Palatini-type variational principle for the SL(2, C) gauge theory of gravitation

An explicitly SL(2, C) gauge-invariant Lagrangian density, equivalent to Hilbert's Lagrangian density, is written, and a Palatini-type variational principle is applied to it. The resulting field equations are Einstein's equations written in dyad notation and a set of equations defining the spin coefficients in terms of the components of the null tetrad vectors and their directional derivatives. The techniques employed throughout this article are those which were developed in the SL(2, C) gauge theory of gravitation.