Quasi-local mass in the covariant Newtonian space-time

TL;DR

This paper explores how various quasi-local mass and energy expressions from general relativity translate into the covariant Newtonian framework, confirming the natural emergence of Newtonian mass from the Komar integral.

Contribution

It demonstrates that the Komar integral directly yields the Newtonian quasi-local mass in the covariant Newtonian space-time, clarifying the connection between relativistic and Newtonian quasi-local quantities.

Findings

Komar integral reproduces Newtonian quasi-local mass naturally.

Brown-York and Dougan-Mason expressions require symmetry conditions.

Provides a covariant approximation linking relativistic and Newtonian theories.

Abstract

In general relativity, quasi-local energy-momentum expressions have been constructed from various formulae. However, Newtonian theory of gravity gives a well known and an unique quasi-local mass expression (surface integration). Since geometrical formulation of Newtonian gravity has been established in the covariant Newtonian space-time, it provides a covariant approximation from relativistic to Newtonian theories. By using this approximation, we calculate Komar integral, Brown-York quasi-local energy and Dougan-Mason quasi-local mass in the covariant Newtonian space-time. It turns out that Komar integral naturally gives the Newtonian quasi-local mass expression, however, further conditions (spherical symmetry) need to be made for Brown-York and Dougan-Mason expressions.