A Quasi-Local Mass

TL;DR

This paper introduces a new gauge-independent quasi-local mass and energy, establishing its positivity, rigidity, and convergence properties, and relating it to existing concepts like the Brown-York and Wang-Yau masses.

Contribution

It proposes a novel quasi-local mass definition that differs from Wang-Yau's, with a proof of positivity, rigidity, and convergence to ADM mass, enriching the understanding of gravitational energy.

Findings

New gauge-independent quasi-local mass defined.

Proof of positivity for certain spacelike 2-surfaces.

Convergence to ADM mass at spatial infinity.

Abstract

We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown-York Hamilton-Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang-Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding.