We begin with a brief description of the work on (a) the regularization of the stress-energy tensor of quantum fields in Schwarzschild spacetime in the 80s and (b) the black hole end-state and information-loss issues in the 80s, the ‘black hole complementarity principle’ of the 90s and the recent ‘firewall’ conjecture and its controversies. We then treat two classes of problems: (1) the backreaction of Hawking radiation on a black hole in the quasi-stationary regime, which occupies the longest span of a black hole’s life, and (2) the metric fluctuations of the event horizon of an evaporating black hole. In (1) the far field case can be solved analytically via the influence functional, highlighting nonlocal dissipation and colored noise; for the near horizon case we describe a strategy by Sinha et al. for treating the backreaction and fluctuations. In (2) we describe Bardeen’s model and discuss the results of Hu and Roura, who reached the same conclusion as Bekenstein, namely, that even for states regular on the horizon the accumulated fluctuations become significant by the time the black hole mass has changed substantially, well before reaching the Planckian regime. These results have direct implications for the end-state issue.

As a second application of stochastic gravity, we discuss in this chapter the backreaction problem in cosmology when the gravitational field couples to a quantum conformal matter field, and derive the Einstein–Langevin equations describing the metric fluctuations on the cosmological background. Conformal matter may be a reasonable assumption, because matter fields in the standard model of particle physics are expected to become effectively conformally invariant in the very early universe. We consider a weakly perturbed spatially flat Friedman–Lemaitre–Robertson–Walker spacetime and derive the Einstein–Langevin equation for the metric perturbations off this spacetime, using the CTP functional formalism described in previous chapters. With this calculation we also obtain the probability for particle creation. The CTP effective action is also used to derive the renormalized expectation value of the quantum stress-energy tensor and the corresponding semiclassical Einstein equation.

In this chapter we describe an important application of stochastic gravity: 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. The results of this calculation show that gravitational fluctuations are negligible at length scales larger than the Planck length and predict that the fluctuations are strongly suppressed at small scales. These results also reveal an important connection between stochastic gravity and the 1/N expansion of quantum gravity. In addition, they are used to study the stability of the Minkowski metric as a solution of semiclassical gravity, which constitutes an application of the validity criterion introduced in the previous chapter. This calculation requires a discussion of the problems posed by the so-called runaway solutions and some of the methods of dealing with them.

This chapter is an overview, placing the body of work described in this book in perspective and describing its overarching structure, namely, how the three levels of structure are related: quantum field theory in curved spacetime established in the 1970s, semiclassical gravity developed in the 80s and stochastic gravity introduced in the 90s, a manifestation of the almost ubiquitous existence of a semiclassical and a stochastic regime in relation to quantum and classical in the description of physical systems. We describe the main physical issues in semiclassical and stochastic gravity, namely, backreaction and fluctuations, the mathematical tools used, and their applications to physical problems in early universe cosmology and black hole physics. In terms of connection to related disciplines, it is pointed out that the popular Newton–Schrödinger equation cherished in alternative quantum theories does not belong to semiclassical gravity, as it is not derivable from quantum field theory and general relativity. However, stochastic gravity is needed for quantum information issues involving gravity. These theories enter even in the low-energy, weak-gravity realm where laboratory experiments are carried out. We finish with a summary of the contents of each chapter and a guide to readers.