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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., [187Jump To The Next Citation Point] 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.

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