Conformal structures of static vacuum data

TL;DR

This paper investigates the conditions under which static vacuum data can be extended smoothly to null infinity, focusing on conformal invariants and the behavior of the Cotton tensor at space-like infinity.

Contribution

It characterizes the gap between conformally invariant conditions on the Cotton tensor and asymptotic staticity, providing criteria for conformal staticity near space-like infinity.

Findings

Identifies a conformally invariant one-form related to the Cotton tensor.

Provides criteria for conformal staticity based on analyticity and asymptotic properties.

Establishes conditions under which data is conformally static near space-like infinity.

Abstract

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$. We provide two criteria: If $h$ is real analytic, $\kappa$ is closed, and one of it integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.