On ``hyperboloidal'' Cauchy data for vacuum Einstein equations and obstructions to smoothness of ``null infinity''

Various works have suggested that the Bondi–Sachs–Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space–times. We have made a detailed analysis of the constraint equations for asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of non–generic’’ solutions of the constraint equations, the evolution of which leads to space–times satisfying the Bondi–Sachs–Penrose smoothness conditions.