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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 ˆ R ⟨Tab[η]⟩ = 0; this is equivalent to assuming that the cosmological constant is zero. Then the Minkowski metric ηab is a solution of the semiclassical Einstein equation (8View Equation). 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 hab(x ) over the Minkowski background, which are given by Equations (16View Equation) and (17View Equation). As we have emphasized before, these metric fluctuations separate in two parts: the first term on the right-hand side of Equation (17View Equation), which corresponds to the intrinsic metric fluctuations, and the second term, which corresponds to the induced metric fluctuations.

6.5.1 Intrinsic metric fluctuations

Let us first consider the intrinsic metric fluctuations,

⟨hab(x)hcd(y)⟩int = ⟨h0ab(x )h0cd(y)⟩s, (147 )
where 0 hab are the homogeneous solutions of the Einstein–Langevin equation (15View Equation), or equivalently the linearly-perturbed semiclassical equation, and where the statistical average is taken with respect to the Wigner distribution that describes the initial quantum state of the metric perturbations. Since these solutions are described by the linearized semiclassical equation around flat spacetime, we can make use of the results derived in [10Jump To The Next Citation Point, 11Jump To The Next Citation Point, 110Jump To The Next Citation Point, 169Jump To The Next Citation Point]. The solutions for the case of a massless scalar field were first discussed in [169] and an exhaustive description can be found in Appendix A of [110Jump To The Next Citation Point]. It is convenient to decompose the perturbation around Minkowski spacetime into scalar, vectorial and tensorial parts, as
( ) hab = ¯φ ηab + ∇ (a∇b) − ηab∇c ∇c ψ + 2∇ (avb) + hTTab , (148 )
where a v is a transverse vector and TT hab is a transverse and traceless symmetric tensor, i.e., a ∇av = 0, a TT ∇ hab = 0 and TT a (h )a = 0. A vector field a ζ characterizes the gauge freedom due to infinitesimal diffeomorphisms as hab → hab + ∇a ζb + ∇bζa. We may use this freedom to choose a gauge; a convenient election is the Lorentz or harmonic gauge defined as
( ) ∇a hab − 1-ηabhcc = 0. (149 ) 2
When this gauge is imposed we have the following conditions on the metric perturbations ∇ ∇avb = 0 a and ¯ ∇bφ = 0, which implies ¯ φ = const. A remaining gauge freedom compatible with the Lorentz gauge is still possible provided the vector field a ζ satisfies the condition a b ∇a ∇ ζ = 0. One can easily see [203Jump To The Next Citation Point] that the vectorial and scalar part ¯φ can be eliminated, as well as the contribution of the scalar part ψ, which corresponds to Fourier modes &tidle;ψ(p) with p2 = 0. Thus, we will assume that we impose the Lorentz gauge with additional gauge transformations, which leave only the tensorial component and the modes of the scalar component ψ with 2 p ⁄= 0 in Fourier space.

Using the metric decomposition (148View Equation) we may compute the linearized Einstein tensor G (a1b). It is found that the vectorial part of the metric perturbation gives no contribution to this tensor, and the scalar and tensorial components give rise, respectively, to scalar and tensorial components: (1)(S) G ab and (1)(T ) Gab. Thus, let us now write the Fourier transform of the homogeneous Einstein–Langevin equation (121View Equation), which is equivalent to the linearized semiclassical Einstein equation,

μν &tidle;(1)αβ F αβ(p) G (p) = 0. (150 )
Using the previous decomposition of the Einstein tensor this equation can be re-written in terms of its scalar and tensorial parts as
[ ] F1(p) + 3p2F2 (p ) &tidle;G (1μ)ν(S)(p) = 0, (151 ) F1 (p) &tidle;G(1)(T)(p) = 0. (152 ) μν

where F1(p) and F2(p) are given by Equations (123View Equation), and G&tidle;(μ1ν)(S) and G&tidle;(μ1)ν(T) denote, respectively, the Fourier-transformed scalar and tensorial parts of the linearized Einstein tensor. To simplify the problem and to illustrate, in particular, how the runaway solutions arise, we will consider the case of a massless and conformally coupled field (see [110Jump To The Next Citation Point] for the massless case with arbitrary coupling and [10Jump To The Next Citation Point, 259Jump To The Next Citation Point] for the general massive case). Thus substituting m = 0 and ξ = 1∕6 into the functions F1(p) and F2(p), and using Equation (117View Equation), the above equations become

