Quantum gravitodiamagnetic interaction

In the framework of linearized quantum gravity, we investigate the quantum gravitational interaction induced by the gravitodiamagnetic coupling of two massive objects to vacuum fluctuations of the gravitational field. Starting from the Lagrangian of a particle in a gravitational field and employing the formalism of Weyl gravitoelectromagnetism, we derive the interaction Hamiltonian associated with gravitodiamagnetic coupling. Unlike the linear couplings that arise in gravitoelectric and gravitomagnetic interactions, the gravitodiamagnetic coupling depends quadratically on the gravitomagnetic field. Based on this Hamiltonian, we show that, for a spherically symmetric gravitational hydrogen-like system in its ground state, the induced quadrupole moment has the opposite sign to the applied gravitomagnetic field, which is the defining signature of gravitodiamagnetism. Using leading-order perturbation theory, we further obtain an explicit expression for the resulting interaction potential, which is attractive and scales as $r^{-11}$ at all separations, where $r$ denotes the distance between the two objects.