The quantum interaction between a gravitationally polarizable object and a gravitational Neumann boundary

There should be quantum vacuum uctuations of gravitational fields if gravitation is quantized. Consequently in analogy to the electromagnetic Casimir-Polder force, one also expects the quantum gravitational interaction generated between a gravitationally polarizable object and a gravitational boundary. In this paper, we study this interaction between a static object and an infinite gravitational Neumann boundary, by a generalization of the DDC (Dalibard, Dupont-Roc and Cohen-Tannoudji) formalism. We find that, when the object-to-boundary distance L is very small compared to the characteristic transition wavelength of the object, the contribution of the radiation reaction of the object dominates over that of vacuum gravitational uctuations, and it corresponds to an attractive interaction force scaling as L -6 , regardless of different polarizations of the object. But quantitatively, the magnitudes of the force are obviously affected, and when the mass of the object is distributed along the vertical-to-boundary axis the magnitude is slightly larger. For extremely large L, the interaction force, as a result of the retardation effect, behaves characteristically one-order higher than the near-region force. Remarkably, although the total interaction force is also attractive, the contribution of the radiation reaction can be either attractive or repulsive, depending on the exact values of L, and in contrast to the near-region case, this contribution can be potentially smaller than the vacuum-uctuation contribution.