Explicit Riemannian Manifolds with Unexpectedly Behaving Center of Mass

The (relativistic) center of mass (CoM) of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e.g. by Arnowitt, Deser, and Misner (ADM). There are also what we call abstract definitions of the CoM in terms of a foliation near infinity itself, going back to the constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give rise to surface integral expressions when equipped with suitable systems of coordinates. We discuss subtle asymptotic convergence issues regarding the ADM- and the coordinate expressions related to the CMC-CoM. In particular, we give explicit examples demonstrating that both can diverge—in a setting where Einstein’s equation is satisfied. We also give explicit examples of the same asymptotic order of decay with prescribed mass and CoM. We illustrate both phenomena by providing analogs in Newtonian gravity. Our examples conflict with some results in the literature.