A Possible Explanation of Dark Matter and Dark Energy Involving a Vector Torsion Field

Round 1

Reviewer 1 Report

Report on universe-1676265

Author(s): Graeme W. Milton
Title: A POSSIBLE EXPLANATION OF DARK MATTER AND DARK ENERGY INVOLVING A VECTOR TORSION FIELD

The author have presented the simplest gravitational model involving torsion for empty space and addressed a possible explanation of dark matter and dark energy. The analysis and the results are clearly presented in the paper. I would like to request the author for some minor modifications and comments on some points of the manuscript. After that, I believe the manuscript would be suitable for publication in Universe.

1. Regarding to dark energy and dark matter interacting model, I would like to ask the author to cite the following works:

   a) H. A . Borges and S. Carneiro, Friedmann cosmology with decaying vacuum density, Gen. Rel. Grav. 37 (2005) 1385-1394;

   b) C. Pigozzo et. al, Evidence for cosmological particle creation? JCAP 1605 (2016) 022;

   c) Humberto A. Borges and David Wands, Growth of structure in in- teracting vacuum cosmologies, Phys. Rev. D (2020) 10, 103519.

2. Possible typos:

   a) In equation (11) would be there is an equal to zero (=0);

   b) On the last term of equation (23) use calligraphic style for the subindex l.

   c) In the line between equations (28) and (29) there is a typo: may be it is any and not ant.

   d) Is there a term Tab in (54)? e) In (62) multiply k_1 by n_1^2; f) In (64) multiply k_4′ by n_4^2; g) In (68) multiply k_r by n_r^2; h) Include d/dr in (84);

   i) In the metric (97) the term sin^2θdφ^2 should be multiplied by σ^2;

3. Are the quantities k_4, k_z and k_θ defined in (69) the same as defined in (67)?

4. I suggest modify Robertson-Walker by Friedmann-Lemaître-Robertson- Walker.

5. Is the factor k, in the line-element (97) that describes a homogeneous and isotropic space-time, the usual constant curvature with the values k = −1, k = 0 and k = 1? Is the coordinates (r,θ,φ) time- independent? If the coordinates (r, θ, φ) are the comoving coordinates of a point in space and k is the constant curvature, the terms on the bracket should be time-independent metric of the 3-dimensional space with a constant curvature, as usual. But according to (97) it is not. Note that the spatial metric does not evolve with time when the uni- verse is flat k = 0. Is the metric (97) in the correct form?

Comments for author File: Comments.pdf

Author Response

I truly appreciate your thorough reading of the manuscript, your positive assessment, and for the many valuable comments especially those that went beyond simple typos. To respond:

(1) Thank-you for bringing my attention these interesting papers. I have referenced (b) and (c) as these seem most relevant to the paper.

(2) Picking up these typos shows how carefully you read the manuscript, and these have been corrected. Actually (e), (f) and (g) went beyond simple typos and subsequent conclusions had to be adjusted.

(3) Thanks! Indeed this could confuse the reader as they are different. Now primes have been added to the second set of constants, k' replacing k.

(4) Done.

(5) I'm glad you found this contradiction with the results for a flat universe (k=0). On closer inspection I found that I had made a sign mistake in the standard Ricci tensor (resulting from using a source that used a different sign convention). The correct analysis shows that k=-1 is the only option (within the unrealistic framework of a homogeneous universe).

There are other corrections made in response to the comment of another referee. Notably, at the beginning of section 7 I look at perturbations and find the predicted instability occurs even without allowing for "gravitational attraction".

Reviewer 2 Report

This paper is devoted to the study of the gravitational effects provided by a pseudovector or axial-vector torsion field. In particular, the author analyzes its possible role in the dark matter and dark energy problems.

Unfortunately, the theory with torsion presented in the paper is incomplete but its completion is well-known. Specifically, any theory of gravity formulated within a Riemann-Cartan space-time with curvature and torsion must provide a complete system of field equations describing the existing correspondence of the curvature and torsion tensors with their matter sources. In this regard, the field equation (33) presented in the paper only refers to one of the field equations of the well-known Einstein-Cartan model with curvature and non-propagating torsion (e.g. see F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976)), but the additional fundamental field equation related to torsion and the spin tensor of matter is missing. The consideration of this field equation is essential since the curvature and torsion tensors are independent quantities and accordingly they have their corresponding field equations. The same holds for the study of the gravitational effects in other relevant models, such as the Einstein-Maxwell or (more generally) the Einstein-Yang-Mills model, where the Einstein equations are not the only field equations of the model but it is also essential to consider the corresponding Maxwell or Yang-Mills equations.

