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 spacetimes. 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 ''nongeneric'' solutions of the constraint equations, the evolution of which leads to spacetimes satisfying the Bondi-Sachs-Penrose smoothness conditions.