Static solutions with spherical symmetry in $f(T)$ theories

The spherically symmetric static solutions are searched for in some $f(T)$ models of gravity theory with a Maxwell term. To do this, we demonstrate that reconstructing the Lagrangian of $f(T)$ theories is sensitive to the choice of frame, and then we introduce a particular frame based on the conformally Cartesian coordinates. In this particular frame, the existence conditions of various solutions are presented. Our results imply that only a limited class of $f(T)$ models can be solved in this frame. For more general models, the search for spherically symmetric static solutions is still an open and challenging problem, hopefully solvable in other frames.

Figure 1

The radius $\tilde{r}$ of ${\mathbf{S}}^2$ (50) decreases monotonically as coordinate $z$ (48) increases.