EinsteinCartanSciamaKibble

Einstein-Cartan-Sciama-Kibble theory (ECSK) is a classical theory of gravity that extends general relativity by including spacetime torsion, a geometric property sourced by the intrinsic spin of matter. The theory arose from the idea that the Poincaré symmetry of matter fields can be gauged, leading to a Riemann-Cartan spacetime in which curvature and torsion coexist. The name reflects contributions by Cartan (early development of torsion) and by Sciama and Kibble, who showed how spin density can couple to torsion.

In ECSK theory, the fundamental variables are the tetrad field and the spin connection, or equivalently a metric and an independent connection with nonzero torsion. The torsion tensor T^λ_{ μν} measures the antisymmetric part of the connection. The field equations resemble Einstein’s equations but include spin contributions: the curvature is sourced not only by energy-momentum but also by spin density of matter. Importantly, torsion is determined algebraically by the spin current, so it does not propagate as a dynamical field in the standard formulation.

Consequences and regime of validity: In regions with negligible spin density, ECSK reduces to general relativity, yielding essentially identical predictions. At extremely high densities, such as in the early universe or inside collapsing objects, the spin-induced torsion can produce a repulsive interaction that can soften or avert singularities in principle. However, in ordinary astrophysical and laboratory conditions, torsion effects are vanishingly small and indistinguishable from GR.

Status: ECSK is a well-established geometric extension and a special case of broader gauge theories of gravity. There is currently no experimental evidence for torsion, and predictions converge to those of general relativity under typical conditions.