Explicit Riemannian manifolds with unexpectedly behaving center of mass

TL;DR

This paper investigates the behavior of the center of mass in asymptotically flat Riemannian manifolds, revealing divergence issues and providing explicit examples that challenge existing results, with insights applicable to both relativistic and Newtonian gravity.

Contribution

It introduces explicit examples demonstrating divergence of ADM and CMC center of mass definitions, highlighting subtle asymptotic issues and conflicting with prior literature.

Findings

Both ADM and CMC center of mass can diverge in Einstein's gravity.

Explicit examples with prescribed mass and center of mass are constructed.

Analogous Newtonian gravity examples illustrate similar phenomena.

Abstract

The (relativistic) center of mass 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 center of mass 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-center of mass. 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 center of mass. We illustrate both phenomena by providing analogous examples in Newtonian gravity. Our examples conflict with some results in the literature.