Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theories

Teleparallel geometry utilizes Weitzenböck connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi–Civita connection. In extended teleparallel theories, for instance in $f\left(T\right)$ or scalar-torsion gravity, the connection must obey its antisymmetric field equations. Thus far, only a few analytic solutions were known. In this note, we solve the $f(T,\varphi )$ gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformation, we present the solution also in Boyer–Lindquist coordinates, often used to study rotating solutions in general relativity. The result hints for the existence of another branch of rotating solutions besides the Kerr family in extended teleparallel gravities.