Active and Passive Conformal Transformations in Scalar-Tensor Gravitational Theories

Through considering the conformal transformations as coordinate transformations in the configuration space, where the different fields, including the metric $g_{μν}$ and the Brans-Dicke scalar field $ϕ$, are assumed as ``generalized coordinates,'' we introduce the notion of active and passive conformal transformations. We then apply both complementary approaches to the conformal frames issue, arising in the context of scalar-tensor gravity theories, to get a better understanding of the problem. Special focus is on the coupling of matter fields to gravity. The recent result that the Lagrangian density of fundamental matter fields and perfect fluids, in its standard form, is form-invariant under the conformal transformations, is taken into consideration in the discussion. We show that the passive conformal transformations do not provide a suitable framework to search for the physical consequences of conformal symmetry; meanwhile, the active conformal transformations do. We also discuss the physical and phenomenological implications of the scalar-tensor theory in the Jordan frame Brans-Dicke parametrization, which is form-invariant under the conformal transformations complemented with a suitable transformation of the coupling parameter $ω$. Although the vacuum case was known for a long time, here we generalize this framework to include matter fields. We argue that the conformal frames issue arises as consequence of two factors: 1) omission of the transformation of field-dependent parameters, as the coupling function $ω=ω(ϕ)$, under the conformal transformation of the fields, in particular of the scalar field $ϕ\rightarrowΩ^{-2}ϕ$, and 2) ignorance of the linear dependence of variations of the metric and of the Brans-Dicke scalar field, which is implicit in the invariance under infinitesimal conformal transformations.