Scalar-tensor representation of $f(R)$ gravity and Birkhoff's theorem

TL;DR

This paper investigates the validity of Birkhoff's theorem in f(R) gravity using scalar-tensor representation and conformal transformations, comparing results in Einstein and Jordan frames with perturbation methods.

Contribution

It introduces a novel analysis of Birkhoff's theorem in f(R) gravity through scalar-tensor representation and frame comparison using perturbation techniques.

Findings

Birkhoff's theorem validity varies between frames.

Differences between f(R) and Brans-Dicke theories are clarified.

Perturbation methods reveal conditions for theorem validity.

Abstract

Birkhoff's theorem is discussed in the frame of f(R) gravity by using its scalar-tensor representation. Modified gravity has become very popular at recent times as it is able to reproduce the unification of inflation and late-time acceleration with no need of a dark energy component or an inflaton field. Here, another aspect of modified f(R) gravity is studied, specifically the range of validity of Birkhoff's theorem, compared with another alternative to General Relativity, the well known Brans-Dicke theory. As a novelty, here both theories are studied by using a conformal transformation and writing the actions in the Einstein frame, where spherically symmetric solutions are studied by using perturbation techniques. The differences between both theories are analyzed as well as the validity of the theorem within the Jordan and Einstein frames, where interesting results are obtained.