Is the Hawking Quasilocal Energy "Newtonian"?

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 feature 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 contribution and a contribution coming solely from the Weyl tensor. The Weyl tensor is then decomposed into an electric part (which has a Newtonian counterpart) and a magnetic one (which does not), which further splits the quasilocal mass into “Newtonian” and “non-Newtonian” parts. Only the electric (Newtonian) part contributes to the quasilocal mass.