Solutions to Einstein equations to simple cases and geodesic eqns predict either "Wall of death trick" (non zero angular momentum) motion or oscilatory motion to origins. The later has not so much been observed i.e in solar system motions.
The first one resembles Newtonian gravity orbits.
Is there a coordinated effort to characterizes our universe by uccurance of motions as De Sitter space or not i.e negative curvature?