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

TL;DR

This paper introduces a new quasi-local energy expression in general relativity inspired by Newtonian theory, demonstrating its properties and behavior in static and asymptotically flat spacetimes, including positivity and flatness characterization.

Contribution

It presents a novel effective quasi-local energy expression with proven positivity and monotonicity properties, bridging Newtonian and relativistic gravitational energy concepts.

Findings

Expression tends to ADM energy at infinity

Positivity proven under certain conditions

Vanishing characterizes flat spacetime

Abstract

An `effective' quasi-local energy expression, motivated by the (relativistically corrected) Newtonian theory, is introduced in exact GR 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 ADM energy at the spatial infinity as a {\em 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.