Vol. 7 (2004) > lrr-2004-3

doi: 10.12942/lrr-2004-3
Living Rev. Relativity 7 (2004), 3

Stochastic Gravity: Theory and Applications

1 Department of Physics, University of Maryland, College Park, Maryland 20742-4111, U.S.A.
2 Departament de Física Fonamental and C.E.R. in Astrophysics, Particles and Cosmology Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain

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Article Abstract

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 bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, 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 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 systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field's Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. 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, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole (enclosed in a box). We derive a fluctuation-dissipation relation between the fluctuations in the radiation and the dissipative dynamics of metric fluctuations.

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

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Article Citation

Since a Living Reviews in Relativity article may evolve over time, please cite the access <date>, which uniquely identifies the version of the article you are referring to:

Bei Lok Hu and Enric Verdaguer,
"Stochastic Gravity: Theory and Applications",
Living Rev. Relativity 7,  (2004),  3. URL (cited on <date>):
http://www.livingreviews.org/lrr-2004-3

Article History

ORIGINAL http://www.livingreviews.org/lrr-2004-3
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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