Active and Passive Conformal Transformations in Scalar-Tensor Gravitational Theories

TL;DR

This work clarifies the conformal frames problem in scalar-tensor gravity by distinguishing active versus passive conformal transformations within a configuration-space formulation. It shows that matter Lagrangians are conformal form-invariant and that a covariant STG description is possible only if field-dependent parameters like ω(φ) transform appropriately under CT; neglecting this transformation leads to the conformal frames issue. The authors demonstrate that the active conformal approach yields physical consequences, including gauge freedom in conformal general relativity (ω=−3/2) that can reproduce or relate to GR in different gauges, and they derive a non-homogeneous continuity equation and a conformal-invariant formulation in terms of physical quantities. These results offer a coherent, covariant framework for interpreting conformal transformations and suggest observational tests via redshift modifications and galactic dynamics to probe conformal invariance. Overall, the paper resolves historical ambiguities by showing CFI stems from incomplete symmetry treatment and highlights the physical role of active CT in STG theories.

Abstract

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.

Figures (1)

• Figure 1: Drawings of the configuration space manifold ${\cal M}_\text{fields}$ (oval region in the plane $\phi-g_{\mu\nu}$). Each point in the manifold represents a vacuum gravitational state. In the left figure, the passive standpoint on the CT is illustrated. In this case the conformal transformation may be viewed as a "rotation" of the coordinate system $R:(\phi,g_{\mu\nu})$ in the plane $\phi-g_{\mu\nu},$ which leaves invariant the gravitational state ${\cal S}_\mathfrak{g}:(\mathfrak{g}_{\mu\nu})$, where the invariant metric reads $\mathfrak{g}_{\mu\nu}=(\phi/M^2_\text{pl})\,g_{\mu\nu}$. According to the active point of view on the CT (right figure), the conformal transformation represents an actual "motion" of the point $(\phi,g_{\mu\nu})$ in ${\cal M}_\text{fields}$. That is, it represents a real change of the gravitational state ${\cal S}_{\bf g}:(g_{\mu\nu},\phi)$$\rightarrow{\bar{\cal S}}_{\bf g}:(\bar{g}_{\mu\nu},\bar{\phi})$.