Scale Transformations in Metric-Affine Geometry
Round 1
Reviewer 1 Report
Gravitational theories with torsion and nonmetricity have been the subject of renewed attention in recent times. When the connection is treated as independent from the metric and/or frame, there is much freedom in choosing its behavior under Weyl transformations. Different authors choose different realizations of the Weyl transformations, often without justifying their choice. The present paper examines this issue in a more systematic way and would be a useful addition to the literature, were it not for some incorrect or at least misleading statements made in section II.B.
Specifically, in equation (18d) it is postulated that the metric \eta remain invariant under a change in the frame field. This is a very bizarre transformation that does not correspond to the elementary notion of a linear transformation in a vector space. If the frame is subjected to a transformation, the components of the metric relative to that frame should also change accordingly. This would leave the metric g in (14), and the coordinate components of the connection, invariant. If one insists on transforming the frame and keeping eta_{ab} fixed, one could transform a metric g to any other metric. A theory that was invariant under such transformations would be a topological theory. I doubt that this is what the authors have in mind.
Another minor point is this: in equation (11) the transformation parameter is a vector. If this is to be viewed as a Weyl transformation, the vector has to be the gradient of a function.
These issues will have to be addressed before the paper can be accepted for publication.
Author Response
We would like to thank the referee for reading the manuscript and for the useful comments. Clarifications have been added to the manuscript accordingly.
After Eqs.(18) when introducing the general kind of transformation, we added the following explanation: "In this paper we focus on scale invariance, but one could generalise these considerations to more general transformations. However, a theory that would be postulated to be invariant under a completely general transformation such as (18), with no restrictions on $ \Lambda^a{}_b$, would have to be topological in the sense of being invariant under arbitrary transformations of the metric.
After equation (10) we mention that the projective transformation reduces to the (Weyl) gauge transformation only in the special case that the vector is a gradient. We return to this point in section 2.2.
Reviewer 2 Report
This is a paper dealing with the unfairly forgotten scale invariance in the community. The paper contains many technical details, but it is well explained and the physical impact of its results clearly communication. Overall, I think the paper would be a very good addition to Universe and should de published on its current form.
Author Response
We thank the referee for reading and assessing the manuscript.
Reviewer 3 Report
This manuscript discusses general scale transformations in metric-affine gravity. By scale transformations the authors mean the projective and the conformal transformations, as well as the frame rescaling.
Obviously, the subject is not completely original, since a scale-invariant theory of gravity has been pursued before in the literature, see e.g. Ref 7. However, there are novel features in this paper, that make it worth publishing.
The main part of the paper is written in great detail, providing a sufficient introduction to scale transformations (section 2) and an illustrative example (section 3) before the more complicated derivations and results in sections 4 and 5. The authors also provide clarifications with the notation (see several footnotes and the appendix), as well as very useful calculations in the three appendices, which will probably be used from now on, as a catalog for the scale invariant quantities.
Furthermore, they considered all the quadratic terms in torsion and non-metricity together with all their possible couplings and they derived the necessary conditions for them to be scale invariant.
Last but not least, the paper is written in the Palatini formalism, in contrast with others in the literature, making itself accessible to a more physically emanating audience, since it is much easier to use it in applications.
I totally recommend this paper for publication in its present form.
Author Response
We thank the referee for reading and assessing the manucript.
Round 2
Reviewer 1 Report
The authors have made minimal changes to answer the points I raised. Although I think that a more extensive revision would have been preferable, I consider the present form acceptable.
