Static vacuum Einstein equations and 3-manifold geometry
The static vacuum Einstein equations are the simplest
equations for Ricci-flat 4-manifolds (Lorentzian or Riemannian), and play
an important role in general relativity. After introducing the basic form
and facts on these equations, we discuss the corresponding Black Hole Uniqueness
Theorem. The usual formulation of this result requires the strong assumption
that the 3-metric is asymptotically flat; we show that this assumption
is in fact not necessary, in almost all cases.
We then discuss relations with degenerations of metrics on 3-manifolds.
Specifically, the static vacuum equations describe the degenerations of
Yamabe metrics (or metrics with bounds on scalar curvature) in regions
where the full curvature goes to infinity. This phenomenon will be illustrated
in specific examples, and its relation with the sphere decomposition of
3-manifolds will be discussed.
Possibly we will also indicate the relations of these ideas with singularity
formation in general relativity.
Spring 1999 meeting of the PNGS