum effects. Therefore, I ask you, why is it so important to study such kinds of wormholes?
2. Why is there the need to modify the Casimir wormholes with a Yukawa (shielding) function?
3. You propose to modify the shape function b(r) by multiplying the shielding function f(r) for different terms by exploring different possibilities. However, have you checked whether these new wormholes are still solutions of the Einstein field equations (2)? This is a fundamental aspect. 
4. It is not clear from the text what are your original contributions with respect to the literature. This should be strongly clarified and highlighted in the text.

 

Below, you can find more detailed comments on all sections of the paper.

 

Abstract

1. The acronyms must be not used, so please remove WHs and also “D”.
2. The sentence “In the manuscript published by the authors, we show that this solution is possible for all space-time of dimension D > 3.” is not clear. The authors you refer are Garattini or you?
3. The abstract is very technical and confusing. You are mentioning concepts that you will introduce later in the paper, so it is difficult to follow and understand what you are proposing and which are your original contributions. It must be not only considerably shortened, but it must be also made more general and comprehensible. 

 

Introduction

4. The acronyms must be used the first time you see a word you want to abbreviate. Therefore, wormholes, General Relativity, Einstein field equations, must be done already at the beginning. Some acronyms are not replaced everywhere, like WHs, EFEs. Please do not use capital letters for example for field equations, zero tidal conditions., and so on.
5. Black holes are astrophysical objects and not cosmological
6. Exotic matter must be explained. It is actually connected with the source of negative energy. In the text it is not clear.
7. The Casimir effect must be better introduced with respect to the previous paragraph, otherwise it is not clear why you are mentioning it.
8. The explanation of the Casimir effect must be improved.
9. “D” must be defined as the acronym of dimension.
10. The sentence “Motivated by his other works involving terms of the Yukawa type” is not clear.
11. It is not clear the part of the shielding function. This must be better introduced and explained.
12. It is not clear what Garattini proposed. This should be generally explained without being technical and must be general. Later in the text, you can be more specific.
13. The organization of the article must be better rephrased. The different cases you mention (global, constant etc) are not clear how they will be applied, making the text difficult to be read and understood.

 

Section 2

14. The title must be changed in “A Review of Casimir Wormholes” in “Casimir wormhole”
15. After “:” you must use lowercase initial letters in the list.
16. It has been not mentioned that the field equations (2) are obtained in the tetrad formalism, please check Eq. (12) in Ref. [13].
17. You did not mention that you are using c=1.
18. It is not true that b’(r_0)<0, but b’(r_0)<1. Therefore the implication of rho(r_0)<0 is not correct. Please review this part carefully
19. This sentence “Due the fact that exotic matter is necessary to construct Morris-Thorne wormhole, in 2019 [10], Garanttini showed that, by promoting the separation distance between the plates to coordinate status, the Casimir energy and pressure” is awkward and not clear.
20. Under Eqs. (6), why r_1 should have the form you mentioned, could you please explain which condition must be imposed in order to obtain r_1?
21. The word “fixation” must be changed in for example “after having fixed the constants r_0 and r_1”. This must be done everywhere in the text.
22. “Incorporation procedure” must be changed, it is not the correct word.

 

Section 3

23. It is not cited the recent manuscript of Garattini.
24. It should be described why the shielding function is used. What are the physical meaning and mathematical purpose? It is also not clear that this shielding function is multiplied by the shape function. Everything should be better stated.
25. The three contexts you mention are not clear, they seem to be related to \mu. Everything should be better clarified.
26. Equation (10) must be better justified.

 

Section 3.1

27. The work of Garattini you mention must be cited.
28. Sentence “Since the multiplicative factor is less than or equal to unity, been the equality at r = r0, the modified shape function satisfies the properties (i), (ii) and (iii)” should be rephrased.
29. You must better explain the double limit imposed by Garattini and what instead your proposal is avoiding.
30. \Omega(r) is a typo, do you mean \omega(r)?
31. This sentence “Due it is made only numerically the exotic behavior for \mu= 0.1 can not be understood.” Is not clear, please explain it.
32. Fig 2 can be merged with Fig. 1. If you want to keep separate, in the caption of Fig. 2 can be stated that the same notations of Fig. 1 are employed.

