Continuity of the torsionless limit as a selection rule for gravity theories with torsion

While one can in principle augment gravity theory with torsion, it is generally thought that any such torsion affects would be too small to be of consequence. Here we show that this cannot in general be the case. We show that the limit of vanishing torsion is not necessarily a continuous one, with the theory obtained in the limit not necessarily coinciding with the theory in which torsion had never been present at all. However, for a standard torsion tensor that is antisymmetric in two of its indices we have found two cases in which the vanishing torsion limit is in fact continuous, namely Einstein gravity and conformal gravity. For other gravity theories of common interest to possess a continuous limit the torsion tensor would need to be antisymmetric in all three of its indices.