In the first part of this review on the fundamentals of the 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. We have also discussed the problem of the validity of semiclassical gravity, a central issue, which stochastic gravity is in a unique position to address.
We have pointed out a number of ongoing research projects related to the topics discussed in this review, such as the equivalence of the correlation functions to the metric perturbations obtained using the Einstein–Langevin equations and the quantum-correlation functions that follow from a pure quantum-field-theory calculation up to leading order in the large limit, the calculation of the spectrum of metric fluctuations in inflationary models driven by the trace anomaly due to conformally-coupled fields, the related problem of runaway solutions in backreaction equations and the issue of the coincidence limit in the noise kernel for black-hole fluctuations.
Theoretically, stochastic gravity is at the front line of the ‘bottom-up’ approach to quantum gravity [185, 187, 188, 190]. Its pathway or angle starts from the well-defined and well-understood theory of semiclassical gravity. 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 fluctuation problems of dynamical spacetimes interacting with quantum fields. We have seen applications to cosmological-structure formation and black-hole backreaction from particle creation. A more complete understanding of the backreaction of Hawking radiation in a fully-dynamical black-hole situation will enable one to address fundamental issues such as the black-hole end state and information-loss puzzles. 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’s function or the 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 the microscopic structures of spacetime.
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