Is the Hawking quasilocal energy "Newtonian"?

TL;DR

This paper investigates whether the Hawking quasilocal energy has a Newtonian nature by decomposing it into electric and magnetic parts, finding that only the electric (Newtonian) component contributes.

Contribution

It generalizes the Misner-Sharp-Hernandez mass to non-spherical spacetimes and decomposes the Hawking mass to identify the Newtonian component.

Findings

Only the electric part of the Weyl tensor contributes to the quasilocal mass.

The Newtonian aspect of the mass is associated with the electric Weyl component.

The magnetic part does not contribute to the Newtonian character.

Abstract

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.