# Flat connection for rotating vacuum spacetimes in extended teleparallel gravity theories

## TL;DR

This paper derives new rotating vacuum solutions in extended teleparallel gravity theories, revealing potential alternative rotating spacetime configurations beyond the Kerr solution.

## Contribution

It provides the first analytic solutions for rotating spacetimes in $f(T,)$ gravity, expanding the understanding of possible geometries in extended teleparallel theories.

## Key findings

- Derived explicit spin connection coefficients for rotating solutions.
- Presented solutions in both Weyl canonical and Boyer-Lindquist coordinates.
- Indicates the possible existence of non-Kerr rotating solutions in extended teleparallel gravity.

## Abstract

Teleparallel geometry utilizes Weitzenb\"ock 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 $f(T)$ or scalar-torsion gravity, the connection must obey its antisymmetric field equations. So far only a few analytic solutions were known. In this note we solve the $f(T,\phi)$ 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.