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

TL;DR

This paper analyzes the asymptotic behavior of gravitational fields near null and spatial infinity for conformally flat, time asymmetric data, revealing conditions for smoothness and the potential for different properties at future and past null infinity.

Contribution

It introduces a recursive method to compute asymptotic expansions of gravitational fields using the conformal Einstein equations and identifies obstructions affecting smoothness at infinity.

Findings

Logarithmic divergences occur at critical sets where null infinity meets spatial infinity.

Obstructions to smoothness are expressed in terms of initial data and are generally time asymmetric.

Vanishing obstructions imply initial data is asymptotically Schwarzschildean.

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 in a recursive way asymptotic expansions for the gravitational field. It is found that the the solutions to these transport equations develop logarithmic divergences at certain critical sets where null infinity meets spatial infinity. Associated to 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 different, and do not generically imply 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 has to be asymptotically Schwarzschildean to some degree.