The semiclassical approximation for the Hamiltonian of Dirac particles
interacting with an arbitrary gravitational field is investigated. The time
dependence of the metrics leads to new contributions to the in-band energy
operator in comparison to previous works on the static case. In particular we
find a new coupling term between the linear momentum and the spin, as well as
couplings which contribute to the breaking of the particle - antiparticle
symmetry