Torsion

Torsion is the antisymmetric part of an affine connection, measuring the failure of an infinitesimal parallelogram to close under parallel transport. In the presence of torsion, a vector transported along one edge of a small loop and then the other does not arrive at the same point as the vector transported in the reverse order. This twisting of spacetime is not merely a mathematical curiosity: in Einstein-Cartan theory, torsion is generated by the spin density of matter and becomes a dynamical variable on equal footing with curvature.

Torsion-free connections — the Levi-Civita connection in particular — are the default in general relativity because standard matter (scalar fields, unpolarized fluids, electromagnetic radiation) carries no macroscopic spin density. But fermionic matter has intrinsic spin, and when spins are aligned at high density, torsion becomes physically significant. The torsion tensor T^λ_μν is related to spin density S^λ_μν through the Einstein-Cartan field equations, making torsion a contact interaction rather than a propagating field.

Torsion also appears in the geometry of Lie groups and homogeneous spaces, where the canonical connection of a reductive space naturally carries torsion. In these contexts, torsion encodes the algebraic structure of the space rather than a physical field, demonstrating that the same geometric object can arise from either physical or algebraic necessity.

Torsion is the geometric shadow of spin — the twist that matter impresses on space when its rotational degrees of freedom are not averaged away. General relativity's assumption of torsion-freedom is not a theorem but a prejudice inherited from the macroscopic world. The question is not whether torsion exists; it is whether we will ever build an experiment sensitive enough to feel the twist.