Transformation properties and general relativity regime in scalar-tensor theories

TL;DR

This paper analyzes how scalar-tensor theories of gravity transform between different parametrizations, especially in the context of the general relativity regime, ensuring the invariance of key physical concepts despite potential singularities.

Contribution

It provides a detailed analysis of the transformation properties of scalar-tensor theories and establishes conditions for consistent correspondence between different parametrizations.

Findings

The general relativity regime remains invariant under certain transformations.

Perturbed cosmological equations are consistent across parametrizations.

Singular transformations do not necessarily break the correspondence between theories.

Abstract

We consider first generation scalar-tensor theories of gravitation in a completely generic form, keeping the transformation functions of the local rescaling of the metric and the scalar field redefinition explicitly distinct from the coupling functions in the action. It is well known that in the Jordan frame Brans-Dicke type parametrization the diverging kinetic coupling function $\omega \rightarrow \infty$ can lead to the general relativity regime, however then the transformation functions to other parametrizations typically become singular, possibly spoiling the correspondence between different parametrizations. We give a detailed analysis of the transformation properties of the field equations with arbitrary metric and also in the Friedmann cosmology, and provide sufficient conditions under which the correspondence between different parametrizations is retained, even if the transformation is singular. It is interesting to witness the invariance of the notion of the general relativity regime and the correspondence of the perturbed cosmological equations as well as their solutions in different parametrizations, despite the fact that in some cases the perturbed equation turns out to be linear in one parametrization and nonlinear in some other.