Charged black holes and unusual wormholes in scalar-tensor gravity

We consider static, spherically symmetric, electrically or/and magnetically charged configurations of a minimally coupled scalar field with an arbitrary potential $V(\phi)$ in general relativity. Using the inverse problem method, we obtain a four-parameter family of asymptotically dS, flat and AdS solutions, including those with naked singularities and both extreme and non-extreme black-hole (BH) solutions. The parameters are identified as the asymptotic cosmological constant, an arbitrary length scale, mass and charge. In all asymptotically flat BH solutions, the potential $V(\phi)$ is partly negative, in accord with Bekenstein and Mayo's no-hair theorem. The well-known conformal mapping extends the BH solutions to Jordan's pictures of a general class of scalar-tensor theories (STT) of gravity under the condition that the nonminimal coupling function $f(\phi)$ is everywhere positive. Relaxing the latter condition and assuming $f=0$ at some value of $\phi$, we obtain wormhole solutions in a particular subclass of STT. In such solutions, the double horizon of an extreme BH in Einstein's picture maps into the second spatial infinity in Jordan's. However, at this second infinity, the effective gravitational coupling infinitely grows.