Extended Theories of Gravitation

TL;DR

This paper reviews the equivalence between Palatini-$f( ilde{ ext{R}})$ theories and Brans-Dicke theories, focusing on action principles, transformations, and the impact of conformal factors on physical quantities.

Contribution

It provides a detailed analysis of the equivalence at the action level and introduces transformations to connect different frames in extended theories of gravitation.

Findings

Establishes the equivalence between Palatini-$f( ilde{ ext{R}})$ and Brans-Dicke theories.

Defines the Helmholtz Lagrangian for Palatini-$f( ilde{ ext{R}})$ theories.

Shows how conformal factors influence physical quantities.

Abstract

In this paper we shall review the equivalence between Palatini$-f(\mathcal R)$ theories and Brans- Dicke (BD) theories at the level of action principles. We shall define the Helmholtz Lagrangian associated to Palatini$-f(\mathcal R)$ theory and we will define some transformations which will be useful to recover Einstein frame and Brans-Dicke frame. We shall see an explicit example of matter field and we will discuss how the conformal factor affects the physical quantities.