Initial data for stationary space-times near space-like infinity

TL;DR

This paper analyzes initial data for stationary vacuum space-times near space-like infinity, proving they have analytic expansions and filling a gap in the understanding of their compactification at null infinity.

Contribution

It demonstrates that stationary initial data admit analytic expansions in powers of a radial coordinate, clarifying their regularity properties at space-like infinity.

Findings

Initial data have analytic expansions in powers of a radial coordinate.

Coefficients of the expansion are analytic functions of angles.

Fills a gap in the proof of analytic compactification at null infinity.

Abstract

We study Cauchy initial data for asymptotically flat, stationary vacuum space-times near space-like infinity. The fall-off behavior of the intrinsic metric and the extrinsic curvature is characterized. We prove that they have an analytic expansion in powers of a radial coordinate. The coefficients of the expansion are analytic functions of the angles. This result allow us to fill a gap in the proof found in the literature of the statement that all asymptotically flat, vacuum stationary space-times admit an analytic compactification at null infinity. Stationary initial data are physical important and highly non-trivial examples of a large class of data with similar regularity properties at space-like infinity, namely, initial data for which the metric and the extrinsic curvature have asymptotic expansion in terms of powers of a radial coordinate. We isolate the property of the stationary data which is responsible for this kind of expansion.