Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II

TL;DR

This paper solves the Einstein constraint equations for 3+1 vacuum spacetimes with a translational Killing field, reducing the problem to 2+1 dimensions and extending the asymptotic expansion of solutions to include global charge equivalents.

Contribution

It advances the understanding of asymptotic solutions for Einstein equations with symmetries by extending the expansion to include global charge-like quantities.

Findings

Extended asymptotic expansion involving global charges.

Reduced 3+1 problem to 2+1 dimensions using symmetry.

Improved understanding of asymptotic behavior in vacuum spacetimes.

Abstract

We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum spacetime with a space-like translational Killing field in the asymptotically flat case.. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1 dimensional problem to a 2+1 dimensional one. The aim of this paper is to go further in the asymptotic expansion of the solutions than in [14]. In particular the expansion we construct involves quantities which are the 2-dimensional equivalent of the global charges.