Casimir–Polder-Like Force for an Atom in Hartle–Hawking Vacuum outside a Schwarzschild Black Hole

We calculate the energy level shift of a static ground state two-level atom interacting with fluctuating massless scalar fields in the Hartle–Hawking vacuum outside a Schwarzschild black hole. It is found that the energy level shift is position-dependent and gives rise to a force on the atom besides the classical gravitational force. This force behaves like r^?1(r?2M)^?1 for r approaching 2M, whereas it behaves like r^?2 for large r.