Gravitational Casimir-Polder interaction in a thermal bath
We have investigated, by separating the contributions from thermal fluctuations (tf) and the radiation reaction (rr), the gravitational Casimir-Polder interaction between a gravitationally polarizable two-level object and an infinite gravitational Dirichlet boundary in a thermal bath at a temperature $T$. The results indicate that the rr-contribution to the interaction potential is independent of the temperature, whereas the tf-contribution is generally governed by a nontrivial interplay between the thermal corrections and the polarization effect. Here, the object-to-boundary distance, the characteristic transition wavelength of the object and the thermal wavelength of gravitons are denoted by $L$, $λ$ and $β$, respectively. In contrast to the vacuum case, where the interaction potential scales as $L^{-5}$ for $L\llλ$ and $L^{-6}$ for $L\ggλ$, corresponding to an always repulsive force, qualitatively new behaviors emerge at high temperatures. Particularly, when $\sqrt[4]{βλ^3}\ll L\llλ$ and the object is polarizable within the plane perpendicular to the boundary, a novel scaling of $TL^{-1}$ arises; when $\sqrt{βλ}\ll L\llλ$ and the object is polarizable along the vertical-to-boundary axis, the interaction force becomes surprisingly attractive. At extremely high temperatures and large distances, i.e. when $β\ll λ\ll L$, the potential oscillates with the distance $L$ and thus an attractive or repulsive and even vanishing force can be resulted, depending on the exact values of $L$. Our work demonstrates that thermal gravitons can act as an active control mechanism for quantum gravitational interactions, and temperature, polarization configuration, and object-to-boundary distance jointly determine the magnitude, scaling law, and even the attractive or repulsive nature of the interaction force.