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- A cosmic dispiration is described in, e.g., http://arxiv.org/abs/hep-th/0301194v2, as a specific type of a topological defect that is a combination of a disclination and a screw dislocation. If flat spacetime is characterized by cylindrical coordinates such that the metric is given by

ds^2 = dt^2 - dr^2 - r^2dφ^2 - dZ,

then the defect is given by identifying points with coordinates

(t,r,φ,Z) ~ (t,r,φ + 2πα, Z + 2πκ),

where α and κ are parameters that characterize the disclination and the screw dislocation, respectively.

There are no singularities involved. The spacetime is flat (Minkowski), but with non-trivial topology. The paper above actually attempts to calculate the behavior of a scalar field in the presence of such a defect. - Surely this metric has a curvature singularity at r=0: the circumference of any circle in the (r, phi) plane centred on r=0 must have circumference different from 2pi r, at least as long as alpha is not 1. Or am I misinterpreting the metric?
- Could you explain for a non-cosmologist the use or significance of cosmic dispirations in cosmology? Are they purely theoretical constructs or do they have potential physical reality?
- One idea was that these topological defects (known in this context as cosmic strings) helped to cause the formation of galaxies in the early universe, by focussing matter together. It turned out that this idea wasn't necessary, so that motivation for having them has gone away. Nevertheless, some theoretical models do predict that they should exist. There is a little, but not much, observational evidence for the existence of cosmic strings: http://en.wikipedia.org/wiki/Cosmic_string
- Robert, I think you are interpreting the metric correctly. Any circle that encircles the cosmic string will have a circumference of 2παr (|α| < 1). That is the nature of this topological defect. I am visualizing it as a cone formed from a flat sheet of paper. The apex corresponds to the cosmic string. Any circle that includes the apex will have a circumference less than 2πr. A non-zero κ, in addition, contributes a spiral structure in the Z direction.
- Thanks, to both of you. I get the picture. Singular points like that at the apex of the cone in a sheet of paper occur in virus capsids and in the cell walls of some bacteria; the wedge removed is 60 degrees. (The picture is essentially that of the 12 pentagons among hexagons making up a closed topological sphere.) The point is a wedge disclination, The translational component, which would make it a wedge dispiration, is zero. The relative displacement associated with the singularity in these biological and polymeric structures is a symmetry operation of the material. In the case of dispirations the screw displacement is a symmetry of the material (polymers and liquid crystals) while the component rotation and translation are not, so in a sense the wedge dispiration (a line singularity) is not a combination of a wedge disclination and a screw dislocation though in a continuum it would be. Is there any underlying screw symmetry that quantises or otherwise defines what cosmic dispirations are possible? If not, what sort of angles are contemplated? Presumably very small? And are the cosmic singularities lines in a 4-dimensional medium?
- I think that the deficit angles are generally expected to be small, and the observational evidence, such as it is, is for small deficit angles. They are not lines in a 4-d medium. At any given time the singularity is a line in 3-dimensional space, so in the 4-d space-time, they are surfaces. Interestingly, you can make the translational component along the t-axis instead of (or as well as) along the z-axis: this can be interpreted as a spinning cosmic string. If you do this then weird phenomena like time machines arise.
- Very nice, thanks, Robert. Anther question: It seems that cosmic dispirations are wedge in character, that is, the screw axis is parallel to the line singularity. In polymers one also gets twist dispirations in which the line is orthogonal to the line. The line is a small loop around the screw axis and screw displacement of polymer molecules occurs by means of moving twist dispirations; polymer chains move in a screw caterpillar-caterpillar-like fashion. Wedge and twist dispirations are two special cases of the general case. Is anything similar contemplated in cosmology? And could strings be pure disclinations, that is, without the translational component?
- I don't know about the more general situation where the screw axis is not along the line singularity: but yes, the simplest case for a cosmic string is a pure disclination (no translational component).

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