 Einstein–Cartan theory

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Einstein–Cartan theory is an extension of general relativity that incorporates an intrinsic connection between geometry and matter, specifically by allowing for the presence of torsion alongside curvature in the spacetime.