Generalized Einstein-Cartan field equations

Abstract

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