Fluctuations in a Thermal Field and Dissipation of a Black Hole Spacetime: Far-Field Limit

TL;DR

This paper develops a quantum field and gravity interaction framework using the influence functional method, deriving an Einstein-Langevin equation to describe non-equilibrium black hole back-reaction effects and establishing a fluctuation-dissipation relation.

Contribution

It introduces a self-consistent approach to back-reaction problems using the influence functional and derives an Einstein-Langevin equation for non-equilibrium gravitational dynamics.

Findings

Established a fluctuation-dissipation relation at all temperatures

Derived an Einstein-Langevin equation for gravitational perturbations

Connected the method to linear response theory and showed its advantages

Abstract

We study the back reaction of a thermal field in a weak gravitational background depicting the far-field limit of a black hole enclosed in a box by the Close Time Path (CTP) effective action and the influence functional method. We derive the noise and dissipation kernels of this system in terms of quantities in quasi-equilibrium, and formally prove the existence of a Fluctuation-Dissipation Relation (FDR) at all temperatures between the quantum fluctuations 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 back-reaction problems. To emphasize this point we derive an Einstein-Langevin equation which describes the non-equilibrium dynamics of the gravitational perturbations under the influence of the thermal field. We show the connection between our method and the linear response theory (LRT), and indicate how the functional method can provide more accurate results than prior derivations of FDRs via LRT in the test-field, static conditions. This method is in principle useful for treating fully non-equilibrium cases such as back reaction in black hole collapse.