Unconventional stringlike singularities in flat spacetime

Abstract

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 string-like 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 string-like 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.

Geometry of massless cosmic strings

“…Given a holonomy one can always construct a conical defect with that holonomy by "cutting" away a wedge of spacetime and "pasting" the opposite sides together. The procedure is straightforward (see [17] for a rigorous treatment). Consider a Lorentz transformation Q that preserves a linear codimension 2 subspace L of Minkoswki space M .…”

Unwrapping Closed Timelike Curves

“…3.2. See also [31,35]: The unwrapped spacetime (M, g) is singular by construction (we excised a set of regular points from its base space), and inextendible, provided (M, g) is inextendible. We cannot use the regularity criterion (6) to show inextendibility, as it only applies to axisymmetric spacetimes, andM is not axisymmetric, because the rotation is lifted to the translation¯ .…”

Even the Minkowski space is holed

“…But after that moment some observers (A and B in the figure) will discover that without experiencing any acceleration they started to move towards each other. Figure 1(b) shows how in the otherwise Minkowskian space a time machine or a wormhole may appear with no visible cause (more of bizarre examples can be found in [3]). "Thus general relativity, which seemed at first as though it would admit a natural and powerful statement at prediction, apparently does not" [4].…”

“…In looking for such a requirement it is desirable, first, to comprehend what exactly is to be prohibited. In particular, each of the singular spacetimes mentioned above can be viewed as a quasiregular singularity (i. e., a singularity with bounded curvature, see [6] for a rigorous definition), or as an "absolutely mild singularity" [3] (it has a finite covering by open sets, each of which can be extended to a singularity-free spacetime). It is also a "locally extendible" spacetime (i. e., it contains an open set U ⊂ M isometric to a subset U ′ of some other spacetime M ′ such that the closure of U ′ in M ′ is compact while the closure of U in M is not [7]).…”