Stochastic Gravity: Theory and Applications

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 c-number stress-energy tensor that one may construct [359, 360] with the stress-energy 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 stress-energy 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 self-consistent 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 X₁ and another configuration of the same mass with the center of mass at X₂. 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 (X₁ + X₂)/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 X₁ and half of the time to the field centered at X₂. The two predictions are clearly different; note that the fluctuation in the position of the center of masses is on the order of (X₁ − X₂)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) Einstein-Langevin equations were derived [52, 58, 73, 74, 192, 206, 248]. The key technical factor to most of these results was the use of the influence-functional method of Feynman and Vernon [108], when only the coarse-grained 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 Einstein-Langevin 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 consistent-histories 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 coarse-graining of the environmental degrees of freedom and a further coarse-graining 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 Born-Oppenheimer approximation in quantum cosmology [297, 298].

An alternative axiomatic approach to the Einstein-Langevin equation, without invoking the open-system paradigm, was later suggested based on the formulation of a self-consistent dynamical equation for a perturbative extension of semiclassical gravity able to account for the lowest-order stress-energy 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 c-number. The important new element in the derivation of the Einstein-Langevin equation, and of stochastic-gravity theory, is the physical observable that measures the stress-energy fluctuations, namely, the expectation value of the symmetrized bitensor constructed with the stress-energy tensor operator: the noise kernel. It is interesting to note that the Einstein-Langevin equation can also be understood as a useful intermediary tool to compute symmetrized two-point correlations of the quantum metric perturbations on the semiclassical background, independent of a suitable classicalization mechanism [317].

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 c-number 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 stress-energy operator is evaluated in this state.

For a free quantum field this theory is robust in the sense that it is self-consistent 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 stress-energy tensor and for a given quantum state the only physically observable c-number stress-energy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantum-gravity 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 order-of-magnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Planck-size 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 stress-energy 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 zeta-function 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 point-separation 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 linear-response 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 two-point 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 stress-energy 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 stress-energy 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 c-number 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 point-separation method is given in [304, 305], see also Section 5.

Note that we refer to the Einstein-Langevin equation as a first-order extension to the semi-classical Einstein equation of semiclassical gravity and the lowest-level representation of stochastic gravity. However, stochastic gravity has a much broader meaning; it refers to the range of theories based on second and higher-order correlation functions. Noise can be defined in effectively-open systems (e.g., correlation noise [61] in the Schwinger-Dyson 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 Einstein-Langevin 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 left-hand 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 left-hand 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 Langevin-like equations are obtained to describe the quantum to classical transition in open quantum systems, when quantum decoherence takes place by coarse-graining of the environment as well as by suitable coarse-graining 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 well-understood 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 full-averaged backreaction of the matter fields is not included self-consistently 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 higher-order 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 order-reduction prescription first introduced by Parker and Simon [295]. More recently Anderson, Molina-París 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 higher-order 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 quantum-gravity effects in three-dimensional quantum-gravity 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 stress-tensor 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 two-point quantum-correlation functions of the metric perturbations in the large N expansion are equivalent to the two-point stochastic metric-correlation 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 quantum-correlation 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 n-point correlation functions.

It is important to emphasize that the above validity criterion incorporates in a unified and self-consistent way the two main ingredients of the criteria exposed above. Namely, the criteria based on the quantum stress-tensor 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 intrinsic-metric 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 induced-metric 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.

The large N expansion has been successfully used in quantum chromodynamics to compute some nonperturbative results. This expansion re-sums and rearranges Feynman perturbative series, including self-energies. For gravity interacting with N matter fields, it shows that graviton loops are of higher order than matter loops. To illustrate the large N expansion, let us first consider the following toy model of gravity, which we will simplify as a scalar field h, interacting with a single scalar field Φ described by the action

$$\begin{array}{*{20}c} {S = {1 \over \kappa}\int {{d^4}} x({\partial _a}h{\partial ^a}h + h{\partial _a}h{\partial ^a}h + \ldots)\quad \quad \quad \quad \quad \quad} \\ {- \int {{d^4}} x({\partial _a}\phi {\partial ^a}\phi + {m^2}\phi) + \int {{d^4}} x(h{\partial _a}\phi {\partial ^a}\phi + \ldots)}, \\ \end{array}$$

(20)

where κ = 8πG and we have assumed that the interaction is linear in the (dimensionless) scalar gravitational field h and quadratic in the matter field Φ to simulate in a simplified way the coupling of the metric with the stress tensor of the matter fields. We have also included a self-coupling graviton term of O(h³), which also appears in perturbative gravity beyond the linear approximation.

