Gravitating opposites attract

Generalizing previous work by two of the authors, we prove the non-existence of certain stationary configurations in general relativity having a spatial reflection symmetry across a non-compact surface disjoint from the matter region. Our results cover cases such as that of two symmetrically arranged rotating bodies with anti-aligned spins in n + 1 (n >= 3) dimensions, or two symmetrically arranged static bodies with opposite charges in (3 + 1) dimensions. They also cover certain symmetric configurations in (3 + 1)-dimensional gravity coupled to a collection of scalars and Abelian vector fields, such as those that arise in supergravity and Kaluza-Klein models. We also treat the bosonic sector of simple supergravity in 4 + 1 dimensions.