(1 + 12 κ¯βp2) &tidle;G(1)(S)(p) = 0, (153 ) μν [ κp2 ( − (p0 + iε)2 + ⃗p2) ] lim 1 + ------ln ---------------- G&tidle;(μ1ν)(T)(p) = 0, (154 ) ε→0+ 960π2 μ2
where κ = 8πG. Let us consider these two equations separately.

For the scalar component when ¯ β = 0 the only solution is &tidle;(1)(S) G μν (p) = 0. When ¯ β > 0 the solutions for the scalar component exhibit an oscillatory behavior in spacetime coordinates, which corresponds to a massive scalar field with m2 = (12 κ|β¯|)− 1; for β¯ < 0 the solutions correspond to a tachyonic field with m2 = − (12κ|¯β|)−1. In spacetime coordinates they exhibit an exponential behavior in time, growing or decreasing, for wavelengths larger than 4π(3κ |β¯|)1∕2 and an oscillatory behavior for wavelengths smaller than 1∕2 4π(3κ|¯β|). On the other hand, the solution (1)(S) &tidle;G μν (p) = 0 is completely trivial since any scalar metric perturbation &tidle;hμν(p) giving rise to a vanishing linearized Einstein tensor can be eliminated by a gauge transformation.

For the tensorial component, when −1 1∕2 γ μ ≤ μcrit = lp (120π) e, where lp is the Planck length (2 lp ≡ κ∕8π), the first factor in Equation (154View Equation) vanishes for four complex values of p0 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 half-plane of the complex p0 plane and two poles in the lower half-plane. We will consider here the case in which μ < μ crit; a detailed description of the situation for μ ≥ μcrit can be found in Appendix A of [110Jump To The Next Citation Point]. 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 well-defined. They are commonly referred to as runaway solutions and for −1 μ ∼ lp they grow exponentially in time scales comparable to the Planck time.

Consequently, in addition to the solutions with G (a1)b (x) = 0, there are other solutions that in Fourier space take the form &tidle;(1) 2 2 G ab (p) ∝ δ(p − p0) for some particular values of p0, but all of them exhibit exponential instabilities with characteristic Planckian time scales. In order to deal with those unstable solutions, one possibility is to make use of the order-reduction prescription [295], which we will briefly summarize in Section 6.5.3. Note that the p2 terms in Equations (153View Equation) and (154View Equation) come from two spacetime derivatives of the Einstein tensor, moreover, the p2lnp2 term comes from the nonlocal term of the expectation value of the stress tensor. The order-reduction prescription amounts here to neglecting these higher derivative terms. Thus, neglecting the terms proportional to p2 in Equations (153View Equation) and (154View Equation), we are left with only the solutions, which satisfy (1) G&tidle;ab (p) = 0. The result for the metric perturbation in the gauge introduced above can be obtained by solving for the Einstein tensor, which in the Lorentz gauge of Equation (149View Equation) reads:

1 ( 1 ) G&tidle;(1a)b (p) =-p2 h&tidle;ab(p) − --ηab&tidle;hcc(p) . (155 ) 2 2
These solutions for &tidle;h (p) ab simply correspond to free linear gravitational waves propagating in Minkowski spacetime expressed in the transverse and traceless (TT) gauge. When substituting back into Equation (147View Equation) and averaging over the initial conditions we simply get the symmetrized quantum correlation function for free gravitons in the TT gauge for the state given by the Wigner distribution. As far as the intrinsic fluctuations are concerned, it seems that the order-reduction prescription is too drastic, at least in the case of Minkowski spacetime, since no effects due to the interaction with the quantum matter fields are left.

