Round 1

Reviewer 1 Report

Referee report on article entitled "Physical Justifications and Possible Astrophysical Manifestations of the Projective Theory of Relativity" by Jacques L. Rubin

The article is devoted to justification of an alternative approach to General Relativity. The author suggests to apply a projective approach which is based on geometrical reduction from higher-dimensional spaces.

So the topic of the study is completely relevant for the current theoretical research. In fact, models with higher dimensions are rather popular especially from the string theory point of view. And the reduction to the 4-dimensional space can be performed in different ways. The paper consists of several Sections which are not strongly enough connected to each other. There is no any Introduction at all. It should include some general words about the statement of the problem and, in particular, explain how different sections of the article are connected.

The first Section is devoted to the description of relativistic systems of references used in modern GPS systems. It looks like a mini-review of the known material. But it is completely unclear whether anything new is there. And the relation of thissections to the rest of the paper is not explained at all.
Sections 2 and 3 are devoted to geometrical foliations in higher-dimensions. New results here are not clearly specified and comparisons with alternative approaches existing in the Literature are not done. These sections might be interesting for readers, if some more physical motivation and consequences are provided. In Section 4 the author introduces Eq.(4) which represents a modification of the Newton's gravitation law. First of all, the origin of the equation is completely unclear. It is just said that it can be shown within the give projective approach. But such
things should be shown explicitly within the paper. Moreover, such a form of gravitational force is simply impossible, because the force does not fall when the distance goes to infinity. This behavior of gravity contradicts everything that we know about forces in Nature. Obviously the introduced
modification can help to describe the rotational curves of spiral galaxies (as shown in Fig. 4). But formula (4) with parameters fixed from the rotational curves (with nonzero value for \beta) leads to completely impossible gravitational effects at larger distances. The authors should either prove that the so strange asymptotic behavior doesn't make problems with astrophysical observables, or make the proper correction to formula (4).

For the above reasons I can not recommend the paper for publication.

Author Response

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Replies and Comments to Reviewer 1 report on article entitled "Physical Justifications and Possible Astrophysical Manifestations of the Projective Theory of Relativity" by Jacques L. Rubin 

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Reviewer 1: The article is devoted to justification of an alternative approach to General Relativity. The author suggests to apply a projective approach which is based on geometrical reduction from higher-dimensional spaces. 

Author's reply: Reviewer 1 wrote: "The author suggests to apply a projective approach which is based on geometrical reduction from higher-dimensional spaces." The claim in this sentence is incorrect. 

First, because projective geometrical approach does not define any geometrical or physical properties of spacetime associated with any processes of "geometrical  reduction;" even if projective spaces are defined from quotient spaces which are not reduced spaces from higher-dimensional spaces but defined from equivalence relations. The process of "quotienting" is associated with no physical or geometrical processes in spacetime. Actually, we consider sets of lines in five-dimensional spaces which are clearly and strongly distinct from sets of points in that same five-dimensional spaces. And theses sets of lines are sets of points in a four-dimensional spaces, namely, the four-dimensional real projective spaces. "Reduction" would mean to consider sets of points rather than lines in five-dimensional spaces.

Second, by projective approach we means "projective geometry" approach of spacetime. Most of the models of spacetime, if not all,  consider basically that spacetime has a Euclidean geometry, i.e., a geometry satisfying the axioms of Euclidean geometry. And then, providing a metric field we get a pseudo-Riemannian structure on spacetime. In the present projective geometry approach,  spacetime has a projective geometry, meaning that the axioms of projective geometry are satisfied. And then, providing metric fields we can defined also pseudo-Riemannian structures compatible with this projective axiomatics. Then spacetime remains a four-dimensional manifold thought five-dimensional "intermediate" (not-projective) spaces are necessary to implement via their lines its four-dimensional projective axiomatics; in particular the four-dimensional projective frames differing strongly from the four-dimensional Euclidean frames. This projective geometry in the four-dimensional spacetime manifold of events differs completely from the well-known projective geometry in the three-dimensional projective space of velocities in relativity.

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Reviewer 1: So the topic of the study is completely relevant for the current theoretical research. In fact, models with higher dimensions are rather popular especially from the string theory point of view. And the reduction to the 4-dimensional space can be performed in different ways. 

Author's reply: Reviewer 1 wrote: "In fact, models with higher dimensions are rather popular especially from the string theory point of view. And the reduction to the 4-dimensional space can be performed in different ways." From previous author's reply above, the present model is not a higher-dimensional model and not a particular string theory. Moreover, string models are based on pseudo-Riemannian spacetime manifolds satisfying Euclidean axiomatics, not projective geometry axiomatics. Then, none of them are associated with projective geometries of spacetime.

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Reviewer 1: "The paper consists of several Sections which are not strongly enough connected to each other. There is no any Introduction at all. It should include some general words about the statement of the problem and, in particular, explain how different sections of the article are connected."

Author's reply: The document has been improved according to the instructions given by the referee.

