Ž .Physics Letters B 465 1999 113–118

On the semiclassical Einstein-Langevin equation

Rosario Martın, Enric Verdaguer 1´Departament de Fısica Fonamental, UniÕersitat de Barcelona, AÕ. Diagonal 647, 08028 Barcelona, Spain´

Received 10 August 1999; accepted 13 September 1999Editor: L. Alvarez-Gaume

Abstract

We introduce a semiclassical Einstein-Langevin equation as a consistent dynamical equation for a first order perturbativecorrection to semiclassical gravity. This equation includes the lowest order quantum stress-energy fluctuations of matterfields as a source of classical stochastic fluctuations of the gravitational field. The Einstein-Langevin equation is explicitlysolved around one of the simplest solutions of semiclassical gravity: Minkowski spacetime with a conformal scalar field inits vacuum state. We compute the two-point correlation function of the linearized Einstein tensor. This calculation illustratesthe possibility of obtaining some ‘‘non-perturbative’’ behavior for the induced gravitational fluctuations. q 1999 Publishedby Elsevier Science B.V. All rights reserved.

PACS: 04.62.qv; 05.40.q jKeywords: Stochastic gravity; Semiclassical Einstein equation; Einstein-Langevin equation; Stress-energy fluctuations

1. Introduction

Semiclassical gravity describes the interaction ofthe gravitational field, which is treated classically,with quantum matter fields. The theory is mathemati-cally well defined and fairly well understood, at least

w xfor linear matter fields 1,2 . The equation of motionfor the classical metric is the semiclassical Einsteinequation, which gives the back reaction of the matterfields on the spacetime; it is a generalization of theEinstein equation where the source is the expectationvalue in some quantum state of the matter stress-en-ergy tensor operator.

In the absence of a complete quantum theory ofgravity interacting with matter fields from which thesemiclassical theory can be derived, the scope and

1 Corresponding author; E-mail: verdague@ffn.ub.es.

limits of semiclassical gravity are less well under-stood. It seems clear, however, that it should not bevalid unless gravitational fluctuations are negligibly

w xsmall 3,4 . This condition may break down when thematter stress-energy has appreciable quantum fluctu-

w xations 1,2 , given that a quantum metric operatorshould couple to the stress-energy operator of matterand, thus, fluctuations in the stress-energy of matter

w xwould induce gravitational fluctuations 5 . In recentyears, a number of examples have been studied,including some quantum fields in cosmological mod-els and even in flat spacetimes with non-trivial topol-ogy, where, for some states of the fields, the stress-

w xenergy tensor have significant fluctuations 6,7 . Itthus seems of interest to try to generalize the semi-classical theory to account for such fluctuations.

In this paper we propose a generalization of semi-classical gravity based on a semiclassical Einstein-

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 01068-0

Langevin equation, which describes the back reac-tion on the spacetime metric of the lowest orderstress-energy fluctuations. This results in an effectivetheory which predicts linear stochastic corrections tothe semiclassical metric and may be applicable whengravitational fluctuations of genuine quantum naturecan be neglected. It should be stressed that thegravitational fluctuations predicted by this stochasticsemiclassical theory are ‘‘passive’’ rather than ‘‘ac-tive’’, i.e., they are induced by the matter field

w xstress-energy fluctuations 8 . Thus, we cannot ex-pect that stochastic semiclassical gravity gives asatisfactory description of gravitational fluctuationsin all situations. In spite of that, this theory may havea number of interesting applications in the physics ofthe early universe and of black holes, where one mayexpect significant matter stress-energy fluctuations.

The idea which motivates such stochastic theoryis that, in the transition from the full quantum regimeto the semiclassical one, there should be some mech-anism which effectively suppresses the quantum in-terference effects in the gravitational field. Oncesuch decoherence process has taken place, gravitymight undergo an intermediate regime in which itwould fluctuate, but it would do so classically. Ifsuch a regime exists, an effective probabilistic de-scription of the gravitational field by a stochastic

w xmetric field may be possible 9–11 .Another motivation for our work is to connect

with some recent results on equations of theLangevin-type that have appeared in the context ofsemiclassical cosmology and which predict stochas-tic perturbations around some cosmological back-

w x w xgrounds 12–18 . Although physically motivated 19 ,the derivation of these equations, obtained by func-tional methods, is formal and doubts may be raisedon the physical significance of the predicted fluctua-tions. Our approach is complementary to these func-tional methods, it allows to write these equations in ageneral form and it links the source of such metricfluctuations to the matter stress-energy fluctuations.

