Relations between Newtonian and Relativistic Cosmology

Round 1

Reviewer 1 Report (Previous Reviewer 2)

Comments and Suggestions for Authors

The author added the references I requested, so I recommend the paper for publication in its present form.

Comments on the Quality of English Language

Minor editing.

Reviewer 2 Report (Previous Reviewer 1)

Comments and Suggestions for Authors

I recommend the manuscript for publication in its present form. No further review is required.

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

To my understanding, the article aims to point out some apparent connections (or formal analogies?) between certain equations in Newtonian mechanics and some corresponding equations in relativity. The article is not a single story, but a collection of disconnected observations. In this report, I will focus on the first section, where the author claims to have an alternative "derivation" of the Schwarzschild metric. In relation to this, they state the following:

 

"This innovative approach elucidates how gravity influences the trajectories of objects within regions of intense gravitational fields. By integrating the tenets of special relativity into our analysis, we have deepened our comprehension of the intricate interplay between matter, energy, and the fabric of spacetime."

 

I disagree with this conclusion. The author did not provide a derivation, in the mathematical or physical sense of the term. A derivation is a sequence of logical steps that should INEVITABLY lead to the conclusion. This is not the case for the sequence of steps followed by the author, which are motivated mostly by aesthetics and notation. 

Here are some ambiguities and inconsistencies.

 

 

1) The result of the author depends on the choice of coordinates, in particular, of the coordinate r. Suppose that someone expresses the metric (4) using a different definition for r. For example, one may notice that the coordinate r in the metric (4) is not the distance from the center, but the radius of the spheres at constant distance. Thus, they may feel that it is more natural to change coordinates, and redefine r so that B=1. Then, they may use this definition of r in equations (1,2,3), and they would get the wrong result. The author did not provide any fundamental criterion for deciding which r is the "correct one". 

 

2) The way in which special relativity is included is also ambiguous. Between equations (1) and (2), simply replacing the Newtonian acceleration with the proper acceleration is not an obvious step. For example, let us replace the gravitational potential with the electric potential. Then, from relativistic electrodynamics we know that, if on the left-hand side we use the proper acceleration, on the right-hand side we must multiply the gradient of the electric potential by a Lorentz factor. Thus, someone could include a factor dot{t} on the right-hand side of (2), and they would obtain the wrong result. The author did not clarify why such factor dot{t} should not be included.

 

3) The author is looking for solutions that describe the metric of a "point mass". However, the actual nature of the mass distribution only enters in the last step, namely in equation (14). If we trust the reasoning of the author, then the whole argument should remain true also in the interior of a relativistic star up until equation (13). In particular, one should be able to conclude that, inside any astrophysical object, AB=1. However, that is not the case. The product AB inside a relativistic star does not equal 1.

 

4) Ultimately, we know that the whole argument MUST be flawed. Here is a proof by contradiction. Suppose that the argument is right. Nowhere in the argument, the validity of Einstein's equations has been assumed. Hence, the main result must hold also within alternative theories of gravity, where deviations from general relativity are included. However, it is well known that most alternative theories of gravity predict different black holes. Hence, we have a contradiction.

 

 

The rest of the manuscript discusses similar arguments in cosmology. I could not understand this part, and the author did not single out any particularly relevant achievement. However, it should be noted that I am not a cosmologist, and thus I may not be able to appreciate certain subtleties. For these reasons, my initial recommendation is rejection. However, if the other referees find the results interesting enough to warrant publication, I may reconsider my decision, provided that the author removes the part on black holes.

 

 

Author Response

Dear referee I've attached my reply.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The author provides a comparative study between Newtonian cosmology and relativistic cosmology. The author shows how one can extract the Friedmann equations from Newtonian dynamics. The case with a scalar field, open systems, and perturbations in the presence of a perfect fluid have all been discussed. The research topic is an active one in cosmology, and the paper is well written. However, in contrast to what is stated in the conclusion, where it is said: "... we have deepened our comprehension of the intricate interplay between matter, energy, and the fabric of spacetime," the results provided here do not seem to bring any substantial novelty into the physics of modern cosmology. Having only showed how the main equations of cosmology could be obtained using Newtonian dynamics, a procedure that was pioneered long ago by Milne and McCrea, does not in itself bring any new understanding of spacetime. Still, I would like to recommend the manuscript for publication as it might give rise to useful generalizations in the future and a deeper study of its limitations by interested readers. Therefore, the only suggestion I offer for improvement is to make the historical background provided by the author more complete by citing the classic work of Milne and McCrea: "W. H. McCrea and E. A. Milne, Quart. J. Mathematics, old series 5, 73 (1934)."

Comments on the Quality of English Language

A few typos here and there need to be fixed.

Author Response

Dear referee in the new version I've cited the following works related with Newtonian cosmology:

1.-W.H McCrea and E. A. MIlne "Newtonian universe and the curvature of space", Q.J. Mathematics, old series 5, 73 (1934).

2.- C. Callan, R.H.  Dicke and J.P. Peebles, "Cosmology and Newtonian mechanics",  Am. J. Phys. 33,  105-108 (1965).

Reviewer 3 Report

Comments and Suggestions for Authors

Fine paper for a) approximation needs and b) prune the GR geometry for any needs of quantum gravity.

Author Response

Dear Referee, thanks for your comments.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author did not address any of my previous concerns. Hence, I cannot recommend the paper for publication. See my comments below.

 

1) Introducing a new coordinate R(r) such that dR=B(r)dr would still lead to the Schwarzschild solution, but in a different coordinate system. In such a coordinate system, stating that the Newtonian potential is Phi(R) would lead to wrong results. The problem still holds.

 

2) The author did not really address my concern. In my previous report, I offered a clear counterexample to the author's reasoning. The author did not attempt to provide a response to it.

 

3) The author did not address my point. I am aware that their goal is to study the exterior of the star. However, it should be clear from the argument itself why I cannot apply the same reasoning to the interior of the star. As it turns out, no matter what the original intention of the author was, one can still decide to apply exactly the same logic all the way up to equation (13) also inside the star. They would get the wrong result. This shows that the reasoning must be flawed.

 

4) The reply of the author is incorrect. The geodesic equation is the same in most theories of gravity. What changes in alternative theories of gravity is the way we compute Phi for a given distribution of mass. Hence, my concern remains.

 

 

Author Response

According to the referee's concerns and considering the last paragraph of the first report, I've removed the entire section on the Schwarzschild metric.