Stochastic Gravity: Theory and Applications
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Abstract
Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stressenergy tensor of quantum fields, stochastic semiclassical gravity is based on the EinsteinLangevin equation, which has, in addition, sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operatorvalued) stressenergy bitensor, which describes the fluctuations of quantummatter 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 stressenergy tensor to the correlation functions. The functional approach uses the FeynmanVernon influence functional and the SchwingerKeldysh closedtimepath 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 twopoint correlation functions of these perturbations and prove that Minkowski spacetime is a stable solution of semiclassical gravity. Second, we discuss structure formation from the stochasticgravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, using the EinsteinLangevin equation, we discuss the backreaction of Hawking radiation and the behavior of metric fluctuations for both the quasiequilibrium condition of a blackhole 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.
1 Overview
Stochastic semiclassical gravity is a theory developed in the 1990s using semiclassical gravity (quantum fields in classical spacetimes, the dynamics of both matter and spacetime are solved selfconsistently) as the starting point and aiming at a theory of quantum gravity as the goal. While semiclassical gravity is based on the semiclassical Einstein equation with the source given by the expectation value of the stressenergy tensor of quantum fields, stochastic semiclassical gravity, or stochastic gravity for short, also includes its fluctuations in a new stochastic semiclassical EinsteinLangevin equation (we will often use the shortened term stochastic gravity as there is no confusion as to the nature and source of stochasticity in gravity being induced from the quantum fields and not a priori from the classical spacetime). If the centerpiece in semiclassicalgravity theory is the vacuum expectation value of the stressenergy tensor of a quantum field and the central issues are how well the vacuum is defined and how the divergences can be controlled by regularization and renormalization, the centerpiece in stochastic semiclassicalgravity theory is the stressenergy bitensor and its expectation value known as the noise kernel. The mathematical properties of this quantity and its physical content in relation to the behavior of fluctuations of quantum fields in curved spacetimes are the central issues of this new theory. How they induce metric fluctuations and seed the structures of the universe, how they affect the blackhole horizons and the backreaction of Hawking radiance in black hole dynamics, including implications for transPlanckian physics, are new horizons to explore. On theoretical issues, stochastic gravity is the necessary foundation to investigate the validity of semiclassical gravity and the viability of inflationary cosmology based on the appearance and sustenance of a vacuum energydominated phase. It is also a useful beachhead supported by wellestablished lowenergy (subPlanckian) physics from which to explore the connection with highenergy (Planckian) physics in the realm of quantum gravity.
In this review we summarize the major work and results of this theory since 1998. It is in the nature of a progress report rather than a review. In fact we will have room only to discuss a handful of topics of basic importance. A review of ideas leading to stochastic gravity and further developments originating from it can be found in [181, 187], a set of lectures, which include a discussion of the issue of the validity of semiclassical gravity in [207] and a pedagogical introduction of stochasticgravity theory with a more detailed treatment of backreaction problems in cosmology and black holes in quasiequilibrium in [208]. A comprehensive formal description of the fundamentals is given in [257, 258], while that of the noise kernel in arbitrary spacetimes can be found in [258, 304, 305]. Here we will try to mention related work so the reader can at least trace out the parallel and sequential developments. The references at the end of each topic below are representative work in which one can seek out further treatments.
Stochastic gravity theory is built on three pillars: general relativity, quantum fields and nonequilibrium statistical mechanics. The first two uphold semiclassical gravity, the last two span statistical field theory. Strictly speaking one can understand a great deal without appealing to statistical mechanics, and we will try to do so here. But concepts such as quantum open systems [88, 246, 370] and techniques such as the influence functional [107, 108] (which is related to the closedtimepath effective action [14, 54, 56, 82, 87, 94, 222, 223, 227, 296, 323, 343]) were a great help in our understanding of the physical meaning of issues involved in the construction of this new theory. Foremost because quantum fluctuations and correlation have ascended the stage and become the focus of attention. Quantum statistical field theory and the statistical mechanics of quantum field theory [55, 57, 59, 61] also aided us in searching for the connection with quantum gravity through the retrieval of correlations and coherence.
We show the scope of stochastic gravity as follows:
 1Ingredients:
 (a)
 (b)Quantum Field Theory in Curved Spacetimes [34, 121, 135, 362]:
 i.
Stressenergy tensor: Regularization and renormalization.
 ii.
Selfconsistent solution: Backreaction problems in early universe and black holes [3, 4, 5, 109, 137, 147, 148, 153, 154, 165, 193, 194, 251], and analog gravity [15, 16, 252, 320, 321].
 iii.
Effective action: Closed time path, initial value formulation [14, 54, 56, 82, 87, 94, 223, 227, 296, 323, 343].
 iv.
Equation of motion: Real and causal [222].
 i.
 (c)Nonequilibrium Statistical Mechanics (see [62] and references therein):
 (d)
Decoherence in Quantum Cosmology and Emergence of Classical Spacetimes [50, 51, 143, 182, 195, 231, 283].
 (a)
 2Theory:
 3
 4Applications: Early Universe and Black Holes:
 5
 6Ideas:
For lack of space we list only the latest work in the respective topics above, describing ongoing research. The reader should consult the references therein for earlier work and background material. We do not seek a complete coverage here, but will discuss only those selected topics in theory, issues and applications. We use the (+, +, +) sign conventions of [266, 361], and units in which c = ℏ =1.
2 From Semiclassical to Stochastic Gravity
There are three main steps that led to the recent development of stochastic gravity. The first step begins with quantum field theory in curved spacetime [34, 93, 121, 135, 362], which describes the behavior of quantum matter fields propagating in a specified (not dynamically determined by the quantum matter field as source) background gravitational field. In this theory the gravitational field is given by the classical spacetime metric determined from classical sources by the classical Einstein equations and the quantum fields propagate as test fields in such a spacetime. Some important processes described by quantum field theory in curved spacetime are particle creation from the vacuum, effects of vacuum fluctuations and polarizations in the early universe [29, 30, 31, 81, 93, 117, 179, 293, 327, 386, 387], and Hawking radiation in black holes [155, 156, 213, 294, 358].
The second step in the description of the interaction of gravity with quantum fields is backreaction, i.e., the effect of quantum fields on spacetime geometry. The source here is the expectation value of the stressenergy operator for matter fields in some quantum state in the spacetime, a classical observable. However, since this object is quadratic in the field operators, which are only welldefined as distributions on the spacetime, it involves illdefined quantities. It contains ultraviolet divergences, the removal of which requires a renormalization procedure [83, 84, 93]. The final expectation value of the stressenergy operator using a reasonable regularization technique is essentially unique, modulo some terms, which depend on the spacetime curvature and which are independent of the quantum state. This uniqueness was proved by Wald [359, 360] who investigated the criteria that a physically meaningful expectation value of the stressenergy tensor ought to satisfy.
The theory obtained from a selfconsistent solution of the geometry of the spacetime and the quantum field is known as semiclassical gravity. Incorporating the backreaction of the quantum matter field into the spacetime is thus the central task in semiclassical gravity. One assumes a general class of spacetime, in which the quantum fields live and on which they act and seek a solution that satisfies simultaneously the Einstein equation for the spacetime and the field equations for the quantum fields. The Einstein equation, which has the expectation value of the stressenergy operator of the quantum matter field as its source, is known as the semiclassical Einstein equation. Semiclassical gravity was first investigated in cosmological backreaction problems [3, 4, 109, 137, 147, 148, 153, 154, 193, 194, 251]; an example is the damping of anisotropy in Bianchi universes by the backreaction of vacuum particle creation. The effect of quantum field processes, such as particle creation, was used to explain why the universe is so isotropic at the present in the context of chaotic cosmology [27, 28, 265] in the late 1970s prior to the inflationarycosmology proposal of the 1980s [2, 140, 243, 244], which assumes the vacuum expectation value of an inflaton field as the source, another, perhaps more wellknown, example of semiclassical gravity.
2.1 The importance of quantum fluctuations
For a free quantum field, semiclassical gravity is fairly well understood. The theory is in some sense unique, since the only reasonable cnumber stressenergy tensor that one may construct [359, 360] with the stressenergy operator is a renormalized expectation value. However, the scope and limitations of the theory are not so well understood. It is expected that the semiclassical theory will break down at the Planck scale. One can conceivably assume that it will also break down when the fluctuations of the stressenergy operator are large [111, 237]. Calculations of the fluctuations of the energy density for Minkowski, Casimir and hot flat spaces as well as Einstein and de Sitter universes are available [86, 198, 237, 258, 259, 282, 302, 303, 304, 305, 313, 316]. It is less clear, however, how to quantify what a large fluctuation is, and different criteria have been proposed [10, 11, 113, 115, 198, 237, 303, 384, 385]. The issue of the validity of semiclassical gravity viewed in the light of quantum fluctuations was discussed in our Erice lectures [207]. More recently in [203, 204] a new criterion has been proposed for the validity of semiclassical gravity. It is based on quantum fluctuations of the semiclassical metric and incorporates, in a unified and selfconsistent way, previous criteria that have been used [10, 111, 169, 237]. One can see the essence of the validity problem in the following example inspired by Ford [111].
Let us assume a quantum state formed by an isolated system, which consists of a superposition with equal amplitude of one configuration of mass M with the center of mass at X_{1} and another configuration of the same mass with the center of mass at X_{2}. The semiclassical theory, as described by the semiclassical Einstein equation, predicts that the center of mass of the gravitational field of the system is centered at (X_{1} + X_{2})/2. However, one would expect that if we send a succession of test particles to probe the gravitational field of the above system, half of the time they would react to a gravitational field of mass M centered at X_{1} and half of the time to the field centered at X_{2}. The two predictions are clearly different; note that the fluctuation in the position of the center of masses is on the order of (X_{1} − X_{2})^{2}. Although this example raises the issue of the importance of fluctuations to the mean, a word of caution should be added to the effect that it should not be taken too literally. In fact, if the previous masses are macroscopic, the quantum system decoheres very quickly [392, 393] and, instead of being described by a pure quantum state, it is described by a density matrix, which diagonalizes in a certain pointer basis. For observables associated with such a pointer basis, the density matrix description is equivalent to that provided by a statistical ensemble. The results will differ, in any case, from the semiclassical prediction.
In other words, one would expect that a stochastic source that describes the quantum fluctuations should enter into the semiclassical equations. A significant step in this direction was made in [181] where it was proposed that one view the backreaction problem in the framework of an open quantum system: the quantum fields acting as the “environment” and the gravitational field as the “system”. Following this proposal a systematic study of the connection between semiclassical gravity and open quantum systems resulted in the development of a new conceptual and technical framework in which (semiclassical) EinsteinLangevin equations were derived [52, 58, 73, 74, 192, 206, 248]. The key technical factor to most of these results was the use of the influencefunctional method of Feynman and Vernon [108], when only the coarsegrained effect of the environment on the system is of interest. Note that the word semiclassical put in parentheses refers to the fact that the noise source in the EinsteinLangevin equation arises from the quantum field, while the background spacetime is classical; generally we will not carry this word since there is no confusion that the source, which contributes to the stochastic features of this theory, comes from quantum fields.
In the language of the consistenthistories formulation of quantum mechanics [43, 99, 100, 101, 126, 136, 144, 145, 146, 149, 209, 210, 211, 212, 228, 229, 230, 275, 276, 277, 278, 279, 280, 299, 350], for the existence of a semiclassical regime for the dynamics of the system, one has two requirements. The first is decoherence, which guarantees that probabilities can be consistently assigned to histories describing the evolution of the system, and the second is that these probabilities should peak near histories, which correspond to solutions of classical equations of motion. The effect of the environment is crucial, on the one hand, to provide decoherence and, on the other hand, to produce both dissipation and noise in the system through backreaction, thus inducing a semiclassical stochastic dynamic in the system. As shown by different authors [46, 127, 131, 221, 352, 389, 390, 391, 392, 393], indeed over a long history predating the current revival of decoherence, stochastic semiclassical equations are obtained in an open quantum system after a coarsegraining of the environmental degrees of freedom and a further coarsegraining in the system variables. It is expected, but has not yet been shown, that this mechanism could also work for decoherence and classicalization of the metric field. Thus far, the analogy could only be made formally [256] or under certain assumptions, such as adopting the BornOppenheimer approximation in quantum cosmology [297, 298].
An alternative axiomatic approach to the EinsteinLangevin equation, without invoking the opensystem paradigm, was later suggested based on the formulation of a selfconsistent dynamical equation for a perturbative extension of semiclassical gravity able to account for the lowestorder stressenergy fluctuations of matter fields [257]. It was shown that the same equation could be derived, in this general case, from the influence functional of Feynman and Vernon [258]. The field equation is deduced via an effective action, which is computed assuming that the gravitational field is a cnumber. The important new element in the derivation of the EinsteinLangevin equation, and of stochasticgravity theory, is the physical observable that measures the stressenergy fluctuations, namely, the expectation value of the symmetrized bitensor constructed with the stressenergy tensor operator: the noise kernel. It is interesting to note that the EinsteinLangevin equation can also be understood as a useful intermediary tool to compute symmetrized twopoint correlations of the quantum metric perturbations on the semiclassical background, independent of a suitable classicalization mechanism [317].
3 The EinsteinLangevin Equation: Axiomatic Approach
In this section we introduce stochastic semiclassical gravity, or stochastic gravity for short, in an axiomatic way. It is introduced as an extension of semiclassical gravity motivated by the search for selfconsistent equations, which describe the backreaction of the quantum stressenergy fluctuations on the gravitational field [257].
