Raychaudhuri equations and gravitational collapse in Einstein-Cartan theory

The Raychaudhuri equations for the expansion, shear, and vorticity are generalized in a spacetime with torsion for timelike as well as null congruences. These equations are purely geometrical like the original Raychaudhuri equations and could be reduced to them when there is no torsion. Using the Einstein-Cartan-Sciama-Kibble field equations, the effective stress-energy tensor is derived. We also consider an Oppenheimer-Snyder model for the gravitational collapse of dust. It is shown that the null energy condition is violated before the density of the collapsing dust reaches the Planck density, hinting that the spacetime singularity may be avoided if there is a nonzero torsion, i.e., if the collapsing dust particles possess intrinsic spin.