 

Section 3.2

33. It is important to underline if this part is new and not treated before.
34. Which are the limits on \mu? Can you calculate them? Can you briefly comment it?
35. This sentence “Meanwhile, the tangential pressure is obtained by the conservation of energy. Expressions, due to size, will be omitted but some graphs indicating their behavior will be displayed.” is not clear.
36. Could you please comment this sentence “the existence of a critical point in the tangential pressure”, because it is not clear.
37. The symbol \mu_c has been not defined and it is not clear which is its value.
38. Can you briefly comment “blueshift deviation” what this implication means?

 

Section 3.3

39. It is important to underline if this part is new and not treated before.
40. This sentence “therefore, this correction ignores the term which is already naturally disregarded in the evaluation of most quantities of physical importance, since they depend on the derivative of the shape function. Thus, it is intuitive to assume that variations in the shielding parameter cause only subtle variations in the quantities of interest.” must be improved.
41. This sentence “Note that the only possibility of having \omega(r0) = 3 is having \mu= 0, which invalidates the correction, or r0 = 0, which makes the solution unfeasible as a wormhole” is not clear.
42. When you say “defined radial coordinate scales and redefinitions” to which part you refer? Please cite the equations of the section you refer.

 

Section 3.4

43. Which are “the cases considered” those of Sec. 3.3 and 3.4, please be precise.
44. Could you please justify why you can expand in Taylor series? Please discuss why \mu should be small.
45. In Eq. (40) must be better specified to which cases the redshift functions refer?
46. Please number the equations under Eqs. (40). In the former equations, \Phi_radial does not provide an asymptotically flat metric. This is a fundamental property that has been not mentioned in Sec. 2.
47. Please rephrase “This change occurs by fixing \Phi(r0), so that the redshift is set to less than an additive constant, with no physical change.”, because it is not clear.
48. Please rephrase “As for first order the global and constant cases result in the same and the radial case becomes independent of it is suggestive to consider corrections in second order”.
49. This sentence “In this order of approximation, the global and radial cases differ from each other, although they are subtle.” must be extended and better explained.

 

Conclusions

50. In Eq. (44), what is the function f(r)? Do you refer to Eq. (9) or to a general function?
51. It is not clear what are your contributions with respect to the Garattini paper. The conclusions are too short and does not explain well the improvements with respect to the literature.
52. There are no future perspectives and applications of the achieved results. 

Author Response

Dear referee, initially we would like to thank you for the points raised, incorporating them has improved the quality of our manuscript.

 

Major Issues:

1. Casimir wormholes are relevant because they allow the construction of these objects with a field already known in nature, unlike several others whose matter is of a purely mathematical nature. Although this class of wormholes does not have considerable effects in an astrophysical scenario, some applications in Condensed Matter have been provided in the literature. Furthermore, a study of these wormholes provides a better understanding of the nature of these solutions of General Relativity.

 

2. Although Garattini initially presents this modification as a way to obtain wormholes that satisfy the ZTC, he presents another motivation. In Garattini's own words "The motivation for this choice stands in the attempt to detect signals of variations of the ordinary gravitational field even for traversable wormhole and to obtain the possibility of having the negative energy density more concentrated in proximity of the throat".

 

3. The shape functions proposed in our manuscript are consistent with Einstein's field equations by construction. These shape functions are introduced into the Einstein field equations and give the corresponding energy density. After that, the choice of the Equation of State that relates energy and pressure gives us the redshift behavior. In addition, all involved quantities of geometry (shape and redshift), as well as matter, have expected behaviors to be categorized in the Morris-Thorne type solution.

 

4. Casimir wormholes do not have constant redshift, so there is no reason to build Yukawa-Casimir WH considering the ZTC. Our original contributions are a more consistent way to create these types of wormholes. In the original article, the author chooses some assumptions that are questionable, such as not guaranteeing the Casimir-type energy profile, in which the radial pressure is given as the derivative of the energy. This condition revealed in our article characteristics not foreseen in the initial article, such as values of the shielding parameter that violate the flare-out condition, as well as an inversion in the redshift behavior, causing such objects to be slightly repulsive. 

# Abstract

 

1.We made the proposed changes

2. We mentioned ourselves. To clarify this point, we rewrite the sentence as " We have recently shown that this can be generalized to higher dimensional spacetimes."