3.1 Semiclassical gravity
Semiclassical gravity describes the interaction of a classical gravitational field with quantum matter fields. This theory can be formally derived as the leading 1/N approximation of quantum gravity interacting with N independent and identical free quantum fields [152, 175, 176, 348], which interact with gravity only. By keeping the value of NG finite, where G is Newton’s gravitational constant, one arrives at a theory in which formally the gravitational field can be treated as a cnumber field (i.e., quantized at tree level) and matter fields are fully quantized. The semiclassical theory may be summarized as follows.
A solution of semiclassical gravity consists of a spacetime \(({\mathcal M},{g_{ab}})\), a quantum field operator \(\hat \phi [g]\), which satisfies the evolution Equation (2) and a physically acceptable state ψ[g]⟩ for this field, such that Equation (8) is satisfied when the expectation value of the renormalized stressenergy operator is evaluated in this state.
For a free quantum field this theory is robust in the sense that it is selfconsistent and fairly well understood. As long as the gravitational field is assumed to be described by a classical metric, the above semiclassical Einstein equations seem to be the only plausible dynamical equation for this metric: the metric couples to matter fields via the stressenergy tensor and for a given quantum state the only physically observable cnumber stressenergy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantumgravity theory, the scope and limits of the theory are not so well understood. It is assumed that the semiclassical theory will break down at Planck scales, which is when simple orderofmagnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Plancksize region, as determined by the Heisenberg uncertainty principle, is comparable to the gravitational energy of that fluctuation.
The theory is expected to break down when the fluctuations of the stressenergy operator are large [111]. A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford [237] (see also work over zetafunction methods [86, 302]). This proposal was questioned by Phillips and Hu [198, 303, 304] because it does not contain a scale at which the theory is probed or how accurately the theory can be resolved. They suggested the use of a smearing scale or pointseparation distance for integrating over the bitensor quantities, which is equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [113, 115]. A different criterion was recently suggested by Anderson et al. [10, 11] based on linearresponse theory. A partial summary of this issue can be found in our Erice Lectures [207].
More recently, in collaboration with A. Roura [203, 204], we have proposed a criterion for the validity of semiclassical gravity, which is based on the stability of the solutions of the semiclassical Einstein equations with respect to quantum metric fluctuations. The twopoint correlations for the metric perturbations can be described in the framework of stochastic gravity, which is closely related to the quantum theory of gravity interacting with N matter fields, to leading order in a 1/N expansion. We will describe these developments in the following sections.
3.2 Stochastic gravity
Once the fluctuations of the stressenergy operator have been characterized, we can perturbatively extend the semiclassical theory to account for such fluctuations. Thus we will assume that the background spacetime metric g_{ ab } is a solution of the semiclassical Einstein equations (8) and we will write the new metric for the extended theory as g_{ ab } + h_{ ab }, where we will assume that h_{ ab } is a perturbation to the background solution. The renormalized stressenergy operator and the state of the quantum field may now be denoted by \(\hat T_{ab}^R[g + h]\) and ψ[g + h]⟩, respectively, and \(\langle \hat T_{ab}^R[g + h]\rangle\) will be the corresponding expectation value.
An important property of this stochastic tensor is that it is covariantly conserved in the background spacetime ∇^{ a }ξ_{ ab }[g; x) = 0. In fact, as a consequence of the conservation of \(\hat T_{ab}^R[g]\) one can see that \(\nabla _x^a{N_{abcd}}(x,y) = 0\). Taking the divergence in Equation (14) one can then show that ⟨∇^{ a }ξ_{ ab }⟩_{ s } = 0 and \({\langle \nabla _x^a{\xi _{ab}}(x){\xi _{cd}}(y)\rangle _{\mathcal S}} = 0\) so that ∇^{ a }ξ_{ ab } is deterministic and represents with certainty the zero vector field in \({\mathcal M}\).
For a conformal field, i.e., a field whose classical action is conformally invariant, ξ_{ ab } is traceless: g^{ ab }ξ_{ ab }[g; x) = 0; so that, for a conformal matter field the stochastic source gives no correction to the trace anomaly. In fact, from the trace anomaly result, which states that \({g^{ab}}\hat T_{ab}^R[g]\) is, in this case, a local cnumber functional of g_{ ab } times the identity operator, we have that g^{ ab }(x)N_{ abcd }[g; x, y) = 0. It then follows from Equation (14) that ⟨g^{ ab }ξ_{ ab }⟩_{ s } = 0 and ⟨g^{ ab }(x)ξ_{ ab }(x)ξ_{ cd }(y)⟩_{ s } = 0; an alternative proof based on the pointseparation method is given in [304, 305], see also Section 5.
Note that we refer to the EinsteinLangevin equation as a firstorder extension to the semiclassical Einstein equation of semiclassical gravity and the lowestlevel representation of stochastic gravity. However, stochastic gravity has a much broader meaning; it refers to the range of theories based on second and higherorder correlation functions. Noise can be defined in effectivelyopen systems (e.g., correlation noise [61] in the SchwingerDyson equation hierarchy) to some degree, but one should not expect the Langevin form to prevail. In this sense we say stochastic gravity is the intermediate theory between semiclassical gravity (a mean field theory based on the expectation values of the energy momentum tensor of quantum fields) and quantum gravity (the full hierarchy of correlation functions retaining complete quantum coherence [187, 188].
The renormalization of the operator \({{\hat T}_{ab}}[g + h]\) is carried out exactly as in the previous case, now in the perturbed metric g_{ ab } + h_{ ab }. Note that the stochastic source ξ_{ ab }[g; x) is not dynamical; it is independent of h_{ ab }, since it describes the fluctuations of the stress tensor on the semiclassical background g_{ ab }.
An important property of the EinsteinLangevin equation is that it is gauge invariant under the change of h_{ ab } by \(h_{ab}{\prime} = {h_{ab}} + {\nabla _a}{\zeta _b} + {\nabla _b}{\zeta _a}\), where ζ^{ a } is a stochastic vector field on the background manifold \({\mathcal M}\). Note that a tensor such as R_{ ab }[g+h] transforms as \({R_{ab}}[g + {h{\prime}}] = {R_{ab}}[g + h] + {{\mathcal L}_\zeta}{R_{ab}}[g]\) to linear order in the perturbations, where \({{\mathcal L}_\zeta}\) is the Lie derivative with respect to ζ^{ a }. Now, let us write the source tensors in Equations (15) and (8) to the lefthand sides of these equations. If we substitute h with h′ in this new version of Equation (15), we get the same expression, with h instead of h′, plus the Lie derivative of the combination of tensors, which appear on the lefthand side of the new Equation (8). This last combination vanishes when Equation (8) is satisfied, i.e., when the background metric g_{ ab } is a solution of semiclassical gravity.
From the statistical average of Equation (15) we have that g_{ ab } + ⟨h_{ ab }⟩_{ s } must be a solution of the semiclassical Einstein equation linearized around the background g_{ ab }; this solution has been proposed as a test for the validity of the semiclassical approximation [10, 11] a point that will be further discussed in Section 3.3.
We should, however, also emphasize that Langevinlike equations are obtained to describe the quantum to classical transition in open quantum systems, when quantum decoherence takes place by coarsegraining of the environment as well as by suitable coarsegraining of the system variables [101, 127, 144, 146, 150, 374]. In those cases the stochastic correlation functions describe actual classical correlations of the system variables. Examples can be found in the case of a moving charged particle in an electromagnetic field in quantum electrodynamics [219] and in several quantum Brownian models [64, 65, 66].
3.3 Validity of semiclassical gravity
As we have emphasized earlier, the scope and limits of semiclassical gravity are not well understood because we still lack a fully wellunderstood quantum theory of gravity. From the semiclassical Einstein equations it also seems clear that the semiclassical theory should break down when the quantum fluctuations of the stress tensor are large. Ford [111] was among the first to have emphasized the importance of these quantum fluctuations. It is less clear, however, how to quantify the size of these fluctuations. Kuo and Ford [237] used the variance of the fluctuations of the stress tensor operator compared to the mean value as a measure of the validity of semiclassical gravity. Hu and Phillips pointed out [198, 303] that such a criterion should be refined by considering the backreaction of those fluctuations on the metric. Ford and collaborators also noticed that the metric fluctuations associated with the matter fluctuations can be meaningfully classified as active [114, 384, 385] and passive [111, 112, 115, 237], which correspond to our intrinsic and induced fluctuations, respectively, and have studied their properties in different contexts [35, 36, 37]. However, these fluctuations are not treated in a unified way and their precise relation to the quantum correlation function for the metric perturbations is not discussed. Furthermore, the fullaveraged backreaction of the matter fields is not included selfconsistently and the contribution from the vacuum fluctuations in Minkowski space is discarded.
A different approach to the validity of semiclassical gravity was pioneered by Horowitz [169, 170], who studied the stability of a semiclassical solution with respect to linear metric perturbations. In the case of a free quantum matter field in its Minkowski vacuum state, flat spacetime is a solution of semiclassical gravity. The equations describing those metric perturbations involve higherorder derivatives and Horowitz found unstable runaway solutions that grow exponentially with characteristic timescales comparable to the Planck time; see also the analysis by Jordan [223]. Later, Simon [329, 330] argued that those unstable solutions lie beyond the expected domain of validity of the theory and emphasized that only those solutions, which resulted from truncating perturbative expansions in terms of the square of the Planck length, are physically acceptable [329, 330]. Further discussion was provided by Flanagan and Wald [110], who advocated the use of an orderreduction prescription first introduced by Parker and Simon [295]. More recently Anderson, MolinaParís and Mottola have taken up the issue of the validity of semiclassical gravity [10, 11] again. Their starting point is the fact that the semiclassical Einstein equation will fail to provide a valid description of the dynamics of the mean spacetime geometry whenever the higherorder radiative corrections to the effective action, involving loops of gravitons or internal graviton propagators, become important. Next, they argue qualitatively that such higherorder radiative corrections cannot be neglected if the metric fluctuations grow without bound. Finally, they propose a criterion to characterize the growth of the metric fluctuations, and hence the validity of semiclassical gravity, based on the stability of the solutions of the linearized semiclassical equation. Following these approaches, the Minkowski metric is shown to be a stable solution of semiclassical gravity with respect to small metric perturbations.
As emphasized in [10, 11] the above criteria may be understood as based on semiclassical gravity itself. It is certainly true that stability is a necessary condition for the validity of a semiclassical solution, but one may also look for criteria within extensions of semiclassical gravity. In the absence of a quantum theory of gravity, such criteria may be found in some more modest extensions. Thus, Ford [111] considered graviton production in linearized quantum gravity and compared the results with the production of gravitational waves in semiclassical gravity. Ashtekar [13] and Beetle [22] found large quantumgravity effects in threedimensional quantumgravity models. In a more recent paper [203] (see also [204]), we advocate for a criteria within the stochastic gravity approach and since stochastic gravity extends semiclassical gravity by incorporating the quantum stresstensor fluctuations of the matter fields, this criteria is structurally the most complete to date.
It turns out that this validity criteria is equivalent to the validity criteria that one might advocate within the large N expansion; that is the quantum theory describing the interaction of the gravitational field with N identical free matter fields. In the leading order, namely the limit in which N goes to infinity and the gravitational constant is appropriately rescaled, the theory reproduces semiclassical gravity. Thus, a natural extension of semiclassical gravity is provided by the next to leading order. It turns out that the symmetrized twopoint quantumcorrelation functions of the metric perturbations in the large N expansion are equivalent to the twopoint stochastic metriccorrelation functions predicted by stochastic gravity. Our validity criterion can then be formulated as follows: a solution of semiclassical gravity is valid when it is stable with respect to quantum metric perturbations. This criterion involves the consideration of quantumcorrelation functions of the metric perturbations, since the quantum field describing the metric perturbations ĥ_{ ab }(x) is characterized not only by its expectation value but also by its npoint correlation functions.
It is important to emphasize that the above validity criterion incorporates in a unified and selfconsistent way the two main ingredients of the criteria exposed above. Namely, the criteria based on the quantum stresstensor fluctuations of the matter fields, and the criteria based on the stability of semiclassical solutions against classical metric perturbations. The former is incorporated through the induced metric fluctuations, and the later through the intrinsic fluctuations introduced in Equation (17). Whereas information on the stability of intrinsicmetric fluctuations can be obtained from an analysis of the solutions of the perturbed semiclassical Einstein equation (the homogeneous part of Equation (15)), the effect of inducedmetric fluctuations is accounted for only in stochastic gravity (the full inhomogeneous Equation (15)). We will illustrate these criteria in Section 6.5 by studying the stability of Minkowski spacetime as a solution of semiclassical gravity.
3.3.1 The large N expansion
To illustrate the relation between the semiclassical, stochastic and quantum theories, a simplified model of scalar gravity interacting with N scalar fields is considered here.
Next, there is a diagram with one graviton loop and two graviton legs. Let us count the order of this diagram: it contains four graviton propagators and two vertices, the propagators contribute as \({(\bar \kappa/N)^4}\) and the vertices as \({(N/\bar \kappa)^2}\), thus this diagram is of \(O({{\bar \kappa}^2}/{N^2})\). Therefore graviton loops contribute to a higher order in the 1/N expansion than matter loops. Similarly there are N diagrams with one loop of matter with an internal graviton propagator and two external graviton legs. Thus we have three graviton propagators and, since there are N of them, their sum is of order \(O({{\bar \kappa}^3}/{N^2})\). To summarize, we have that when N → ∞ there are no graviton propagators and gravity is classical, yet the matter fields are quantized. This is semiclassical gravity as was first described in [152]. Then we go to the nexttoleading order 1/N. Now the graviton propagator includes all matterloop contributions, but no contributions from graviton loops or internal graviton propagators in matter loops. This is what stochastic gravity reproduces.
