Static vacuum solutions from convergent null data expansions at space-like infinity

TL;DR

This paper develops a method to characterize all asymptotically flat static vacuum solutions using null data expansions at space-like infinity, establishing convergence criteria linked to Geroch multipoles.

Contribution

It introduces a novel null data formalism for static vacuum solutions and provides necessary and sufficient growth conditions for convergence.

Findings

Established a 1:1 correspondence between null data and Geroch multipoles.

Derived growth estimates ensuring absolute convergence of formal expansions.

Provided a complete characterization of asymptotically flat static vacuum solutions.

Abstract

We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations.