Topological rigidity of dimensional reduction to three dimensions

TL;DR

This paper proves that dimensional reduction of certain 4D spacetimes to 3D imposes strict topological constraints, limiting the possible higher-dimensional topologies to two specific types.

Contribution

It establishes topological rigidity results for dimensional reduction in gravity, showing only two product topologies are possible for physically reasonable 3D reductions.

Findings

Higher-dimensional spacetimes must have specific topologies when reduced to 3D.

Reductions of non-allowed topologies lead to pathological spacetimes.

Results are independent of field equations and reduction methods.

Abstract

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.