That stochastic gravity is connected to the large N expansion can be seen from the stochastic correlations of linear metric perturbations on the Minkowski background computed in [259]. These correlations are in exact agreement with the imaginary part of the graviton propagator found by Tomboulis in the large N expansion for the quantum theory of gravity interacting with N Fermio fields [348]. This has been proven in detail in [203]; see also [204], where the case of a general background is also briefly discussed.
4 The EinsteinLangevin Equation: Functional Approach
The EinsteinLangevin equation (15) may also be derived by a method based on functional techniques [258]. Here we will summarize these techniques starting with semiclassical gravity.
In semiclassical gravity functional methods were used to study the backreaction of quantum fields in cosmological models [109, 147, 153]. The primary advantage of the effectiveaction approach is, in addition to the wellknown fact that it is easy to introduce perturbation schemes like loop expansion and nPI formalisms, that it yields a fully selfconsistent solution. (For a general discussion in the semiclassical context of these two approaches, equation of motion versus effective action, contrast, e.g., [137, 193, 194, 251] with the above references and [3, 4, 148, 154]. See also comments in Section 8 on the blackhole backreaction problem comparing the approach by York et al. [380, 381, 382] to that of Sinha, Raval and Hu [332].
The well known inout effectiveaction method treated in textbooks, however, led to equations of motion, which were not real because they were tailored to compute transition elements of quantum operators rather than expectation values. The correct technique to use for the backreaction problem is the SchwingerKeldysh [14, 56, 82, 87, 227, 323, 343] closedtimepath (CTP) or ‘inin’ effective action. These techniques were adapted to the gravitational context [54, 72, 94, 222, 223, 296] and applied to different problems in cosmology. One could deduce the semiclassical Einstein equation from the CTP effective action for the gravitational field (at tree level) with quantum matter fields.
Furthermore, in this case the CTP functional formalism turns out to be related [58, 69, 70, 73, 134, 239, 247, 256, 258, 267, 343] to the influencefunctional formalism of Feynman and Vernon [108], since the full quantum system may be understood as consisting of a distinguished subsystem (the “system” of interest) interacting with the remaining degrees of freedom (the environment). Integrating out the environment variables in a CTP path integral yields the influence functional, from which one can define an effective action for the dynamics of the system [58, 134, 191, 206]. This approach to semiclassical gravity is motivated by the observation [181] that in some open quantum systems classicalization and decoherence [46, 131, 221, 352, 389, 390, 391, 392, 393] on the system may be brought about by interaction with an environment, the environment being in this case the matter fields and some “highmomentum” gravitational modes [50, 51, 143, 182, 195, 231, 283, 374]. Unfortunately, since the form of a complete quantum theory of gravity interacting with matter is unknown, we do not know what these “highmomentum” gravitational modes are. Such a fundamental quantum theory might not even be a field theory, in which case the metric and scalar fields would not be fundamental objects [187]. Thus, in this case, we cannot attempt to evaluate the influence action of Feynman and Vernon starting from the fundamental quantum theory and performing the path integrations in the environment variables. Instead, we introduce the influence action for an effective quantum field theory of gravity and matter [95, 96, 97, 98, 297, 298, 331], in which such “highmomentum” gravitational modes are assumed to have already been “integrated out.”
4.1 Influence action for semiclassical gravity
Let us formulate semiclassical gravity in this functional framework. Adopting the usual procedure of effective field theories [63, 95, 96, 97, 98, 366, 367], one has to take the effective action for the metric and the scalar field of the most general local form compatible with general covariance: S[g, Φ] ≡ S_{g}[g] + S_{m}[g, Φ] + …, where S_{g}[g] and S_{m}[g, Φ] are given by Equations (9) and (1), respectively, and the dots stand for terms of order higher than two in the curvature and in the number of derivatives of the scalar field. Here, we shall neglect the higherorder terms as well as selfinteraction terms for the scalar field. The second order terms are necessary to renormalize oneloop ultraviolet divergences of the scalar field stressenergy tensor, as we have already seen. Since \({\mathcal M}\) is a globally hyperbolic manifold, we can foliate it by a family of t = constant Cauchy hypersurfaces Σ_{ t }, and we will indicate the initial and final times by t_{i}; and t_{f}, respectively.
Expression (22) contains ultraviolet divergences and must be regularized. We shall assume that dimensional regularization can be applied, that is, it makes sense to dimensionally continue all the quantities that appear in Equation (22). For this we need to work with the ndimensional actions corresponding to S_{m} in Equation (22) and S_{g} in Equation (9). For example, the parameters G, Λ, α, and β of Equation (9) are the bare parameters G_{B}, Λ_{B}, α_{B}, and β_{B}, and in S_{g}[g], instead of the square of the Weyl tensor in Equation (9), one must use \({2 \over 3}({R_{abcd}}{R^{abcd}}  {R_{ab}}{R^{ab}})\), which by the GaussBonnet theorem leads to the same equations of motion as the action (9) when n = 4. The form of S_{g} in n dimensions is suggested by the SchwingerDeWitt analysis of the ultraviolet divergences in the matter stressenergy tensor using dimensional regularization. One can then write the FeynmanVernon effective action S_{eff} [g^{±}] in Equation (24) in a form suitable for dimensional regularization. Since both S_{m} and S_{g} contain secondorder derivatives of the metric, one should also add some boundary terms [206, 361]. The effect of these terms is to cancel out the boundary terms, which appear when taking variations of S_{eff} [g^{±}], keeping the value of \(g_{ab}^ +\) and \(g_{ab}^ \) fixed at \({\Sigma _{{t_{\rm{i}}}}}\) and \({\Sigma _{{t_{\rm{f}}}}}\). Alternatively, in order to obtain the equations of motion for the metric in the semiclassical regime, we can work with the action terms without boundary terms and neglect all boundary terms when taking variations with respect to \(g_{ab}^ \pm\). From now on, all the functional derivatives with respect to the metric will be understood in this sense.
4.2 Influence action for stochastic gravity
4.3 Explicit form of the EinsteinLangevin equation
4.3.1 The kernels for the vacuum state
Finally, the causality of the EinsteinLangevin equation (41) can be explicitly seen as follows. The nonlocal terms in that equation are due to the kernel H(x, y), which is defined in Equation (29) as the sum of H_{S}(x, y) and H_{A}(x, y). Now, when the points x and y are spacelike separated, \({{\hat \phi}_n}(x)\) and \({{\hat \phi}_n}(y)\) commute and, thus, \(G_n^ + (x,y) = i{G_{{F_n}}}(x,y) = {1 \over 2}\langle 0\vert \{{{\hat \phi}_n}(x),{{\hat \phi}_n}(y)\} \vert 0\rangle\), which is real. Hence, from the above expressions, we have that \(H_{{{\rm{A}}_n}}^{abcd}(x,y) = H_{{{\rm{S}}_n}}^{abcd}(x,y) = 0\), and thus \(H_n^{abcd}(x,y) = 0\). This fact is expected since, from the causality of the expectation value of the stressenergy operator [359], we know that the nonlocal dependence on the metric perturbation in the EinsteinLangevin equation, see Equation (15), must be causal. See [208] for an alternative proof of the causal nature of the EinsteinLangevin equation.
5 Noise Kernel and Point Separation

the validity of semiclassical gravity [237], e.g., whether the fluctuations to mean ratio is a correct criterion [10, 11, 113, 115, 198, 303];

whether the fluctuations in the vacuum energy density, which drive some models of inflationary cosmology, violate the positiveenergy condition;

the physical effects of blackhole horizon fluctuations and Hawking radiation backreaction — to begin with, are the fluctuations finite or infinite?

general relativity as a lowenergy effective theory in the geometrohydrodynamic limit towards a kinetictheory approach to quantum gravity [185, 187, 188].
Thus, for comparison with ordinary phenomena at low energy, we need to find a reasonable prescription for obtaining a finite quantity of the noise kernel in the limit of ordinary (pointdefined) quantum field theory. It is wellknown that several regularization methods can work equally well for the removal of ultraviolet divergences in the stressenergy tensor of quantum fields in curved spacetime. Their mutual relations are known and discrepancies explained. This formal structure of regularization schemes for quantum fields in curved spacetime should remain intact when applied to the regularization of the noise kernel in general curved spacetimes; it is the meaning and relevance of regularization of the noise kernel, which is of more concern (see comments below). Specific considerations will, of course, enter for each method. But for the methods employed so far (such as zetafunction, point separation, dimensional and smearedfield) applied to simple cases (Casimir, Einstein, thermal fields) there is no new inconsistency or discrepancy.
Regularization schemes used in obtaining a finite expression for the stressenergy tensor have been applied to the noise kernel. This includes the simple normalordering [237, 378] and smearedfield operator [303] methods applied to the Minkowski and Casimir spaces, zetafunction [68, 106, 232] for spacetimes with an Euclidean section, applied to the Casimir effect [86] and the Einstein universe [302], or the covariant pointseparation methods applied to the Minkowski [303], hot flat space and Schwarzschild spacetime [305]. There are differences and deliberations on whether it is meaningful to seek a pointwise expression for the noise kernel, and if so what is the correct way to proceed, e.g., regularization by a subtraction scheme or by integrating over a test field. Intuitively the smearfield method [303] may better preserve the integrity of the noise kernel, as it provides a sampling of the twopoint function rather than using a subtraction scheme, which alters its innate properties by forcing a nonlocal quantity into a local one. More investigation is needed to clarify these points, which bear on important issues like the validity of semiclassical gravity. We shall set a more modest goal here, to derive a general expression for the noise kernel for quantum fields in an arbitrary curved spacetime in terms of Green’s functions and leave the discussion of pointwise limit to a later date. For this purpose the covariant pointseparation method that highlights the bitensor features, when used not as a regularization scheme, is perhaps closest to the spirit of stochastic gravity.
The task of finding a general expression for the noisekernel for quantum fields in curved spacetimes was carried out by Phillips and Hu in two papers using the “modified” pointseparation scheme [1, 358, 360]. Their first paper [304] begins with a discussion of the procedures for dealing with the quantum stresstensor bioperator at two separated points, and ends with a general expression for the noise kernel, defined at separated points expressed as products of covariant derivatives up to the fourth order of the quantum field’s Green’s function. (The stress tensor involves up to two covariant derivatives.) This result holds for x ≠ y without recourse to renormalization of the Green’s function, showing that N_{ abc′d′ } (x, y) is always finite for x ≠ y (and off the light cone for massless theories). In particular, for a massless conformallycoupled free scalar field on a fourdimensional manifold, they computed the trace of the noise kernel at both points and found this double trace vanishes identically. This implies that there is no stochastic correction to the trace anomaly for massless conformal fields, in agreement with results arrived at in [58, 73, 258] (see also Section 3). In their second paper [305] a Gaussian approximation for the Green’s function (which is what limits the accuracy of the results) is used to derive finite expressions for two specific classes of spacetimes, ultrastatic spacetimes, such as the hot flat space, and the conformallyultrastatic spacetimes, such as the Schwarzschild spacetime. Again, the validity of these results may depend on how we view the relevance and meaning of regularization. We will only report the result of their first paper here.
5.1 Point separation
The point separation scheme introduced in the 1960s by DeWitt [92] was brought to more popular use in the 1970s in the context of quantum field theory in curved spacetimes [83, 84, 93] as a means for obtaining a finite quantum stress tensor. Since the stressenergy tensor is built from the product of a pair of field operators evaluated at a single point, it is not welldefined. In this scheme, one introduces an artificial separation of the single point x into a pair of closely separated points x and x′. The problematic terms involving field products such as \(\hat \phi {(x)^2}\) become \(\hat \phi (x)\hat \phi ({x{\prime}})\), whose expectation value is welldefined. If one is interested in the lowenergy behavior captured by the pointdefined quantum field theory — as the effort in the 1970s was directed — one takes the coincidence limit. Once the divergences present are identified, they may be removed (regularization) or moved (by renormalizing the coupling constants), to produce a welldefined, finite stress tensor at a single point.
Thus, the first order of business is the construction of the stress tensor and then to derive the symmetric stressenergy tensor twopoint function, the noise kernel, in terms of the Wightman Green’s function. In this section we will use the traditional notation for index tensors in the pointseparation context.
5.1.1 ntensors and endpoint expansions
The bitensor of parallel transport \({g_a}^{{b\prime}}\) is defined such that when it acts on a vector v_{ b′ } at y, it parallel transports the vector along the geodesics connecting x and y. This allows us to add vectors and tensors defined at different points. We cannot directly add a vector v_{ a } at x and vector w_{ a′ } at y. But by using \({g_a}^{{b{\prime}}}\), we can construct the sum \({\upsilon ^a} + {g_a}^{{b{\prime}}}{w_{{b{\prime}}}}\). We will also need the obvious property \(\left[ {{g_a}^{{b\prime}}} \right] = {g_a}^b\).
Further details on these objects and discussions of the definitions and properties are contained in [83, 84] and [90, 301]. There it is shown how the defining equations for σ and Δ^{1/2} are used to determine the coincident limit expression for the various covariant derivatives of the world function ([σ_{;a}], [σ_{;ab}], etc.) and how the defining differential equation for Δ^{1/2} can be used to determine the series expansion of Δ^{1/2}. We show how the expansion tensors \(A_{{a_{1 \cdots}}{a_n}}^{(n)}\) are determined in terms of the coincident limits of covariant derivatives of the biscalar S(x, y). ([301] details how point separation can be implemented on the computer to provide easy access to a wider range of applications involving higher derivatives of the curvature tensors.)
