Invariant quantities in the scalar-tensor theories of gravitation

TL;DR

This paper develops a framework for constructing invariant quantities in scalar-tensor theories of gravity, enabling consistent formulation and comparison of physical predictions across different conformal frames and parametrizations.

Contribution

It introduces a scheme to identify invariants under transformations, facilitating frame-independent analysis of scalar-tensor gravitational theories.

Findings

Invariant functions of the scalar field are constructed.

Formulation of the theory in terms of invariants is demonstrated.

Cosmological solutions are compared across Einstein and Jordan frames.

Abstract

We consider the general scalar-tensor gravity without derivative couplings. By rescaling of the metric and reparametrization of the scalar field, the theory can be presented in different conformal frames and parametrizations. In this work we argue, that while due to the freedom to transform the metric and the scalar field, the scalar field itself does not carry a physical meaning (in a generic parametrization), there are functions of the scalar field and its derivatives which remain invariant under the transformations. We put forward a scheme how to construct these invariants, discuss how to formulate the theory in terms of the invariants, and show how the observables like parametrized post-Newtonian parameters and characteristics of the cosmological solutions can be neatly expressed in terms of the invariants. In particular, we describe the scalar field solutions in Friedmann-Lema\^itre-Robertson-Walker cosmology in Einstein and Jordan frames, and explain their correspondence despite the approximate equations turning out to be linear and non-linear in different frames.