The Einstein-Dirac-Yang/Mills equations: Black holes

We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills Held. We prove that the only black-hole solutions of the corresponding Einstein-Dirae-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU(2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.