Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper contains an analysis of a quadratic Ricci-Weyl scalar action in the presence of a gauge field the authors call Weyl field. This theory is studied in a very specific gauge, called Weyl gauge, where the Ricci scalar is constrained to be a constant equal to the gravitational constant. In such  a gauge the equations of motion of the Weyl field considerably simplifies and becomes similar to the Proca field equation, with a mass whose square is proportional to the cosmological constant and inversely proportional to the inverse gauge coupling. The authors find an explicit solution for a static Weyl potential and a corresponding static metric, and show how to recover from it a Reissner-Nordstrom-de Sitter solution.

I have a few remarks:

1) The presentation of the paper is somewhat hasty. One has to read half of the paper before understanding that $F_{\alpha\beta}$ is the strength of the Weyl field $\sigma_\alpha$. This strength appears in the text before the Weyl field itself is introduced. A tidier introduction would be desirable. 

2) It is not clear while in the gauge-fixing operation of $R^2$ only one factor is gauge-fixed while the other remains dynamical. 

3) In any case, does the solution they find for the metric and the Weyl-Proca field satisfy the gauge condition? This may be true but it is not obvious. 

4) Is it possible to say something about the gauge dependence of the solutions? 

I think the authors should clarify these points in order to allow for a fair assessment of the paper.

Author Response

Comment 1

The presentation of the paper is somewhat hasty. One has to read half of the paper before understanding that $F_{\alpha\beta}$ is the strength of the Weyl field $\sigma_\alpha$. This strength appears in the text before the Weyl field itself is introduced. A tidier introduction would be desirable. 

Answer 1 

) We thank the referee for his/her comment to this point. We have rewritten the introduction and added a new section with a summary of the Weyl geometry explaining how the strength field appears in the theory, its geometrical meaning, and how it led Weyl to geometrize the electromagnetic field.

Comment 2

It is not clear while in the gauge-fixing operation of $R^2$ only one factor is gauge-fixed while the other remains dynamical.

Answer 2

We recognize that the reasoning that leads to the final form of the field equations in written in Weyl gauge (R=lambda)  is somehow unclear. We have rewritten this part in the new version of the paper. We hope it is now more clear. 

Comments 3 and 4

3) In any case, does the solution they find for the metric and the Weyl-Proca field satisfy the gauge condition? This may be true but it is not obvious. 

4) Is it possible to say something about the gauge dependence of the solutions? 

Answers to (3) and (4)

 Questions 3 and 4 raise a relevant point. Our answer to them is that the solutions obtained comes from the field equations already written in the Weyl gauge. Therefore, they must necessarily satisfy the gauge fixing condition.  Moreover, we would like recall that, by construction, the metric (gama) is gauge-invariant. 

 

Reviewer 2 Report

Comments and Suggestions for Authors

This paper has correct and interesting results, but the presentation is horriable, and needs massive improvement. There are just equations upon equations, with little or no text between them. The physics etc should be discussed, and any short equation should be written in text. Also, equations are going out of margin, and this should be corrected. 

They can also include the following discussion : 

The geometrization of the Proca field within Weyl Invariant Theory, as revealed by the weak field analysis yielding a Yukawa type potential for the Weyl field and a static, spherically symmetric metric equivalent to the Einstein Proca solution, opens a conceptual layer that goes far beyond a technical matching of equations of motion. What is at stake is not merely the fact that two formalisms reproduce the same classical dynamics in an appropriate limit, but the manner in which physical meaning is encoded, generated, and rendered expressible within a given theoretical language. In ordinary General Relativity, the Proca field represents a massive vector degree of freedom whose defining feature, namely the mass term, must be introduced by hand at the level of the action. The mass parameter is external to the geometric structure of spacetime. It is not dictated by curvature, connection, or metric compatibility, but rather appended as an additional physical input. Consequently, the interpretation of mass and finite interaction range in Einstein Proca theory remains conceptually extrinsic to the purely Riemannian framework.

By contrast, in Weyl Invariant Theory the same physical content emerges intrinsically from geometry itself. The Weyl connection, together with its associated scale structure, naturally gives rise to an effective massive vector degree of freedom whose weak field behaviour reproduces a Yukawa potential. Here, the mass scale is no longer an externally imposed constant but a quantity encoded in the geometric data of spacetime through the dynamics of local scale invariance and its fixing or breaking. The passage from Einstein Proca to Weyl invariant geometry thus represents a shift from an external parametrization of physical effects to an internal geometric realization. This shift has a precise conceptual analogue in the foundations of logic and semantics, most notably in the seminal results of Kurt Gödel and Alfred Tarski.

