In the first part of this review on the fundamentals of theory we have given two routes to the establishment of stochastic gravity and derived a general (finite) expression for the noise kernel. In the second part we gave three applications, the correlation functions of gravitons in a perturbed Minkowski metric, structure formation in stochastic gravity theory, and the outline of a program for the study of black hole fluctuations and backreaction. A central issue which stochastic gravity can perhaps best address is the validity of semiclassical gravity as measured by the fluctuations of stress-energy compared to the mean. We will include a review of this topic in a future update.

There is ongoing research related to the topics discussed in this review. On the theory side, Roura and Verdaguer [255] have recently shown how stochastic gravity can be related to the large limit of quantum metric fluctuations. Given free matter fields interacting with the gravitational field, Hartle and Horowitz [128], and Tomboulis [277] have shown that semiclassical gravity can be obtained as the leading order large limit (while keeping times the gravitational coupling constant fixed). It is of interest to find out where in this setting can one place the fluctuations of the quantum fields and the metric fluctuations they induce; specifically, whether the inclusion of these sources will lead to an Einstein–Langevin equation [44, 157, 167, 58, 202], as it was derived historically in other ways, as described in the first part of this review. This is useful because it may provide another pathway or angle in connecting semiclassical to quantum gravity (a related idea is the kinetic theory approach to quantum gravity described in [155]).

Theoretically, stochastic gravity is at the frontline of the ‘bottom-up’ approach to quantum gravity [152, 154, 155]. Structurally, as can be seen from the issues discussed and the applications given, stochastic gravity has a very rich constituency because it is based on quantum field theory and nonequilibrium statistical mechanics in a curved spacetime context. The open systems concepts and the closed-time-path/influence functional methods constitute an extended framework suitable for treating the backreaction and fluctuations problems of dynamical spacetimes interacting with quantum fields. We have seen it applied to cosmological backreaction problems. It can also be applied to treat the backreaction of Hawking radiation in a fully dynamical black hole collapse situation. One can then address related issues such as the black hole end state and information loss puzzles (see, e.g., [231, 151] and references therein). The main reason why this program has not progressed as swiftly as desired is due more to technical rather than programatic difficulties (such as finding reasonable analytic approximations for the Green function or numerical evaluation of mode-sums near the black hole horizon). Finally, the multiplex structure of this theory could be used to explore new lines of inquiry and launch new programs of research, such as nonequilibrium black hole thermodynamics and statistical mechanics.

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