Static Ricci-flat 5-manifolds admitting the 2-sphere

Abstract

We examine, in a purely geometrical way, static Ricci-flat 5-manifolds admitting the 2-sphere and an additional hypersurface-orthogonal Killing vector. These are widely studied in the literature, from different physical approaches, and are known variously as the Kramer–Gross–Perry–Davidson–Owen solutions. The two-fold infinity of cases that result are studied by way of new coordinates (which are in most cases global) and the cases likely to be of interest in any physical approach are distinguished on the basis of the nakedness and geometrical mass of their associated singularities. It is argued that the entire class of solutions has to be considered unstable about the exceptional solutions: the black string and soliton cases. Any physical theory which admits the non-exceptional solutions as the external vacua of a collapsing object has to accept the possibility of collapsing to zero volume leaving behind the weakest possible, albeit naked, geometrical singularities at the origin. Finally, it is pointed out that these types of solutions generalize, in a straightforward way, to higher dimensions.