A Remark on Boundary Effects in Static Vacuum Initial Data sets

TL;DR

This paper proves that certain static vacuum initial data sets with boundary conditions are isometric to Schwarzschild slices and provides bounds on their mass based on boundary properties, extending classical results.

Contribution

It generalizes the Bunting and Masood-ul-Alam result by relaxing boundary conditions and introduces mass bounds related to boundary mean curvature and area.

Findings

(M, g) is isometric to a Schwarzschild slice under zero mean curvature boundary.

Derived an upper bound on ADM mass based on boundary area and mean curvature.

Extended classical boundary effect results to more general boundary conditions.

Abstract

Let (M, g) be an asymptotically flat static vacuum initial data set with non-empty compact boundary. We prove that (M, g) is isometric to a spacelike slice of a Schwarzschild spacetime under the mere assumption that the boundary of (M, g) has zero mean curvature, hence generalizing a classic result of Bunting and Masood-ul-Alam. In the case that the boundary has constant positive mean curvature and satisfies a stability condition, we derive an upper bound of the ADM mass of (M, g) in terms of the area and mean curvature of the boundary. Our discussion is motivated by Bartnik's quasi-local mass definition.