  In this paper we study some global properties of static potentials on
asymptotically flat $3$-manifolds $(M,g)$ in the nonvacuum setting.
Heuristically, a static potential $f$ represents the (signed) length along $M$
of an irrotational timelike Killing vector field, which can degenerate on
surfaces corresponding to the zero set of $f$. Assuming a suitable version of
the null energy condition, we prove that a noncompact component of the zero set
must be area minimizing. From this we obtain some rigidity results for static
potentials that have noncompact zero set components, or equivalently, that are
unbounded. Roughly speaking, these results show, at the pure initial data
level, that `boost-type' Killing vector fields can exist only under special
circumstances.
