Topological Rigidity of Dimensional Reduction to Three Dimensions

Studying spacetimes with continuous symmetries by dimensional reduction to a lower dimensional spacetime is a well known technique in field theory and gravity. Recently, its use has been advocated in numerical relativity as an efficient computational technique for the numerical study of axisymmetric asymptotically flat 4-dimensional spacetimes. We prove here that if the dimensionally reduced spacetime is a physically reasonable 3-dimensional asymptotically flat or asymptotically anti-de Sitter spacetime, then, surprisingly, the topology of the higher dimensional spacetime must be one of two product topologies. Reductions of other topologies result in physically pathological spacetimes. In particular, reduction of asymptotically flat 4-dimensional spacetimes must lead to pathologies. These results use only the topological censorship theorem and topological methods and consequently are independent of the field equations and reduction method.