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Reviewer 1: "The first Section is devoted to the description of relativistic systems of references used in modern GPS systems. It looks like a mini-review of the known material. But it is completely unclear whether anything new is there. And the relation of this sections to the rest of the paper is not explained at all."

Author's reply: Reviewer 1 wrote: "The first Section is devoted to the description of relativistic systems of references used in modern GPS systems." There are no 'relativistic positioning systems' technologically realized and maintained by any international institution or state at present days. 'GPS' is the official name of a particular (not relativistic) positioning system; the one realized and maintained by the Department of Defence of the USA. The term GPS is not generic and it is not a 'reference system,' equivalentely, 'system of reference' (which could be relativistic or not). The expression "relativistic systems of references used in modern GPS systems" cannot be correct because there are radical, strong differences between 'positioning systems' and 'reference systems.' Moreover the letter "S" in "GPS" means "System" and the expression "GPS systems" is a non-sense. Also, the present days positioning systems, i.e., "modern positioning systems" are not relativistic; none of them are relativistic. And, moreover, write "relativistic systems of references used in modern GPS systems" is clearly again a non-sense because positioning systems such as GPS are not 'systems of reference.' 

The first section is devoted to new positioning systems not implemented nowadays and explicitly relativistic contrarily to the present day ones. 'Relativistic positioning systems' might be implemented in a few ten years by certain states, federations or communities of states. It is the next, second generation of 'positioning systems.' GPS, GALLEO, GLONASS, BEIDOU, etc. which are not relativistic constitute the first generation of 'positioning systems.'

The author has succeeded for the first time in designing 'relativistic localizing systems ' that are operational extensions of 'relativistic positioning systems.' These new localizing systems not only provide positioning but also allow the localization of events in space-time, which positioning systems cannot do as a matter of principle. And from this result, it turned out from the very fundamental viewpoint that the geometry of space-time cannot be Euclidean (with a compatible pseudo-Riemannian structure) but must necessarily and mandatorily be a projective geometry instead of the Euclidean. This result is demonstrated by the exclusive use of causal axiomatics for spacetime such as the one designed historically by Kronheimer and Penrose.

May be one or two pages in the paper for a longer introduction should be necessary to explain all the details but the editorial constraints to limit the number of pages to ten, references included,  makes it difficult, as usual, if not impossible to make a clearer article for readers without significant pre-requisites on the subject.

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Reviewer 1: "Sections 2 and 3 are devoted to geometrical foliations in higher-dimensions."

Author's reply: This assertion is incorrect. These sections are devoted to geometrical foliations in the four-dimensional (and not in higher-dimensions) spacetime manifolds due to the group action of homographies compatible with the pseudo-Riemannian structure.

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Reviewer 1: "New results here are not clearly specified and comparisons with alternative approaches existing in the Literature are not done. These sections might be interesting for readers, if some more physical motivation and consequences are provided."

Author's reply: The presentation of comparisons with alternative approaches to this theory would be the subject of an article of certainly more than ten pages in length. The question  is whether or not spacetime satisfies or not the axiomatics of projective geometry: either this axiomatics is verified via the actions of homographies or it is not. The people who initiated this theory are indicated in the article references and in the introduction (Veblen, Hoffmann, etc.). Then, the theories differ only in the way they are made compatible with (pseudo-)Riemannian structures.  I have added references on alternative theories to those of Veblen et al, in particular those due to Arcidiacano-Fantappié and then those of Chiatti and Licata et al.

We can note that the great mathematician Oswald Veblen created the axiomatics of projective geometry and that he is one of the founders of the 'projective theory of relativity' which is clearly not a string theory. This projective theory of relativity was completely forgotten mainly because of its relative mathematical complexity and because the physical observables were not determined at the time this theory was developed. The 'relativistic localizing systems' the author designed provide explicitly the physical observables; and this is completely new and makes it possible to get this theory out of strict mathematical theory.

Moreover, physical motivations are not the only ones to be taken into account in theoretical physics. Mathematical motivations or simple mathematical exploration allow us to discover properties that can have consequences in physics. Our approach was exploratory at the level of the foliation in the four-dimensional spacetime manifold induced by the homographies to finally obtain as a consequence a structure similar to that of black holes; which is clearly a possible physical consequence.

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Reviewer 1: "In Section 4 the author introduces Eq.(4) which represents a modification of the Newton's gravitation law. First of all, the origin of the equation is completely unclear. It is just said that it can be shown within the give projective approach. But such things should be shown explicitly within the paper. "

Author's reply: The origin of equation (4) and its proof have been given in ref. 21: Rubin, J.L. Applications of a Particular Four-Dimensional Projective Geometry to Galactic Dynamics. Galaxies 2018, 6, 83,  DOI:10.3390/galaxies6030083. 

