  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.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory