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.
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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.