A second possibility, proposed by Hawking et al. [161Jump To The Next Citation Point, 162Jump To The Next Citation Point], 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 0 p complex plane. Following that procedure we get, for example, that for a massless conformally-coupled matter field with β¯> 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 extra-massive 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 [162Jump To The Next Citation Point].

6.5.2 Induced metric fluctuations

Induced metric fluctuations are described by the second term in Equation (17View Equation). They are dependent on the noise kernel that describes the stress-tensor fluctuations of the matter fields,

2 ∫ ¯κ-- 4 ′4 ′∘ ----′---′- ret ′ efgh ′ ′ ret ′ ⟨hab(x)hcd(y )⟩ind = N d xd y g (x )g(y )Gabef(x,x )N (x ,y )G cdgh(y, y), (156 )
where here we have written the expression in the large N limit, so that κ¯= N κ, where κ = 8 πG and N is the number of independent free scalar fields. The contribution corresponding to the induced quantum fluctuations is equivalent to the stochastic correlation function obtained by considering just the inhomogeneous part of the solution to the Einstein–Langevin equation. We can make use of the results for the metric correlations obtained in Sections 6.3 and 6.4 for solving the Einstein–Langevin equation. In fact, one should simply take N = 1 to transform our expressions here to those of Sections 6.3 and 6.4 or, more precisely, one should multiply the noise kernel in these expressions by N in order to use those expressions here, as follows from the fact that we now have N independent matter fields.

As we have seen in Section 6.4, following [259Jump To The Next Citation Point], the Einstein–Langevin equation can be entirely written in terms of the linearized Einstein tensor. The equation involves second spacetime derivatives of that tensor and, in terms of its Fourier components, is given in Equation (121View Equation) as

μν &tidle;(1)αβ &tidle;μν F αβ(p)G (p) = ¯κ ξ (p), (157 )
where we have now used the rescaled coupling ¯κ. The solution for the linearized Einstein tensor is given in Equation (133View Equation) in terms of the retarded propagator D μνρσ(p) defined in Equation (129View Equation). Now this propagator, which is written in Equation (130View Equation), exhibits two poles in the upper half complex p0 plane and two poles in the lower half-plane, as we have seen analyzing the zeros in Equations (153View Equation) and (154View Equation) for the massless and conformally coupled case. The retarded propagator in spacetime coordinates is obtained, as usual, by taking the appropriate integration contour in the p0 plane. It is convenient in this case to deform the integration path along the real p0 axis so as to leave the two poles of the upper half-plane below that path. In this way, when closing the contour by an upper half-circle, in order to compute the anti-causal part of the propagator, there will be no contribution. The problem now is that when closing the contour on the lower half-plane, in order to compute the causal part, the contribution of the upper half-plane poles gives an unbounded solution, a runaway instability. If we adopt the Hawking et al. [161Jump To The Next Citation Point, 162Jump To The Next Citation Point] criterion of imposing final boundary conditions, which discard solutions growing unboundedly in time, this implies that we just need to take the integral along the real axis, as was done in Section 6.4.2. But now that the propagator is no longer strictly retarded, there are causality violations in time scales on the order of √ --- N lp, which should have no observable consequences. This propagator, however, has a well-defined Fourier transform.

Following the steps after Equation (133View Equation), the Fourier transform of the two-point correlation for the linearized Einstein tensor can be written in our case as,

&tidle;(1) &tidle;(1) ′ ¯κ2- 4 4 ′ &tidle; ρσλγ ⟨Gμν (p )G αβ(p )⟩ind = N (2π) δ(p + p )D μνρσ(p)D αβλγ(− p)N (p), (158 )
where the noise kernel &tidle; ρσλγ N (p) is given by Equation (125View Equation). Note that these correlation functions are invariant under gauge transformations of the metric perturbations because the linearized Einstein tensor is invariant under those transformations.

