Statistical Gravity and Entropy of Spacetime

Round 1

Reviewer 1 Report (Previous Reviewer 2)

Comments and Suggestions for Authors

The inquiries I had are now addressed, and the paper can be accepted for publication.

Author Response

Dear Editor,

Let me thank all three Referees. Apart from Referee 1 that is now happy about the first round of revision I found the comments/suggestions of the other two Referees (Referee 2 and Referee 3) very helpful to further improve the manuscript. 

I will answer below to all the points organically raised by all Referees, indicating the 
changes made in the second revised version point by point:

----------------------------------------------------------------------------------------------------------
Referee 1
----------------------------------------------------------------------------------------------------------

The inquiries I had are now addressed, and the paper can be accepted for publication.

----------------------------------------------------------------------------------------------------------
Referee 2
----------------------------------------------------------------------------------------------------------

In the manuscript entitled ``Statistical Gravity and entropy of spacetime'' the author, Riccardo Fantoni, tries to explain the origins of gravity from the statistical approach.
Despite the fact that the topic is interesting and novel, I would say that the author fails to prove the connection between the two. 

The derivations seems valid, but I do not see how the actual metrics used in general relativity enters Eq. (8) - indeed, Eq. (8) holds for some field denoted as g_{\mu \nu} and 
the operator pi defined there, but the author (in my opinion) had not demonstrated that it is the same metric field as in GR. The same is true for Eq (27) - yes, there are
functions R and g but their relation to the Ricci scalar and metric function in GR is not rigorously demonstrated.

Last but not least - generally speaking, GR does not require matter fields to be present and is well-formulated in their absence - I might be wrong but from the derivations
presented in the manuscript it seems that for the derivations to be valid either matter fields or non-zero Ricci scalar are required, so that the connection to the vacuum GR 
cannot be demonstrated.

Overall, I would say that the results presented in the manuscript hold some interest but either the relation between functions used and their GR counterparts should be rigorously
demonstrated or the formulations used through the manuscript should be changed accordingly (i.e., like, ``if we associate g_munu with the metric tensor from GR, then...'' and 
similar); also, the question of vacuum GR posed above should be addressed as well. 

So I would recommend major revision until the changes mentioned above are implemented.

----------------------------------------------------------------------------------------------------------
Referee 3
----------------------------------------------------------------------------------------------------------

Dear editor/author  
I studied the manuscript: Statistical Gravity and entropy of spacetime which is extension of the published work of the author: `Quantum Reports, 6, 706 (2024)`
Although this version of the manuscript has good novelty and is written as professional but this version of the manuscript needs to improve by regarding the following comments before to publish in my opinion:
As we know there are some essential problems for realistic physical phenomenon for instance in early cosmology which are not solved still ( See for instance : Astro-ph/0602280) . Every alternative quantum gravity model should be have some predictions about these essential problems. For instance such models should  give out a suitable answer to the following questions:  
1.    What is dynamics of metric signature transition from Euclidean (++++) (before the Big Bang) to the present metric signature in which we live , i.e., The Lorentzian (-+++)? In all of quantum gravity models we changed real time with an imaginary time (The Wick rotation) by hand to control boundary conditions of the system. This actually gives the weakness of the existing models. This should be resolve  by an understandable alternative. 
2.    How a quantum gravity model can be resolve naked singularity with infinite energy density of the universe at the begin of born? 
After to add some comments to the paper by regarding at least the above problems the revised manuscript can be consider for review process again in my opinion.
 Sincerely yours
Reviewer

----------------------------------------------------------------------------------------------------------
My Reply to Referee 1
----------------------------------------------------------------------------------------------------------

Even if the Referee is happy I felt the need to add the following third paragraph to section "II. Gentropy" 
to better visualize the conceptual point he raised in the first round of revision:

"
In order to formulate a statistical theory of gravity we need to determine the {\sl statistical 
distribution} of a subregion of a macroscopic spacetime region. We know from General Relativity 
that each spacetime subregion has a metric so our statistical distribution will describe the 
statistical properties of these metric tensors $g_{\mu\nu}$.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 2
----------------------------------------------------------------------------------------------------------

I agree with both comments of the Referee and I therefore added the following paragraph just after 
footnote 2:

"
Here we are associating $g_{\mu\nu}(x)$ with the metric tensor from General Relativity entering 
our Eq. (\ref{eq:ltdm}). A delicate point is that of a consistent description of the vacuum of 
General Relativity where both matter fields and the Ricci scalar vanish for which our high 
temperature density matrix of Eq. (\ref{eq:ltdm}) reduces to a functional Dirac delta.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 3
----------------------------------------------------------------------------------------------------------

