Born-Infeld and Charged Black Holes with non-linear source in $f(T)$ Gravity

We investigate $f(T)$ theory coupled with a nonlinear source of electrodynamics, for a spherically symmetric and static spacetime in $4D$. We re-obtain the Born-Infeld and Reissner-Nordstrom-AdS solutions. We generalize the no-go theorem for any content that obeys the relationship ${\mathcal{T}}_0^0={\mathcal{T}}_1^1$ for the energy-momentum tensor and a given set of tetrads. Our results show new classes of solutions where the metrics are related through $b(r)=-Na(r)$. We do the introductory analysis showing that solutions are that of asymptotically flat black holes, with a singularity at the origin of the radial coordinate, covered by a single event horizon. We also reconstruct the action for this class of solutions and obtain the functional form $f(T)=f_0{(-T)}^{(N+3)\mathrm{/}[2(N+1)]}$ and ${\mathcal{L}}_{NED}={\mathcal{L}}_0{(-F)}^{(N+3)\mathrm{/}[2(N+1)]}$. Using the Lagrangian density of Born-Infeld, we obtain a new class of charged black holes where the action reads $f(T) = -16\beta_{BI} \left[1 - \sqrt{1 + (T/4\beta_{BI})}\right]$.