The idea, when implemented in the framework ofa perturbative approach around semiclassical gravity,is quite simple. We start realizing that, for a givensolution of semiclassical gravity, one can associatethe lowest order matter stress-energy fluctuations toa classical stochastic tensor field. Then, we seek aconsistent equation which incorporates this stochas-

tic tensor as a source of a first order correction tosemiclassical gravity. It is important to remark that,even if this approach differs from those based onfunctional methods, the semiclassical Einstein-Lan-gevin equation introduced here can actually be for-mally derived using those methods. The details of

w xsuch derivation are given in Ref. 20 .As a simple application of stochastic semiclassical

gravity, we have explicitly solved the Einstein-Lan-gevin equation around a solution of semiclassicalgravity consisting on Minkowski spacetime with aconformal scalar field in the Minkowskian vacuumstate. We have computed the two-point correlationfunction for the induced linearized Einstein tensor.Even if, as expected, such correlation function hasnegligible values for points separated by scales largerthan the Planck scales, the result hints at the possibil-ity that induced gravitational fluctuations are non-analytic in some characteristic correlation lengths.

Ž .Throughout we use the qqq sign conventionsand work in units in which cs"s1.

2. The semiclassical Einstein-Langevin equation

We start with the semiclassical Einstein equation.Ž .Let M, g be a globally hyperbolic four-dimen-ab

sional spacetime and consider a linear quantum fieldF on this background. Working in the Heisenberg

ˆ w x w xpicture, let F g and r g be respectively the fieldˆoperator and the density operator describing the stateof the field. Such state will be assumed to be physi-

w xcally acceptable in the sense of Ref. 1 .ˆŽ .The set M, g ,F ,r is a solution of semiclassi-ˆab

cal gravity if it satisfies the semiclassical Einsteinequation:

1w x w xG g qLg y2 a A qbB gŽ .Ž .ab ab ab ab8p G

ˆ R² :w xs T g , 1Ž .ab

ˆ R w xwhere T g is the renormalized stress-energy ten-abˆ w xsor operator for the field F g , which satisfies the

corresponding field equation on the spacetimeŽ .M, g , and the expectation value is taken in theab

w x w xstate described by r g 1,2 . In the above equation,ˆ1rG, LrG, a and b are renormalized couplingconstants, G is the Einstein tensor, and A andab ab

B are the local curvature tensors obtained by func-ab

tional derivation with respect to the metric of theaction terms corresponding to the Lagrangian densi-ties C C abcd and R2, respectively, where C isabcd abcd

the Weyl tensor and R is the scalar curvature. Aclassical stress-energy tensor can also be added to

Ž .the right hand side of Eq. 1 , but, for simplicity, weshall ignore this term.

Given a solution of semiclassical gravity, thematter stress-energy tensor will in general havequantum fluctuations. To lowest order, such fluctua-tions may be described by the following bi-tensor,which we call noise kernel,

ˆ ˆ w x8 N x , y ' t x , t y g , 2² :Ž . Ž . Ž . Ž .� 4abcd ab cd

ˆ� 4 ˆwhere , means the anticommutator and t 'Tab abˆ ˆ² : w xy T , where T g denotes the unrenormalizedab ab

wstress-energy ‘‘operator’’ we use the word ‘‘oper-ˆator’’ for T in a formal sense; it should be under-ab

stood that the matrix elements of this ‘‘operator’’ aresuitably regularized and that the regularization is

Ž .xremoved after computing the right hand side of 2 .For a linear matter field, this noise kernel is free of

