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

TL;DR

This paper explores new solutions for charged black holes in $f(T)$ gravity coupled with nonlinear electrodynamics, generalizing known solutions and deriving new classes with specific functional forms of $f(T)$ and the Lagrangian.

Contribution

It introduces new classes of charged black hole solutions in $f(T)$ gravity with nonlinear electrodynamics, generalizing no-go theorems and deriving explicit functional forms of $f(T)$.

Findings

Reproduction of Born-Infeld and Reissner-Nordstrom-AdS solutions.

Derivation of a new class of charged black holes with specific $f(T)$ forms.

Asymptotically flat black holes with a singularity at the origin.

Abstract

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 $f(T) = f_0\left(-T\right)^{(N+3)/[2(N+1)]}$ and $\mathcal{L}_{NED} = \mathcal{L}_0\left(-F\right)^{(N+3)/[2(N+1)]}$. Using the Lagrangian density of Born-Infeld, we obtain a new class of charged black holes where the action reads $f(T) = -16\beta_{BI} \left[1 - \sqrt{1 + (T/4\beta_{BI})}\right]$.