Volume 27 · Part Three · Chapter 13 of 19
Defects, Vortices, and Torsion
Where the braided picture meets the Riemannian one, and the distinction a twisted thread tends to blur.
Curvature is not torsion
Curvature is what a vector notices when it is carried around a closed loop and comes back rotated. Torsion is what happens when the loop fails to close: transport one vector along another and vice versa, and the two paths do not meet. General Relativity sets torsion to zero by choosing a symmetric connection. That is a choice, not a derivation, and Einstein–Cartan theory keeps torsion and sources it with intrinsic spin — a theory indistinguishable from GR at all densities so far reached.
The figure of a twisting thread invites the reader to feel that twist and curvature are the same thing. They are not, and the volume's own braid imagery is a repeat offender. A helix in flat space has torsion in the curve-geometry sense while the space it sits in has no curvature at all. Two different torsions, two different objects, one word.
Defects as geometry
The productive version of the analogy is the defect picture from continuum mechanics: in a crystal, dislocations behave as sources of torsion and disclinations as sources of curvature in an effective geometry built from the deformation field. This is a real and quantitative correspondence, used in materials science. Vortex lines in a superfluid, with their quantised circulation, are the closest thing in this volume to a line defect one can point at.
What is not established is any claim that spacetime itself contains such defects. That has been proposed — cosmic strings are the best-developed version — and not observed.
Equations borrowed
- Riemann curvature tensor R^ρ_σμν and torsion tensor T^ρ_μν as independent objects
- Einstein–Cartan theory with spin-sourced torsion
- The defect/geometry correspondence in continuum mechanics
- Quantised circulation of superfluid vortex lines
Validity band
The defect correspondence holds within continuum elasticity. Torsion in spacetime remains unobserved; Einstein–Cartan deviations require densities far beyond current reach.
Falsifier
Detection of a spacetime torsion effect, or of gravitational lensing signatures of cosmic strings, would move this from analogy to physics.
Where this chapter is weakest
This chapter names a confusion the volume itself commits elsewhere. Its usefulness depends on the rest of the book being audited against it, which has not yet been done systematically.
The volume-wide audit of these weak points is collected in Where This Volume Is Weak.