  Using the nonrelativistic approximation in the curved-space Dirac equation,
the analog of the Pauli equation is derived for a weak, but otherwise
arbitrary, gravitational field, in the presence of an electromagnetic field. On
top of that, are obtained the equations of motion for the massive spin-$1/2$
charged particle. In the two particular cases, we consider the previously
explored backgrounds of the plane gravitational wave and the homogeneous static
gravitational field. Different from the previous works which were employing
either the exact or conventional Foldy-Wouthuysen transformations, here we
perform calculations in the simplest way, in particular, aiming to clean up the
existing discrepancies in the literature.
