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 dAe is nonzero, that is, have torsion. See Einstein-Cartan theory.
Read more about this topic: Cartan Formalism (physics)
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