### 8.4 Other work on metric fluctuations but without backreaction

In closing we mention some work on metric fluctuations where no backreaction is considered.
Barrabes et al. [20, 21] have considered the propagation of null rays and massless fields in a black hole
fluctuating geometry and have shown that the stochastic nature of the metric leads to a modified dispersion
relation and helps to confront the trans-Planckian frequency problem. However, in this case the stochastic
noise is put in by hand and does not naturally arise from coarse-graining as in a quantum open systems
approach, in terms of which stochastic gravity can be interpreted. It also does not take backreaction into
account. It will be interesting to explore how a stochastic black hole metric, arising as a solution to the
Einstein–Langevin equation, hence fully incorporating backreaction, would affect the trans-Planckian
problem.
As mentioned earlier, Ford and his collaborators [114, 115, 377] have also explored the issue of metric
fluctuations in detail and in particular have studied the fluctuations of the black hole horizon induced by
metric fluctuations. However, the fluctuations they considered are in the context of a fixed background and
do not relate to the backreaction.

Another work on metric fluctuations with no backreaction is that of Hu and Shiokawa [205], who study
effects associated with electromagnetic wave propagation in a Robertson–Walker universe and Schwarzschild
spacetime with a small amount of given metric stochasticity. They find that time-independent randomness
can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside
the black hole and gives rise to event horizon fluctuations and fluctuations in the Hawking
temperature. The stochasticity in the background metric in their work is assumed rather than derived
(as induced by quantum-field fluctuations). But it is interesting to compare their results with
those obtained in stochastic gravity with backreaction, as one can begin to get a sense of the
different sources of stochasticity and their weights (see, e.g., [187] for a list of possible sources of
stochasticity.)

In a subsequent paper, Shiokawa [328] shows that the scalar and spinor waves in a stochastic spacetime
behave similarly to the electrons in a disordered system. Viewing this as a quantum-transport problem, he
expresses the conductance and its fluctuations in terms of a nonlinear sigma model in the closed time-path
formalism and shows that the conductance fluctuations are universal, independent of the volume of the
stochastic region and the amount of stochasticity. This result has significant importance in
characterizing the mesoscopic behavior of spacetimes resting between the semiclassical and the quantum
regimes.