On Static n-body Configurations in Relativity

The static n-body problem of General Relativity states that there are, under a reasonable energy condition, no static $n$-body configurations for $n>1$, provided the configuration of the bodies satisfies a suitable separation condition. In this paper we solve this problem in the case that there exists a closed, noncompact, totally geodesic surface disjoint from the bodies. This covers the situation where the configuration has a reflection symmetry across a noncompact surface disjoint from the bodies.