Gravitational spin Hall effect of electrons in Schwarzschild metric

In this study, we derive the non-relativistic Hamiltonian for electrons within the Schwarzschild metric from covariant Dirac equations, using both the weak field approximation and the Foldy-Wouthuysen transformation. This Hamiltonian incorporates a gravitational spin-orbit coupling term, resulting in the gravitational spin Hall effect (SHE), which separates electrons by their spin. By solving the Schr\"odinger equation for these electrons, we investigate the gravitational SHE as they orbit a non-rotating gravitational source. Our findings reveal that the spin-dependent separation of electrons increases in proportion to their orbital periods, significantly improving the detectability of gravitational SHE. Specifically, for electrons in a low Earth orbit, the separation is estimated to be $3.0\times 10^{-12}\, \text{m}$ annually. These results indicate the practicality of detecting the gravitational SHE in electrons orbiting Earth, especially with prolonged orbital durations, underscoring the potential for quantum test of the Weak Equivalence Principle.