  It is shown by several authors going back to Huisken-Yau that asymptotically
Schwarzschildean time-slices possess a unique foliation by stable constant mean
curvature (CMC) spheres defining the so-called CMC center of mass. We analyze
how the leaves of this foliation evolve in time under the Einstein equations.
More precisely, we prove that, asymptotically, their time evolution is a
translation induced by the quotient of their linear momentum and mass, as to be
expected from the corresponding Newtonian setting. In particular, the
definitions of mass and linear momentum defined by Arnowitt-Deser-Misner (ADM)
are compatible with the interpretation of the CMC foliation as the center of
mass of the time-slice by Huisken-Yau. Furthermore, we prove that the
coordinate version of the center of mass by Arnowitt-Deser-Misner and the
coordinate version of the CMC center of mass coincide - without additional
conditions on the scalar curvature. This is even true in the sense of
existence, i.e. if one of the two exists then so does the other.