We may also use the order-reduction prescription, which amounts in this case to neglecting terms in the propagator, which are proportional to 2 p, corresponding to two spacetime derivatives of the Einstein tensor. The propagator then becomes a constant, and we have

2 ⟨ &tidle;G(μ1ν)(p) &tidle;G(1)(p′)⟩ind = ¯κ-(2π)4δ4(p+p ′)N&tidle;μναβ(p). (159 ) αβ N
Finally, we may derive the correlations for the metric perturbations from Equations (158View Equation) or (159View Equation). In the Lorentz or harmonic gauge the linearized Einstein tensor takes the particularly simple form of Equation (155View Equation) in terms of the metric perturbation. One may derive the correlation functions for &tidle; hμν(p) as it was done in Section 6.4.2 to get
&tidle; &tidle; ′ 4 ′ ⟨¯hμν(p)¯hαβ(p )⟩ind = --2-2 ⟨G&tidle;μν(p) &tidle;G αβ(p)⟩ind. (160 ) (p )
There will be one possible expression for the two-point metric correlation, which corresponds to the Einstein-tensor correlation of Equation (158View Equation), and another expression corresponding to Equation (158View Equation), when the order-reduction prescription is used. We should note that, contrary to the correlation functions for the Einstein tensor, the two-point metric correlation is not gauge invariant (it is given in the Lorentz gauge). Moreover, when taking the Fourier transform to get the correlations in spacetime coordinates, there is an apparent infrared divergence when p2 = 0 in the massless case. This can be seen from the expression for the noise kernel N&tidle;μναβ(p) defined in Equation (125View Equation). For the massive case no such divergence due to the factor 2 2 θ(− p − 4m ) exists, but as one takes the limit m → 0 it will show up. This infrared divergence, however, is a gauge artifact that has been enforced by the use of the Lorentz gauge. A gauge different from the Lorentz gauge should be used in the massless case; see [203Jump To The Next Citation Point] for a more detailed discussion of this point.

Let us now write the two-point metric correlation function in spacetime coordinates for the massless and conformally coupled fields. In order to avoid runaway solutions we use the prescription that the propagator should have a well-defined Fourier transform by integrating along the real axis in the complex p0 plane. This was, in fact, done in Section 6.4.3 and we may now write Equation (146View Equation) as

2 ∫ 4 ip(x− y) 2 ⟨&tidle;¯h (x)&tidle;¯h (y)⟩ = --¯κ---- -d-p--e-------Pμναβ θ-(−-p-)-, (161 ) μν αβ ind 720πN (2π)4 |1 + (¯κ∕2)p2H&tidle;(p;μ¯2 )|2
where the projector P μναβ is defined in Equation (107View Equation). This correlation function for the metric perturbations is in agreement with the real part of the graviton propagator obtained by Tomboulis in [348] using a large N expansion with Fermion fields. Note that when the order-reduction prescription is used the terms in the denominator of Equation (161View Equation) that are proportional to 2 p are neglected. Thus, in contrast to the intrinsic metric fluctuations, there is still a nontrivial contribution to the induced metric fluctuations due to the quantum matter fields in this case.

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 lp. For space separations |r| ≫ lp we have that the two-point correlation (161View Equation) goes as 4 4 ∼ N lp∕|r| , and for √ --- |r| ∼ N lp we have that it goes as √ --- ∼ exp(− |r|∕ N lp)lp∕|r|. 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 l2p∕|r|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 Order-reduction prescription and large N

Runaway solutions are a typical feature of equations describing backreaction effects, such as in classical electrodynamics, and are due to higher than two time derivatives in the dynamical equations. Here we will give a qualitative analysis of this problem in semiclassical gravity. In a very schematic way the semiclassical Einstein equations have the form

