Quadratic scale-invariant gravity non minimally coupled to a scalar field
provides a competitive model for inflation, characterized by the transition
from an unstable to a stable fixed point, both characterized by constant scalar
field configurations. We provide a complementary analysis of the same model in
the static, spherically symmetric setting, obtaining two Schwarzschild-de
Sitter solutions in parallel to the fixed points existing in the cosmological
scenario. The stability of such solutions is thoroughly investigated from two
different perspectives. First, we perform a semi-classical analysis based on
the Euclidean path integral formulation. By studying the difference between the
Euclidean on-shell actions evaluated on both solutions, we prove that the
unstable one has a meta-stable character and is spontaneously decaying into the
stable fixed point which is always favoured. Then, the system is further
studied at the classical level by the analysis of linear perturbations. In
particular, we provide both analytical and numerical results for the late-time
behavior of the perturbations, confirming the stable and unstable character of
the two solutions.
