Fluctuations in a thermal field and dissipation of a black hole spacetime:Far-field limit

We study the backreaction of a thermal field in a weak gravitational backgr ound depicting the far-field limit of a black hole enclosed in a box by the closed time path (CTP) effective action and the influence functional metho d. We derive the noise and dissipation kernels of this system in terms of q uantities in quasiequilibrium, and formally prove the existence of a fluctu ation-dissipation relation (FDR) at all temperatures between the quantum fl uctuations of the thermal radiance and the dissipation of the gravitational field. This dynamical self-consistent interplay between the quantum field and the classical spacetime is, we believe, the correct way to treat backre action problems. To emphasize this point we derive an Einstein-Langevin equ ation which describes the nonequilibrium dynamics of the gravitational pert urbations under the influence of the thermal field. We show the connection between our method and the linear response theory (LRT), and indicate how t he functional method can provide more accurate results than prior derivatio ns of FDRs via LRT in the test-field, static conditions. This method is in principle useful for treating fully nonequilibrium cases such as backreacti on in black hole collapse.