I added the Reference of George Ellis suggested by the Referee in the introduction with some comments related to the two points raised by the Referee:

1. I do not agree with the Referee. We use signature (-+++) for Quantum Gravity and signature (++++) for Statistical Gravity. As usual you turn Quantum Physics to Statistical Physics by a Wick rotation. That has nothing to do with before or after the birth of our Universe. Even if from a philosophycal point of view we may think that before creation it was only chaos for which I could agree that between the 2 probably Statistical Physics would be the most appropriate. It would be interesting to discuss these matters in a round table "mountain" with George Ellis.

2. At creation, between before and after creation, something like what the Referee imagine could as well be going on. From a description point of view we are already accustomed to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a diffusion process. But there are many others. So I would not find this "At" problem as an issue.  

In particular I added the following third paragraph to the introduction:

"
From a philosophycal point of view \cite{Ellis2006} we should ask about the mathematical issues of 
existence and unicity of the Universe as well as some anthropic questions like the fine tuning for 
life in our Universe or the natures of existence. We may think that before creation it was only chaos 
for which one could agree that between the two signatures of the metric of spacetime 
(the Euclidean and the Lorentzian) the one describing statistical physics (the Euclidean) would be 
the most appropriate. At creation, between before and after, it could be that one has to deal with
infinite energy densities or maybe density. From a description point of view we are already accustomed 
to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a 
diffusion process. But there are many others.
"

 

I hope that the revised version of the manuscript will be suitable for publication on Stats of MDPI,

 

best regards,
the Author

Reviewer 2 Report (New Reviewer)

Comments and Suggestions for Authors

In the manuscript entitled ``Statistical Gravity and entropy of spacetime'' the author, Riccardo Fantoni, tries to explain the origins of gravity from the statistical approach.
Despite the fact that the topic is interesting and novel, I would say that the author fails to prove the connection between the two. 

The derivations seems valid, but I do not see how the actual metrics used in general relativity enters Eq. (8) - indeed, Eq. (8) holds for some field denoted as g_{\mu \nu} and 
the operator pi defined there, but the author (in my opinion) had not demonstrated that it is the same metric field as in GR. The same is true for Eq (27) - yes, there are
functions R and g but their relation to the Ricci scalar and metric function in GR is not rigorously demonstrated.

Last but not least - generally speaking, GR does not require matter fields to be present and is well-formulated in their absence - I might be wrong but from the derivations
presented in the manuscript it seems that for the derivations to be valid either matter fields or non-zero Ricci scalar are required, so that the connection to the vacuum GR 
cannot be demonstrated.

Overall, I would say that the results presented in the manuscript hold some interest but either the relation between functions used and their GR counterparts should be rigorously
demonstrated or the formulations used through the manuscript should be changed accordingly (i.e., like, ``if we associate g_munu with the metric tensor from GR, then...'' and 
similar); also, the question of vacuum GR posed above should be addressed as well. 

So I would recommend major revision until the changes mentioned above are implemented.

Author Response

Dear Editor,

Let me thank all three Referees. Apart from Referee 1 that is now happy about the first round of revision I found the comments/suggestions of the other two Referees (Referee 2 and Referee 3) very helpful to further improve the manuscript. 

I will answer below to all the points organically raised by all Referees, indicating the 
changes made in the second revised version point by point:

----------------------------------------------------------------------------------------------------------
Referee 1
----------------------------------------------------------------------------------------------------------

The inquiries I had are now addressed, and the paper can be accepted for publication.

----------------------------------------------------------------------------------------------------------
Referee 2
----------------------------------------------------------------------------------------------------------

In the manuscript entitled ``Statistical Gravity and entropy of spacetime'' the author, Riccardo Fantoni, tries to explain the origins of gravity from the statistical approach.
Despite the fact that the topic is interesting and novel, I would say that the author fails to prove the connection between the two. 

The derivations seems valid, but I do not see how the actual metrics used in general relativity enters Eq. (8) - indeed, Eq. (8) holds for some field denoted as g_{\mu \nu} and 
the operator pi defined there, but the author (in my opinion) had not demonstrated that it is the same metric field as in GR. The same is true for Eq (27) - yes, there are
functions R and g but their relation to the Ricci scalar and metric function in GR is not rigorously demonstrated.