It is impossible to prove this formula in less than few pages. Its origin comes from invariance with respect to the non-linear homographic actions compatible with rotational invariance and time and space splitting. It is the simplest possible expression of a modified Newton force compatible with projective geometry. There are much complex expressions. We give the simplest one. Besides, we can note that because projective geometry is locally an affine geometry to which points at  infinity are added, most of the definitions of projective geometrical objects are dependent on a common, given and arbitrarily fixed origin contrarily to geometrical objects in Euclidean geometry. It is also one of the reasons that makes physical interpretations in the projective framework complex compared to Euclidean framework. 

In other words, this formula (4) is not universal.  This lost of universality manifests with parameters alpha and beta that depend on geometric/physical characteristics at the origin of the Cartesian coordinates chosen to express this formula (4). This also means that this origin has a physical content. The fits of galaxies were done considering that this origin coincides with the galactic center. In other words, these parameters depend on large scale structures. The latter can be mass distributions or spacetime curvature at the galactic center which could be considered somehow as a "geometrical synthesis" of the mass distribution around this center.

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Reviewer 1: "Moreover, such a form of gravitational force is simply impossible, because the force does not fall when the distance goes to infinity. This behavior of gravity contradicts everything that we know about forces in Nature. Obviously the introduced modification can help to describe the rotational curves of spiral galaxies (as shown in Fig. 4). But formula (4) with parameters fixed from the rotational curves (with nonzero value for \beta) leads to completely impossible gravitational effects at larger distances. The authors should either prove that the so strange asymptotic behavior doesn't make problems with astrophysical observables, or make the proper correction to formula (4)."

Author's reply: Theory must be submitted to observations not the contrary. In the present case, the observations are those we obtain providing among the best possible fits of galactic rotation curves. Wojnar et al. (ref.22) obtained also extremely good fits and, moreover, without any a priori theoretical starting points. These observations cannot therefore be dismissed as being nothing. The framework is different, namely, it is a projective framework with new concepts and new geometrical principles/interpretations. In particular,  considering that the alpha and beta parameters depend on, for instance, the curvature, this means that we have a kind of feedback process: first, the universal Newton's law bends spacetime, and then, the resulting non-vanishing local curvature modifies the Newton's initial law to remain compatible with projective geometry. It is rather the opposite that could surprise us: that the change in spacetime curvature has no effects on the universality of certain physical laws. Indeed, why does the Newton's law not change with topological/geometrical changes induced by the curvature changes it produces? Roughly speaking, universality would refer to Euclidean aspects and Newton's law is clearly expressed only within the framework of Euclidean geometry.

Moreover, projective spaces are topologically compact spaces meaning that the projective geometry cannot be considered with an infinite extension from the chosen origin imposing a limit on the modifications of the Newton's law. Therefore, such modifications could be unavailable at very large scales/distances. As I indicated in previous papers, we have a 'generalised Cartan space(time)' locally homeomorphic to local four-dimensional projective spaces.

To finish, the "astrophysical observables'' are particular spacetime observables, the latter defined from 'relativistic localizing systems' indicating that the spacetime geometry must be projective and not Euclidean.

Reviewer 2 Report

Well written, a paper of great interest on a theme unfortunately little known. I sggest to the author for completeness, to add in the historical background, in  biblio also the projective de Sitter Relativity, developèed in the 50s and  60s by Italian mathematicians, see for ex:

De Sitter Projective Relativity

  Licata, Ignazio, Chiatti, Leonardo, BENEDETTO, ELMO

Springer 2017

and outline the essential differences between the two approaches.

Author Response

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Replies and Comments to Reviewer 2 report on article entitled "Physical Justifications and Possible Astrophysical Manifestations of the Projective Theory of Relativity" by Jacques L. Rubin 

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Reviewer 2: "Well written, a paper of great interest on a theme unfortunately little known. I suggest to the author for completeness, to add in the historical background, in  biblio also the projective de Sitter Relativity, developed in the 50s and  60s by Italian mathematicians, see for ex:

De Sitter Projective Relativity

  Licata, Ignazio, Chiatti, Leonardo, BENEDETTO, ELMO, Springer 2017

and outline the essential differences between the two approaches."

Author's reply: This reference was added in the bibliography as well as a little note in Section 1 about de Sitter Relativity. Other references are added for Fantappié, Arcidiacono works. In addition, the last section 5 is considerably improved and sentences are added at the end of the previous sections to improve the links between them.

Reviewer 3 Report

It is not clear what is added to the published paper in Galaxies August 2018?

Author Response

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Replies and Comments to Reviewer 3 report on article entitled "Physical Justifications and Possible Astrophysical Manifestations of the Projective Theory of Relativity" by Jacques L. Rubin 

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Reviewer 3: "It is not clear what is added to the published paper in Galaxies August 2018?"

Author's reply:  What is new is about the physical interpretations of the modified Newton's law which losses its universality because of projective geometry. This is indicated in the last Section 5 which has been modified. Moreover, the principles of relativistic localizing systems are presented Section 1 in a more synthetic way to understand the deep meaning of the localizing process.

Round 2

Reviewer 1 Report

Yes, the second version of the manuscript is improved. Main remarks of the first report were taken into account. The presentation of results is still somewhat obscure. But the article might be interesting for readers.