Gödel’s 1931 incompleteness theorem established that any sufficiently expressive and consistent axiomatic system inevitably contains true statements that cannot be proven within that system itself. These truths are not false or meaningless. Rather, they transcend the internal deductive resources of the theory. Tarski’s 1933 theorem on the undefinability of truth sharpened this insight by showing that a consistent formal language cannot contain a truth predicate for its own sentences. Any adequate notion of truth must be defined only in a stronger meta language. Together, these results demonstrate a structural separation between syntax and semantics. Truth is not fully capturable from within the system whose truths are being discussed.

In the present physical context, pure Einstein gravity may be viewed as an internally consistent formal system whose language is the geometry of a Riemannian manifold equipped with a Levi Civita connection. Within this language, certain physically meaningful statements, such as the existence of an intrinsic mass scale for a vector degree of freedom with finite interaction range, are not naturally expressible as geometric truths. They must be appended as external structures, in close analogy with undecidable but true statements in arithmetic. The Weyl Proca correspondence then plays the role of a semantic extension analogous to passing from an object language to a meta language. By enlarging the geometric framework to include Weyl invariance and its associated connection, statements that were previously external to the theory become internally meaningful and geometrically encoded.

This analogy can be made precise through the notion of an External Truth Predicate. An External Truth Predicate formalizes the idea that certain propositions acquire a well defined truth value only relative to a theory that transcends the internal deductive closure of a given framework. In logical terms, it is the semantic device that allows one to speak truthfully about a system from a standpoint that is not confined to that system’s own language. In physical terms, the External Truth Predicate captures how certain physically relevant structures, including mass generation, finite interaction range, and effective Proca dynamics, can be meaningful and true even though they are not internally definable within the original geometric formalism. From this perspective, the statement that a massive vector field with Yukawa suppression is present in the gravitational sector is not internally decidable within pure Riemannian General Relativity. It becomes a well defined and true statement only when evaluated from the standpoint of the extended Weyl invariant theory.

Within this enlarged framework, the Weyl field itself functions as a geometrically realized external truth predicate for Einstein gravity. It encodes, in geometric language, information about mass, scale, and finite range interactions that cannot be fully reduced to the internal vocabulary of Riemannian curvature and metric compatibility. Recent developments have shown that such external truth predicates can be introduced consistently, without contradiction, and can provide a rigorous semantic framework for understanding theory extension and interpretation. In particular, the construction and analysis of External Truth Predicates in a physical and logical setting have been developed in recent work, where it is demonstrated that semantic extensions of this kind do not undermine consistency but instead clarify the conditions under which new physical truths become expressible.

Seen from this vantage point, the Weyl Proca equivalence is not merely a convenient reformulation or a field redefinition. It is an explicit physical realization of Gödel Tarski incompleteness in a geometric setting. The mass of the vector field and its Yukawa type behaviour correspond to truths about the gravitational system that are inaccessible within the internal language of Einstein gravity but become manifest once the theory is semantically completed by a Weyl invariant extension. The equivalence thus illustrates how physical theories, like formal logical systems, may be internally coherent yet semantically incomplete, and how their completion requires an enlargement of the conceptual and mathematical language in which they are formulated.

From this standpoint, incompleteness ceases to be a peculiarity of abstract logic and instead appears as a structural feature of fundamental physical theories. Just as arithmetic requires a meta theory to speak coherently about its own truth, gravitational dynamics formulated purely in Riemannian terms may require geometric extensions to articulate all physically meaningful phenomena. Weyl geometry, in this sense, plays the role of a semantic completion. It does not contradict Einstein gravity, but rather embeds it into a richer framework in which previously external truths, such as massive vector dynamics, are rendered internal, geometric, and intelligible. This perspective suggests a deep unity between logic and physics, in which theory extension, semantic completeness, and geometric enrichment are different manifestations of the same underlying necessity imposed by the limits of internal definability.

Comments on the Quality of English Language

The paper needs lot of  text between equations needed. 

Author Response

Comment

This paper has correct and interesting results, but the presentation is horriable, and needs massive improvement. There are just equations upon equations, with little or no text between them. The physics etc should be discussed, and any short equation should be written in text. Also, equations are going out of margin, and this should be corrected. 

