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 $L(g, \partial g,\partial \partial g, \Gamma , \partial \Gamma )$ 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 $\frac{\delta L_{\mathrm{matter}}}{\delta g_{\mathrm{}\left(\mathrm{ij}\right)\mathrm{}}}$ evaluated with $\delta ({{\Gamma }^l}_{\mathrm{mn}}-\left\{\left\{l\right\}\left\{\mathrm{mn}\right\}\right\})=0\mathrm{}$ 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 ${{\lambda }^i}_{\mathrm{jk}}\equiv {{\Gamma }^i}_{\mathrm{jk}}-\left\{\left\{i\right\}\left\{\mathrm{jk}\right\}\right\}$ is derived.