ˆultraviolet divergencies, and the ‘‘operator’’ t inabˆ R ˆ RŽ . ² :2 can be replaced by the operator T y T .ab ab

We want now to introduce an equation in whichŽ .the stress-energy fluctuations described by 2 are

the source of classical gravitational fluctuations, as aperturbative correction to semiclassical gravity. Thus,we assume that the gravitational field is described byg qh , where h is a linear perturbation to theab ab ab

Ž .metric g , solution of Eq. 1 . The renormalizedab

stress-energy operator and the density operator de-scribing the state of the field will be denoted asˆ R ˆ Rw x w x ² :wT gqh and r gqh , respectively, and T gˆab ab

xqh will be the corresponding expectation value.In order to write an equation which describes the

dynamics of the metric perturbation h , let us intro-ab

duce a Gaussian stochastic tensor field j charac-ab

terized by the following correlators:

² : ² :j x s0, j x j y sN x , y ,Ž . Ž . Ž . Ž .ab ab cd abcdc c

3Ž .

²:where means statistical average. Note that thec

two-point correlation function of a stochastic tensorfield j must be a symmetric positive semi-ab

Ždefinite real bi-tensor field since, obviously,

ˆ R.² : ² :j x j y s j y j x . Since T isŽ . Ž . Ž . Ž .ab cd cd ab abc cŽ .self-adjoint, it is easy to see from the definition 2

Ž .that N x, y satisfies all these conditions. There-abcdŽ .fore, relations 3 , with the cumulants of higher order

being zero, do truly characterize a stochastic tensorfield j . One could also seek higher order correc-ab

tions which would take into account higher orderstress-energy fluctuations, but we stick, for simplic-ity, to the lowest order. The simplest equation whichcan incorporate in a consistent way the stress-energy

Ž .fluctuations described by N x, y as the source ofabcd

metric fluctuations is

1w xG gqh qL g qhŽ .Ž .ab ab ab8p G

w xy2 a A qbB gqhŽ .ab ab

ˆ R² :w xs T gqh q2j , 4Ž .ab ab

which must be understood to linear order in h .ab

This is the semiclassical Einstein-Langevin equation,which gives a first order correction to semiclassical

Ž .gravity. Notice that, in writing Eq. 4 , we are im-plicitly assuming that h is also a stochastic tensorab

field.We must now ensure that this equation is consis-

tent and, thus, it can truly describe the dynamics ofmetric perturbations. Note that the term j in Eq.abŽ .4 is not dynamical, i.e., it does not depend on h ,ab

since it is defined through the semiclassical metricŽ .g by the correlators 3 . Being the source of theab

metric perturbation h , this term is of first order inab

perturbation theory around semiclassical gravity. Letus now see that j is covariantly conserved up toab

first order in perturbation theory, in the sense that thestochastic vector field , aj is deterministic andab

represents with certainty the zero vector field on MŽ a, means the covariant derivative associated to the

a ˆ R. w xmetric g . Using that , T g s0, we haveab aba Ž ., N x, y s0 and, from the covariant deriva-x abcd

Ž . ² a :tive of the correlators 3 , we get , j s0 andcab² a Ž . c Ž .:, j x , j y s0. It is thus consistent tocx ab y cd

include the term j in the right hand side of Eq.abŽ .4 . Note that for a conformal matter field, i.e., afield whose classical action is conformally invariant,the stochastic source j is ‘‘traceless’’ up to firstab

order in perturbation theory. That is, g abj is deter-ab

ministic and represents with certainty the zero scalar

field on M. In fact, from the trace anomaly result,ab ˆ R w xwhich states that g T g is in this case a localab

c-number functional of g times the identity opera-cdabŽ . Ž .tor, we have that g x N x, y s0. It thenabcd

Ž . ² a b :follows from 3 that g j s 0 andca b² abŽ . Ž . cdŽ . Ž .:g x j x g y j y s0. Hence, in thecab cd

case of a conformal matter field, the stochastic sourcegives no correction to the trace anomaly.

