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

TL;DR

This paper investigates the asymptotic behavior of gravitational fields near spatial infinity for conformally flat initial data, combining two approaches to analyze decay properties, peeling behavior, and conserved quantities in general relativity.

Contribution

It integrates null and spatial infinity formalisms to identify conditions for peeling behavior and relates Newman-Penrose constants at future and past null infinity.

Findings

Decay of the Weyl tensor under certain conditions

Conditions for peeling behavior of gravitational fields

Equality of Newman-Penrose constants at future and past null infinity

Abstract

Certain aspects of the behaviour of the gravitational field near null and spatial infinity for the developments of asymptotically Euclidean, conformally flat initial data sets are analysed. Ideas and results from two different approaches are combined: on the one hand the null infinity formalism related to the asymptotic characteristic initial value problem and on the other the regular Cauchy initial value problem at spatial infinity which uses Friedrich's representation of spatial infinity as a cylinder. The decay of the Weyl tensor for the developments of the class of initial data under consideration is analysed under some existence and regularity assumptions for the asymptotic expansions obtained using the cylinder at spatial infinity. Conditions on the initial data to obtain developments satisfying the Peeling Behaviour are identified. Further, the decay of the asymptotic shear on null infinity is also examined as one approaches spatial infinity. This decay is related to the possibility of selecting the Poincar\'e group out of the BMS group in a canonical fashion. It is found that for the class of initial data under consideration, if the development peels, then the asymptotic shear goes to zero at spatial infinity. Expansions of the Bondi mass are also examined. Finally, the Newman-Penrose constants of the spacetime are written in terms of initial data quantities and it is shown that the constants defined at future null infinity are equal to those at past null infinity.