Nonrelativistic approximation of the Dirac equation for slow fermions in static metric spacetimes

We analyze the nonrelativistic approximation of the Dirac equation for slow fermions, having small kinetic energies compared to their rest energy $m$ and moving in spacetimes with a static metric, caused by the weak gravitational field of the Earth and a chameleon field, and derive the most general effective gravitational potential to order $1/m$, induced by a static metric of spacetime excluding possible rotations of the coordinate frame. The derivation of the nonrelativistic Hamilton operator of the Dirac equation is carried out by using a standard Foldy–Wouthuysen transformation. We discuss the chameleon field as source of a torsion field and torsion–matter interactions.