Stochastic gravity: a primer with applications

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Abstract
Stochastic semiclassical gravity of the 1990s is a theory naturally evolved from semiclassical gravity of the 1970s and 1980s. It improves on the semiclassical Einstein equation with source given by the expectation value of the stress–energy tensor of quantum matter fields in curved spacetime by incorporating an additional source due to their fluctuations. In stochastic semiclassical gravity the main object of interest is the noise kernel, the vacuum expectation value of the (operator-valued) stress–energy bi-tensor, and the centrepiece is the (semiclassical) Einstein–Langevin equation. We describe this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the energy–momentum tensor to their correlation functions. The functional approach uses the Feynman–Vernon influence functional and the Schwinger–Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open system concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise and decoherence. We then describe the applications of stochastic gravity to the backreaction problems in cosmology and black-hole physics. In the first problem, we study the backreaction of conformally coupled quantum fields in a weakly inhomogeneous cosmology. In the second problem, we study the backreaction of a thermal field in the gravitational background of a quasi-static black hole (enclosed in a box) and its fluctuations. These examples serve to illustrate closely the ideas and techniques presented in the first part. This topical review is intended as a first introduction providing readers with some basic ideas and working knowledge. Thus, we place more emphasis here on pedagogy than completeness. (Further discussions of ideas, issues and ongoing research topics can be found in Hu (1999 Int. J. Theor. Phys. 38 2987), Hu and Verdaguer (2002 Advances in the Interplay between Quantum and Gravity Physics ed V De Sabbata (Dordrecht: Kluwer)) and Hu and Verdaguer (2003 Living Rev. Rel. in preparation), respectively.)