The Bartnik-Bray outer mass of small metric spheres in time-symmetric 3-slices

TL;DR

This paper calculates the first-order approximation of the Bartnik-Bray outer mass for small metric spheres in time-symmetric 3-slices, confirming physical expectations and exploring higher-order terms related to scalar curvature.

Contribution

It provides the first-order computation of the Bartnik-Bray outer mass for near-round spheres and extends to fifth order for flat points, linking mass to scalar curvature derivatives.

Findings

Mass-to-volume ratio converges to energy density at the center.

Upper and lower bounds for mass are established using static vacuum extensions.

Higher-order terms relate to the Laplacian of scalar curvature.

Abstract

Given a sphere with Bartnik data close to that of a round sphere in Euclidean 3-space, we compute its Bartnik-Bray outer mass to first order in the data's deviation from the standard sphere. The Hawking mass gives a well-known lower bound, and an upper bound is obtained by estimating the mass of a static vacuum extension. As an application we confirm that in a time-symmetric slice concentric geodesic balls shrinking to a point have mass-to-volume ratio converging to the energy density at their center, in accord with physical expectation and the behavior of other quasilocal masses. For balls shrinking to a flat point we can also compute the outer mass to fifth order in the radius---the term is proportional to the Laplacian of the scalar curvature at the center---but our estimate is not refined enough to identify this term at a point which is merely scalar flat. In particular it cannot discern gravitational contributions to the mass.