Ž .Since Eq. 4 gives a linear stochastic equation forh with an inhomogeneous term j , a solution canab ab

w xbe formally written as a functional h j of theab

stochastic source j . Such a solution can be charac-cd

terized by the whole family of its correlation func-Ž .tions. By taking the average of Eq. 4 , one sees that

² :the metric g q h must be a solution of theab ab c

semiclassical Einstein equation linearized aroundŽ .g . For the solutions of Eq. 4 we have the gaugeab

freedom h ™hX'h q, z q, z , where z a

ab ab ab a b b a

is any stochastic vector field on M which is afunctional of the Gaussian stochastic field j , andcd

z 'g z b. It is easy to see that h and hX area ab ab ab

physically equivalent solutions provided thatˆŽ w x w x.M, g ,F g ,r g satisfies the semiclassical Eq.ˆab

Ž .1 .

3. Vacuum fluctuations in flat spacetime

As an example, we shall now consider the class oftrivial solutions of semiclassical gravity. Each ofsuch solutions consists of Minkowski spacetime,Ž 4 .R ,h , a linear matter field, and the usualab

w xMinkowskian vacuum state for this field, r h sˆ< : ² <0 0 . As it is well known, we can always choose a

ˆ R² < < :w xrenormalization scheme in which 0 T 0 h s0.abŽ .Thus, each of the above sets is a solution to Eq. 1

< :with Ls0. Note that, although the vacuum 0 is aneigenstate of the total four-momentum operator whichcan be defined in Minkowski spacetime, such state is

ˆ R w xnot an eigenstate of T h . Hence, even in theseab

trivial solutions, quantum fluctuations of the matterstress-energy are present, and the noise kernel de-

Ž .fined in 2 does not vanish. This fact leads toconsider the Einstein-Langevin correction to the triv-

w xial solutions of semiclassical gravity 21 . For amassless field, these vacuum stress-energy fluctua-tions should be characterized by the only scale avail-able, the Planck length, and, thus, they should be

very small on macroscopic scales, unlike the casesw xconsidered in Refs. 6,7 .

The corresponding Einstein-Langevin equationbecomes simpler when the matter field is a masslessconformally coupled real scalar field. In the global

� m4inertial coordinate system x , the components ofŽ .the flat metric are simply h sdiag y1,1,1,1 . Wemn

shall use GŽ1., AŽ1. and BŽ1. to denote, respectively,mn mn mn

the components of the tensors G , A and Bab ab ab

linearized around the flat metric. These tensor com-ponents can be written in terms of GŽ1. asmn

2Ž1. Ž1. a a Ž1.A s FF G yFF G ,Ž .mn mn a a mn3

BŽ1.s2 FF GŽ1. a , 5Ž .mn mn a

where FF is the differential operator FF 'h Imn mn mn

yE E . It is also convenient to introduce the Fourierm n

˜ y4Ž . Ž . Ž . Ž .transform, f p , of a field f x as f x ' 2p4 i p x Ž . Ž .=Hd p e f p . Eq. 4 reduces in this case to

1Ž .Ž1. 1 Ž1.G x y2 a A qbB xŽ . Ž .ž /mn mn mn8p G

q d4 y H xyy ;m2 AŽ1. yŽ .Ž .H mn

s2j x , 6Ž . Ž .mn

2Ž .where a ' a q 1r 3600p , b ' b y 1r2 ˜ 2 2 2Ž . Ž . Ž < <34560p , and H p;m ' ln p rm y ip =

0 Ž 2 .. Ž 2 .sign p u yp r 1920p , being m a renormaliza-w xtion mass scale 18 . The components j of themn

stochastic source tensor satisfy in this case

1 x² :j x j y s FF N xyy , 7Ž . Ž . Ž . Ž .cmn a b mna b6

˜Ž .where FF '3FF FF yFF FF and N p 'mna b mŽa b .n mn a b

Ž 2 . Ž .u yp r 1920p .Ž . Ž1.Eq. 6 can be solved for the components G ofmn

the linearized Einstein tensor. Using a procedurew xsimilar to that described in the Appendix of Ref. 3 ,

we find the family of solutions which can be writtenas a linear functional of the stochastic source and

