Attractive spin-spin contact interactions in the Einstein-Cartan-Sciama-Kibble torsion theory of gravitation

Kerlick has obtained the unexpected result that the spin-spin contact interactions, which are characteristic of the Einstein-Cartan-Sciama-Kibble theory of gravitation, are attractive for the case of a totally antisymmetric spin angular momentum density ${\tau }_{\mathrm{ijk}}$ which, in particular, is appropriate for the Dirac field. Using our previous techniques—where the emphasis is on the use of Lagrangian densities as opposed to energy-momentum densities—we present a simple and explicit verification of this result.