They can also include the following discussion : 

The geometrization of the Proca field within Weyl Invariant Theory, as revealed by the weak field analysis yielding a Yukawa type potential for the Weyl field and a static, spherically symmetric metric equivalent to the Einstein Proca solution, opens a conceptual layer that goes far beyond a technical matching of equations of motion. What is at stake is not merely the fact that two formalisms reproduce the same classical dynamics in an appropriate limit, but the manner in which physical meaning is encoded, generated, and rendered expressible within a given theoretical language. In ordinary General Relativity, the Proca field represents a massive vector degree of freedom whose defining feature, namely the mass term, must be introduced by hand at the level of the action. The mass parameter is external to the geometric structure of spacetime. It is not dictated by curvature, connection, or metric compatibility, but rather appended as an additional physical input. Consequently, the interpretation of mass and finite interaction range in Einstein Proca theory remains conceptually extrinsic to the purely Riemannian framework.

By contrast, in Weyl Invariant Theory the same physical content emerges intrinsically from geometry itself. The Weyl connection, together with its associated scale structure, naturally gives rise to an effective massive vector degree of freedom whose weak field behaviour reproduces a Yukawa potential. Here, the mass scale is no longer an externally imposed constant but a quantity encoded in the geometric data of spacetime through the dynamics of local scale invariance and its fixing or breaking. The passage from Einstein Proca to Weyl invariant geometry thus represents a shift from an external parametrization of physical effects to an internal geometric realization. This shift has a precise conceptual analogue in the foundations of logic and semantics, most notably in the seminal results of Kurt Gödel and Alfred Tarski.

Gödel’s 1931 incompleteness theorem established that any sufficiently expressive and consistent axiomatic system inevitably contains true statements that cannot be proven within that system itself. These truths are not false or meaningless. Rather, they transcend the internal deductive resources of the theory. Tarski’s 1933 theorem on the undefinability of truth sharpened this insight by showing that a consistent formal language cannot contain a truth predicate for its own sentences. Any adequate notion of truth must be defined only in a stronger meta language. Together, these results demonstrate a structural separation between syntax and semantics. Truth is not fully capturable from within the system whose truths are being discussed.

In the present physical context, pure Einstein gravity may be viewed as an internally consistent formal system whose language is the geometry of a Riemannian manifold equipped with a Levi Civita connection. Within this language, certain physically meaningful statements, such as the existence of an intrinsic mass scale for a vector degree of freedom with finite interaction range, are not naturally expressible as geometric truths. They must be appended as external structures, in close analogy with undecidable but true statements in arithmetic. The Weyl Proca correspondence then plays the role of a semantic extension analogous to passing from an object language to a meta language. By enlarging the geometric framework to include Weyl invariance and its associated connection, statements that were previously external to the theory become internally meaningful and geometrically encoded.

This analogy can be made precise through the notion of an External Truth Predicate. An External Truth Predicate formalizes the idea that certain propositions acquire a well defined truth value only relative to a theory that transcends the internal deductive closure of a given framework. In logical terms, it is the semantic device that allows one to speak truthfully about a system from a standpoint that is not confined to that system’s own language. In physical terms, the External Truth Predicate captures how certain physically relevant structures, including mass generation, finite interaction range, and effective Proca dynamics, can be meaningful and true even though they are not internally definable within the original geometric formalism. From this perspective, the statement that a massive vector field with Yukawa suppression is present in the gravitational sector is not internally decidable within pure Riemannian General Relativity. It becomes a well defined and true statement only when evaluated from the standpoint of the extended Weyl invariant theory.

Within this enlarged framework, the Weyl field itself functions as a geometrically realized external truth predicate for Einstein gravity. It encodes, in geometric language, information about mass, scale, and finite range interactions that cannot be fully reduced to the internal vocabulary of Riemannian curvature and metric compatibility. Recent developments have shown that such external truth predicates can be introduced consistently, without contradiction, and can provide a rigorous semantic framework for understanding theory extension and interpretation. In particular, the construction and analysis of External Truth Predicates in a physical and logical setting have been developed in recent work, where it is demonstrated that semantic extensions of this kind do not undermine consistency but instead clarify the conditions under which new physical truths become expressible.

Seen from this vantage point, the Weyl Proca equivalence is not merely a convenient reformulation or a field redefinition. It is an explicit physical realization of Gödel Tarski incompleteness in a geometric setting. The mass of the vector field and its Yukawa type behaviour correspond to truths about the gravitational system that are inaccessible within the internal language of Einstein gravity but become manifest once the theory is semantically completed by a Weyl invariant extension. The equivalence thus illustrates how physical theories, like formal logical systems, may be internally coherent yet semantically incomplete, and how their completion requires an enlargement of the conceptual and mathematical language in which they are formulated.

