Evaluation of Bartnik's quasilocal mass function

TL;DR

This paper analyzes Bartnik's quasilocal mass function, showing it often equals the ADM mass of extensions and examining its behavior in specific geometries, revealing singularities at small volumes.

Contribution

It provides a detailed analysis of Bartnik's quasilocal mass, including explicit calculations and behavior in Schwarzschild geometry, highlighting singularities at small scales.

Findings

Bartnik's mass often equals the ADM mass of certain extensions.

The mass-to-volume ratio becomes singular as volume approaches zero.

Explicit calculation of mass for a non-central ball in Schwarzschild geometry.

Abstract

Bartnik's definition of gravitational quasilocal energy is analyzed. For a wide class of systems Bartnik's function is given by the ADM mass of some vacuous extension. As an example we calculate mass of a non central ball in Schwarzschild geometry. The ratio mass to volume becomes singular in the limit of small volumes.