5.2 Stressenergy bitensor operator and noise kernel
Even though we believe that the pointseparated results are more basic in the sense that it reflects a deeper structure of the quantum theory of spacetime, we will nevertheless start with quantities defined at one point, because they are what enter in conventional quantum field theory. We will use point separation to introduce the biquantities. The key issue here is thus the distinction between pointdefined (pt) and pointseparated (bi) quantities.
5.2.1 Finiteness of the noise kernel
5.2.2 Explicit form of the noise kernel
5.2.3 Trace of the noise kernel
One of the most interesting and surprising results to come out of the investigations of the quantum stress tensor undertaken in the 1970s was the discovery of the trace anomaly [77, 102]. When the trace of the stress tensor T = g^{ ab }T_{ ab } is evaluated for a field configuration that satisfies the field equation (2), the trace is seen to vanish for massless conformallycoupled fields. When this analysis is carried over to the renormalized expectation value of the quantum stress tensor, the trace no longer vanishes. Wald [360] showed that this was due to the failure of the renormalized Hadamard function G_{ren}(x, x′) to be symmetric in x and x′, implying that it does not necessarily satisfy the field equation (2) in the variable x′. (The definition of G_{ren}(x, x′) in the context of point separation will come next.)
6 Metric Fluctuations in Minkowski Spacetime
Although the Minkowski vacuum is an eigenstate of the total fourmomentum operator of a field in Minkowski spacetime, it is not an eigenstate of the stressenergy operator. Hence, even for those solutions of semiclassical gravity, such as the Minkowski metric, for which the expectation value of the stressenergy operator can always be chosen to be zero, the fluctuations of this operator are nonvanishing. This fact leads us to consider the stochastic metric perturbations induced by these fluctuations.
Here we derive the EinsteinLangevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated twopoint correlation functions, as well as the twopoint correlation functions for the metric perturbations. Even though in this case we expect to have negligibly small values for these correlation functions for points separated by lengths larger than the Planck length, there are several reasons why it is worth carrying out this calculation.
On the one hand, these are the first backreaction solutions of the full EinsteinLangevin equation. There are analogous solutions to a “reduced” version of this equation inspired in a “minisuperspace” model [52, 74], and there is also a previous attempt to obtain a solution to the EinsteinLangevin equation in [73], but there the nonlocal terms in the EinsteinLangevin equation are neglected.
On the other hand, the results of this calculation, which confirm our expectations that gravitational fluctuations are negligible at length scales larger than the Planck length, but also predict that the fluctuations are strongly suppressed on small scales, can be considered a first test of stochastic semiclassical gravity. These results also reveal an important connection between stochastic gravity and the large N expansion of quantum gravity. In addition, they are used in Section 6.5 to study the stability of the Minkowski metric as a solution of semiclassical gravity, which constitutes an application of the validity criterion introduced in Section 3.3. This calculation also requires a discussion of the problems posed by the socalled runaway solutions, which arise in the backreaction equations of semiclassical and stochastic gravity, and some of the methods to deal with them. As a result we conclude that Minkowski spacetime is a stable and valid solution of semiclassical gravity.
We advise the reader that Section 6 is rather technical since it deals with an explicit non trivial backreaction computation in stochastic gravity. We tried to make it reasonable selfcontained and detailed, however a more detailed exposition can be found in [259].
6.1 Perturbations around Minkowski spacetime
The Minkowski metric η_{ ab } in a manifold \({\mathcal M}\), which is topologically ℝ^{4}, together with the usual Minkowski vacuum, denoted as 0⟩, is the simplest solution to the semiclassical Einstein equation (8), the socalled trivial solution of semiclassical gravity [110]. It constitutes the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value \(\langle 0\vert \hat T_{\rm{R}}^{ab}[\eta ]\vert 0\rangle = 0\). Thus, the Minkowski spacetime (ℝ^{4},η_{ ab }) plus the vacuum state 0⟩ is a solution to the semiclassical Einstein equation with renormalized cosmological constant Λ = 0. The fact that the vacuum expectation value of the renormalized stressenergy operator in Minkowski spacetime should vanish was originally proposed by Wald [359], and it may be understood as a renormalization convention [121, 135]. Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stressenergy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations, which induce neither particle creation nor vacuum polarization [91, 125, 128].
As we have already mentioned, the vacuum 0⟩ is an eigenstate of the total fourmomentum operator in Minkowski spacetime, but not an eigenstate of \(\hat T_{ab}^R[\eta ]\). Hence, even in the Minkowski background there are quantum fluctuations in the stressenergy tensor and, as a result, the noise kernel does not vanish. This fact leads us to consider stochastic corrections to this class of trivial solutions of semiclassical gravity. Since in this case the Wightman and Feynman functions (44), their values in the twopoint coincidence limit and the products of derivatives of two of such functions appearing in expressions (45) and (46) are known in dimensional regularization, we can compute the EinsteinLangevin equation using the methods outlined in Sections 3 and 4.
6.2 The kernels in the Minkowski background
6.3 The EinsteinLangevin equation
It is interesting to consider the massless conformallycoupled scalar field, i.e., the case Δξ = 0, which is of particular interest because of its similarities with the electromagnetic field, and also because of its interest to cosmology; massive fields become conformally invariant when their masses are negligible compared to the spacetime curvature. We have already mentioned that, for a conformally coupled field, the stochastic source tensor must be traceless (up to first order perturbations around semiclassical gravity), in the sense that the stochastic variable \(\xi _\mu ^\mu \equiv {\eta _{\mu \nu}}{\xi ^{\mu \nu}}\) behaves deterministically as a vanishing scalar field. This can be directly checked by noticing from Equations (102) and (115) that when Δξ = 0, one has \({\langle \xi _\mu ^\mu (x){\xi ^{\alpha \beta}}(y)\rangle _{\rm{S}}} = 0\), since \({\mathcal F}_\mu ^\mu = 3\square\) and \({{\mathcal F}^{\mu \alpha}}{\mathcal F}_\mu ^\beta = \square{{\mathcal F}^{\alpha \beta}}\). The EinsteinLangevin equations for this particular case (and generalized to a spatiallyflat RobertsonWalker background) were first obtained in [73], where the coupling constant β was fixed to be zero. See also [208] for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [162, 339, 355].
Note that the expectation value of the renormalized stressenergy tensor for a scalar field can be obtained by comparing Equation (118) with the EinsteinLangevin equation (15); its explicit expression is given in [259]. The results agree with the general form found by Horowitz [169, 170] using an axiomatic approach and coincide with that given in [110]. The particular cases of conformal coupling, Δξ = 0, and minimal coupling, Δξ = −1/6, are also in agreement with the results for these cases given in [72, 169, 170, 223, 340], modulo local terms proportional to A^{(1)μν} and B^{(1)μν} due to different choices of the renormalization scheme. For the case of a massive minimallycoupled scalar field, \(\Delta \xi =  {1 \over 6}\), our result is equivalent to that of [347].
6.4 Correlation functions for gravitational perturbations
Here we solve the EinsteinLangevin equations (118) for the components G^{(1)μν} of the linearized Einstein tensor. Then we use these solutions to compute the corresponding twopoint correlation functions, which give a measure of the gravitational fluctuations predicted by the stochastic semiclassical theory of gravity in the present case. Since the linearized Einstein tensor is invariant under gauge transformations of the metric perturbations, these twopoint correlation functions are also gauge invariant. Once we have computed the twopoint correlation functions for the linearized Einstein tensor, we find the solutions for the metric perturbations and compute the associated twopoint correlation functions. The procedure used to solve the EinsteinLangevin equation is similar to the one used by Horowitz [169] (see also [110]) to analyze the stability of Minkowski spacetime in semiclassical gravity.
6.4.1 Correlation functions for the linearized Einstein tensor
6.4.2 Correlation functions for the metric perturbations
6.4.3 Conformallycoupled field
For a conformally coupled field, i.e., when m = 0 and Δξ = 0, the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [259]; here we outline the main features.
6.5 Stability of Minkowski spacetime
In this section we apply the validity criterion for semiclassical gravity introduced in Section 3.3 to flat spacetime. The Minkowski metric is a particularly simple and interesting solution of semiclassical gravity. In fact, as we have seen in Section 6.1, when the quantum fields are in the Minkowski vacuum state, one may take the renormalized expectation value of the stress tensor as \(\left\langle {\hat T_{ab}^R[\eta ]} \right\rangle = 0\); this is equivalent to assuming that the cosmological constant is zero. Then the Minkowski metric η_{ ab } is a solution of the semiclassical Einstein equation (8). Thus, we can look for the stability of Minkowski spacetime against quantum matter fields. According to the criteria we have established, we have to look for the behavior of the twopoint quantum correlations for the metric perturbations h_{ ab }(x) over the Minkowski background, which are given by Equations (16) and (17). As we have emphasized before, these metric fluctuations separate in two parts: the first term on the righthand side of Equation (17), which corresponds to the intrinsic metric fluctuations, and the second term, which corresponds to the induced metric fluctuations.
6.5.1 Intrinsic metric fluctuations
For the scalar component when \(\bar \beta = 0\) the only solution is \(\tilde G_{\mu \nu}^{(1)\,({\rm{S}})}(p) = 0\). When \(\bar \beta > 0\) the solutions for the scalar component exhibit an oscillatory behavior in spacetime coordinates, which corresponds to a massive scalar field with \({m^2} = {(12\kappa \vert \bar \beta \vert)^{ 1}}\); for \(\bar \beta < 0\) the solutions correspond to a tachyonic field with \({m^2} =  {(12\kappa \vert \bar \beta \vert)^{ 1}}\). In spacetime coordinates they exhibit an exponential behavior in time, growing or decreasing, for wavelengths larger than \(4\pi {(3\kappa \vert \bar \beta \vert)^{1/2}}\) and an oscillatory behavior for wavelengths smaller than \(4\pi {(3\kappa \vert \bar \beta \vert)^{1/2}}\). On the other hand, the solution \(\tilde G_{\mu \nu}^{(1)\,({\rm{S}})}(p) = 0\) is completely trivial since any scalar metric perturbation \({{\tilde h}_{\mu \nu}}(p)\) giving rise to a vanishing linearized Einstein tensor can be eliminated by a gauge transformation.
For the tensorial component, when \(\mu \leq {\mu _{{\rm{crit}}}} = l_p^{ 1}{(120\pi)^{1/2}}{e^\gamma}\), where l_{ p } is the Planck length \((l_p^2 \equiv \kappa/8\pi)\), the first factor in Equation (154) vanishes for four complex values of p^{0} of the form ±ω and ±ω*, where ω is some complex value. This means that, in the corresponding propagator, there are two poles on the upper halfplane of the complex p^{0} plane and two poles in the lower halfplane. We will consider here the case in which μ < μ_{crit}; a detailed description of the situation for μ ≥ μ_{crit} can be found in Appendix A of [110]. The two zeros on the upper half of the complex plane correspond to solutions in spacetime coordinates, which exponentially grow in time, whereas the two on the lower half correspond to solutions exponentially decreasing in time. Strictly speaking, these solutions only exist in spacetime coordinates, since their Fourier transform is not welldefined. They are commonly referred to as runaway solutions and for \(\mu \sim l_p^{ 1}\) they grow exponentially in time scales comparable to the Planck time.
A second possibility, proposed by Hawking et al. [161, 162], is to impose boundary conditions, which discard the runaway solutions that grow unbounded in time. These boundary conditions correspond to a special prescription for the integration contour when Fourier transforming back to spacetime coordinates. As we will discuss in more detail in Section 6.5.2, this prescription reduces here to integrating along the real axis in the p^{0} complex plane. Following that procedure we get, for example, that for a massless conformallycoupled matter field with \(\bar \beta > 0\) the intrinsic contribution to the symmetrized quantum correlation function coincides with that of free gravitons plus an extra contribution for the scalar part of the metric perturbations. This extramassive scalar renders Minkowski spacetime stable, but also plays a crucial role in providing a graceful exit in inflationary models driven by the vacuum polarization of a large number of conformal fields. Such a massive scalar field would not be in conflict with present observations because, for the range of parameters considered, the mass would be far too large to have observational consequences [162].
6.5.2 Induced metric fluctuations
To estimate the above integral let us follow Section 6.4.3 and consider spacelike separated points x − y = (0,r) and introduce the Planck length l_{ p }. For space separations ∣r∣ ≫ l_{ p } we have that the twopoint correlation (161) goes as \(\sim Nl_p^4\vert r{\vert ^4}\), and for \(\vert r\vert \sim \sqrt N {l_p}\) we have that it goes as \(\sim \exp ( \vert r\vert/\sqrt N lp){l_p}/\vert r\vert\). Since these metric fluctuations are induced by the matter stress fluctuations we infer that the effect of the matter fields is to suppress metric fluctuations at small scales. On the other hand, at large scales the induced metric fluctuations are small compared to the free graviton propagator, which goes like \(l_p^2/\vert r{\vert ^2}\).
We thus conclude that, once the instabilities giving rise to the unphysical runaway solutions have been discarded, the fluctuations of the metric perturbations around the Minkowski spacetime induced by the interaction with quantum scalar fields are indeed stable (instabilities lead to divergent results when Fourier transforming back to spacetime coordinates). We have found that, indeed, both the intrinsic and the induced contributions to the quantum correlation functions of metric perturbations are stable, and consequently Minkowski spacetime is stable.
