Some remarks concerning invariant quantities in scalar-tensor gravity

TL;DR

This paper clarifies the formalism of scalar-tensor gravity by analyzing invariant quantities under Weyl rescaling and scalar field redefinition, providing a theorem linking invariants to specific functional choices.

Contribution

It introduces a theorem connecting invariant objects to particular functional choices in scalar-tensor theories, and discusses the existence of translation rules.

Findings

Identifies objects invariant under Weyl rescaling and scalar redefinition.

Proves a theorem linking invariants to specified functionals.

Discusses the conceptual basis of translation rules in scalar-tensor gravity.

Abstract

The aim of the current paper is to clarify some aspects of the formalism used for describing the scalar-tensor gravity characterized by four arbitrary local functionals of the scalar field. We recall the objects that are invariant with respect to a spacetime point under the local Weyl rescaling of the metric and under the scalar field redefinition. We phrase and prove a theorem that allows to link such an object to each quantity in a theory where two out of the four arbitrary local functionals of the scalar field are specified in a suitable manner. Based on these results we phrase and reason the existence of the so called translation rules.