Equivalence between Palatini and metric formalisms of f(R)-gravity by divergence free current

TL;DR

This paper demonstrates the equivalence between metric and Palatini formalisms in f(R)-gravity theories through a divergence-free current, linking a symmetry to a conservation law and scalar field dynamics.

Contribution

It establishes a general framework showing the equivalence of the two formalisms via a conserved current and scalar field redefinition.

Findings

Equivalence holds in theories with divergence-free current.

Conservation equation can be recast as a massless scalar field.

Symmetry underpins the formal equivalence.

Abstract

The equivalence between metric and Palatini formalisms in f(R)-gravity can be achieved in the general context of theories with divergence free current. This equivalence is a necessary result of a symmetry which is included in a particular conservation equation of the current. In fact the conservation equation, by an appropriate redefinition of the introduced auxiliary field, may be encoded in a massless scalar field equation.