# Stochastic Gravity: Theory and Applications

**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 Física Fonamental and

Institut de Ciències del Cosmos

Universitat de Barcelona

Av. Diagonal 647, 08028 Barcelona

Spain

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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 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.