2013/12/22 by Carla Cederbaum, Cederbaum, Carla, Christopher Nerz +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #gr-qc #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1312.6391
examples with prescribed mass and center of mass included; asymptotic decay described in more detail; references updated
openalex publication_date 2013/12/22 · arxiv created 2013/12/30 · arxiv updated 2013/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.