The Unknown Face of Scalar-Tensor Gravitational Theories

Abstract

It has long been demonstrated that the vacuum scalar-tensor theory in the Jordan-frame Brans-Dicke parametrization is form-invariant under conformal transformations, provided that a suitable transformation of the coupling parameter $ω$ is applied. Here, we generalize this framework to include the coupling of matter fields to gravity. We take into consideration the recent result that, for point-dependent masses transforming as $m\rightarrowΩ^{-1}m$ under the conformal transformations, the Lagrangian density of fundamental matter fields and perfect fluids is conformal form-invariant. We demonstrate that the conformal frame issue, that arises in the context of scalar-tensor gravity theories, is a consequence of two factors: i) the omission of the transformation of field-dependent parameters, such as the coupling function $ω=ω(ϕ)$, under the conformal transformation of the fields, and ii) the ignorance of the Ward identity due to conformal form-invariance of the Lagrangian density of matter, which leads to an incorrect Klein-Gordon-type equation of motion for the Brans-Dicke field. By considering the conformal transformations as coordinate transformations in the configuration space, where the metric $g_{μν}$, the Brans-Dicke scalar $ϕ$, and $N$ matter fields $χ=\{χ_1,χ_2,...,χ_N\}$, which are coupled to gravity, are assumed as ``generalized coordinates,'' we introduce the notion of active and passive conformal transformations. We demonstrate that passive conformal transformations do not represent a suitable framework for exploring the physical consequences of conformal symmetry; in contrast, active conformal transformations do.