doi: 10.12942/lrr-2008-3

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein–Langevin equation, which has, in addition, sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operator-valued) stress-energy bitensor, which describes the fluctuations of quantum-matter fields in curved spacetimes. A new improved criterion for the validity of semiclassical gravity may also be formulated from the viewpoint of this theory. In the first part of this review we describe the fundamentals of 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 stress-energy tensor to the correlation functions. The functional approach uses the Feynman–Vernon influence functional and the Schwinger–Keldysh closed-time-path effective action methods. In the second part, we describe three applications of stochastic gravity. First, we consider metric perturbations in a Minkowski spacetime, compute the two-point correlation functions of these perturbations and prove that Minkowski spacetime is a stable solution of semiclassical gravity. Second, we discuss structure formation from the stochastic-gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, using the Einstein–Langevin equation, we discuss the backreaction of Hawking radiation and the behavior of metric fluctuations for both the quasi-equilibrium condition of a black-hole in a box and the fully nonequilibrium condition of an evaporating black hole spacetime. Finally, we briefly discuss the theoretical structure of stochastic gravity in relation to quantum gravity and point out directions for further developments and applications.

Keywords: stochastic gravity, semiclassical gravity, quantum gravity, quantum fields in curved space, inflation, structure formation, black holes

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Since a *Living Reviews in Relativity* article may
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Bei Lok Hu and Enric Verdaguer,

"Stochastic Gravity: Theory and Applications",Living Rev. Relativity11, (2008), 3. URL (cited on <date>):

http://www.livingreviews.org/lrr-2008-3

ORIGINAL | http://www.livingreviews.org/lrr-2004-3 |
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Title | Stochastic Gravity: Theory and Applications |

Author | Bei Lok Hu / Enric Verdaguer |

Date | accepted 27 February 2004, published 11 March 2004 |

UPDATE | http://www.livingreviews.org/lrr-2008-3 |

Title | Stochastic Gravity: Theory and Applications |

Author | Bei Lok Hu / Enric Verdaguer |

Date | accepted 11 April 2008, published 29 May 2008 |

Changes | 1. Abstract and sections 1, 2, 4, 5, 7 and 9 revised and updated.
2. Major changes in subsection 3.2 3. New subsection 3.3 on the "Validity of semiclasscal gravity" and the "Large N expansion". 4. New subsection 6.5 on the "Stability of Minkowski spacetime". 5. Major changes in section 8. 6. New subsection 8.3 on "Metric fluctuations of an evaporating black hole". 7. 86 new references. |

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