Birkhoff's Theorem in f(T) Gravity up to the Perturbative Order

TL;DR

This paper examines the validity of Birkhoff's theorem in f(T) gravity using perturbative methods, confirming its validity at zero order and presenting new solutions at first order with diagonal tetrads.

Contribution

It provides a perturbative analysis of Birkhoff's theorem in f(T) gravity and introduces a new spherically symmetric solution at first order for diagonal tetrads.

Findings

Birkhoff's theorem holds at zero order in f(T) gravity.

A new spherically symmetric solution is found at first order.

Physical equivalence between Jordan and Einstein frames is maintained up to first order.

Abstract

f(T) gravity, a generally modified teleparallel gravity, has become very popular in recent times as it is able to reproduce the unification of inflation and late-time acceleration without the need of a dark energy component or an inflation field. In this present work, we investigate specifically the range of validity of Birkhoff's theorem with the general tetrad field via perturbative approach. At zero order, Birkhoff's theorem is valid and the solution is the well known Schwarzschild-(A)dS metric. Then considering the special case of the diagonal tetrad field, we present a new spherically symmetric solution in the frame of f(T) gravity up to the perturbative order. The results with the diagonal tetrad field satisfy the physical equivalence between the Jordan and the so-called Einstein frames, which are realized via conformal transformation, at least up to the first perturbative order.