We may now compute the graviton dressed propagator perturbatively as the following series of Feynman diagrams (Figure 1). The first diagram is just the free graviton propagator, which is of O(κ), as one can see from the kinetic term for the graviton in Equation (20). The next diagram is one loop of matter with two external legs, which are the graviton propagators. This diagram has two vertices with one graviton propagator and two matter field propagators. Since the vertices and the matter propagators contribute with 1 and each graviton propagator contributes with a κ this diagram is of order O(κ²). The next diagram contains two loops of matter, three gravitons and consequently it is of order O(κ³). There will also be terms with one graviton loop and two graviton propagators as external legs and three graviton propagators at the two vertices due to the O(h³) term in the action (20). Since there are four graviton propagators, which carry a κ⁴, but two vertices, which have κ⁻², this diagram is of order O(κ²), like the term with one matter loop. Thus, in this perturbative expansion, a graviton loop and a matter loop both contribute at the same order to the dressed graviton propagator.

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Let us now consider the large N expansion. We assume that the gravitational field is coupled with a large number of identical fields Φ _j , j = 1,…, N, which couple only with h. Next we rescale the gravitational coupling in such a way that \(\bar \kappa = \kappa N\) is finite even when N goes to infinity. The action of this system is:

$$\begin{array}{*{20}c} {S = {N \over {\bar \kappa}}\int {{d^4}} x({\partial _a}h{\partial ^a}h + h{\partial _a}h{\partial ^a}h + \ldots)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- \sum\limits_j^N {\int {{d^4}} x} ({\partial _a}{\phi _j}{\partial ^a}{\phi _j} + {m^2}{\phi ^2}) + \sum\limits_j^N {\int {{d^4}} x(h{\partial _a}{\phi _j}{\partial ^a}{\phi _j} + \ldots).}} \\ \end{array}$$

(21)

Now an expansion in powers of 1/N of the dressed graviton propagator is given by the following series of Feynman diagrams (Figure 2). The first diagram is the free graviton propagator, which is now of order \(O(\bar \kappa/N)\). The following diagrams are N identical Feynman diagrams with one loop of matter and two graviton propagators as external legs, each diagram, due to the two graviton propagators, is of order \(O({{\bar \kappa}^2}/{N^2})\), but since there are N of them the sum can be represented by a single diagram with a loop of matter of weight N, and therefore this diagram is of order \(O({{\bar \kappa}^2}/N)\). This means that it is of the same order as the first diagram in an expansion in 1/N. Then there are the diagrams with two loops of matter and three graviton propagators, as before we can assign a weight of N to each loop and taking into account the three graviton propagators this diagram is of order \(O({{\bar \kappa}^3}/N)\), and so on. This means that to order 1/N the dressed graviton propagator contains all the perturbative series in powers of \({\bar \kappa}\) of the matter loops.

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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 next-to-leading order 1/N. Now the graviton propagator includes all matter-loop 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.

4.1 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 higher-order terms as well as self-interaction terms for the scalar field. The second order terms are necessary to renormalize one-loop ultraviolet divergences of the scalar field stress-energy 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 n-dimensional 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 Gauss-Bonnet 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 Schwinger-DeWitt analysis of the ultraviolet divergences in the matter stress-energy tensor using dimensional regularization. One can then write the Feynman-Vernon effective action S_eff [g^±] in Equation (24) in a form suitable for dimensional regularization. Since both S_m and S_g contain second-order 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 Einstein-Langevin equation

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 stress-energy tensor is built from the product of a pair of field operators evaluated at a single point, it is not well-defined. 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 well-defined. If one is interested in the low-energy behavior captured by the point-defined 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 well-defined, 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 stress-energy tensor two-point 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 point-separation context.

5.2 Stress-energy bitensor operator and noise kernel

Even though we believe that the point-separated 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 bi-quantities. The key issue here is thus the distinction between point-defined (pt) and point-separated (bi) quantities.

6.1 Perturbations around Minkowski spacetime

The Minkowski metric η _ab in a manifold \({\mathcal M}\), which is topologically ℝ⁴, together with the usual Minkowski vacuum, denoted as |0⟩, is the simplest solution to the semiclassical Einstein equation (8), the so-called 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 (ℝ⁴,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 stress-energy 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 stress-energy 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 four-momentum 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 stress-energy 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 two-point 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 Einstein-Langevin equation using the methods outlined in Sections 3 and 4.

6.2 The kernels in the Minkowski background

6.3 The Einstein-Langevin equation

It is interesting to consider the massless conformally-coupled 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 Einstein-Langevin equations for this particular case (and generalized to a spatially-flat Robertson-Walker 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 stress-energy tensor for a scalar field can be obtained by comparing Equation (118) with the Einstein-Langevin 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 minimally-coupled 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 Einstein-Langevin equations (118) for the components G^(1)μν of the linearized Einstein tensor. Then we use these solutions to compute the corresponding two-point correlation functions, which give a measure of the gravitational fluctuations predicted by the stochastic semi-classical theory of gravity in the present case. Since the linearized Einstein tensor is invariant under gauge transformations of the metric perturbations, these two-point correlation functions are also gauge invariant. Once we have computed the two-point correlation functions for the linearized Einstein tensor, we find the solutions for the metric perturbations and compute the associated two-point correlation functions. The procedure used to solve the Einstein-Langevin equation is similar to the one used by Horowitz [169] (see also [110]) to analyze the stability of Minkowski spacetime in semiclassical gravity.