6.5.3 Orderreduction prescription and large N
However, if we have a large number N of matter fields, the estimates for the different terms change in a remarkable way. This is interesting because the large N expansion seems, as we have argued in Section 3.3.1, the best justification for semiclassical gravity. In fact, now the N vacuumpolarization terms involving loops of matter are of order NR^{2} ∼ NL^{−4}. For this reason, the contribution of the graviton loops, which is just of order R^{2} as is any other loop of matter, can be neglected in front of the matter loops; this justifies the semiclassical limit. Similarly, higherorder corrections are of order \(Nl_p^2{R^3} \sim Nl_p^2{L^{ 6}}\). Now there is a regime, when \(L \sim \sqrt {N{l_p}}\), where the EinsteinHilbert term is comparable to the vacuum polarization of matter fields, \((1/l_p^2)R \sim N{R^2}\), and yet the higher correction terms can be neglected because we still have L ≫ l_{ p }, provided N ≫ 1. This is the kind of situation considered in trace anomaly driven inflationary models [162], such as that originally proposed by Starobinsky [339], see also [355], where exponential inflation is driven by a large number of massless conformal fields. The orderreduction prescription would completely discard the effect from the vacuum polarization of the matter fields even though it is comparable to the EinsteinHilbert term. In contrast, the procedure proposed by Hawking et al. keeps the contribution from the matter fields. Note that here the actual physical Planck length l_{ p } is considered, not the rescaled one, \(\bar l_p^2 = \bar \kappa/8\pi\), which is related to l_{ p } by \(l_p^2 = \kappa/8\pi = \bar l_p^2/N\).
6.5.4 Summary
An analysis of the stability of any solution of semiclassical gravity with respect to small quantum perturbations should include not only the evolution of the expectation value of the metric perturbations around that solution, but also their fluctuations encoded in the quantum correlation functions. Making use of the equivalence (to leading order in 1/N, where N is the number of matter fields) between the stochastic correlation functions obtained in stochastic semiclassical gravity and the quantum correlation functions for metric perturbations around a solution of semiclassical gravity, the symmetrized twopoint quantum correlation function for the metric perturbations can be decomposed into two different parts: the intrinsic metric fluctuations due to the fluctuations of the initial state of the metric perturbations itself, and the fluctuations induced by their interaction with the matter fields. From the linearized perturbations of the semiclassical Einstein equation, information on the intrinsic metric fluctuations can be retrieved. On the other hand, the information on the induced metric fluctuations naturally follows from the solutions of the EinsteinLangevin equation.
We have analyzed the symmetrized twopoint quantum correlation function for the metric perturbations around the Minkowski spacetime interacting with N scalar fields initially in the Minkowski vacuum state. Once the instabilities that arise in semiclassical gravity, which are commonly regarded as unphysical, have been properly dealt with by using the orderreduction prescription or the procedure proposed by Hawking et al. [161, 162], both the intrinsic and the induced contributions to the quantum correlation function for the metric perturbations are found to be stable [203]. Thus, we conclude that Minkowski spacetime is a valid solution of semiclassical gravity.
7 Structure Formation in the Early Universe

The sources: Instead of a classical whitenoise source arbitrarily specified, the seeds of structures of the new theory are from quantum fluctuations, which obey equations derivable from the dynamics of the inflaton field, which is responsible for driving inflation.

The spectrum: The almost scaleinvariant spectrum (masses of galaxies as a function of their scales) has a more natural explanation from the almost exponential expansion of the inflationary universe than from the powerlaw expansion of the FRW universe in the traditional theory.
Stochastic gravity provides a sound and natural formalism for the derivation of the cosmological perturbations generated during inflation. In [316] it was shown that the correlation functions that follow from the EinsteinLangevin equation, which emerges in the framework of stochastic gravity, coincide with that obtained with the usual quantization procedures [270] when both the metric perturbations and the inflaton fluctuations are linearized. Stochastic gravity, however, can naturally deal with the fluctuations of the inflaton field even beyond the linear approximation. In Section 7.4 we will enumerate possible advantages of the stochasticgravity treatment of this problem over the usual methods based on the quantization of the linear cosmological and linear inflaton perturbations.
We should point out that the equivalence at the linearized level is proved in [316] directly from the field equations of the perturbations and by showing that the stochastic and the quantum correlations are both given by identical expressions. Within the stochastic gravity framework an explicit computation of the curvature perturbation correlations was performed by Urakawa and Maeda [353]. A convenient approximation for that computation, used by these authors, leads only to a small discrepancy with the usual approach for the observationally relevant part of the spectrum. We think the deviation from the standard result found for superhorizon modes would not arise if an exact calculation were used.
Here we illustrate the equivalence with the conventional approach with one of the simplest chaotic inflationary models in which the background spacetime is a quasi de Sitter universe [315, 316].
7.1 The model
7.2 EinsteinLangevin equation for scalar metric perturbations
7.3 Correlation functions for scalar metric perturbations
The assumption of a massless field for the computation of the Hadamard function is made because massless modes in de Sitter are much simpler to deal with than massive modes. We can see that this is nonetheless a reasonable approximation as follows: For a given mode, the m = 0 approximation is reasonable when its wavelength λ is shorter than the Compton wavelength, λ_{ c } = 1/m. In our case we have a very small mass m and the horizon size H^{−1}, where H is the Hubble constant H = ȧ/a (here a(t) with t the physical time dt = adη), satisfies H^{−1} < λ_{ c }. Thus, for modes inside the horizon, λ < λ_{c} and m = 0 are a good approximation. Outside the horizon, massive modes decay in amplitude as ∼ exp(−m^{2}t/3H) whereas massless modes remain constant. Thus, when modes leave the horizon, the approximation will eventually break down. However, we only need to ensure that the approximation is still valid after 60 efolds, i.e., HΔt ∼ 60 (Δt being the time between horizon exit and the end of inflation). But this is the case provided 3H^{2}/m^{2} > 60, since the decay factor ∼ exp[−(m^{2}/3H^{2})HΔt] will not be too different from unity for those modes that left the horizon during the last sixty efolds of inflation. This condition is indeed satisfied given that m ≪ H in most slowroll inflationary models [233, 284] and in particular for the model considered here, in which m ∼ 10^{−6}m_{ P }.
We now comment on some differences with [53, 60, 262, 263], which used a selfinteracting scalar field or a scalar field interacting nonlinearly with other fields. In these works an important relaxation of the ratio m/m_{ P } was found. The long wavelength modes of the inflaton field were regarded as an open system in an environment made out of the shorter wavelength modes. Then, Langevin type equations were used to compute the correlations of the long wavelength modes driven by the fluctuations of the shorter wavelength modes. In order to get a significant relaxation on the above ratio, however, one had to assume that the correlations of the free longwavelength modes, which correspond to the dispersion of the system’s initial state, were very small. Otherwise they dominate by several orders of magnitude those fluctuations that come from the noise of the environment. This would require a great amount of finetuning for the initial quantum state of each mode [316].
We should remark that in the linear model discussed here there is no environment for the inflaton fluctuations. When one linearizes with respect to both the scalar metric perturbations and the inflaton perturbations, the system cannot be regarded as a true open quantum system. The reason is that Fourier modes decouple and the dynamical constraints due to diffeomorphism invariance link the metric perturbations of scalar type with the perturbations of the inflaton field so that only one true dynamical degree of freedom is left for each Fourier mode. Nevertheless, the inflaton fluctuations are responsible for the noise that induces the metric perturbations.
7.4 Summary and outlook
Stochastic gravity provides an alternative framework to study the generation of primordial inhomogeneities in inflationary models. Besides the interest of the problem in its own right, there are also other reasons that make this problem worth discussing from the point of view of stochastic gravity. The EinsteinLangevin equation is not restricted by the use of linearized perturbations of the inflaton field. In practice this may not be very important for inflationary models, which are driven by an inflaton field, which takes a nonzero expectation value, because the linear perturbations will give the leading contribution. However, the importance of considering corrections due to oneloop contributions from the inflatonfield perturbations, beyond the tree level of the linear cosmological perturbation theory, has recently been emphasized by Weinberg [368, 369] as a means to understand the consequences of the theory even where in practice its predictions may not be observed. Note also [225, 226] for the effect of the quantum gravitational loop corrections on the dynamics of the inflaton field.
In the stochastic gravity approach some insights on the exact treatment of the inflaton scalar field perturbations have been discussed in [316, 353, 354]. The main features that would characterize an exact treatment of the inflaton perturbations are the following. First, the three types of metric perturbations (scalar, vectorial and tensorial perturbations) couple to the perturbations of the inflaton field. Second, the corresponding EinsteinLangevin equation for the linear metric perturbations will explicitly couple to the scalar and tensorial metric perturbations. Furthermore, although the Fourier modes (with respect to the spatial coordinates) for the metric perturbations will still decouple in the EinsteinLangevin equation, any given mode of the noise and dissipation kernels will get contributions from an infinite number of Fourier modes of the inflaton field perturbations. This fact will imply, in addition, the need to properly renormalize the ultraviolet divergences arising in the dissipation kernel, which actually correspond to the divergences associated with the expectation value of the stresstensor operator of the quantum matter field evolving on the perturbed geometry.
We should remark that although the gravitational fluctuations are here assumed to be classical, the correlation functions obtained correspond to the expectation values of the symmetrized quantum metric perturbations [66, 316]. This means that even in the absence of decoherence, the fluctuations predicted by the EinsteinLangevin equation still give the correct symmetrized quantum twopoint correlation functions. In [66] it was explained how a stochastic description based on a Langevintype equation could be introduced to gain information on fully quantum properties of simple linear open systems. In a forthcoming paper [317] it will be shown that, by carefully dealing with the gauge freedom and the consequent dynamical constraints, this result can be extended to the case of N free quantum matter fields interacting with the metric perturbations around a given background. In particular, the correlation functions for the metric perturbations obtained using the EinsteinLangevin equation are equivalent to the correlation functions that would follow from a purely quantum field theory calculation up to the leading order contribution in the large N limit. This will generalize the results already obtained on a Minkowski background [203, 204].
These results have important implications on the use of the EinsteinLangevin equation to address situations in which the background configuration for the scalar field vanishes. This includes not only the case of a Minkowski background spacetime, but also the remarkably interesting case of the trace anomalyinduced inflation. That is, inflationary models driven by the vacuum polarization of a large number of conformal fields [162, 339, 355], in which the usual approaches based on the linearization of both the metric perturbations and the scalar field perturbations and their subsequent quantization can no longer be applied. More specifically, the semiclassical Einstein equations (8) for massless quantum fields conformally coupled to the gravitational field admit an inflationary solution that begins in an almost de Sitterlike regime and ends up in a matterdominatedlike regime [339, 355]. In these models the standard approach based on the quantization of the gravitational and the matter fields to linear order cannot be used because the calculation of the metric perturbations correspond to having only the last term in the noise kernel in Equation (171), since there is no homogeneous field ϕ(η) as the expectation value \(\langle \hat \phi \rangle = 0\) and linearization becomes trivial.
In the trace anomaly induced inflation model Hawking et al. [162] were able to compute the twopoint quantum correlation function for scalar and tensorial metric perturbations in a spatiallyclosed de Sitter universe, making use of the antide Sitter/conformal field theory correspondence. They find that shortscale metric perturbations are strongly suppressed by the conformal matter fields. This is similar to what we obtained in Section 6 for the induced metric fluctuations in Minkowski spacetime. In the stochastic gravity context, the noise kernel in a spatiallyclosed de Sitter background was derived in [314], and in a spatiallyflat arbitrary FriedmannRobertsonWalker model the EinsteinLangevin equations describing the metric perturbations were first obtained in [73]. The computation of the corresponding twopoint correlation functions for the metric perturbations is now in progress.
8 Black Hole Backreaction and Fluctuations
As another illustration of the application of stochastic gravity we now consider the backreaction and fluctuations in black hole spacetimes. Backreaction refers to the quantum effects of matter fields such as vacuum polarization, quantum fluctuations and particle creation on the spacetime structure and dynamics. Studying the dynamics of quantum fields in a fixed background spacetime, Hawking found that black holes emit thermal radiation with a temperature inversely proportional to their mass [156, 213, 294, 358]. When the backreaction of the quantum fields on the spacetime dynamics is included, one expects that the mass of the black hole decreases as thermal radiation at higher and higher temperatures is emitted. The reduction of the mass of a black hole due to particle creation is often referred to as the black hole ‘evaporation’ process. Backreaction of Hawking radiation [9, 18, 142, 166, 167, 380, 381, 382] could alter the evolution of the background spacetime and change the nature of its end state, more drastically so for Plancksize black holes.
Backreaction is a technically challenging but conceptually rewarding problem. Progress is slow in this long standing problem, but it cannot be ignored because existing results from testfield approximations or semiclassical analysis are not trustworthy when backreaction becomes strong enough as to alter the structure and dynamics of the background spacetime. At the least one needs to know how strong the backreaction effects are, and under what circumstances the existing predictions make sense. Without an exact quantum solution of the blackholeplusquantumfield system or at least a full backreaction consideration including the intrinsic and induced effects of metric fluctuations, much of the long speculation on the endstate of black hole collapse — remnants, naked singularity, baby universe formation or complete evaporation (see, e.g., [12, 47, 130, 168, 184, 250, 306, 318, 319, 375, 376]) — and the information loss issue [157, 158, 160, 286] (see, e.g., [288, 307] for an overview and recent results from quantum information [40, 333]) will remain speculation and puzzles. This issue also enters into the extension of the wellknown blackhole thermodynamics [23, 25, 26, 171, 172, 214, 224, 253, 254, 337, 342, 346, 363, 364] to nonequilibrium conditions [105] and can lead to new inferences on the microscopic structure of spacetime and the true nature of Einstein’s equations [216] from the viewpoint of general relativity as geometrohydrodynamics and gravity as emergent phenomena. (See the nontraditional views of Volovik [356, 357]and Hu [185, 186, 190] on spacetime structure, Wen [139, 240, 372, 373] on quantum order, Seiberg [326], Horowitz and Polchinsky [173] on emergent gravity, Herzog on the hydrodynamics of Mtheory [164] and the seminal work of Unruh and Jacobson [215, 351] leading to analog gravity [15, 16, 320].)
