The Momentum Constraints of General Relativity and Spatial Conformal Isometries

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.