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

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.

The final version will appear in Communications in Mathematical Physics

References in corpus (4)

• The limiting behavior of the Liu-Yau quasi-local energy
• Asymptotically flat extensions of CMC Bartnik data
• Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity
• Evaluating small sphere limit of the Wang-Yau quasi-local energy

Cited by in corpus (8)

• The small sphere limit of quasilocal energy in higher dimensions along lightcone cuts
• Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture
• Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity
• Quasi-local mass on unit spheres at spatial infinity
• Second-order mass estimates for static vacuum metrics with small Bartnik data
• On Hawking mass and Bartnik mass of CMC surfaces
• Small sphere limit of the quasi-local energy with anti de-Sitter space reference
• Local gravitational energy in higher dimensions