Therefore, I suggest the author to address this fundamental incompleteness and make a major revision. Otherwise I cannot recommend the paper for publication in Universe.

Author Response

Thank-you for looking at the paper. As acknowledged, while it is incomplete, it does provide equations for empty space and this is perhaps the first starting point as Einstein's equations in empty space could have easily been guessed by a similar approach. While you say the completion is well known I do not believe that is the case in the sense that other theories do not have these equations in empty space.

Reviewer 3 Report

In the manuscript the author investigates a gravitational theory in the presence of the torsion. A system of generalized Einstein equations is proposed in Eqs. (32), and their implications are investigated in some detail. The manuscript may be publishable in Universe if the author would fully consider the following points:

1. In Section 5, when discussing the weak field approximation, the authors does not recover the Poisson equation of the Newtonian gravity. Is it possible to obtain it in the present theory? Does the g_{00} component of the metric tensor have a Newtonian limit that can be related to the non-relativistic gravitational potential?

2. In Eq. (101) the author obtains a linear time dependence of the scale factor A. However, such a scale factor cannot account for the accelerated expansion of the Universe. The discussions in Section 10 are not particularly enlightening about this aspect, and it is clear at all if the present model incorporates dark energy or dark matter, This issue must be discussed in more detail, and the advantages/limitations of the model must be clearly stated. 

Author Response

Thank-you so much for your positive assessment and for the helpful comments you made. In response:

(1) This is an important point. One that also may not be clear to other readers. Accordingly, after (35) I mention the point that if the torsion field is small enough we recover Einstein's original equations to a good approximation and hence those of Newtonian gravity. Note the remark at the end of section 10 that one can expect the strength of the torsion field to be small. It is even small near the black holes in the extended Schwarzschild solutions (but not in figure 4d which seems unrealistic and does not have a critical black hole radius but rather just a singularity at the origin).

(2) Indeed, I agree that the discussion in section 10 was rather handwaving. Accordingly I have now, at the beginning of section 7, before section 7.1, examined the stability of the torsion field equations in the weak field approximation, and find that instability occurs both in subluminal and superluminal regions. This confirms the expectation that spacetime is inhomogeneous within the model, even without accounting for ordinary gravitation acting on fluctuations in the effective energy and mass associated with the torsion field. I believe that going further would require numerical calculations that are beyond my expertise.

Round 2

Reviewer 2 Report

The author has not accomplished a major revision concerning the fundamental issues pointed out in the first report, but still displays an incomplete description of the torsion field, which turns out to be crucial for the analysis performed in the manuscript. In particular, "the new equation" presented in the Expression (32) of the manuscript is just the well-known first equation of the Einstein-Cartan theory (also known as the Einstein-Cartan-Sciama-Kibble or just the Cartan-Sciama-Kibble theory) in the presence of an axial-vector torsion field, e.g. see the first field equation of Expression (4.110) on page 133 of "T. Ortin, Gravity and Strings, Cambridge University Press (2004)" for a torsion tensor determined by an axial-vector. The joint field equation associates the torsion tensor with the intrinsic angular momentum of matter, and it has a deep implication: torsion is inextricably tied to matter and vanishes in empty space-time. In other words, there is no difference between General Relativity and the Einstein-Cartan theory in empty space-time (see also F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976) and A. Trautman, Encyclopedia of Mathematical Physics, Elsevier, vol. 2, 189 (2006)). Thus, the consideration of the first part of the Einstein-Cartan theory without the corresponding description of the torsion field can give rise to the fundamental unconsistency of a torsion field providing physical effects in empty space-time, as is claimed and performed in the manuscript. This unconsistency is clearly in opposition to all the aforementioned works.

Consequently, I cannot recommend the paper for publication in Universe.

Author Response

Thank-you for clarifying the remarks you made in your first report, and for providing additional references (that are now incorporated in the manuscript). I have now drawn attention to the fact that the Einstein-Cartan-Sciama-Kibble theory implies no torsion in empty space, but that other theories allow for torsion in empty space.