Geometric characterizations of asymptotic flatness and linear momentum in general relativity

TL;DR

This paper characterizes asymptotic flatness in three-dimensional Riemannian manifolds through the existence of a special CMC-cover with specific geometric properties, linking it to linear momentum definitions in general relativity.

Contribution

It provides a geometric, coordinate-free characterization of asymptotic flatness and establishes the equivalence of this with the existence of a suitable CMC-foliation, connecting to ADM linear momentum.

Findings

Asymptotic flatness characterized by CMC-cover existence

Geometric definition of linear momentum consistent with ADM

Extension of previous results on CMC-foliations

Abstract

In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-Metzger, and the author. In this work, we prove the reverse implication, i.e. any three-dimensional Riemannian manifold is asymptotically flat if it possesses a CMC-cover satisfying certain geometric curvature estimates, a uniqueness property, a weak foliation property, and each surface has weakly controlled instability. With the author's previous result that every asymptotically flat manifold possesses a CMC-foliation, we conclude that asymptotic flatness is characterized by existence of such a CMC-cover. Additionally, we use this characterization to give a geometric (i.e. coordinate-free) definition of a (CMC-)linear momentum and prove its compatibility with the linear momentum defined by Arnowitt-Deser-Misner.