Last but not least - generally speaking, GR does not require matter fields to be present and is well-formulated in their absence - I might be wrong but from the derivations
presented in the manuscript it seems that for the derivations to be valid either matter fields or non-zero Ricci scalar are required, so that the connection to the vacuum GR 
cannot be demonstrated.

Overall, I would say that the results presented in the manuscript hold some interest but either the relation between functions used and their GR counterparts should be rigorously
demonstrated or the formulations used through the manuscript should be changed accordingly (i.e., like, ``if we associate g_munu with the metric tensor from GR, then...'' and 
similar); also, the question of vacuum GR posed above should be addressed as well. 

So I would recommend major revision until the changes mentioned above are implemented.

----------------------------------------------------------------------------------------------------------
Referee 3
----------------------------------------------------------------------------------------------------------

Dear editor/author  
I studied the manuscript: Statistical Gravity and entropy of spacetime which is extension of the published work of the author: `Quantum Reports, 6, 706 (2024)`
Although this version of the manuscript has good novelty and is written as professional but this version of the manuscript needs to improve by regarding the following comments before to publish in my opinion:
As we know there are some essential problems for realistic physical phenomenon for instance in early cosmology which are not solved still ( See for instance : Astro-ph/0602280) . Every alternative quantum gravity model should be have some predictions about these essential problems. For instance such models should  give out a suitable answer to the following questions:  
1.    What is dynamics of metric signature transition from Euclidean (++++) (before the Big Bang) to the present metric signature in which we live , i.e., The Lorentzian (-+++)? In all of quantum gravity models we changed real time with an imaginary time (The Wick rotation) by hand to control boundary conditions of the system. This actually gives the weakness of the existing models. This should be resolve  by an understandable alternative. 
2.    How a quantum gravity model can be resolve naked singularity with infinite energy density of the universe at the begin of born? 
After to add some comments to the paper by regarding at least the above problems the revised manuscript can be consider for review process again in my opinion.
 Sincerely yours
Reviewer

----------------------------------------------------------------------------------------------------------
My Reply to Referee 1
----------------------------------------------------------------------------------------------------------

Even if the Referee is happy I felt the need to add the following third paragraph to section "II. Gentropy" 
to better visualize the conceptual point he raised in the first round of revision:

"
In order to formulate a statistical theory of gravity we need to determine the {\sl statistical 
distribution} of a subregion of a macroscopic spacetime region. We know from General Relativity 
that each spacetime subregion has a metric so our statistical distribution will describe the 
statistical properties of these metric tensors $g_{\mu\nu}$.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 2
----------------------------------------------------------------------------------------------------------

I agree with both comments of the Referee and I therefore added the following paragraph just after 
footnote 2:

"
Here we are associating $g_{\mu\nu}(x)$ with the metric tensor from General Relativity entering 
our Eq. (\ref{eq:ltdm}). A delicate point is that of a consistent description of the vacuum of 
General Relativity where both matter fields and the Ricci scalar vanish for which our high 
temperature density matrix of Eq. (\ref{eq:ltdm}) reduces to a functional Dirac delta.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 3
----------------------------------------------------------------------------------------------------------

I added the Reference of George Ellis suggested by the Referee in the introduction with some comments related to the two points raised by the Referee:

1. I do not agree with the Referee. We use signature (-+++) for Quantum Gravity and signature (++++) for Statistical Gravity. As usual you turn Quantum Physics to Statistical Physics by a Wick rotation. That has nothing to do with before or after the birth of our Universe. Even if from a philosophycal point of view we may think that before creation it was only chaos for which I could agree that between the 2 probably Statistical Physics would be the most appropriate. It would be interesting to discuss these matters in a round table "mountain" with George Ellis.

2. At creation, between before and after creation, something like what the Referee imagine could as well be going on. From a description point of view we are already accustomed to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a diffusion process. But there are many others. So I would not find this "At" problem as an issue.  

