  The coordinate freedom of General Relativity makes it challenging to find
mathematically rigorous and physically sound definitions for physical
quantities such as the center of mass of an isolated gravitating system. We
will argue that a similar phenomenon occurs in Newtonian Gravity once one
ahistorically drops the restriction that one should only work in Cartesian
coordinates when studying Newtonian Gravity. This will also shed light on the
nature of the challenge of defining the center of mass in General Relativity.
Relatedly, we will give explicit examples of asymptotically Euclidean
relativistic initial data sets which do not satisfy the Regge--Teitelboim
parity conditions often used to achieve a satisfactory definition of center of
mass. These originate in our joint work with Jan Metzger. This will require
appealing to Bartnik's asymptotic harmonic coordinates.
