The variation of G in a negatively curved space-time
  Scalar-tensor (ST) gravity theories provide an appropriate theoretical framework for the variation of Newton's fundamental constant, conveyed by the dynamics of a scalar-field non-minimally coupled to the space-time geometry. The experimental scrutiny of scalar-tensor gravity theories has led to a detailed analysis of their post-newtonian features, and is encapsulated into the so-called parametrised post-newtonian formalism (PPN). Of course this approach can only be applied whenever there is a newtonian limit, and the latter is related to the GR solution that is generalized by a given ST solution under consideration. This procedure thus assumes two hypothesis: On the one hand, that there should be a weak field limit of the GR solution; On the other hand that the latter corresponds to the limit case of given ST solution. In the present work we consider a ST solution with negative spatial curvature. It generalizes a general relativistic solution known as being of a degenerate class (A) for its unusual properties. In particular, the GR solution does not exhibit the usual weak field limit in the region where the gravitational field is static. The absence of a weak field limit for the hyperbolic GR solution means that such limit is also absent for comparison with the ST solution, and thus one cannot barely apply the PPN formalism. We therefore analyse the properties of the hyperbolic ST solution, and discuss the question o defining a generalised newtonian limit both for the GR solution and for the purpose of contrasting it with the ST solution. This contributes a basic framework to build up a parametrised pseudo-newtonian formalism adequate to test ST negatively curved space-times. 
Jos\'e P. Mimoso, Francisco S. N. Lobo
gr-qc astro-ph.CO hep-th