˜Ž1.Ž .whose Fourier transform G p depends locally onmn

˜ Ž .j p . Each of such solutions is a Gaussianab

stochastic field and, thus, it can be completely char-acterized by its one-point and two-point correlationfunctions. The one-point correlation functions, i.e.,

² Ž1.:the averages G , are solution of the linearizedcmn

semiclassical Einstein equations obtained by averag-Ž .ing Eq. 6 ; solutions to these equations were first

w xfound by Horowitz 22 . We can then compute theŽ X .two-point correlation functions GG x, x 'mn a b

² Ž1.Ž . Ž1. Ž X .: ² Ž1.Ž .: ² Ž1. Ž X .: w xG x G x y G x G x 21 .c c cmn a b mn a b

These correlation functions are invariant under gaugetransformations of the metric perturbations and givea measure of the induced gravitational fluctuations inthe present context, to the extent that these fluctua-tions can be described by solutions to the full equa-

Ž . Žtions 6 we have not attempted any ‘‘reduction ofw x.order’’ procedure 3,23–25 . We get

pX X2 xGG x , x s G FF GG xyx , 8Ž . Ž . Ž .mna b mna b45

y22 2 2˜ ˜Ž . Ž . < Ž . <where GG p s u yp 1 q 16p Gp H p;m ,2Ž .with m'm exp 1920p a . Introducing the function

2 2 2Ž .w x ;l ' 1yx ln lxre qp x , with xG0Ž .Ž .and l)0, GG x can be written as

3r2120p 1Ž .GG x sŽ . 3 3 < <x2p LP

='` 120p

< < < < < < < <d q q sin x qHL0 P

=

2'` 120p u yqŽ .0 0 0dq cos x q ,H 2L w yq ;lŽ .0 P

9Ž .

m Ž 0 . m Ž 0 .where x s x , x and q s q ,q , l'120p er2 2 'Ž .L m , and L ' G is the Planck length. If weP P

y1 3assume that mFL , then l)10 . Let us considerPŽ X . mthe case when xyx is spacelike, then we can

Ž X . m Ž X. Ž X .choose xyx s 0, xyx and GG x, x willmna bX Ž . Ž .be a function of xyx only. From 8 and 9 , one

Ž X . Ž X.can see that GG xyx sGG xyx s0. The000 i 0 i jk

remaining components are non-null and can beexplicitly computed performing the integrals withsome approximations. The result depends on the

Ž .constant k ' ln lx l re , being x l theŽ .Ž .0 0Ž . Žvalue of x where w x ;l has its minimum this

can be found by solving the equation p 2x s0

.1yx ln lx re 1q ln lx re numerically .Ž . Ž .0 0 0

For x/x X, we get

2 ab6 1X ysGG xyx , eŽ .0000 4 23p L sP

=4 12 24 24

1q q q q ,2 3 4s s s s

10Ž .

< X < Ž .where s ' b x y x rL , a ' 1 q 2rp =PŽ . Ž 2 .a rc tan k r p an d b ' 4 a r p =

1r22 2'15p k qp yk . We should remark thatŽ .

this is not an expansion in s . Similar results areobtained for the other non-null components. Fromthese results, one can conclude that, as expected inthis case, there are negligibly small correlations forthe Einstein tensor at points separated by distanceslarge compared to the Planck length. Thus, at suchscales, the semiclassical approach is satisfactoryenough to describe the dynamics of gravitational

w xperturbations to Minkowski spacetime 22,3,23–25 .Deviations from semiclassical gravity start to beimportant at Planckian scales. At such scales, how-ever, gravitational fluctuations of genuine quantumnature cannot be neglected and, thus, the classicaldescription based on the Einstein-Langevin equationwould break down. Note, however, the factor eys inour result, which is non-analytic in the Planck lengthand gives a characteristic correlation length of theorder of L . This kind of behavior cannot be ob-P

tained from a perturbative approach to quantumgravity, in which one expands physical quantities as

w xa power series in " 26–28 . Actually, if, followingw xRefs. 23–25 , we had tried to find solutions of Eq.

