6 Metric Fluctuations in Minkowski Spacetime

Although the Minkowski vacuum is an eigenstate of the total four-momentum operator of a field in Minkowski spacetime, it is not an eigenstate of the stress-energy operator. Hence, even for those solutions of semiclassical gravity such as the Minkowski metric, for which the expectation value of the stress-energy operator can always be chosen to be zero, the fluctuations of this operator are non-vanishing. This fact leads to consider the stochastic metric perturbations induced by these fluctuations.

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].