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 two-point 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 right-hand side of Equation (17), which corresponds to the intrinsic metric fluctuations, and the second term, which corresponds to the induced metric fluctuations.

7.1 The model

7.2 Einstein-Langevin 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(−m²t/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 e-folds, i.e., HΔt ∼ 60 (Δt being the time between horizon exit and the end of inflation). But this is the case provided 3H²/m² > 60, since the decay factor ∼ exp[−(m²/3H²)HΔt] will not be too different from unity for those modes that left the horizon during the last sixty e-folds of inflation. This condition is indeed satisfied given that m ≪ H in most slow-roll inflationary models [233, 284] and in particular for the model considered here, in which m ∼ 10⁻⁶m _P .

We now comment on some differences with [53, 60, 262, 263], which used a self-interacting 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 long-wavelength 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 fine-tuning 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 Einstein-Langevin 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 non-zero expectation value, because the linear perturbations will give the leading contribution. However, the importance of considering corrections due to one-loop contributions from the inflaton-field 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 Einstein-Langevin 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 Einstein-Langevin 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 stress-tensor 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 Einstein-Langevin equation still give the correct symmetrized quantum two-point correlation functions. In [66] it was explained how a stochastic description based on a Langevin-type 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 Einstein-Langevin 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 Einstein-Langevin 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 anomaly-induced 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 Sitter-like regime and ends up in a matter-dominated-like 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 two-point quantum correlation function for scalar and tensorial metric perturbations in a spatially-closed de Sitter universe, making use of the anti-de Sitter/conformal field theory correspondence. They find that short-scale 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 spatially-closed de Sitter background was derived in [314], and in a spatially-flat arbitrary Friedmann-Robertson-Walker model the Einstein-Langevin equations describing the metric perturbations were first obtained in [73]. The computation of the corresponding two-point correlation functions for the metric perturbations is now in progress.

8.1 General issues of backreaction

Backreaction studies of quantum field-processes in cosmological spacetimes have progressed further than the corresponding black-hole problems, partly because of the relative technical simplicity associated with the higher symmetry of relevant cosmological-background geometries. (For a summary of the cosmological backreaction problem treated in stochastic-gravity theory, see [208].) How the problem is set up and approached, e.g., via effective action, in these previously-studied models can be carried over to black-hole problems. In fact, since the interior of a black hole can be described by a cosmological model (e.g., the Kantowski-Sachs 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.

8.2 Backreaction on black holes under quasi-static conditions

As an illustration of the application of stochastic-gravity theory we outline the essential steps in a black-hole backreaction calculation, focusing on a more manageable quasi-static class. We adopt the Hartle-Hawking picture [151] where the black hole is bathed eternally — actually in quasi-thermal 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 quasi-equilibrium 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 far-field case and the near-horizon case. Campos and Hu [69, 70] have treated a relativistic thermal plasma in a weak gravitational field. Since the far-field 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 anti-de 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 Einstein-Langevin equation one needs to begin with a self-consistent solution of the semiclassical Einstein equation for the thermal field and the perturbed background spacetime. For a black-hole background, a semiclassical gravity solution is provided by York [380, 381, 382]. For a Robertson-Walker 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 quasi-static black holes. For the near-horizon 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 Einstein-Langevin equation from it. As a by-product, from the fluctuation-dissipation relation, one can derive the vacuum-susceptibility function and the isothermal-compressibility function for black holes, two quantities of fundamental interest in characterizing the nonequilibrium thermodynamic properties of black holes.

8.3 Metric fluctuations of an evaporating black hole

At the semiclassical gravity level of description, black-hole 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 mean-field description that neglects the fluctuations of spacetime geometry. A number of studies have suggested the existence of large fluctuations near black-hole 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 angular-momentum particles near the horizon, while Sorkin [336, 338] calculates fluctuations of the shape of the horizon induced by the quantum-field 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 apparently-contradictory claims and the fact that most claims on black-hole horizon fluctuations were based on qualitative arguments and/or semi-quantitative estimates indicate that a more quantitative and self-consistent description is needed. This is what stochastic-gravity 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 black-hole 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.

8.4 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 trans-Planckian frequency problem. However, in this case the stochastic noise is put in by hand and does not naturally arise from coarse-graining 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 Einstein-Langevin equation, hence fully incorporating backreaction, would affect the trans-Planckian 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 Robertson-Walker universe and Schwarzschild spacetime with a small amount of given metric stochasticity. They find that time-independent 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 quantum-field 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 quantum-transport problem, he expresses the conductance and its fluctuations in terms of a nonlinear sigma model in the closed time-path 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.