Ž .6 as a Taylor expansion in ", we would have˜Ž1.Ž .obtained a series for G p which, as the abovemn

˜ Ž .solutions, would be linear and local in j p , butab

whose corresponding two-point correlation functionsŽ . Žwould not converge to 8 for a discussion of this

w x.point in similar contexts, see Ref. 3 .For solutions of semiclassical gravity in which

some scales larger than the Planck length are alsopresent, induced gravitational fluctuations may becharacterized by correlation lengths of the order ofsuch scales. In those cases, the above results suggestthat stochastic semiclassical gravity might yieldphysically relevant results which cannot be obtained

from a calculation involving a perturbative expan-sion in these correlation lengths.

Acknowledgements

We are grateful to Enrique Alvarez, EstebanCalzetta, Jaume Garriga, Bei-Lok Hu, Ted Jacobsonand Albert Roura for very helpful suggestions anddiscussions. This work has been partially supportedby the CICYT Research Project number AEN98-0431, and the European Project number CI1-CT94-1180004. R.M. also acknowledges support of a FPIgrant from the Spanish MEC.

References

w x1 R.M. Wald, Quantum Field Theory in Curved Spacetime andŽBlack Hole Thermodynamics The University of Chicago

.Press, Chicago, 1994 , and references therein.w x Ž .2 R.M. Wald, Commun. Math. Phys. 54 1977 1.

´ ´w x Ž .3 E.E. Flanagan, R.M. Wald, Phys. Rev. D 54 1996 6233.w x Ž .4 A. Ashtekar, Phys. Rev. Lett. 77 1996 4864.

w x Ž .5 L.H. Ford, Ann. Phys. 144 1982 238.w x Ž .6 C.-I. Kuo, L.H. Ford, Phys. Rev. D 47 1993 4510.w x Ž .7 N.G. Phillips, B.-L. Hu, Phys. Rev. D 55 1997 6123.w x Ž .8 L.H. Ford, Phys. Rev. D 51 1995 1692.w x Ž .9 M. Gell-Mann, J.B. Hartle, Phys. Rev. D 47 1993 3345.

w x Ž .10 J.J. Halliwell, Phys. Rev. D 57 1998 2337.w x Ž .11 L. Diosi, J.J. Halliwell, Phys. Rev. Lett. 81 1998 2846.´w x Ž .12 E. Calzetta, B.-L. Hu, Phys. Rev. D 49 1994 6636.w x Ž .13 B.-L. Hu, A. Matacz, Phys. Rev. D 51 1995 1577.w x Ž .14 B.-L. Hu, S. Sinha, Phys. Rev. D 51 1995 1587.w x Ž .15 F.C. Lombardo, F.D. Mazzitelli, Phys. Rev. D 55 1997

3889.w x16 E. Calzetta, A. Campos, E. Verdaguer, Phys. Rev. D 56

Ž .1997 2163.w x Ž .17 A. Campos, B.-L. Hu, Phys. Rev. D 58 1998 125021.w x Ž .18 A. Campos, E. Verdaguer, Phys. Rev. D 53 1996 1927.w x Ž .19 B.-L. Hu, Physica A 158 1989 399.w x Ž .20 R. Martın, E. Verdaguer, Phys. Rev. D 60 1999 0640XX.´w x Ž .21 R. Martın, E. Verdaguer, in preparation 1999 .´w x Ž .22 G.T. Horowitz, Phys. Rev. D 21 1980 1445.w x Ž .23 J.Z. Simon, Phys. Rev. D 41 1990 3720.w x Ž .24 J.Z. Simon, Phys. Rev. D 43 1991 3308.w x Ž .25 L. Parker, J.Z. Simon, Phys. Rev. D 47 1993 1339.w x Ž .26 G.’t Hooft, M. Veltman, Ann. Inst. H. Poincare A 20 1974´

69.w x Ž .27 J.F. Donoghue, Phys. Rev. Lett. 72 1994 2996.w x Ž .28 J.F. Donoghue, Phys. Rev. D 50 1994 3874.