Cosmologies with quasiregular singularities. I. Spacetimes and test waves

The nature of spacetimes with quasiregular singularities is discussed. Such singularities are the end points of incomplete, inextendible geodesics at which the Riemann tensor and its derivatives remain at least bounded in all parallel-propagated orthonormal frames; observers approaching such a singularity would find that their world lines come to an end in a finite proper time, without encountering infinite tidal forces. Particular attention is paid to Taub-NUT-(Newman-Unti-Tamburino) type cosmologies, which are an interesting class of spacetimes containing quasiregular singularities. These cosmologies are characterized by incomplete geodesics which spiral infinitely around a topologically closed spatial dimension: They include the $R^3\times S^1$ and $R^1\times T^3$ flat Kasner universes, the two-parameter family of Taub-NUT universes, and an infinite-dimensional subclass of Einstein-Rosen-Gowdy spacetimes studied by Moncrief. The global structure of each of these spacetimes is described. The flat Kasner and Moncrief universes both possess a null hypersurface which is a Cauchy and a Killing horizon and which contains a quasiregular singularity; the Taub-NUT universes possess two such null hypersurfaces. Timelike geodesics exhibit two sorts of behavior in the vicinity of any one of these null hypersurfaces: They may pass right through the hypersurface or they may approach it asymptotically, spiraling around a closed spatial dimension. The behavior of scalar test fields in each of the Taub-NUT-type cosmologies is also very similar in the vicinity of a singularity-containing null hypersurface. In each case there are three types of wave modes: There are modes whose amplitude diverges logarithmically in the vicinity of the null hypersurface; there are other modes whose phase diverges logarithmically; and there are modes without any divergent behavior. It is shown that generic finite data on an initial Cauchy hypersurface lead to divergent test fields at the null hypersurface. The possibility of the existence of additional Taub-NUT-type cosmologies among spatially homogeneous spacetimes of the various Bianchi types and among inhomogeneous spacetimes is also discussed. The geodesic and scalar field behavior in these spacetimes is used in a subsequent paper to investigate the stability of Taub-NUT-type cosmologies.