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.