Conformal transformations and the conformal equivalence principle

TL;DR

This paper clarifies the meaning of conformal transformations in gravity theories, formulates a conformal equivalence principle, and explores its implications in a modified Brans-Dicke theory, emphasizing the physical equivalence of conformal frames.

Contribution

It introduces a conformal equivalence principle and analyzes its implications for conformally invariant gravity theories, clarifying the physical significance of different conformal frames.

Findings

Vacuum Brans-Dicke theory is not conformally invariant.

Different conformal frames are just different representations of the same physical phenomenon.

Conformal equivalence requires a consistent interpretation of conformal transformations.

Abstract

It is demonstrated that, unless the meaning of conformal transformations for the underlying geometrical structure is discussed on a same footing as it is done for the equations of the given gravity theory, the notion of "conformal equivalence", as it has been mostly used in the bibliography to date, lacks physical and mathematical content. A principle of conformal equivalence is then formulated and its consequences are investigated within a conformally invariant modification of Brans-Dicke (BD) theory of gravity. It is also demonstrated that vacuum BD theory ($\omega\neq -3/2$) is not conformally invariant and, consequently, the different conformal frames in which the theory can be formulated are just different - conformally related - representations of a same phenomenon. Hence these can not be compared: either both are physically correct, or none is it.