A note on the post-Newtonian limit of quasi-local energy expressions

An 'effective' quasi-local energy expression, motivated by the (relativistically corrected) Newtonian theory, is introduced in exact general relativity as the volume integral of all the source terms in the field equation for the Newtonian potential in static spacetimes. In particular, we exhibit a new post-Newtonian correction in the source term in the field equation for the Newtonian gravitational potential. In asymptotically flat spacetimes, this expression tends to the Arnowitt-Deser-Misner energy at spatial infinity as a monotonically decreasing set function. We prove its positivity in spherically symmetric spacetimes under certain energy conditions, and that its vanishing characterizes flatness. We argue that any physically acceptable quasi-local energy expression should behave qualitatively like this 'effective' energy expression in this limit.