3. We rewrote the abstract in order to make the results more understandable.

 

# Introduction

4. We made the proposed changes.

5. We made the proposed changes

6. To explain the exotic matter, we add the sentence “ Such solutions generally require some exotic matter as a source, i.e.,  which violates the standard energy conditions of General Relativity, like negative energy density or negative pressure-energy ratio.”

7. To introduce the Casimir effect, we add the following text at the end of the first paragraph: “Although in the usual context of General Relativity, it is not possible to obtain this type of matter, quantum fields in some specific situations can present states with negative energy. In particular, the Casimir effect occurs when the electromagnetic field produces those states.

8. We removed the mention of the Casimir effect from the first paragraph and introduced the following explanation in the second paragraph: “In 1948, Casimir discovered that when we place two planes, parallel, closely spaced, and uncharged metallic plates in a vacuum, an attractive force between them appears. The Casimir effect occurs due to the zero-point energy of the quantum electromagnetic field distorted by the boundary conditions on the plates. This effect is closely related to the geometry of the boundaries and, as proved by Boyer in 1968, for a conducting spherical shell, the Casimir effect produces a positive force.”

9. We prefer to remove the acronym “D” from the text.

10. We change the sentence “Motivated by his other works involving terms of the Yukawa type”  to  “Garattini, still in 2019,  proposed another method to obtain a traversable WH using a Yukawa-type shape function [12]. ”

11.The shielding factor refers to the Yukawa factor. To make it more clear, we have changed, throughout the text, to the Yukawa factor.

12. To clarify Garattini's proposal , we rewrite the last part of the third paragraph as “The author directly modifies the Casimir shape function in three ways: globally, only in the constant term, and only in the variable. In each case, he deduces the corresponding energy density via Einstein field equations (EFE). On the other hand, he imposes the zero tide condition (ZTC), i.e., a constant redshift factor, to get the corresponding radial pressure via EFE. As he uses two different approaches to obtain the energy density and the radial pressure, he loses the Casimir characteristic, i.e., the relation between both quantities.”

13. We made the proposed changes.

 

# SECTION 2

14. We've revamped the organization of the article.

15. We made the proposed changes.

16. Unlike reference [12], we do not use the tetrad formalism to compute the components of the Einstein field equation. For this, we use the tensor formalism using the orthonormal basis described by the line element (1). To reinforce this procedure, we include the sentence “The independent components of Einstein’s equation in the orthonormal base, described by line element (1), are” before the set of equations (2).

17. It is a standard convention in relativity, but to reinforce this, we add the parentheses “(considering c = 1)”  before the line element (1).

18. Indeed, you are right. We update the text to "If we calculate pr (r)+ρ(r) using Eq.'s (2a) and (2b) we notice that the result, near the throat, is negatively defined by the flare-out condition [16] This suggests that matter must have more resistance to tension (τ(r) = −pr(r)) than it has energy. This result shows us that, at last in the throat, an exotic type of matter, ρ(r ) < 0, is necessary to build a traversable WH.".

19. We change the sentence “Due the fact that exotic matter is necessary to construct Morris-Thorne wormhole, in 2019 [10], Garanttini showed that, by promoting the separation distance between the plates to coordinate status, the Casimir energy and pressure” to “In this sense, Garattini showed in a recent manuscript [11] that promoting Casimir energy and pressure to functions of r-coordinate”

20. r_1^2 is the constant related to the Casimir energy, specifically \kappa\rho = r_1^2/r^4.  We add the sentence “ r_1^2  is the constant related to the Casimir energy, specifically κρ = r_1^2/r^4, with …”  after equation (6b) to better explain this point.

21. We made the proposed changes.

22. We change the word “incorporation procedure” to “embedding procedure”.

 

# SECTION 3

23. We fix this issue by citing Grattini's manuscript.

24. Physically, this modification is related to the existence of massive modes with negative energy. This discussion was introduced in the article.

25. We detail the three cases to be addressed, making their equations explicit, the way Garattini constructed them.

26. To justify equation (10), we add the sentence “Casimir pressure (P) and its energy (E) are related through the equation P = - dE/dL, where L is the separation distance between the plates” before it. Equation (10) is the proper generalization involving p_r(r) and \rho(r). 

