Time asymmetric spacetimes near null and spatial infinity: I. Expansions of developments of conformally flat data

Abstract

The conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen–York ansatz. The conformal Einstein equations imply, upon evaluation on the cylinder at spatial infinity, a hierarchy of transport equations which can be used to calculate asymptotic expansions for the gravitational field in a recursive way. It is found that the solutions to these transport equations develop logarithmic divergences at the critical sets where null infinity meets spatial infinity. Associated with these, there is a series of quantities expressible in terms of the initial data (obstructions), which if zero, preclude the appearance of some of the logarithmic divergences. The obstructions are, in general, time asymmetric. That is, the obstructions at the intersection of future null infinity with spatial infinity are in general different from those obtained at the intersection of past null infinity with spatial infinity. The latter allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. Finally, it is shown that if both sets of obstructions vanish up to a certain order, then the initial data have to be asymptotically Schwarzschildean in a certain sense.