# Stochastic Gravity:

Theory and Applications

**Update available:
http://www.livingreviews.org/lrr-2008-3**

**Bei Lok Hu
**

Department of Physics

University of Maryland

College Park, Maryland 20742-4111

U.S.A.

http://www.physics.umd.edu/people/faculty/hu.html

**Enric Verdaguer
**

Departament de Fisica Fonamental and

C.E.R. in Astrophysics, Particles and Cosmology

Universitat de Barcelona

Av. Diagonal 647

08028 Barcelona

Spain

###
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.