In particular I added the following third paragraph to the introduction:

"
From a philosophycal point of view \cite{Ellis2006} we should ask about the mathematical issues of 
existence and unicity of the Universe as well as some anthropic questions like the fine tuning for 
life in our Universe or the natures of existence. We may think that before creation it was only chaos 
for which one could agree that between the two signatures of the metric of spacetime 
(the Euclidean and the Lorentzian) the one describing statistical physics (the Euclidean) would be 
the most appropriate. At creation, between before and after, it could be that one has to deal with
infinite energy densities or maybe density. From a description point of view we are already accustomed 
to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a 
diffusion process. But there are many others.
"

 

I hope that the revised version of the manuscript will be suitable for publication on Stats of MDPI,

 

best regards,
the Author

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

Dear editor/author  
I studied the manuscript: Statistical Gravity and entropy of spacetime which is extension of the published work of the author: `Quantum Reports, 6, 706 (2024)`
Although this version of the manuscript has good novelty and is written as professional but this version of the manuscript needs to improve by regarding the following comments before to publish in my opinion:
As we know there are some essential problems for realistic physical phenomenon for instance in early cosmology which are not solved still ( See for instance : Astro-ph/0602280) . Every alternative quantum gravity model should be have some predictions about these essential problems. For instance such models should  give out a suitable answer to the following questions:  
1.    What is dynamics of metric signature transition from Euclidean (++++) (before the Big Bang) to the present metric signature in which we live , i.e., The Lorentzian (-+++)? In all of quantum gravity models we changed real time with an imaginary time (The Wick rotation) by hand to control boundary conditions of the system. This actually gives the weakness of the existing models. This should be resolve  by an understandable alternative. 
2.    How a quantum gravity model can be resolve naked singularity with infinite energy density of the universe at the begin of born? 
After to add some comments to the paper by regarding at least the above problems the revised manuscript can be consider for review process again in my opinion.
 Sincerely yours
Reviewer

 

Author Response

Dear Editor,

Let me thank all three Referees. Apart from Referee 1 that is now happy about the first round of revision I found the comments/suggestions of the other two Referees (Referee 2 and Referee 3) very helpful to further improve the manuscript. 

I will answer below to all the points organically raised by all Referees, indicating the 
changes made in the second revised version point by point:

----------------------------------------------------------------------------------------------------------
Referee 1
----------------------------------------------------------------------------------------------------------

The inquiries I had are now addressed, and the paper can be accepted for publication.

----------------------------------------------------------------------------------------------------------
Referee 2
----------------------------------------------------------------------------------------------------------

In the manuscript entitled ``Statistical Gravity and entropy of spacetime'' the author, Riccardo Fantoni, tries to explain the origins of gravity from the statistical approach.
Despite the fact that the topic is interesting and novel, I would say that the author fails to prove the connection between the two. 

The derivations seems valid, but I do not see how the actual metrics used in general relativity enters Eq. (8) - indeed, Eq. (8) holds for some field denoted as g_{\mu \nu} and 
the operator pi defined there, but the author (in my opinion) had not demonstrated that it is the same metric field as in GR. The same is true for Eq (27) - yes, there are
functions R and g but their relation to the Ricci scalar and metric function in GR is not rigorously demonstrated.

Last but not least - generally speaking, GR does not require matter fields to be present and is well-formulated in their absence - I might be wrong but from the derivations
presented in the manuscript it seems that for the derivations to be valid either matter fields or non-zero Ricci scalar are required, so that the connection to the vacuum GR 
cannot be demonstrated.

Overall, I would say that the results presented in the manuscript hold some interest but either the relation between functions used and their GR counterparts should be rigorously
demonstrated or the formulations used through the manuscript should be changed accordingly (i.e., like, ``if we associate g_munu with the metric tensor from GR, then...'' and 
similar); also, the question of vacuum GR posed above should be addressed as well. 

So I would recommend major revision until the changes mentioned above are implemented.

----------------------------------------------------------------------------------------------------------
Referee 3
----------------------------------------------------------------------------------------------------------

Dear editor/author  
I studied the manuscript: Statistical Gravity and entropy of spacetime which is extension of the published work of the author: `Quantum Reports, 6, 706 (2024)`
Although this version of the manuscript has good novelty and is written as professional but this version of the manuscript needs to improve by regarding the following comments before to publish in my opinion:
As we know there are some essential problems for realistic physical phenomenon for instance in early cosmology which are not solved still ( See for instance : Astro-ph/0602280) . Every alternative quantum gravity model should be have some predictions about these essential problems. For instance such models should  give out a suitable answer to the following questions:  
1.    What is dynamics of metric signature transition from Euclidean (++++) (before the Big Bang) to the present metric signature in which we live , i.e., The Lorentzian (-+++)? In all of quantum gravity models we changed real time with an imaginary time (The Wick rotation) by hand to control boundary conditions of the system. This actually gives the weakness of the existing models. This should be resolve  by an understandable alternative. 
2.    How a quantum gravity model can be resolve naked singularity with infinite energy density of the universe at the begin of born? 
After to add some comments to the paper by regarding at least the above problems the revised manuscript can be consider for review process again in my opinion.
 Sincerely yours
Reviewer

