Torsional four-fermion interaction and the Raychaudhuri equation

Abstract

Intrinsic spin of fermions can generate torsion in spacetime. This torsion is a non-propagating field that can be integrated out, leaving an effective non-universal four-fermion interaction. This geometrical interaction affects fermions inside a matter distribution and can be expected to become stronger as the density grows. We investigate the role of this interaction in a gravitationally collapsing fermionic distribution, by considering a statistical average of the interaction term which incorporates the effect of mixed vector and axial currents. We consider a gravitationally collapsing distribution of massive fermions, ignoring other interactions. Using simplified yet reasonable assumptions, we establish that the contribution can be attractive or repulsive depending on how torsion couples with different chiralities. Also, the interaction starts to dominate as the collapse proceeds, accelerating or decelerating the collapse depending on the relative signs of the geometrical interaction between different species of fermions.