Energy Conservation Law in the Closed Universe and a Concept of the Proper Time

Round 1

Reviewer 1 Report

The paper under consideration concerns some dynamical aspects of the homogeneous anisotropic model of the Bianchi-IX universe. The modification of the dynamic of the universe obtained seems to be new and worth to note. In consequence, I consider this work sufficiently interesting to be published in Universe. However, I have some remarks concerning the formal form of the paper. Namely, the Lagrangian in eq. (5) contains terms $\dot s-\dot s$ instead of $\dot s-\epsilon$. Next, the notations $p_\beta\pm$ and $p\pm$ in eqs. (7) and (10) are inconsistent. Please check the signs in eq. (18). Which equation of motion gives formula (19). Finally, there are question marks (at least in my pdf file) instead of references and equations numbers. In view of this the readability of the paper could be improved.

Author Response

1. We corrected a typo in equation (5).
2. We removed an extra subscript beta in (7).
3. The signs in (18) have been fixed; thanks. 
4. We apologize for the question marks instead of references. It was not our fault. We have sent an e-mail to the journal, and they admitted that it was their oversight, so they promised that the correct version would be uploaded. 

Reviewer 2 Report

In the version I received all the references and citations along the paper appear as question marks, probably because some problem with the compilation of the source file. It is difficult to read the paper in this way, so I would be grateful to receive a corrected version in order to be able to check it properly.

Concerning the scientific content of the paper, I have two major questions:

1/ I don't understand completely the derivation of the action (5). In particular, the authors state that "We find the Euler-Lagrange equation for s and subsequently add it to the initial action, multiplying by corresponding Lagrange multiplier." Which Lagrange multiplier is this one? Note that there is not any new variable in (5). It might be helpful if the authors could explain this derivation more explicitly. In addition, in the expression (5) there are a couple of terms that go like (\dot{s}-\dot{s}), is this a typo or, otherwise, why are these terms nonvanishing?

2/ After (18) the authors state that they will deform the dynamics given by general relativity (GR), since they state that "we assume that this mass is nonzero and see what this leads to in classical theory". What is the motivation to do so? Is the theory they are considering better than GR in any sense? And what kind of effects or improvement respect to GR are they seeking?

Author Response

We apologize for the question marks instead of references. It was not our fault. We have written to the journal, and they admitted that it was their oversight, so they promised that the correct version would be uploaded. 

1. The Lagrange multiplier in (5) is an infinitesimal shift of the proper time s, and an additional condition for it – the corresponding Euler-Lagrange equation for s. That is how we calculate EL equations – by means of an infinitesimal variation of a variable in action. Self-evident is the fact that for the time, this equation is the energy conservation law, and the appearance of the multiplier (12) in functional integral is attributable to it. We apologize for the typo in (5), which we removed. 
2. Our goal is an attempt to solve the problem of time, which appears when quantizing just GR. This requires further investigations. In this communication, we discuss some consequences that appear already in classical theory, with the addition of an extra parameter – "mass of the universe." This quantity changes the dynamics of the universe near zero-radius considerably. At the next stage, it is necessary to investigate the influence of this "mass" heterogeneities. At this moment, it is not easy to judge some benefits of introducing a new parameter. However, we emphasize that if we put this parameter equal to zero, we will have the theory's correspondence with the initial GR.  

Round 2

Reviewer 2 Report

The authors have clarified the questions from my previous report.