  We extend complete complementarity relations to curved spacetimes by
considering a succession of infinitesimal local Lorentz transformations, which
implies that complementarity remains valid as the quanton travels through its
world line and the complementarity aspects in different points of spacetime are
connected. This result allows the study of these different complementary
aspects of a quantum system as it travels through spacetime. In particular, we
study the behavior of these different complementary properties of massive
spin-$1/2$ particles in the Schwarzschild spacetime. For geodetic circular
orbits, we find that the spin state of one particle oscillates between a
separable and an entangled state. For non-geodetic circular orbits, we notice
that the frequency of these oscillations gets bigger as the orbit gets nearer
to the Schwarzschild radius $r_s$.
