Here we derive the Einstein-Langevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated two-point correlation functions, as well as, the two-point correlation functions for the metric perturbations. Even though, in this case, we expect to have negligibly small values for these correlation functions for points separated by lengths larger than the Planck length, there are several reasons why it is worth carrying out this calculation.
On the one hand, these are the first backreaction
solutions of the full Einstein-Langevin equation. There are
analogous solutions to a “reduced” version of this equation
inspired in a “mini-superspace” model [59, 38], and there
is also a previous attempt to obtain a solution to the
Einstein-Langevin equation in [58], but there the
non-local terms in the Einstein-Langevin equation were neglected.
On the other hand, the results of this
calculation, which confirm our expectations that gravitational
fluctuations are negligible at length scales larger than the Planck
length, but also predict that the fluctuations are strongly
suppressed on small scales, can be considered a first test of
stochastic semiclassical gravity. In addition, these results reveal
an important connection between stochastic gravity and the large
expansion of quantum gravity. We can also extract
conclusions on the possible qualitative behavior of the solutions
to the Einstein-Langevin equation. Thus, it is interesting to note
that the correlation functions at short scales are characterized by
correlation lengths of the order of the Planck length; furthermore,
such correlation lengths enter in a non-analytic way in the
correlation functions.
We advise the reader that his section is rather
technical since it deals with an explicit non-trivial backreaction
computation in stochastic gravity. We have tried to make it
reasonable self-contained and detailed, however a more detailed
exposition can be found in [209].