The influence functional previously introduced in
Equation (16
) can be written in
terms of the the CTP effective action
derived
in Equation (174
) using
Equation (17
). The
Einstein-Langevin equation follows from taking the functional
derivative of the stochastic effective action (27
) with respect to
and imposing
. This
leads to
As we have seen before and here, the Einstein-Langevin equation is a dynamical equation governing the dissipative evolution of the gravitational field under the influence of the fluctuations of the quantum field, which, in the case of black holes, takes the form of thermal radiance. From its form we can see that even for the quasi-static case under study the backreaction of Hawking radiation on the black hole spacetime has an innate dynamical nature.
For the far field case, making use of the
explicit forms available for the noise and dissipation kernels,
Campos and Hu [54, 55
] formally proved the
existence of a fluctuation-dissipation relation at all temperatures
between the quantum fluctuations of the thermal radiance and the
dissipation of the gravitational field. They also showed the formal
equivalence of this method with linear response theory for lowest
order perturbations of a near-equilibrium system, and how the
response functions such as the contribution of the quantum scalar
field to the thermal graviton polarization tensor can be derived.
An important quantity not usually obtained in linear response
theory, but of equal importance, manifest in the CTP stochastic
approach is the noise term arising from the quantum and statistical
fluctuations in the thermal field. The example given in this
section shows that the backreaction is intrinsically a dynamic
process described (at this level of sophistication) by the
Einstein-Langevin equation.