Approximations of the quasi-local Bartnik mass in general relativity

In this study, we employ eth-operators and spin-weighted spherical harmonics to express the ADM mass of a static space-time based on the mean values of its components over a a radius-$r$ sphere. While initially derived for standard spherical coordinates, we showcase its adaptability by demonstrating its usefulness in expressing a quasilocal mass; specifically, the Bartnik mass, of an almost round 2D-hypersurface in terms of some specific boundary conditions. Additionally, we utilize this formulation to propose a deep learning methodology for numerically constructing static metrics that incorporate 2D-hypersurfaces with specified Bartnik mass.