----------------------------------------------------------------------------------------------------------
My Reply to Referee 1
----------------------------------------------------------------------------------------------------------

Even if the Referee is happy I felt the need to add the following third paragraph to section "II. Gentropy" 
to better visualize the conceptual point he raised in the first round of revision:

"
In order to formulate a statistical theory of gravity we need to determine the {\sl statistical 
distribution} of a subregion of a macroscopic spacetime region. We know from General Relativity 
that each spacetime subregion has a metric so our statistical distribution will describe the 
statistical properties of these metric tensors $g_{\mu\nu}$.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 2
----------------------------------------------------------------------------------------------------------

I agree with both comments of the Referee and I therefore added the following paragraph just after 
footnote 2:

"
Here we are associating $g_{\mu\nu}(x)$ with the metric tensor from General Relativity entering 
our Eq. (\ref{eq:ltdm}). A delicate point is that of a consistent description of the vacuum of 
General Relativity where both matter fields and the Ricci scalar vanish for which our high 
temperature density matrix of Eq. (\ref{eq:ltdm}) reduces to a functional Dirac delta.
"

----------------------------------------------------------------------------------------------------------
My Reply to Referee 3
----------------------------------------------------------------------------------------------------------

I added the Reference of George Ellis suggested by the Referee in the introduction with some comments related to the two points raised by the Referee:

1. I do not agree with the Referee. We use signature (-+++) for Quantum Gravity and signature (++++) for Statistical Gravity. As usual you turn Quantum Physics to Statistical Physics by a Wick rotation. That has nothing to do with before or after the birth of our Universe. Even if from a philosophycal point of view we may think that before creation it was only chaos for which I could agree that between the 2 probably Statistical Physics would be the most appropriate. It would be interesting to discuss these matters in a round table "mountain" with George Ellis.

2. At creation, between before and after creation, something like what the Referee imagine could as well be going on. From a description point of view we are already accustomed to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a diffusion process. But there are many others. So I would not find this "At" problem as an issue.  

In particular I added the following third paragraph to the introduction:

"
From a philosophycal point of view \cite{Ellis2006} we should ask about the mathematical issues of 
existence and unicity of the Universe as well as some anthropic questions like the fine tuning for 
life in our Universe or the natures of existence. We may think that before creation it was only chaos 
for which one could agree that between the two signatures of the metric of spacetime 
(the Euclidean and the Lorentzian) the one describing statistical physics (the Euclidean) would be 
the most appropriate. At creation, between before and after, it could be that one has to deal with
infinite energy densities or maybe density. From a description point of view we are already accustomed 
to deal with infinities. I am here thinking at the evolution of a Dirac delta into a Gaussian in a 
diffusion process. But there are many others.
"

 

I hope that the revised version of the manuscript will be suitable for publication on Stats of MDPI,

 

best regards,
the Author

Round 2

Reviewer 2 Report (New Reviewer)

Comments and Suggestions for Authors With the amendments (and explanations) made in response to mine and other reviewer's criticism I re-evaluated the manuscript and think that now it is ready for publication.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

As far as I can tell, this paper is a sloppy written empty comment. The one sentence abstract is nothing but a self citation… The first sentence in the introduction speaks about horizontal theory, whatever it may refer to… The canonical Gentropy(?) section is quite standard, and in my opinion, is not directly related to gravity whatsoever… The two sentences conclusion section does not contain anything to support the acclaimed ‘logical foundation’ … I have no choice but to recommend total rejection!

Reviewer 2 Report

Comments and Suggestions for Authors

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{\Large \sffamily \textbf{Report on Stats-3481524} }  \\

 

 

In this paper, Stats-3481524, the author presents a study on \textit{Statistical Gravity and Entropy of Spacetime}, a topic that is both intriguing and highly relevant to the ongoing discourse in theoretical physics. While the  manuscript explores a highly interesting intersection of statistical mechanics, quantum principles, and general relativity (GR),  there are several critical points which need to be addressed in order to make the paper more clear, rigorous  and novel. Here are the concerns that I have, and some suggestions for improvement.
\\
\section*{1. Quantum Principles and Classical General Relativity: A Contradiction?}
In Equation (2), the authors propose a subregion governed by quantum principles, with a density matrix that purportedly plays a role in presenting GR. However, this raises a significant conceptual issue: GR is fundamentally a classical theory, while the density matrix is inherently quantum mechanical. The authors must explicitly address this apparent contradiction.

