Torsion singularities

In the Einstein-Cartan-Sciama-Kibble theory of gravity, the torsion of the spacetime is linearly related to the spin density of the source. For many sources (e.g., most tensor fields), the spin density itself has an implicit linear dependence on the torsion. Consequently, in order to obtain the explicit solution for the torsion, one must invert a source-dependent linear operator. For many sources, this linear operator is degenerate for certain values of the source field. If the source field approaches such values, some components of the torsion will approach infinity. We begin the investigation of such ''torsion singularities'' by using the Hamiltonian formalism to study the simplest case in which they can occur: a Proca field coupled to torsion in a Bianchi type-V cosmology. In this case we find that, for all choices of initial data, as the cosmological model collapses the fields approach the critical values. Consequently the torsion increases dramatically and, instead of causing the model to ''bounce,'' drives it harder into a singularity. This singularity is qualitatively different from that which occurs in the corresponding Einstein-Proca (zero torsion) cosmology