The Momentum Constraints of General Relativity and Spatial Conformal Isometries

TL;DR

This paper investigates the structure and constraints of transverse-tracefree tensors in asymptotically flat three-dimensional spaces, focusing on how conformal symmetries influence their multipole expansions and moment realizability.

Contribution

It provides a detailed analysis of TT tensors' multipole expansions and identifies obstructions caused by conformal Killing vectors in constructing arbitrary moments.

Findings

TT tensors admit multipole expansions under certain fall-off conditions.

Absence of conformal Killing vectors allows arbitrary moments to be realized.

Existence of CKVs imposes constraints on the moments that can be realized.

Abstract

Transverse-tracefree (TT-) tensors on $({\bf R}^3,g_{ab})$, with $g_{ab}$ an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. When $g_{ab}$ has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tensor. When CKV's exist there are obstructions -- certain (combinations of) moments have to vanish -- which we study.