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Wikipedia gives an explanation of cosmic strings that I'm sure would be very helpful if I had a major in topology, but alas I do not. I know that a topological defect is any sort of discontinuity in a medium that's hard to remove. Things I know are capable of topological defects include crystal lattices, the orientation of molecules in LCDs, and the direction a telephone cord coils. But how does spacetime/the vacuum do this?

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First, if you need to review what a topological defect is or how it relates to an order parameter, see Order Parameters, Broken Symmetry, and Topology.

Here is an example that gives the flavor of what a cosmological topological defect might look like in a vacuum.

Suppose you have an electric dipole floating in space, a rod with + and - charges at the ends. It does not rotate with respect to an inertial frame of reference. If it has an initial velocity, it will drift to other points in the space.

The orientation of the dipole is an order parameter. Given an appropriate initial velocity or force on the center of mass, it will drift through any point you choose. It orientation when it arrives defines a vector that is the order parameter at that point. The order parameter points in the same direction everywhere.

Now suppose there is a long charged wire in space. The order parameter behaves very differently. The dipole orients toward the wire. If you move the dipole in a loop around the wire, the dipole rotates through a full circle. If you move it in a loop that does not circle the wire, it rotates, but returns to its original orientation without rotation through a circle. The wire is a topological defect.

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  • $\begingroup$ I'm not sure I have the right words for this, but it seems like a cosmic string is a sort of kink or knot in the way spacetime is curved. $\endgroup$
    – zucculent
    Nov 2, 2021 at 3:54
  • $\begingroup$ A line of charge is a kink in the direction of the electric field. Imagine a kink in the nature of space so that the direction an uncharged rod pointed rotated as the rod moved, simply because of an inertial reference frame changing from place to place. Or imagine that straight trajectories bent as they passed near the kink. $\endgroup$
    – mmesser314
    Nov 2, 2021 at 4:14
  • $\begingroup$ This sounds a lot like the way light is bent around a black hole because its straight path is bent. $\endgroup$
    – zucculent
    Nov 2, 2021 at 4:19
  • $\begingroup$ Yes, or any mass. Mass curves spacetime. That is what gravity is. But imagine a kink in space without a mass. $\endgroup$
    – mmesser314
    Nov 2, 2021 at 4:21
  • $\begingroup$ So how does this extend to magnetic monopoles and domain walls? Is the former the same kind of knot except 0-dimensional? $\endgroup$
    – zucculent
    Nov 2, 2021 at 4:26

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