Matter Conditions for Regular Black Holes in $\mathbf{f(T)}$ Gravity

We study the conditions imposed on matter to produce a regular (non-singular) interior of a class of spherically symmetric black holes in the $f(T)$ extension of teleparallel gravity. The class of black holes studied is necessarily singular in general relativity. We derive a tetrad which is compatible with the black hole interior and utilize this tetrad in the gravitational equations of motion to study the black hole interior. It is shown that in the case where the gravitational Lagrangian is expandable in a power series $f(T)=T+\underset{n\neq 1}{\sum} b_{n}T^{n}$ that black holes can be non-singular while respecting certain energy conditions in the matter fields. Thus the black hole singularity may be removed and the gravitational equations of motion can remain valid throughout the manifold. This is true as long as $n$ is positive, but is not true in the negative sector of the theory. Hence, gravitational $f(T)$ Lagrangians which are Taylor expandable in powers of $T$ may yield regular black holes of this type. Although it is found that these black holes can be rendered non-singular in $f(T)$ theory, we conjecture that a mild singularity theorem holds in that the dominant energy condition is violated in an arbitrarily small neighborhood of the general relativity singular point if the corresponding $f(T)$ black hole is regular. The analytic techniques here can also be applied to gravitational Lagrangians which are not Laurent or Taylor expandable.