From this standpoint, incompleteness ceases to be a peculiarity of abstract logic and instead appears as a structural feature of fundamental physical theories. Just as arithmetic requires a meta theory to speak coherently about its own truth, gravitational dynamics formulated purely in Riemannian terms may require geometric extensions to articulate all physically meaningful phenomena. Weyl geometry, in this sense, plays the role of a semantic completion. It does not contradict Einstein gravity, but rather embeds it into a richer framework in which previously external truths, such as massive vector dynamics, are rendered internal, geometric, and intelligible. This perspective suggests a deep unity between logic and physics, in which theory extension, semantic completeness, and geometric enrichment are different manifestations of the same underlying necessity imposed by the limits of internal definability.

Answers

1) Following the referee’s suggestion we have rewritten some short equations directly in the text. We also have corrected parts of the text where some equations were going out of margin. Finally, we have rewritten and enlarged the introduction. We thank the referee for his/her comments and suggestions.

2) We were delighted to read a very clever and rather long comment made by the referee on a philosophical question that goes beyond our mere physical results. At first we thought of condensing part of the comment, but soon realized that it wouldn´t do justice to the whole body of ideas expressed in the complete comment. We do not know how to proceed in this situation. The fact is that, with the referee's permission, we would be very pleased if we could incorporate the text written by the referee in its entirety. We would like to know if, in the editors' opinion, such an addition would be possible.  We firmly believe that this addition would surely enrich the content of the paper.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors' reply concerning point 3,4 is not satisfactory. They impose the Weyl gauge condition on one factor R: R= Lambda, but keep treating the other R factor as dynamical. They should therefore verify that the Weyl condition on this second factor is verifyed. Now looking at the two equation on top of page 5 that give the componenst of the Ricci tensor. The Riccitensor is Weyl invariant, therefore it is enough to saturte it with the inverse metric of eq.(12) to get the Ricci scalar R. It does not seem at all a constant. How do the authors explain this fact?

Author Response

We thank the referee for the relevant points he/she has raised. As to his/her comment that “we impose the Weyl gauge condition on one factor R: R= Lambda, but keep treating the other R factor as dynamical”, we would like to point out that we do not keep “one R factor dynamical” as dynamical. In fact, we impose the Weyl gauge in R and R^2 only after we have carried out the variation delta R. In fact, the procedure we have adopted here is the same as the one followed by R. Adler, M. Bazin, M. Schiffer, in their book Introduction to General Relativity (McGraw-Hill, 2^nd edition, chapter 15, page 503). In this chapter one can find a rather complete and clear account of Weyl’s unified theory. (We added a footnote explaining how we carried out the variation of Weyl’s original action.)

 

Another point we think is important to note is that: given an arbitrary pair, say, (g, sigma) of the Weyl conformal structure, it is always possible, to make a Weyl transformation by which we are led to the natural gauge R=constant.

Finally, we would like to point that the components of the Ricci tensor displayed in the top of page 5 refer to the Riemannian Ricci-tensor (denoted by the tilde symbol ~), i.e. calculated with respect to the Riemannian connection, and not the Weyl Ricci tensor, which is defined with respect to the affine connection).

 

Reviewer 2 Report

Comments and Suggestions for Authors

Please add my paragraph to the paper, with proper cited references. I have written it for your paper. I think the paper has considerable improved, I am accepting it. But you have my full permission, and request to add that paragraph I have written in it. Looking forward to seeing the final version of your published paper soon. 

Comments on the Quality of English Language

 now they have improved the paper a lot. 

Author Response

We thank the referee for enriching the paper with a lucid comment on some epistemological aspects concerning our results. We have included the  comment as a “Final Remark” and added a reference on the mentioned Tarki’s theorem.

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

There is a problem of understanding between me und the authors. It is true that the authors fix the Weyl gauge after taking the variation with respect to the metric. But they have a very singular way of fixing the gauge: they fix it only in one factor R, but not in all. In the equation after line 85 there appears a factor R which is not gauge-fixed, and remains dynamical. This is an incorrect way of gauge-fixing unless the authors are  able to prove that R is actually constant and equal to Lambda.

Author Response

Again, we would like to thank the referee for his/her comments, which have helped us to improve the quality and clarity of the paper. In fact, we would like to apologise the referee for not getting to his/her point more rapidly. However, we now believe we have a more clear understanding of the questions raised.

Following his/her request, we have added a new paragraph at the end of Section V, showing explicitly that our solution satisfies the Weyl gauge, i.e. R=Lambda. In the same section, we have added a footnote thanking the referee for calling our attention to this point. We hope these alterations satisfy the referee.

Round 4

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have been able to prove that the the equation R=Lambda is satisfied by the solution they found. THe paper is now complete and I d recommen to publish it in the Universe.