Quantum gravitational interaction between a polarizable object and a boundary

We investigate the interaction caused by quantum gravitational vacuum fluctuations between a gravitationally polarizable object and a gravitational boundary, and find a position-dependent energy shift of the object, which induces a force in close analogy to the Casimir-Polder force in the electromagnetic case. For a Dirichlet boundary, the explicit form of the quantum gravitational potential for the polarizable object in its ground-state is worked out and is found to behave like $z^{-5}$ in the near regime, and $z^{-6}$ in the far regime, where $z$ is the distance to the boundary. Taking a Bose-Einstein condensate as a gravitationally polarizable object, we find that the relative correction to the radius caused by fluctuating quantum gravitational waves in vacuum is of order $10^{-21}$. Although far too small to observe in comparison with its electromagnetic counterpart, it is nevertheless of the order of the gravitational strain caused by a recently detected black hole merger on the arms of the LIGO.