Extended Birkhoff’s theorem in f(T) gravity

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, Eur. Phys. J. (2011), arXiv:1107.0629v1], we prove that 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 Birkhoff’s theorem in the frame of f(T) gravity via a conformal transformation by regarding the Brans–Dicke-like scalar as effective matter, and study the equivalence between both Einstein frame and Jordan frame.