On Existence of Static Metric Extensions in General Relativity

Details On Existence of Static Metric Extensions in General Relativity On Existence of Static Metric Extensions in General Relativity

2003-09-17

arxiv.org/abs/math-ph/0309041v1

Commun. Math. Phys. 241(1) , 27-46 (2003)

Physics

Mathematical Physics

20 pages

Scientific paper

10.1007/s00220-003-0925-2

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 $\partial B_1$.

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