Conformally related metrics and Lagrangians and their physical interpretation in cosmology

Understand

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.