I read the book The global nonlinear stability of the minkowski space by Christodoulou, D., and Klainerman, S.
In the Chapter I, Page 10 it says:
“However, it remains questionable whether there exists any nontrivial6 solution of the field equations that satisfies the Penrose requirements. Indeed, his regularity assumptions translate into fall-off conditions of the curvature that may be too stringent and thus may fail to be satisfied by any solution that would allow gravitational waves.”
So I am curious about if anyone had found such solutions above that is non-stationary asymptotically flat and geodesic completeness.