Conformally related metrics and Lagrangians and their physical interpretation in cosmology

TL;DR

This paper explores the relationships between conformally related metrics and Lagrangians in scalar-tensor cosmology, establishing conditions for their equivalence and implications for physical interpretation in different conformal frames.

Contribution

It proves that conformally related Lagrangians have related field equations if the Hamiltonian vanishes and shows how non-minimally coupled scalar fields relate to minimally coupled ones in conformally related spaces.

Findings

Field equations are conformally related if Hamiltonian vanishes.

Non-minimally coupled scalar fields can be associated with minimally coupled ones.

Results extend to n-dimensional Riemannian spaces.

Abstract

Conformally related metrics and Lagrangians are considered in the context of scalar-tensor gravity cosmology. After the discussion of the problem, we pose a lemma in which we show that the field equations of two conformally related Lagrangians are also conformally related if and only if the corresponding Hamiltonian vanishes. Then we prove that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. The latter result implies that the field equations of a non-minimally coupled scalar field are the same at the conformal level with the field equations of the minimally coupled scalar field. This fact is relevant in order to select physical variables among conformally equivalent systems. Finally, we find that the above propositions can be extended to a general Riemannian space of n-dimensions.