8.1 General issues of backreaction
Backreaction studies of quantum fieldprocesses in cosmological spacetimes have progressed further than the corresponding blackhole problems, partly because of the relative technical simplicity associated with the higher symmetry of relevant cosmologicalbackground geometries. (For a summary of the cosmological backreaction problem treated in stochasticgravity theory, see [208].) How the problem is set up and approached, e.g., via effective action, in these previouslystudied models can be carried over to blackhole problems. In fact, since the interior of a black hole can be described by a cosmological model (e.g., the KantowskiSachs universe for a spherically symmetric black hole), some aspects even convey directly. The latest important work on this problem is that of Hiscock, Larson and Anderson [165] on backreaction in the interior of a black hole, in which one can find a concise summary of earlier work.
8.1.1 Regularized energymomentum tensor
The first step in a backreaction problem is to find a regularized energymomentum tensor of the quantum fields using reasonable techniques, since the expectation value of this serves as the source in the semiclassical Einstein equation. For this, much work started in the 1980s (and still ongoing sparingly) is concerned with finding the right approximations for the regularized energymomentum tensor [6, 7, 8, 165, 217, 260, 292]. Even in the simplest spherically symmetric spacetime, including the important Schwarzschild metric, it is technically quite involved. To name a few of the important landmarks in this endeavor (this is adopted from [165]), Howard and Candelas [177, 178] have computed the stressenergy of a conformallyinvariant scalar field in the Schwarzschild geometry; Jensen and Ottewill [218] have computed the vacuum stressenergy of a massless vector field in Schwarzschild. Approximation methods have been developed by Page, Brown, and Ottewill [41, 42, 287] for conformallyinvariant fields in Schwarzschild spacetime, by Frolov and Zel’nikov [120] for conformallyinvariant fields in a general static spacetime, and by Anderson, Hiscock and Samuel [7, 8] for massless arbitrarilycoupled scalar fields in a general static sphericallysymmetric spacetime. Furthermore, the DeWittSchwinger approximation has been derived by Frolov and Zel’nikov [118, 119] for massive fields in Kerr spacetime, and by Anderson, Hiscock and Samuel [7, 8] for a general (arbitrary curvature coupling and mass) scalar field in a general, static, sphericallysymmetric spacetime. And they have applied their method to the ReissnerNordström geometry [6]. Though arduous and demanding, the effort continues on because of its importance in finding the backreaction effects of Hawking radiation on the evolution of black holes and the quantum structure of spacetime.
Here we wish to address the black hole backreaction problem with new insights and methods provided by stochastic gravity. (For the latest developments, see, e.g., [183, 187, 207, 208].) It is not our intention to seek better approximations for the regularized energymomentum tensor, but to point out new ingredients lacking in the existing semiclassicalgravity framework. In particular one needs to consider both the dissipation and the fluctuation aspects in the backreaction of particle creation and vacuum polarization.
In a short note Hu, Raval and Sinha [199] first used the stochastic gravity formalism to address the backreaction of evaporating black holes. A more detailed analysis is given by the recent work of Hu and Roura [201, 202]. For the class of quasistatic black holes, the formulation of the problem in this new light was sketched out by Sinha, Raval, and Hu [332]. We follow these two latter works in the stochastic gravity theory approach to the blackhole fluctuations and backreaction problems.
8.1.2 Backreaction and fluctuationdissipation relation
From the statistical fieldtheory perspective provided by stochastic gravity, one can understand that backreaction effect is the manifestation of a fluctuationdissipation relation [48, 49, 103, 104, 274, 365]. This was first conjectured by Candelas and Sciama [76, 324, 325] for a dynamic Kerr black hole emitting Hawking radiation and Mottola [268] for a static black hole (in a box) in quasiequilibrium with its radiation via linearresponse theory [33, 234, 235, 236, 238]. This postulate was shown to hold for fully dynamical spacetimes. From the cosmologicalbackreaction problem Hu and Sinha [206] derived a generalized fluctuationdissipation relation relating dissipation (of anisotropy in Bianchi Type I universes) and fluctuations (measured by particle numbers created in neighboring histories).
While the fluctuationdissipation relation in linearresponse theory captures the response of the system (e.g., dissipation of the black hole) to the environment (in these cases the quantum matter field), linearresponse theory (in the way it is commonly presented in statistical thermodynamics) cannot provide a full description of selfconsistent backreaction on at least two counts:
First, because it is usually based on the assumption of a specified background spacetime (static in this case) and state (thermal) of the matter field(s) (e.g., [268]). The spacetime and the state of matter should be determined in a selfconsistent manner by their dynamics and mutual influence. Second, the fluctuation part represented by the noise kernel is amiss, e.g., [10, 11]. This is also a problem in the fluctuationdissipation relation proposed by Candelas and Sciama [76, 324, 325] (see below). As demonstrated by many authors [73, 206] backreaction is intrinsically a dynamic process. The EinsteinLangevin equation in stochastic gravity overcomes both of these deficiencies.
For Candelas and Sciama [76, 324, 325], the classical formula they showed relating the dissipation in area linearly to the squared absolute value of the shear amplitude is suggestive of a fluctuationdissipation relation. When the gravitational perturbations are quantized (they choose the quantum state to be the Unruh vacuum) they argue that it approximates a flux of radiation from the hole at large radii. Thus the dissipation in area due to the Hawking flux of gravitational radiation is allegedly related to the quantum fluctuations of gravitons. The criticism in [199] is that their’s is not a fluctuationdissipation relation in the truly statistical mechanical sense because it does not relate dissipation of a certain quantity (in this case, horizon area) to the fluctuations of the same quantity. To do so would require one to compute the twopoint function of the area, which, being a fourpoint function of the graviton field, is related to a twopoint function of the stress tensor. The stress tensor is the true “generalized force” acting on the spacetime via the equations of motion, and the dissipation in the metric must eventually be related to the fluctuations of this generalized force for the relation to qualify as a fluctuationdissipation relation.
8.1.3 Noise and fluctuations — the missing ingredient in older treatments
From this reasoning, we see that the vacuum expectation value of the stressenergy bitensor, known as the noise kernel, is the necessary new ingredient in addition to the dissipation kernel, and that stochastic gravity as an extension of semiclassical gravity is the appropriate framework for backreaction considerations. The noise kernel for quantum fields in Minkowski and de Sitter spacetime has been carried out by Martin, Roura and Verdaguer [257, 259, 316], and for thermal fields in blackhole spacetimes and scalar fields in general spacetimes by Campos, Hu and Phillips [69, 70, 304, 305].
8.2 Backreaction on black holes under quasistatic conditions
As an illustration of the application of stochasticgravity theory we outline the essential steps in a blackhole backreaction calculation, focusing on a more manageable quasistatic class. We adopt the HartleHawking picture [151] where the black hole is bathed eternally — actually in quasithermal equilibrium — in the Hawking radiance it emits. It is described here by a massless scalar quantum field at the Hawking temperature. As is well known, this quasiequilibrium condition is possible only if the black hole is enclosed in a box of size suitably larger than the event horizon. We can divide our consideration into the farfield case and the nearhorizon case. Campos and Hu [69, 70] have treated a relativistic thermal plasma in a weak gravitational field. Since the farfield limit of a Schwarzschild metric is just the perturbed Minkowski spacetime, one can perform a perturbation expansion of hot flat space using the thermal Green’s functions [129]. Strictly speaking, the location of the box holding the black hole in equilibrium with its thermal radiation is as far as one can go, thus the metric may not reach the perturbed Minkowski form. But one can also put the black hole and its radiation in an antide Sitter space [163], which contains such a region. Hot flat space has been studied before for various purposes. See, e.g., [39, 89, 138, 311, 312]. Campos and Hu derived a stochastic CTP effective action and from it an equation of motion, the Einstein Langevin equation, for the dynamical effect of a scalar quantum field on a background spacetime. To perform calculations leading to the EinsteinLangevin equation one needs to begin with a selfconsistent solution of the semiclassical Einstein equation for the thermal field and the perturbed background spacetime. For a blackhole background, a semiclassical gravity solution is provided by York [380, 381, 382]. For a RobertsonWalker background with thermal fields it is given by Hu [180].
We follow the strategy outlined by Sinha, Raval and Hu [332] for treating the near horizon case, following the same scheme of Campos and Hu. In both cases two new terms appear, which are absent in semiclassical gravity considerations: a nonlocal dissipation and a (generally colored) noise kernel. When one takes the noise average one recovers York’s [380, 381, 382] semiclassical equations for radially perturbed quasistatic black holes. For the nearhorizon case one cannot obtain the full details yet, because the Green’s function for a scalar field in the Schwarzschild metric comes only in an approximate form, e.g., Page’s approximation [287], which, though reasonably accurate for the stress tensor, fails badly for the noise kernel [305]. In addition, a formula is derived in [332] expressing the CTP effective action in terms of the Bogoliubov coefficients. Since it measures not only the number of particles created, but also the difference of particle creation in alternative histories, this provides a useful avenue to explore the wider set of issues in black hole physics related to noise and fluctuations.
Since backreaction calculations in semiclassical gravity have been under study for a much longer time than in stochastic gravity we will concentrate on explaining how the new stochastic features arise from the framework of semiclassical gravity, i.e., noise and fluctuations and their consequences. Technically the goal is to obtain an influence action for this model of a black hole coupled to a scalar field and to derive an EinsteinLangevin equation from it. As a byproduct, from the fluctuationdissipation relation, one can derive the vacuumsusceptibility function and the isothermalcompressibility function for black holes, two quantities of fundamental interest in characterizing the nonequilibrium thermodynamic properties of black holes.
8.2.1 The model
8.2.2 CTP effective action for the black hole
8.2.3 Near flat case
Using the property T^{ μν,αβ }(q,k) = T^{ μν,αβ }(−q, −k), it is easy to see that the kernel N^{ μν,αβ }(x − x′) is symmetric and D^{ μν,αβ }(x − x′) is antisymmetric in its arguments; that is, N^{ μν,αβ }(x) = N^{ μν,αβ }(−x) and D^{ μν,αβ }(x) = −D^{ μν,αβ }(−x).
The physical meanings of these kernels can be extracted if we write the renormalized CTP effective action at finite temperature (195) in an influencefunctional form [45, 133, 196, 197]. N, the imaginary part of the CTP effective action, can be identified with the noise kernel and D, the antisymmetric piece of the real part, with the dissipation kernel. Campos and Hu [69, 70] have shown that these kernels identified as such indeed satisfy a thermal fluctuationdissipation relation.
Finally, as defined above, N^{ μν,αβ }(x) is the noise kernel representing random fluctuations of thermal radiance and D^{ μν,αβ }(x) is the dissipation kernel, describing the dissipation of energy of the gravitational field.
8.2.4 Nearhorizon case
In this case, since the perturbation is taken around the Schwarzschild spacetime, exact expressions for the corresponding unperturbed propagators \(G_{ab}^\beta [h_{\mu \nu}^ \pm ]\) are not known. Therefore, apart from the approximation of computing the CTP effective action to certain order in perturbation theory, an appropriate approximation scheme for the unperturbed Green’s functions is also required. This feature manifested itself in York’s calculation of backreaction as well, where, in writing 〈T_{ μν }〉 on the righthand side of the semiclassical Einstein equation in the unperturbed Schwarzschild metric, he had to use an approximate expression for 〈T_{ μν }〉 in the Schwarzschild metric, given by Page [287]. The additional complication here is that, while to obtain 〈T_{ μν }〉 as in York’s calculation, the knowledge of only the thermal Feynman Green’s function is required, to calculate the CTP effective action one needs the knowledge of the full matrix propagator, which involves the Feynman, Schwinger and Wightman functions.
It is indeed possible to construct the full thermal matrix propagator \(G_{ab}^\beta [h_{\mu \nu}^ \pm ]\) based on Page’s approximate Feynman Green’s function by using identities relating the Feynman Green’s function with the other Green’s functions with different boundary conditions. One can then proceed to explicitly compute a CTP effective action and hence the influence functional based on this approximation. However, we desist from delving into such a calculation for the following reason. Our main interest in performing such a calculation is to identify and analyze the noise term, which is the new ingredient in the backreaction. We have mentioned that the noise term gives a stochastic contribution ξ^{ μν } to the EinsteinLangevin equation (15). We had also stated that this term is related to the variance of fluctuations in T_{ μν }, i.e, schematically, to \(\langle T_{\mu \nu}^2\rangle\). However, a calculation of \(\langle T_{\mu \nu}^2\rangle\) in the HartleHawking state in a Schwarzschild background using the Page approximation was performed by Phillips and Hu [304, 305] and it was shown that, though the approximation is excellent as far as 〈T_{ μν }〉 is concerned, it gives unacceptably large errors for 〈T_{ μν }〉 at the horizon. In fact, similar errors will be propagated in the nonlocal dissipation term as well, because both terms originate from the same source, that is, they come from the last trace term in (195), which contains terms quadratic in the Green’s function. However, the influence functional or CTP formalism itself does not depend on the nature of the approximation, so we will attempt to exhibit the general structure of the calculation without resorting to a specific form for the Green’s function and conjecture on what is to be expected. A more accurate computation can be performed using this formal structure once a better approximation becomes available.
