Good and bad tetrads in f(T) gravity

TL;DR

This paper emphasizes the importance of selecting appropriate tetrads in $f(T)$ gravity, demonstrating how good tetrads can lead to consistent solutions like Schwarzschild-de Sitter, and providing a method to construct such tetrads for various symmetries.

Contribution

The authors introduce a systematic approach to identify good tetrads in $f(T)$ gravity that do not restrict the form of $f(T)$, ensuring consistent field equations and solutions.

Findings

Good tetrads can be constructed using local rotations.

Good tetrads lead to consistent Schwarzschild-de Sitter solutions.

The method applies to homogeneity, isotropy, and spherical symmetry.

Abstract

We investigate the importance of choosing good tetrads for the study of the field equations of $f(T)$ gravity. It is well known that this theory is not invariant under local Lorentz transformations, and therefore the choice of tetrad plays a crucial role in such models. Different tetrads will lead to different field equations which in turn have different solutions. We suggest to speak of a good tetrad if it imposes no restrictions on the form of $f(T)$. Employing local rotations, we construct good tetrads in the context of homogeneity and isotropy, and spherical symmetry, where we show how to find Schwarzschild-de Sitter solutions in vacuum. Our principal approach should be applicable to other symmetries as well.