\subsection*{Suggestions for Improvement:}
\begin{itemize}
    \item Clarify how the quantum density matrix can be reconciled with the classical framework of GR. Is there an underlying assumption of semi-classical gravity, or does the model propose a new framework that bridges the quantum-classical divide?
    \item Discuss the problem of hierarchy and the non-renormalizability of GR in the context of this model. How does the proposed approach handle these well-known challenges in quantum gravity?
\end{itemize}

\section*{2. Uncertainty Principles in the Model}
The manuscript claims that the proposed model follows statistical and quantum mechanical principles. However, it is unclear how the uncertainty principles—specifically the Heisenberg uncertainty principle and the Károlyházy uncertainty relation \cite{Karoly1966,Author2007} are satisfied within this framework.

\subsection*{Suggestions for Improvement:}
\begin{itemize}
    \item Provide a detailed discussion on how the uncertainty principles are incorporated into the model. Are there modifications or extensions to these principles to accommodate the statistical nature of spacetime?
    \item If the model deviates from standard quantum mechanics, justify these deviations and explain their physical implications.
\end{itemize}

\section*{3. Novelty and Contribution to the Field}
The manuscript should clearly articulate its novelty and distinguish itself from prior work. References such as \cite{Rovelli1993} and \cite{Banerjee2010} address similar themes, making it essential for the authors to highlight how their work advances the field.

\subsection*{Suggestions for Improvement:}
\begin{itemize}
    \item Explicitly state the novel aspects of the manuscript. Does the model propose a new mathematical framework, a unique interpretation of spacetime entropy, or an innovative approach to unifying quantum mechanics and GR?
    \item Compare and contrast the proposed model with existing literature, emphasizing its unique contributions and potential implications for future research.
\end{itemize}

\section*{4. Technical and Grammatical Issues}
The manuscript contains several typographical and grammatical errors that detract from its readability and professionalism. For instance, the title does not follow a consistent capitalization scheme, and there are likely other errors throughout the text.

\subsection*{Suggestions for Improvement:}
\begin{itemize}
    \item Ensure consistent capitalization in the title and throughout the manuscript. For example, either capitalize all major words (e.g., \textit{Statistical Gravity and Entropy of Spacetime}) or only the first word and proper nouns (e.g., \textit{Statistical gravity and entropy of spacetime}).
    \item Perform thorough proofreading to eliminate grammatical errors, improve sentence structure, and enhance clarity.
\end{itemize}

\section*{5. Additional Recommendations}
To further strengthen the manuscript, the authors may consider addressing the following points:
\begin{itemize}
    \item \textbf{Mathematical Rigor:} Ensure that all equations are derived and presented with sufficient detail. Provide physical interpretations for key mathematical results.
    \item \textbf{Physical Interpretation:} Discuss the physical implications of the proposed model. How does it impact our understanding of spacetime, entropy, and the quantum-gravity interface?
    \item \textbf{Future Directions:} Suggest potential applications or extensions of the model. Are there experimental or observational tests that could validate or refute the proposed framework?
\end{itemize}

The manuscript addresses a fascinating and challenging topic at the intersection of quantum mechanics and general relativity. However, to make a meaningful contribution to the field, the author must address the concerns outlined above. Specifically, he should clarify the reconciliation of quantum and classical aspects, demonstrate how uncertainty principles are satisfied, highlight the novelty of their work, and correct technical and grammatical errors.

Once these issues are addressed, the manuscript has the potential to make an impact on the ongoing discourse in theoretical physics. I look forward to reviewing the revised version and making a final decision based on the author's responses.

\begin{thebibliography}{9}

\bibitem{Karoly1966}
F. Károlyházy, \textit{Nuovo Cim. A}\textbf{42}, 390 (1966).

\bibitem{Author2007}
 M Maziashvili , \textit{Title of the Article}, \textit{Phys. Lett. B} \textbf{652}, 165–168 (2007).

\bibitem{Rovelli1993}
C. Rovelli, \textit{Class. Quantum Grav.} \textbf{10}, 1549 (1993).

\bibitem{Banerjee2010}
R. Banerjee, B. R. Majhi, \textit{Phys. Rev. D} \textbf{81}, 124006 (2010).
\end{thebibliography}

 

\end{document}

 

Comments for author File: Comments.pdf