The general structure of the CTP effective action arising from the calculation of the traces in Equation (195) remains the same. But to write down explicit expressions for the nonlocal kernels, one requires the input of the explicit form of \(G_{ab}^\beta [h_{\mu \nu}^ \pm ]\) in the Schwarzschild metric, which is not available in closed form. We can make some general observations about the terms in there. The first line containing L does not have an explicit Fourier representation as given in the farfield case; neither will \(T_{(\beta)}^{\mu \nu}\) in the second line representing the zerothorder contribution to 〈T_{ μν }〉 have a perfect fluid form. The third and fourth terms containing the remaining quadratic component of the real part of the effective action will not have any simple or even complicated analytic form. The symmetry properties of the kernels H^{ μν,αβ }(x, x′) and D^{ μν,αβ }(x, x′) remain intact, i.e., they are respectively even and odd in x, x′. The last term in the CTP effective action gives the imaginary part of the effective action and the kernel N(x,x′) is symmetric.
8.2.5 EinsteinLangevin equation
In this section we show how a semiclassical EinsteinLangevin equation can be derived from the previous thermal CTP effective action. This equation depicts the stochastic evolution of the perturbations of the black hole under the influence of the fluctuations of the thermal scalar field.
As we have seen before and here, the EinsteinLangevin equation is a dynamical equation governing the dissipative evolution of the gravitational field under the influence of the fluctuations of the quantum field, which, in the case of black holes, takes the form of thermal radiance. From its form we can see that, even for the quasistatic case under study, the backreaction of Hawking radiation on the blackhole spacetime has an innate dynamical nature.
For the farfield case, making use of the explicit forms available for the noise and dissipation kernels, Campos and Hu [69, 70] formally prove the existence of a fluctuationdissipation relation at all temperatures between the quantum fluctuations of the thermal radiance and the dissipation of the gravitational field. They also show the formal equivalence of this method with linearresponse theory for lowest order perturbations of a nearequilibrium system and how the response functions, such as the contribution of the quantum scalar field to the thermal graviton polarization tensor, can be derived. An important quantity not usually obtained in linearresponse theory, but of equal importance, manifest in the CTP stochastic approach is the noise term arising from the quantum and statistical fluctuations in the thermal field. The example given in this section shows that the backreaction is intrinsically a dynamic process described (at this level of sophistication) by the EinsteinLangevin equation. By comparison, traditional linear response theory calculations cannot capture the dynamics as fully and thus cannot provide a complete description of the backreaction problem.
8.2.6 Comments
As remarked earlier, except for the nearflat case, an analytic form of the Green’s function is not available. Even the Page approximation [287], which gives unexpectedly good results for the stressenergy tensor, has been shown to fail in the fluctuations of the energy density [305]. Thus, using such an approximation for the noise kernel will give unreliable results for the EinsteinLangevin equation. If we confine ourselves to Page’s approximation and derive the equation of motion without the stochastic term, we expect to recover York’s semiclassical Einstein’s equation, if one retains only the zerothorder contribution, i.e, the first two terms in the expression for the CTP effective action in Equation (200). Thus, this offers a new route to arrive at York’s semiclassical Einstein’s equations. (Not only is it a derivation of York’s result from a different point of view, but it also shows how his result arises as an appropriate limit of a more complete framework, i.e, it arises when one averages over the noise.) Another point worth noting is that a nonlocal dissipation term arises from the fourth term in Equation (200) in the CTP effective action, which is absent in York’s treatment. This difference exists primarily due to the difference in the way backreaction is treated, at the level of iterative approximations on the equation of motion as in York, versus the treatment at the effectiveaction level, as in the influencefunctional approach. In York’s treatment, the Einstein tensor is computed to first order in perturbation theory, while 〈T_{ μν }〉 on the righthand side of the semiclassical Einstein equation is replaced by the zerothorder term. In the influencefunctional treatment the full effective action is computed to second order in perturbation and hence includes the higherorder nonlocal terms.
The other important conceptual point that comes to light from this new approach is that related to the fluctuationdissipation relation. In the quantum Brownian motion analog (e.g., [45, 133, 196, 197] and references therein), the dissipation of the energy of the Brownian particle as it approaches equilibrium and the fluctuations at equilibrium are connected by the FluctuationDissipation relation. Here the backreaction of quantum fields on black holes also consists of two forms — dissipation and fluctuation or noise — corresponding to the real and imaginary parts of the influence functional as embodied in the dissipation and noise kernels. A fluctuationdissipation relation has been shown to exist for the nearflat case by Campos and Hu [69, 70] and is expected to exist between the noise and dissipation kernels for the general case, as it is a categorical relation [45, 133, 183, 196, 197]. Martin and Verdaguer have also proved the existence of a fluctuationdissipation relation when the semiclassical background is a stationary spacetime and the quantum field is in thermal equilibrium. Their result was then extended to a conformal field in a conformallystationary background [257]. As discussed earlier, the existence of a fluctuationdissipation relation for the blackhole case has previously been suggested by some authors previously [76, 268, 324, 325]. This relation and the relevant physical quantities contained therein, such as the blackhole susceptibility function, which characterizes the statistical mechanical and dynamical responses of a black hole interacting with its quantum field environment, will allow us to study the nonequilibrium thermodynamic properties of the black hole and through it, perhaps, the microscopic structure of spacetime.
There are limitations of a technical nature in the quasistatic case studied, as mentioned above, i.e., there is no reliable approximation to the Schwarzschild thermal Green’s function to explicitly compute the noise and dissipation kernels. Another technical limitation of this example is the following: although we have allowed for backreaction effects to modify the initial state in the sense that the temperature of the HartleHawking state gets affected by the backreaction, our analysis is essentially confined to a HartleHawking thermal state of the field. It does not directly extend to a more general class of states, for example to the case in which the initial state of the field is in the Unruh vacuum. To study the dynamics of a radiating black hole under the influence of a quantum field and its fluctuations a different model and approach are needed, which we now discuss.
8.3 Metric fluctuations of an evaporating black hole
At the semiclassical gravity level of description, blackhole evaporation results from the backreaction of particle production in Hawking effect. This is believed to be valid at least before the Planckian scale is reached [18, 260]. However, as is explained above, semiclassical gravity [34, 110, 362] is a meanfield description that neglects the fluctuations of spacetime geometry. A number of studies have suggested the existence of large fluctuations near blackhole horizons [80, 255, 335, 336] (and even instabilities [264]) with characteristic time scales much shorter than the black hole evaporation time. For example, Casher et al. [80] and Sorkin [336, 338] have concentrated on the issue of fluctuations of the horizon induced by a fluctuating metric. Casher et al. [80] consider the fluctuations of the horizon induced by the “atmosphere” of high angularmomentum particles near the horizon, while Sorkin [336, 338] calculates fluctuations of the shape of the horizon induced by the quantumfield fluctuations under a Newtonian approximation. A relativistic generalization of this vein is given by Marolf [255]. Both groups of authors came to the conclusion that horizon fluctuations become large at scales much larger than the Planck scale (note Ford and Svaiter [114] later presented results contrary to this claim). Though these works do deal with backreaction, the fluctuations considered do not arise as an explicit stochastic noise term as in stochastic gravity. Either states, which are singular on the horizon (such as the Boulware vacuum for Schwarzschild spacetime), were explicitly considered or fluctuations were computed with respect to those states and found to be large near the horizon. Whether these huge fluctuations are of a generic nature or an artifact from the consideration of states singular on the horizon is an issue that deserves further investigation. By contrast, the fluctuations for states regular on the horizon were estimated in [377] and found to be small even when integrated over a time on the order of the evaporation time. These apparentlycontradictory claims and the fact that most claims on blackhole horizon fluctuations were based on qualitative arguments and/or semiquantitative estimates indicate that a more quantitative and selfconsistent description is needed. This is what stochasticgravity theory can provide.
Such a program of research has been pursued rigorously by Hu and Roura (HR) [200, 202] In contrast to the claims made before, they find that even for states regular on the horizon the accumulated fluctuations become significant by the time the blackhole mass has changed substantially, but well before reaching the Planckian regime. This result is different from those obtained in prior studies, but in agreement with earlier work by Bekenstein [24]. The apparent difference from the conclusions drawn in the earlier work of Hu, Raval and Sinha [199], which was also based on stochastic gravity, will be explained later. We begin with the evolution of the mean geometry.
8.3.1 Evolution of the mean geometry of an evaporating black hole
Solving Equations (215)–(217) is not easy. However, one can introduce a useful adiabatic approximation in the regime, in which the mass of the black hole is much larger than the Planck mass, which is, in any case, a necessary condition for the semiclassical treatment to be valid. What this entails is that when M ≫ 1 (remember that we are using Planckian units) for each value of v one can simply substitute 〈T_{ μν }〉 by its “parametric value” — by this we mean the expectation value of the stressenergy tensor of the quantum field in a Schwarzschild black hole with a mass corresponding to M(v) evaluated at that value of v. This is in contrast to its dynamical value, which should be determined by solving selfconsistently the semiclassical Einstein equation for the spacetime metric and the equations of motion for the quantum matter fields. This kind of approximation introduces errors of higher order in L_{ H } ≡ B/M^{2}(B is a dimensionless parameter that depends on the number of massless fields and their spins and accounts for their corresponding greybody factors; it has been estimated to be of order 10^{−4} [285]), which are very small for black holes well above Planckian scales. These errors are due to the fact that M(v) is not constant and that, even for a constant M(v), the resulting static geometry is not exactly Schwarzschild because the vacuum polarization of the quantum fields gives rise to a nonvanishing \({\langle {{\hat T}_{ab}}[g]\rangle _{ren}}\) [381].
The expectation value of the stress tensor for a Schwarzschild spacetime has been found to correspond to a thermal flux of radiation (with \(\langle T_\upsilon ^r\rangle = {L_H}/(4\pi {r^2})\)) for large radii and of order L_{ H } near the horizon [8, 75, 177, 178, 287]. This shows the consistency of the adiabatic approximation for L_{ H } ≪ 1: the righthand side of Equations (215)–(217) contains terms of order L_{ H } and higher, so that the derivatives of m(v,r) and ψ(v,r) are indeed small. We note that the natural quantum state for a black hole formed by gravitational collapse is the Unruh vacuum, which corresponds to the absence of incoming radiation far from the horizon. The expectation value of the stresstensor operator for that state is finite on the future horizon of Schwarzschild, which is the relevant one when identifying a region of the Schwarzschild geometry with the spacetime outside the collapsing matter for a black hole formed by gravitational collapse.
8.3.2 Sphericallysymmetric induced fluctuations
As explained earlier, the symmetrized twopoint function consists of two contributions: intrinsic and induced fluctuations. The intrinsic fluctuations are a consequence of the quantum width of the initial state of the metric perturbations; they are obtained in stochastic gravity by averaging over the initial conditions for the solutions of the homogeneous part of Equation (221), distributed according to the reduced Wigner function associated with the initial quantum state of the metric perturbations. On the other hand, the induced fluctuations are due to the quantum fluctuations of the matter fields interacting with the metric perturbations; they are obtained by solving the EinsteinLangevin equation using a retarded propagator with vanishing initial conditions.
In this section we study the sphericallysymmetric sector, i.e., the monopole contribution, which corresponds to l = 0, in a multipole expansion in terms of spherical harmonics Y_{ lm }(θ, ϕ), of metric fluctuations for an evaporating black hole. Restricting one’s attention to the sphericallysymmetric sector of metric fluctuations necessarily implies a partial description of the fluctuations because, contrary to the case for semiclassicalgravity solutions, even if one starts with sphericallysymmetric initial conditions, the stresstensor fluctuations will induce fluctuations involving higher multipoles. Thus, the multipole structure of the fluctuations is far richer than that of sphericallysymmetric semiclassicalgravity solutions, but this also means that obtaining a complete solution (including all multipoles) for fluctuations, rather than the mean value, is much more difficult. For sphericallysymmetric fluctuations only induced fluctuations are possible. The fact that intrinsic fluctuations cannot exist can be clearly seen if one neglects vacuumpolarization effects, since Birkhoff’s theorem forbids the existence of sphericallysymmetric free metric perturbations in the exterior vacuum region of a sphericallysymmetric black hole that keep the ADM mass constant. (This fact rings an alarm in the approach taken in [383] to the blackhole fluctuation problem. The degrees of freedom corresponding to sphericallysymmetric perturbations are constrained by the Hamiltonian and momentum constraints both at the classical and quantum level. Therefore, they will not exhibit quantum fluctuations unless they are coupled to a quantum matter field.) Even when vacuumpolarization effects are included, sphericallysymmetric perturbations, characterized by m(v, r) and ψ(v, r), are not independent degrees of freedom. This follows from Equations (215)–(217), which can be regarded as constraint equations.
Considering only sphericalsymmetry fluctuations is a simplification but it should be emphasized that it gives more accurate results than twodimensional dilationgravity models resulting from simple dimensional reduction [249, 341, 349]. This is because we project the solutions of the EinsteinLangevin equation just at the end, rather than considering only the contribution of the swave modes to the classical action for both the metric and the matter fields from the very beginning. Hence, an infinite number of modes for the matter fields with l ≠ 0 contribute to the l = 0 projection of the noise kernel, whereas only the swave modes for each matter field would contribute to the noise kernel if dimensional reduction had been imposed right from the start, as done in [289, 290, 291] as well as in studies of twodimensional dilationgravity models.
