Extended Birkhoff's Theorem in the f(T) Gravity

TL;DR

This paper extends Birkhoff's theorem within f(T) gravity, demonstrating its validity in more general tetrad configurations and exploring the implications for vacuum and gravitational fields, including conformal transformations.

Contribution

It proves Birkhoff's theorem in off-diagonal tetrad cases of f(T) gravity and analyzes its validity through conformal transformations and frame equivalence.

Findings

Birkhoff's theorem holds in off-diagonal tetrad cases of f(T) gravity.

The theorem's validity is confirmed in both Einstein and Jordan frames.

Extended meaning of Birkhoff's theorem in f(T) gravity is discussed.

Abstract

The f(T) theory, a generally modified teleparallel gravity, has been proposed as an alternative gravity model to account for the dark energy phenomena. Following our previous work [Xin-he Meng and Ying-bin Wang, EPJC(2011), arXiv:1107.0629v1], we prove that the Birkhoff's theorem holds in a more general context, specifically with the off diagonal tetrad case, in this communication letter. Then, we discuss respectively the results of the external vacuum and internal gravitational field in the f(T) gravity framework, as well as the extended meaning of this theorem. We also investigate the validity of the Birkhoff's theorem in the frame of f(T) gravity via conformal transformation by regarding the Brans-Dicke-like scalar as effective matter, and study the equivalence between both Einstein frame and Jordan frame.