Mathematical Equivalence vs. Physical Equivalence between Extended Theories of Gravitations

TL;DR

This paper explores the differences between extended theories of gravitation, specifically Palatini f(R) and Brans-Dicke models, focusing on their physical equivalence and experimental test outcomes like Mercury's perihelion precession.

Contribution

It demonstrates that Palatini f(R) and Brans-Dicke theories, though mathematically equivalent, differ in physical predictions due to assumptions about free fall and measurement conventions.

Findings

Palatini f(R) passes Mercury's perihelion test

Brans-Dicke does not pass the perihelion test

Physical equivalence depends on measurement assumptions

Abstract

We shall show that although Palatini f(R)-theories are equivalent to Brans-Dicke theories, still the first pass the Mercury precession of perihelia test, while the second do not. We argue that the two models are not physically equivalent due to a different assumptions about free fall. We shall also go through perihelia test without fixing a conformal gauge (clocks or rulers) in order to highlight what can be measured in a conformal invariant way and what cannot. We shall argue that the conformal gauge is broken by choosing a definition of clock, rulers or, equivalently, of masses.