 

## SUBSECTION 3.1

27. We fix this issue by citing Grattini's manuscript.

28. The excerpt was rewritten for “As the multiplicative factor is global in the shape function, the modified shape function directly satisfies properties (i), (ii) and (iii)”.

29. In the case of Garattini's proposal it was necessary lim_{r→r_0} lim_{\mu→ 0}\omega(r) to return to the usual case. Our proposal, on the other hand, only requires \mu=0. This single condition should already map the Yukawa-Casimir wormholes to the Casimir ones. We now specify this double limit in the text.

30. It's not just a typo. \Omega(r) is the EoS factor involving tangential pressure, while \omega(r) is the radial pressure factor. They are different quantities.

31. To make it more clear, we rewrite the sentence “ Due it is made only numerically the exotic behavior for \mu= 0.1 can not be understood.” as “Note that the behavior for \bar{\mu} = 1.0 is very different from the other cases because, near the throat, it exhibits an increasing behavior, reaching a maximum value and decreasing with the increase of r/r0. We can not explain the physical reasons for this behavior in tangential pressure  because the process to obtain it has been numerical.”

32. Figures were kept separate, but we stated the caption of Fig. 2 as the same notations of Fig. 1

 

## SUBSECTION 3.2

33. Clarified at the end of section 3.

34. The limit was obtained and detailed in the text.

35. The excerpt was rewritten for “Finally, the tangential pressure is obtained by the conservation of stress-energy tensor (3). The analytical expression for the tangential pressure will be omitted and its behavior will be shown in Fig. 3d.”.

36. The term 'critical point' refers to the fact that, for mu=1, the tangential pressure has a maximum value, that is, whose derivative is zero.

37. This quantity is what reverses the redshift behavior. We change the redshift plots in both cases to show the effect of an approximate value of \mu_c.

38. The implication of the change in the sign of the redshift function is associated with the Yukawa-Casimir wormholes presenting, for certain values of \mu, a slightly repulsive behavior.

 

## SUBSECTION 3.3

39. Clarified at the end of section 3.

40. The text has been improved to "since most physical quantities depend on the derivatives of the shape function, a modification that ignores the constant term will tend to smoothly diverge from the initial solution. This is because functions that differ only by additive constants have the same derivatives".

41. In order to make the text clearer, we changed it to "Note that the only ways to have $\omega(r_0) = 3$ are if $\mu = 0$ or $r_0 = 0$. The first is the trivial solution of Casimir WH, while the second does not characterize a traversable wormhole".

42. We specify in the text with " by adopting the previously defined radial coordinate scales (u = r/r_0) and redefinition (\Bar{\mu} = \mu r_0), the following dimensionless energy".

 

## SUBSECTION 3.4

43. We made the proposed changes.

44. By construction, the redshift function should be well behaved, this allows for expansion. As the value \mu=0 should model the wormholes for the usual Casimir case, thus small values of mu are on the borderline of what is already known in the literature. This has been clarified better now in the article.

45. We made the proposed changes.

46. The equations were numbered. The first of the equations has been rewritten to factor the dependency on \mu. Since we set \Phi(r0) = 0, the asymptotic limit on the redshift function is a constant, which can be absorbed into a redefinition of the t-coordinate. So, the radial solution is indeed asymptotically flat, as expected for being the Casimir wormhole redshift.

47. The text was changed to "The value \mu = 0 should model the WH for the usual Casimir case, thus small values of $\mu$ are on the borderline of what is already known in the literature. Furthermore, as we saw in the previous subsection, the Yukawa-Casimir WH is stable for small values of the Yukawa parameter".

48. The excerpt was rewritten for “In first-order corrections, we observe that the redshift for the global and constant cases are the same. Furthermore, the redshift for the radial case is independent of the Yukawa parameter \mu. In this way, it is suggestive to consider corrections in the second order.”.

49.We made the proposed changes..

 

# CONCLUSIONS

50. It's a general function. Changed to g(r).

51. We further detail our contributions..

52.We added some study extension possibilities.

Reviewer 2 Report

In this article the authors has studied, "ON THE TRAVERSABLE YUKAWA-CASIMIR WORMHOLES". My comments regarding the publication of this paper is appended below point-wise:

1. The subject content of the paper is very interesting and the authors has managed to written the work in a very crisp fashion.

