  In this paper we investigate wormhole and spherically symmetric solutions in
4D gravity plus a matter source consisting of a ghost scalar field with a
sine-Gordon potential. For the wormhole solutions we also include the
possibility of electric and/or magnetic charges. For both types of solutions we
perform a linear stability analysis and show that the wormhole solutions are
stable and that when one turns on the electric and/or magnetic field the
solution remains stable. The linear stability analysis of the spherically
symmetric solutions indicates that they can be stable or unstable depending on
one of the parameters of the system. This result for the spherically symmetric
solution is nontrivial since a previous investigation of 4D gravity plus a
ghost scalar field with a $\lambda \phi ^4$ interaction found only unstable
spherically symmetric solutions. Both the wormhole and spherically symmetric
solutions presented here asymptotically go to anti-de-Sitter space-time.
