Analytic rotating black hole solutions in $N$-dimensional $f(T)$ gravity

A non-diagonal vielbein ansatz is applied to the $N$-dimension field equations of $f(T)$ gravity. An analytical vacuum solution is derived for the quadratic polynomial $f(T)=T+\epsilon T^2$ in the presence of a cosmological constant $\Lambda$. Since the induced metric has off diagonal components, that cannot be removed by a mere of a coordinate transformation, the solution has a rotating parameter. The curvature and torsion scalars invariants are calculated to study the singularities and horizons of the solution. In contrast to the general relativity (GR), the Cauchy horizon is differ from the horizon which shows the effect of the higher order torsion. The general expression of the energy-momentum vector of $f(T)$ gravity is used to calculate the energy of the system. Finally, we have shown that this kind of solution satisfies the first law of thermodynamics in the framework of $f(T)$ gravitational theories.