2. Plots and the related mathematical analysis is very clear and presented in a very well fashioned way.

3. Computations performed in this paper is correct.

4. The reference list is complete.

Based on the above mentioned points I strongly recommend the publication of this paper in the journal Symmetry.

Author Response

We appreciate the good evaluation of our manuscript. We made changes that took into account the opinions of the three reviewers and improved the writing of it. We hope to have achieved a good result. Thank you for recommending our article.

Reviewer 3 Report

First of all, I would like to emphasize that I read the article with interest, I think it is an interesting study. I also scanned the work for ethical due diligence through the iThenticate system. Before moving on to my thoughts on the problem addressed in the study, I would like to emphasize that I did not encounter any problems as a result of iThenticate scanning. On the other hand, in the present study, the authors have mainly proposed a more consistent method to extend the Casimir wormholes with Yukawa-type shielding terms in shape functions in the three ideas initially introduced by Garattini (Eur. Phys. J. C 81, 2021, 824). The first question that comes to mind in such theoretical studies is whether the proposed model is physically consistent. For this question, the authors underlined that the model (including its all sub-cases) satisfies the QWEC (Quantum Weak Energy Condition). In fact, I think that the work in its current form could be published in this journal. However, I would like to bring a curious point to the attention of the authors. Did the authors examine the behavior of the square of the speed of sound (V_s^2=dp/drho) for stability analysis of the model? The necessary mathematical definitions for energy density and pressure have already been obtained by the authors for all special cases, so this analysis can be done easily. I believe that the article will provide more consistent results when the authors clarify this situation.

Author Response

We appreciate the good evaluation of our manuscript. We made changes that took into account the opinions of the three reviewers and improved the writing of it. In particular, we augmented the manuscript with the stability discussion, as suggested. We hope to have achieved a good result. 

Round 2

Reviewer 1 Report

Dear Authors,

 

You new revised version has been considerably improved with respect to the previous one. My major comments have been generally answered. However, I think that the paper needs more attention on the English styles and in the explanations of some obscure sentences. Please find below my minor comments.

 

 

Abstract

1. There is Wormhole to the fourth line with capital letter, please put in lower case.
2. The abstract is still too long and full of details, especially the new part in red you have written. Please drastically reduce it and highlight clearly your new contributions.

 

Introduction

3. “two planes, parallel, closely spaced, and uncharged metallic plates in a vacuum,” —> put “planes” as follows “metallic plane plates” 
4. The sentence “modifies the Casimir shape function in three ways: globally only in the constant term, and only in the variable.”, which is repeated several times in the paper is still not clear. Please modify in a way it is understandable throughout the paper. This was a point that I have already raised in my previous referee report.
5. Einstein field equations (EFE) —> Einstein field equations (EFEs)
6. Substitute the acronyms everywhere in the text, wormhole is not substituted uniformly, as well as also EFEs. The same holds also for Sec. instead of section. This was also another point I already asked you in my previous referee report
7. “This article is organized….”, the last part of the introduction is not clear in several points, please rephrase it. In addition, “: In Sec.” —> “: in Sec.”.

 

Section 2

8. When you mention Eqs. (2) they are written in the tetrad base adopted to the metric (1), please write it better.
9. Eq.’s, Fig.’s —> Eqs., Figs.
10. “This suggests that matter must have more resistance to tension…” not clear.
11. “are able to obtain a Morris-Thorne type” -> “is able to obtain a Morris-Thorne type”.

 

Section 3

13. “This study, part of Eq. (6a) and imposes multiplicative terms of the type” not clear.
14. You have exchanged the definition (II) with (III), please check the formulas you have written.
15. “As the multiplicative factor is global in the shape function, the modified shape function directly satisfies properties (i), (ii) and (iii).” Please be clear, which properties you refer.
16. “Note that the behavior… has been numerical” it is unclear, please rephrase them.
17. The values of \mu_c and other values you write have several significative digits, why do you write with this high precision? Write them with two or maximum three digits.
18. “In this sense, the flare-out condition is satisfied for all points in the domain if … were approximate.” It is not clear, please rephrase it.

 

Conclusions

19. “This analysis for small parameters, in addition to allowing us to perform the calculations analytically, revealed aspects on the..” Please rephrase this sentence.
20. GUP write it without acronyms.

