Freely falling 2-surfaces and the quasi-local energy

TL;DR

This paper introduces a new geometrically defined effective gravitational mass for closed spacelike 2-surfaces, compares it with existing quasi-local energy measures, and explores its properties in various spacetime scenarios.

Contribution

It provides a novel geometrical expression for gravitational mass based on freely falling 2-surfaces and compares it with established quasi-local energy concepts in General Relativity.

Findings

Effective gravitational mass is positive for small spheres in non-vacuum spacetimes.

It can be negative in vacuum cases.

The effective mass aligns with quasi-local energy expressions in vacuum, including cosmological constant contributions.

Abstract

We derive an expression for effective gravitational mass for any closed spacelike 2-surface. This effective gravitational energy is defined directly through the geometrical quantity of the freely falling 2-surface and thus is well adapted to intuitive expectation that the gravitational mass should be determined by the motion of test body moving freely in gravitational field. We find that this effective gravitational mass has reasonable positive value for a small sphere in the non-vacuum space-times and can be negative for vacuum case. Further, this effective gravitational energy is compared with the quasi-local energy based on the $(2+2)$ formalism of the General Relativity. Although some gauge freedoms exist, analytic expressions of the quasi-local energy for vacuum cases are same as the effective gravitational mass. Especially, we see that the contribution from the cosmological constant is the same in general cases.