  The effect of a classical gravitational field on the spin entanglement of a
system of two spin-1/2 particles moving in the curved spacetime is discussed.
The system is described by a two-particle Gaussian wave packet represented in
the momentum space and both the acceleration of the system and the curvature of
the spacetime cause to produce a Wigner rotation acting on the wave packet as
it moves along a path in the curved spacetime. By calculating the reduced
density operator at a final point, we focus on the spin entanglement of the
system. In a spherically symmetric and static gravitational field, for example
a charged black hole, there can be particular paths on which the Wigner
rotation is trivial and so the initial reduced density matrix remains intact.
This causes the spin entanglement to be invariant during the motion. The spin
entanglement descends to zero by increasing the angular velocity of the mean
centroid of the system as well increasing the proper time during which the
centroid moves on its circular path around the center.