2 Gh + lp ¨Gh = 0, (162 )
where, say, Gh stands for the linearized Einstein tensor over the Minkowski background and we have simplified the equation as much as possible. The second term of the equation is due to the vacuum polarization of matter fields and contains four time derivatives of the metric perturbation. Some specific examples of such an equation are, in momentum space, Equations (153View Equation) and (154View Equation). The order-reduction procedure is based on treating perturbatively the terms involving higher-order derivatives, differentiating the equation under consideration, and substituting back the higher derivative terms in the original equation, keeping only terms up to the required order in the perturbative parameter. In the case of the semiclassical Einstein equation, the perturbative parameter is l2p. If we differentiate Equation (162View Equation) with respect to time twice, it is clear that the second-order derivatives of the Einstein tensor are of order l2 p. Substituting back into the original equation, we get the following equation up to order 4 lp: 4 Gh = 0 + O (lp). Now there are certainly no runaway solutions but also no effect due to the vacuum polarization of matter fields. Note that the result is not so trivial when there is an inhomogeneous term on the right-hand side of Equation (162View Equation), this is what happens with the induced fluctuations predicted by the Einstein–Langevin equation.

Semiclassical gravity is expected to provide reliable results as long as the characteristic length scales under consideration, say L, satisfy that L ≫ lp [110Jump To The Next Citation Point]. This can be qualitatively argued by estimating the magnitude of the different contributions to the effective action for the gravitational field, considering the relevant Feynman diagrams and using dimensional arguments. Let us write the effective gravitational action, again in a very schematic way, as

∫ ( ) S = d4x√ −-g 1R + αR2 + l2R3 + ... , (163 ) eff l2p p
where R is the Ricci scalar. The first term is the usual classical Einstein–Hilbert term. The second stands for terms quadratic in the curvature (square of Ricci and Weyl tensors). These terms appear as radiative corrections due to vacuum polarization of matter fields. Here α is a dimensionless parameter presumably of order 1 and the R3 terms are higher-order corrections, which appear, for instance, when one considers internal graviton propagators inside matter loops. Let us assume that R ∼ L −2; then the different terms in the action are on the order of 2 −4 R ∼ L and 2 3 2 −6 lpR ∼ lpL. Consequently, when 2 L ≫ lp, the term due to matter loops is a small correction to the Einstein–Hilbert term 2 2 (1∕lp)R ≫ R and this term can be treated as a perturbation. The justification for the order-reduction prescription is actually based on this fact. Therefore, significant effects from the vacuum polarization of the matter fields are only expected when their small corrections accumulate in time, as would be the case for an evaporating macroscopic black hole all the way before reaching Planckian scales (see Section 8.3).

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 vacuum-polarization terms involving loops of matter are of order 2 −4 N R ∼ N L. For this reason, the contribution of the graviton loops, which is just of order R2 as is any other loop of matter, can be neglected in front of the matter loops; this justifies the semiclassical limit. Similarly, higher-order corrections are of order N l2R3 ∼ N l2L −6 p p. Now there is a regime, when √ --- L ∼ N lp, where the Einstein–Hilbert term is comparable to the vacuum polarization of matter fields, 2 2 (1∕lp)R ∼ N R, and yet the higher correction terms can be neglected because we still have L ≫ lp, provided N ≫ 1. This is the kind of situation considered in trace anomaly driven inflationary models [162Jump To The Next Citation Point], such as that originally proposed by Starobinsky [339Jump To The Next Citation Point], see also [355Jump To The Next Citation Point], where exponential inflation is driven by a large number of massless conformal fields. The order-reduction prescription would completely discard the effect from the vacuum polarization of the matter fields even though it is comparable to the Einstein–Hilbert 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 lp is considered, not the rescaled one, ¯l2p = ¯κ∕8π, which is related to lp by l2 = κ∕8π = ¯l2∕N p p.

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 two-point 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 Einstein–Langevin equation.

We have analyzed the symmetrized two-point 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 order-reduction prescription or the procedure proposed by Hawking et al. [161, 162Jump To The Next Citation Point], both the intrinsic and the induced contributions to the quantum correlation function for the metric perturbations are found to be stable [203Jump To The Next Citation Point]. Thus, we conclude that Minkowski spacetime is a valid solution of semiclassical gravity.

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