Solutions of special asymptotics to the Einstein constraint equations

TL;DR

This paper develops methods to construct solutions with specific asymptotic properties for the Einstein constraint equations and explores examples where physical quantities like center of mass are ill-defined.

Contribution

It introduces a cut-off technique for solving the Einstein constraint equations with prescribed asymptotics and provides examples of vacuum manifolds with ill-defined physical quantities.

Findings

Constructed solutions with prescribed asymptotics

Identified examples with ill-defined center of mass and angular momentum

Demonstrated limitations in defining physical quantities in certain manifolds

Abstract

We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are ill-defined.