  The Misner-Sharp-Hernandez mass defined in general relativity and in
spherical symmetry has been recognized as having a Newtonian character in
previous literature. In order to better understand this aspect we relax
spherical symmetry and we study the generalization of the
Misner-Sharp-Hernandez mass to general spacetimes, i.e., the Hawking quasilocal
mass. The latter is decomposed into a matter and a pure Weyl contribution. The
decomposition of the Weyl tensor into an electric part (which has a Newtonian
counterpart) and a magnetic one (which does not) further splits the quasilocal
mass into "Newtonian" and "non-Newtonian" parts. It is found that only the
electric (Newtonian) part contributes.
