The conformal transformation's controversy: what are we missing?

TL;DR

This paper explores the interpretation of conformal transformations as mappings between different geometrical spaces, arguing that they represent complementary descriptions of the same physical reality rather than distinct frames, and emphasizes the importance of conformal invariance for equivalence.

Contribution

It introduces a novel interpretation of conformal transformations as mappings between Riemannian and Weyl-integrable spaces, challenging the conventional notion of physical frame equivalence.

Findings

Conformal transformations relate complementary geometrical pictures of the same reality.

Conformal invariance of field equations is key to defining frame equivalence.

Brans-Dicke theory's frames are not equivalent under conformal invariance.

Abstract

An alternative interpretation of the conformal transformations of the metric is discussed according to which the latter can be viewed as a mapping among Riemannian and Weyl-integrable spaces. A novel aspect of the conformal transformation's issue is then revealed: these transformations relate complementary geometrical pictures of a same physical reality, so that, the question about which is the physical conformal frame, does not arise. In addition, arguments are given which point out that, unless a clear statement of what is understood by "equivalence of frames" is made, the issue is a semantic one. For definiteness, an intuitively "natural" statement of conformal equivalence is given, which is associated with conformal invariance of the field equations. Under this particular reading, equivalence can take place only if the metric is defined up to a conformal equivalence class. A concrete example of a conformal-invariant theory of gravity is then explored. Since Brans-Dicke theory is not conformally invariant, then the Jordan's and Einstein's frames of the theory are not equivalent. Otherwise, in view of the alternative approach proposed here, these frames represent complementary geometrical descriptions of a same phenomenon. The different points of view existing in the literature are critically scrutinized on the light of the new arguments.