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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 [111Jump To The Next Citation Point] 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 [237Jump To The Next Citation Point] 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 [198Jump To The Next Citation Point, 303Jump To The Next Citation Point] 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 [114Jump To The Next Citation Point, 384, 385] and passive [111Jump To The Next Citation Point, 112, 115Jump To The Next Citation Point, 237Jump To The Next Citation Point], 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 [169Jump To The Next Citation Point, 170Jump To The Next Citation Point], 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 [223Jump To The Next Citation Point]. Later, Simon [329Jump To The Next Citation Point, 330Jump To The Next Citation Point] 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, 330Jump To The Next Citation Point]. Further discussion was provided by Flanagan and Wald [110Jump To The Next Citation Point], who advocated the use of an order-reduction prescription first introduced by Parker and Simon [295Jump To The Next Citation Point]. More recently Anderson, Molina-París and Mottola have taken up the issue of the validity of semiclassical gravity [10Jump To The Next Citation Point, 11Jump To The Next Citation Point] 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 higher-order 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 [10Jump To The Next Citation Point, 11Jump To The Next Citation Point] 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 [203Jump To The Next Citation Point] (see also [204Jump To The Next Citation Point]), 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 ˆh (x) ab 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 (17View Equation). 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 (15View Equation)), the effect of induced-metric fluctuations is accounted for only in stochastic gravity (the full inhomogeneous Equation (15View Equation)). 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

∫ S = 1- d4x(∂ah ∂ah + h ∂ah∂ah + ...) κ ∫ ∫ 4 ( a 2 2) 4 a − d x ∂aφ ∂ φ + m φ + d x(h ∂aφ∂ φ + ...), (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 3 O (h ), 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 1View Image). 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 (20View Equation). 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 3 O (h ) term in the action (20View Equation). 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.

View Image

Figure 1: Series of Feynman diagrams expanded in powers of κ for the dressed propagator of the graviton interacting with a single matter field according to the toy model described by the action (20View Equation).

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

∫ N-- 4 a a S = ¯κ d x (∂ah∂ h + h ∂ah ∂ h + ...) N ∫ N ∫ ∑ 4 ( a 2 2) ∑ 4 a − d x ∂aφj∂ φj + m φ + d x(h ∂aφj∂ φj + ...) . (21 ) j j
Now an expansion in powers of 1∕N of the dressed graviton propagator is given by the following series of Feynman diagrams (Figure 2View Image). The first diagram is the free graviton propagator, which is now of order O (¯κ∕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 (κ¯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 2 O (¯κ ∕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 (¯κ3∕N ), and so on. This means that to order 1∕N the dressed graviton propagator contains all the perturbative series in powers of ¯κ of the matter loops.
View Image

Figure 2: Series of Feynman diagrams expanded in powers of 1∕N for the dressed propagator of the graviton interacting with N matter fields oaccording to the toy model described by the action (21View Equation).

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 (¯κ∕N )4 and the vertices as (N∕ ¯κ)2, thus this diagram is of O (¯κ2∕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 (¯κ3∕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 [152Jump To The Next Citation Point]. 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 [259Jump To The Next Citation Point]. 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 [348Jump To The Next Citation Point]. This has been proven in detail in [203Jump To The Next Citation Point]; see also [204Jump To The Next Citation Point], where the case of a general background is also briefly discussed.


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