The Bartnik-Bray Outer Mass of Small Metric Spheres in Time-Symmetric 3-Slices

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 point where the Riemann curvature tensor vanishes 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 in general at a point where merely the scalar curvature vanishes. In particular, it cannot discern gravitational contributions to the mass.