Asymptotic properties of the development of conformally flat data near spatial infinity

Abstract

The analysis of the relation between Bondi-type systems (NP-gauge) and a gauge used in the analysis of the structure of spatial infinity (F-gauge) which was carried out by Friedrich and Kánnár (2000 J. Math Phys. 41 2195) is retaken and applied to the development of a suitable class of conformally flat initial data sets with non-vanishing second fundamental form. The calculations presented depend on a certain assumption about the existence and regularity of the solutions to the conformal Einstein field equations close to null and spatial infinity. As a result of the calculations the Newman–Penrose constants of both future and past null infinity are calculated in terms of initial data and are shown to be equal. It is also shown that the asymptotic shear goes to zero as one approaches spatial infinity along the generators of null infinity so that it is possible to select, in a canonical fashion, the Poincaré group out of the BMS group. An expansion—again in terms of initial data quantities—of the Bondi mass close to spatial infinity is calculated. This expansion shows that if the existence and regularity assumptions hold, the Bondi mass approaches the ADM mass. A discussion of possible conditions on the initial data which would render a peeling development is presented.