  We explore the nonadditive three-body quantum gravitational quadrupole
interaction among three nonpointlike objects in their ground states in the
framework of linearized quantum gravity and find that the interaction exhibits
a distance behavior of $r_A^{-5}r_B^{-5}r_C^{-5}$ in the near regime where the
interobject distances are small compared with the characteristic transition
wavelength of the objects, and $r_A^{-5}r_B^{-5}r_C^{-5}(r_A+r_B+r_C)^{-1}$ in
the far regime where the distances are larger than the characteristic
transition wavelength, where $r_\xi $ denotes the distance between any two
objects except object $\xi\;(\xi=A,B,C)$. Compared to the additive quantum
gravitational interaction between two ground-state objects in vacuum which is
always attractive both in the near and far regimes, the nonadditive interaction
among three ground-state objects can be either attractive or repulsive
depending on the geometrical configuration of these objects. That is, in
principle, the attractive or repulsive properties of the nonadditive three-body
quantum gravitational interaction can be manipulated by changing the
geometrical arrangement of the objects.