A more serious issue raised by HR is that in most previous investigations [24, 377] of the problem of metric fluctuations driven by quantum matter field fluctuations of states regular on the horizon (as far as the expectation value of the stress tensor is concerned) most authors assumed the existence of correlations between the outgoing energy flux far from the horizon and a negative energy flux crossing the horizon. (See, however, [290, 291], in which those correlators were shown to vanish in an effectively twodimensional model.) In semiclassical gravity, using energy conservation arguments, such correlations have been confirmed for the expectation value of the energy fluxes, provided that the mass of the black hole is much larger than the Planck mass. However, a more careful analysis by HR shows that no such simple connection exists for energy flux fluctuations. It also reveals that the fluctuations on the horizon are in fact divergent. This requires that one modify the classical picture of the event horizon from a sharply defined threedimensional hypersurface to that possessing a finite width, i.e., a fluctuating geometry. One needs to find an appropriate way of probing the metric fluctuations near the horizon and extracting physically meaningful information. It also testifies to the necessity of a complete reexamination of all cases afresh and that an evaluation of the noise kernel near the horizon seems unavoidable for the consideration of fluctuations and backreaction issues.
Having registered this cautionary note, Hu and Roura [202] first make the assumption that a relation between the fluctuations of the fluxes exists, so as to be able to compare with earlier work. They then show that this relation does not hold and discuss the essential elements required in understanding not only the mathematical theory but also the operational meaning of metric fluctuations.
8.3.2.1 Case 1: Assuming that there is a relation between fluctuations
The result of HR for the growth of the fluctuations in size of the blackhole horizon agrees with the result obtained by Bekenstein in [24] and implies that, for a sufficiently massive black hole (a few solar masses or a supermassive black hole), the fluctuations become important before the Planckian regime is reached.
This growth of the fluctuations, which was found by Bekenstein and confirmed here via the EinsteinLangevin equation, seems to be in conflict with the estimate given by Wu and Ford in [377]. According to their estimate, the accumulated mass fluctuations over a period on the order of the black hole evaporation time (\(\Delta t \sim M_0^3\)) would be on the order of the Planck mass. The discrepancy is due to the fact that the first term on the righthand side of Equation (224), which corresponds to the perturbed expectation value \({\langle \hat T_{ab}^{(1)}[g + h]\rangle _{ren}}\) in Equation (221), was not taken into account in [377]. The larger growth obtained here is a consequence of the secular effect of that term, which builds up in time (slowly at first, during most of the evaporation time, and becoming more significant at late times when the mass has changed substantially) and reflects the unstable nature of the background solution for an evaporating black hole.
As for the relation between HR’s results reported here and earlier results of Hu, Raval and Sinha in [199], there should not be any discrepancy, since both adopted the stochastic gravity framework and performed their analysis based on the EinsteinLangevin equation. The claim in [199] was based on a qualitative argument that focused on the dynamics of the stochastic source alone. If one adds in the consideration that the perturbations around the mean are unstable for an evaporating black hole, their results agree.
All this can be qualitatively understood as follows. Consider an evaporating black hole with initial mass M_{0} and suppose that the initial mass is perturbed by an amount δM_{0} = 1. The mean evolution for the perturbed black hole (without taking into account any fluctuations) leads to a mass perturbation that grows like δM = (M_{0}/M)^{2} δM_{0} = (M_{0}/M)^{2}, so that it becomes comparable to the unperturbed mass M when \(M \sim M_0^{2/3}\), which coincides with the result obtained above. Such a coincidence has a simple explanation: the fluctuations of the Hawking flux, which are on the order of the Planck mass, slowly accumulated during most of the evaporating time, as found by Wu and Ford, and gave a dispersion of that order for the mass distribution at the time when the instability of the small perturbations around the background solution start to become significant.
8.3.2.2 Case 2: When no such relation exists and the consequences
For conformal fields in twodimensional spacetimes, HR shows that the correlations between the energy flux crossing the horizon and the flux far from it vanish. The correlation function for the outgoing and ingoing nullenergy fluxes in an effectively twodimensional model is explicitly computed in [290, 291] and is also found to vanish. On the other hand, in four dimensions the correlation function does not vanish in general and correlations between outgoing and ingoing fluxes do exist near the horizon (at least partially).
For blackhole masses much larger than the Planck mass one can use the adiabatic approximation for the background mean evolution. Therefore, to lowest order in L_{ H } one can compute the fluctuations of the stress tensor in Schwarzschild spacetime. In Schwarzschild, the amplitude of the fluctuations of \({r^2}\langle T_\upsilon ^r\rangle\) far from the horizon is of order 1/M^{2} (= M^{2}/M^{4}) when smearing over a correlation time of order M, which one can estimate for a hot thermal plasma in flat space [69, 70] (see also [377] for a computation of the fluctuations of \({r^2}\langle T_\upsilon ^r\rangle\) far from the horizon). The amplitude of the fluctuations of \({r^2}\langle T_\upsilon ^r\rangle\) is, thus, of the same order as its expectation value. However, their derivatives with respect to v are rather different: since the characteristic variation times for the expectation value and the fluctuations are M^{3} and M, respectively, \(\partial ({r^2}\langle T_\upsilon ^r\rangle)/\partial \upsilon\) is of order 1/M^{5}, whereas \(\partial ({r^2}\xi _\upsilon ^r)/\partial \upsilon\) is of order 1/M^{3}. This implies an additional contribution of order L_{ H } due to the second term in Equation (218) if one radially integrates the same equation applied to stresstensor fluctuations (the stochastic source in the EinsteinLangevin equation). Hence, in contrast to the case of the mean value, the contribution from the second term in Equation (218) cannot be neglected when radially integrating, since it is of the same order as the contributions from the first term, and one can no longer obtain a simple relation between the outgoing energy flux far from the horizon and the energy flux crossing the horizon.
What then? Without this convenience (which almost all earlier researchers have taken for granted), to get a more precise depiction we need to compute the noise kernel near the horizon. However, as shown by Hu and Phillips earlier [305] when they examine the coincidence limit of the noise kernel and confirmed by the careful analysis of HR using smearing functions [202], the noise kernel smeared over the horizon is divergent and so are the induced metric fluctuations. Hence, one cannot study the fluctuations of the horizon as a threedimensional hypersurface for each realization of the stochastic source because the amplitude of the fluctuations is infinite, even when restricting one’s attention to the l = 0 sector. Instead, one should regard the horizon as possessing a finite effective width due to quantum fluctuations. In order to characterize its width one must find a sensible way of probing the metric fluctuations near the horizon and extracting physicallymeaningful information, such as their effect on the Hawking radiation emitted by the black hole. How to probe metric fluctuations is an issue at the root base, which needs be dealt with in all discussions of metric fluctuations.
8.3.3 Summary and prospects
The work of HR [202], based on the stochasticgravity program, found that the sphericallysymmetric fluctuations of the horizon size of an evaporating black hole become important at late times, and even comparable to its mean value when \(M \sim M_0^{2/3}\), where M_{0} is the mass of the black hole at some initial time when the fluctuations of the horizon radius are much smaller than the Planck length (remember that for large blackhole masses this can still correspond to physical distances much larger than the Planck length, as explained in Section 4). This is consistent with the result previously obtained by Bekenstein in [24].
It is important to realize that, for a sufficiently massive black hole, the fluctuations become significant well before the Planckian regime is reached. More specifically, for a solarmass black hole, they become comparable to the mean value when the blackhole radius is on the order of 10 nm, whereas for a supermassive black hole with M ∼ 10^{7} M_{⊙}, that happens when the radius reaches a size on the order of 1 mm. One expects that in those circumstances the lowenergy effectivefield theory approach of stochastic gravity should provide a reliable description.
Finally, we remark on the relation of this finding to earlier wellknown results. Does the existence of significant deviations for the mean evolution mentioned above invalidate the earlier results by Bardeen and Massar based on semiclassical gravity in [18, 260]? First, those deviations start to become significant only after a period on the order of the evaporation time, when the mass of the black hole has decreased substantially. Second, since fluctuations were not considered in those references, a direct comparison cannot be established. Nevertheless, we can compare the average of the fluctuating ensemble. Doing so exhibits an evolution that deviates significantly when the fluctuations become important. However, if one considers a single member of the ensemble at that time, its evolution will be accurately described by the corresponding semiclassical gravity solution until the fluctuations around that particular solution become important again, after a period on the order of the evaporation time associated with the new initial value of the mass at that time.
8.4 Other work on metric fluctuations but without backreaction
In closing we mention some work on metric fluctuations where no backreaction is considered. Barrabes et al. [20, 21] have considered the propagation of null rays and massless fields in a black hole fluctuating geometry and have shown that the stochastic nature of the metric leads to a modified dispersion relation and helps to confront the transPlanckian frequency problem. However, in this case the stochastic noise is put in by hand and does not naturally arise from coarsegraining as in a quantum open systems approach, in terms of which stochastic gravity can be interpreted. It also does not take backreaction into account. It will be interesting to explore how a stochastic black hole metric, arising as a solution to the EinsteinLangevin equation, hence fully incorporating backreaction, would affect the transPlanckian problem.
As mentioned earlier, Ford and his collaborators [114, 115, 377] have also explored the issue of metric fluctuations in detail and in particular have studied the fluctuations of the black hole horizon induced by metric fluctuations. However, the fluctuations they considered are in the context of a fixed background and do not relate to the backreaction.
Another work on metric fluctuations with no backreaction is that of Hu and Shiokawa [205], who study effects associated with electromagnetic wave propagation in a RobertsonWalker universe and Schwarzschild spacetime with a small amount of given metric stochasticity. They find that timeindependent randomness can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside the black hole and gives rise to event horizon fluctuations and fluctuations in the Hawking temperature. The stochasticity in the background metric in their work is assumed rather than derived (as induced by quantumfield fluctuations). But it is interesting to compare their results with those obtained in stochastic gravity with backreaction, as one can begin to get a sense of the different sources of stochasticity and their weights (see, e.g., [187] for a list of possible sources of stochasticity.)
In a subsequent paper, Shiokawa [328] shows that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing this as a quantumtransport problem, he expresses the conductance and its fluctuations in terms of a nonlinear sigma model in the closed timepath formalism and shows that the conductance fluctuations are universal, independent of the volume of the stochastic region and the amount of stochasticity. This result has significant importance in characterizing the mesoscopic behavior of spacetimes resting between the semiclassical and the quantum regimes.
9 Concluding Remarks
In the first part of this review on the fundamentals of the theory, we have given two routes to the establishment of stochastic gravity and derived a general (finite) expression for the noise kernel. In the second part, we gave three applications: the correlation functions of gravitons in a perturbed Minkowski metric, structure formation in stochasticgravity theory and the outline of a program for the study of blackhole fluctuations and backreaction. We have also discussed the problem of the validity of semiclassical gravity, a central issue, which stochastic gravity is in a unique position to address.
We have pointed out a number of ongoing research projects related to the topics discussed in this review, such as the equivalence of the correlation functions to the metric perturbations obtained using the EinsteinLangevin equations and the quantumcorrelation functions that follow from a pure quantumfieldtheory calculation up to leading order in the large N limit, the calculation of the spectrum of metric fluctuations in inflationary models driven by the trace anomaly due to conformallycoupled fields, the related problem of runaway solutions in backreaction equations and the issue of the coincidence limit in the noise kernel for blackhole fluctuations.
Theoretically, stochastic gravity is at the front line of the ‘bottomup’ approach to quantum gravity [185, 187, 188, 190]. Its pathway or angle starts from the welldefined and wellunderstood theory of semiclassical gravity. Structurally, as can be seen from the issues discussed and the applications given, stochastic gravity has a very rich constituency because it is based on quantum field theory and nonequilibrium statistical mechanics in a curvedspacetime context. The open systems concepts and the closedtimepath/influencefunctional methods constitute an extended framework suitable for treating the backreaction and fluctuation problems of dynamical spacetimes interacting with quantum fields. We have seen applications to cosmologicalstructure formation and blackhole backreaction from particle creation. A more complete understanding of the backreaction of Hawking radiation in a fullydynamical blackhole situation will enable one to address fundamental issues such as the blackhole end state and informationloss puzzles. The main reason why this program has not progressed as swiftly as desired is due more to technical rather than programatic difficulties (such as finding reasonable analytic approximations for the Green’s function or the numerical evaluation of modesums near the blackhole horizon). Finally, the multiplex structure of this theory could be used to explore new lines of inquiry and launch new programs of research, such as nonequilibrium blackhole thermodynamics and the microscopic structures of spacetime.
Notes
Acknowledgements
The materials presented here originated from the research of BLH with Antonio Campos, Nicholas Phillips, Alpan Raval, Albert Roura and Sukanya Sinha, and of EV with Rosario Martin and Albert Roura. We thank them as well as Daniel Arteaga, Andrew Matacz, Tom Shiokawa, and Yuhong Zhang for fruitful collaboration and their cordial friendship since their Ph.D. days. We enjoy lively discussions with our friends and colleagues Esteban Calzetta, Diego Mazzitelli and Juan Pablo Paz, whose work in the early years contributed towards the establishment of this field. We acknowledge useful discussions with Paul Anderson, Larry Ford, Ted Jacobson, Emil Mottola, Renaud Parentani, Raphael Sorkin and Richard Woodard. This work is supported in part by NSF grants PHY0601550 and PHY0551164, the MEC Research projects FPA200404582C02 and FPA200766665C02 and by DURSI 2005SGR00082.
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