Are four dimensions enough, a note on ambient cosmology

Abstract

The group of homothetic symmetries in the conformal infinity (the $4$-dimensional "ambient boundary") of a $5$-dimensional spacetime restricts the choice of topology to a topology under which the group of homeomorphisms of a spacetime manifold is the group of homothetic transformations. Since there are such spacetime topologies in the class of Zeeman-G\"obel, under which the formation of basic contradiction present in proofs of singularity theorems is impossible, an important question is raised: why should one construct a $5$-dimensional metric, in order to return back such a topology to its $4$-dimensional conformal boundary, while such topologies, like those ones in the Zeeman-G\"obel class, are already considered as more "natural" topologies for a spacetime, rather than the artificial (according to Zeeman) manifold topology?