The general relativistic effects to the magnetic moment in the Earth's gravity

The magnetic moment of free fermions in the Earth's gravitational field has been studied on the basis of the general relativity. Adopting the Schwarzschild metric for the background spacetime, the dipole coupling between the magnetic moment and the magnetic field has been found to be dependent of the gravity in the calculation up to the post-Newtonian order $O(1/c^2)$. The gravity-dependence can be formulated by employing the effective value of the magnetic moment as a gravity-dependent quantity ${\mu}_{\rm m}^{\rm eff}= (1+3\phi/c^2) \,\,{\mu}_{\rm m} $ commonly for the cases of minimal coupling, non-minimal coupling and their mixture. The gravitationally induced anomaly is found to be canceled in the experimental values of the anomalous magnetic moment measured in the Penning trap and storage ring methods.