Generalized Einstein-Cartan field equations

Field equations are studied for generalized Einstein-Cartan-Sciama-Kibble (EC) theories in which the connection is not necessarily compatible with the metric and the Lagrangian is not necessarily the curvature scalar. The condition that the Euler-Lagrange equations for a general Lagrangian density L(g, ∂g,∂∂g, Γ, ∂Γ) involve no third- or higher-order derivatives of the metric requires that the gravitational field equations be equivalent to those of general relativity with modified sources. The divergence of the symmetric "energymomentum" tensor δL_matter/δg_(ij) evaluated with δ(Γ^l_mn-{{l}{mn}})=0 for a generalized EC theory does not vanish in the presence of spin. The general form of the spin field equation linear in the defect λⁱ_jk≡Γⁱ_jk-{{i}{jk}} is derived.