Cartan Formalism (physics) - The Palatini Action

The Palatini Action

In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection form A over a four-dimensional differential manifold M is given by

where F is the gauge curvature 2-form and ε is the antisymmetric intertwiner of four "vector" reps of SO(3,1) normalized by η.

Note that in the presence of spinor fields, the Palatini action implies that d_Ae is nonzero, that is, have torsion. See Einstein-Cartan theory.