 

Please read carefully all paper and be sure that you apply my comments throughout the article.

 

Author Response

Dear referee,

Thanks for the prompt feedback. We augmented the manuscript with the new recommendations.

 

# ABSTRACT

1. We put ‘Wormhole’ in lowercase.

2. We rewrite the red part of the abstract to contemplate your suggestions.

 

# INTRODUCTION

3. We made the proposed changes.

4. In the introduction we change the sentence “ e generalize these objects by introducing Yukawa-type factors into the shape function in three different ways: i) globally, ii) in the constant term, and iii) in the variable term” to “...we generalize these objects by introducing Yukawa-type factors into the shape function in the same three ways made by Garatinni, but without forcing the ZTC.”

We change the last sentence of section 3  from “Our proposals will modify the shape function in three ways: i) globally, ii) in the constant term and iii) in the variable term. These proposals constituted the following subsections,” to “The three ways to modify the shape function listed above are the subject of the following subsections.” 

5. We made the proposed changes.

6. Changed all 'wormhole' terms to 'WH'. Also, we dropped the term 'Sec.' and use 'section'.

7. We reformulated the presentation of the organization of the article.

 

# SECTION 2

8. Even though we do not use the tetrad formalism to compute the components of Einstein's field equation, we use a tetrad base to describe the line element (1).  To reinforce this, we change the sentence  “The independent components of Einstein’s equation in the orthonormal base, described by line element (1), are” to “In the tetrad base described by the line element (1), Einstein's Field Equation splits into the following set of equations”

9. We made the proposed changes.

10. We prefer to remove the sentence "This suggests that matter must have more resistance to tension (\tau(r) = -p_r(r)) than it has energy."

11. We made the modification from "are able to obtain a Morris-Thorne type" to "is able to obtain a Morris-Thorne type".

 

# SECTION 3

12. Point 12 was not informed in the report.

13. We made the modification from “This study, part of Eq. (6) and imposes multiplicative terms of the type” to “The proposed Casimir WH extension is such that the shape function (6) is multiplied by...”.

14. The definitions were correct, as they are what Garattine made, but the quoted equation was wrong. The correct equation is (7a) instead of (6a). Since we relate the pressure and the density by equation (10), a new relation between r1 and r0 will appear in each case. Due to this fact, we use in our manuscript a modification in eq. (6a). We correct the quote above equation (9) and change the first sentence of subsection (3.1) from "The first modification made by Garattini [14] was a global modification in the shape function, i.e., he multiplies the solution (6a) by the factor (9) to obtain"  to "The first case that we will study is a global modification in the shape function, i.e., to multiply the solution (6a) by the factor (9) to obtain "

15. We specify that properties (i), (ii) and (iii) were defined in section 2.

16. We made the modification from “The numerical result of the tangential pressure is plotted in fig. 1d for some values of μ. Note that the behavior for μ = 1.0 is very different from the other cases because, near the throat it exhibits an increasing behavior, reaching a maximum value and decreasing with the increase of r/r0. We can not explain the physical causes of this unusual behavior.” to “We plot the tangential pressure in fig.  1d for some values of μ, and we can note that the behavior for μ = 1.0 is very different from the other cases since it exhibits a maximum value near the throat. The analytical result of the tangential pressure on the throat shows that this behavior changes around μ = 0.72.”

17. We switched from using three decimal places to three significant figures. In this case, the values that violate the flare-out condition are expressed as 0.618 < \mu < 1.43

18. We chose to emphasize the regime that violates the flare-out condition. That's why we changed "In this sense, the flare-out condition is satisfied for all points in the domain if ... These values were approximate." to "In this sense, the flare-out condition is violated in regime 0.618 < \mu < 1.43".

 

# CONCLUSIONS

19. We changed the text to "Our small parameter analysis not only allowed us to obtain the redshift analytically, but also revealed information about the boundary between the Casimir and Yukawa-Casimir wormholes. This analysis clarifies why the value that causes the redshift shift is similar in the global (as shown in Figure 2a) and constant (as shown in Figure 4a) cases because they differ only in the second order of \mu." – We hope this has made it clearer.

20. We made the modification from “GUP-like quantum corrections” to “a study of the effects of the Generalized Uncertainty Principle”.