We propose a Lagrangian formulation for a varying G Newtonian-like theory inspired by the Brans-Dicke gravity. Rather than imposing an ad hoc dependence for the gravitational coupling, as previously done in the literature, in our proposal the running of G emerges naturally from the internal dynamical structure of the theory. We explore the features of the resulting gravitational field for static and spherically symmetric mass distributions as well as within the cosmological framework.

We classified and studied the charged black hole and wormhole solutions in the Einstein–Maxwell system in the presence of a massless, real scalar field. The possible existence of charged black holes in general scalar–tensor theories was studied in Bronnikov et al, 1999; black holes and wormholes exist for a negative kinetic term for the scalar field. Using a conformal transformation, the static, spherically symmetric possible structures in the minimal coupled system are described. Besides wormholes and naked singularities, only a restricted class of black hole exists, exhibiting a horizon with an infinite surface and a timelike central singularity. The black holes and wormholes defined in the Einstein frame have some specificities with respect to the non-minimal coupling original frame, which are discussed in the text.

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