On Existence of Static Metric Extensions in General Relativity

TL;DR

This paper proves the existence of static metric extensions in general relativity for metrics close to Euclidean with symmetric boundary data, advancing understanding of quasi-local mass and boundary conditions.

Contribution

It establishes the existence of asymptotically flat, scalar-flat static metric extensions satisfying Bartnik's boundary conditions for near-Euclidean metrics with symmetric boundary data.

Findings

Existence of static metric extensions near Euclidean metrics.

Extensions satisfy Bartnik's geometric boundary condition.

Results applicable to metrics with reflection symmetry.

Abstract

Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik \cite{Bartnik_energy}. We show that, for any metric on $\bar{B}_1$ that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat {\em static} metric extension in $M = \R^3 \setminus B_1$ such that it satisfies Bartnik's geometric boundary condition \cite{Bartnik_energy} on $\partial B_1$.