Unconventional stringlike singularities in flat spacetime

The conical singularity in flat spacetime is mostly known as a model of the cosmic string or the wedge disclination in solids. Another, equally important, function is to be a representative of quasiregular singularities. From all these points of view it seems interesting to find out whether there exist other similar singularities. To specify what similar means I introduce the notion of the stringlike singularity, which is, roughly speaking, an absolutely mild singularity concentrated on a curve or on a 2-surface S (depending on whether the space is three- or four-dimensional). A few such singularities are already known: the aforementioned conical singularity, its two Lorentzian versions, the spinning string, the screw dislocation, and Tod's spacetime. In all these spacetimes S is a straight line (or a plane) and one may wonder if this is an inherent property of the stringlike singularities. The aim of this paper is to construct stringlike singularities with less trivial S. These include flat spacetimes in which S is a spiral, or even a loop. If such singularities exist in nature (in particular, as an approximation to gravitational field of strings), their cosmological and astrophysical manifestations must differ drastically from those of the conventional cosmic strings. Likewise, being realized as topological defects in crystals, such loops and spirals will probably also have rather unusual properties.