Measuring Mass via Coordinate Cubes

TL;DR

This paper introduces a novel method to measure the mass of asymptotically flat 3-manifolds using coordinate cubes, connecting scalar curvature, mean curvature, and geometric angles.

Contribution

It develops a new mass formula based on coordinate cube faces and edges, linking scalar curvature to geometric boundary data in Riemannian polyhedra.

Findings

Mass formula relates scalar curvature to boundary geometric data.

The approach connects Gromov's scalar curvature comparison with boundary angle defects.

The method provides a geometric interpretation of mass via curvature and angles.

Abstract

Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat $3$-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss-Bonnet theorem.