External field effect in gravity

In both Newtonian gravity and Einstein there is no force on a test particle located inside spherical cavity cut out of static, spherically symmetric mass distribution. Inside the exterior matter decoupled external field effect that could act particle. However, for potentials other than potential or geometries Ricci flat ones this longer case, then an effect. We explore possibility in various alternate scenarios, suggest such (Machian) effects can serve as diagnostic gravitational theory.