Polyhomogeneous Expansions Close to Null and Spatial Infinity

Abstract

A study of the linearised gravitational field (spin 2 zero-rest-mass field) on a Minkowski background close to spatial infinity is done. To this purpose, a certain representation of spatial infinity in which it is depicted as a cylinder is used. A first analysis shows that the solutions generically develop a particular type of logarithmic divergence at the sets where spatial infinity touches null infinity. A regularity condition on the initial data can be deduced from the analysis of some transport equations on the cylinder at spatial infinity. It is given in terms of the linearised version of the Cotton tensor and symmetrised higher order derivatives, and it ensures that the solutions of the transport equations extend analytically to the sets where spatial infinity touches null infinity. It is later shown that this regularity condition together with the requirement of some particular degree of tangential smoothness ensures logarithm-free expansions of the time development of the linearised gravitational field close to spatial and null infinities.