Let us first consider the intrinsic metric fluctuations,
where Using the metric decomposition (148) we may compute the linearized Einstein tensor
. 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:
and
.
Thus, let us now write the Fourier transform of the homogeneous Einstein–Langevin equation (121
), which
is equivalent to the linearized semiclassical Einstein equation,
where and
are given by Equations (123
), and
and
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 [110
] for the massless case with arbitrary coupling
and [10
, 259
] for the general massive case). Thus substituting
and
into
the functions
and
, and using Equation (117
), the above equations become
For the scalar component when the only solution is
. When
the
solutions for the scalar component exhibit an oscillatory behavior in spacetime coordinates, which
corresponds to a massive scalar field with
; for
the solutions correspond to a
tachyonic field with
. In spacetime coordinates they exhibit an exponential behavior in
time, growing or decreasing, for wavelengths larger than
and an oscillatory behavior for
wavelengths smaller than
. On the other hand, the solution
is completely
trivial since any scalar metric perturbation
giving rise to a vanishing linearized Einstein tensor can
be eliminated by a gauge transformation.
For the tensorial component, when , where
is the Planck
length (
), the first factor in Equation (154
) vanishes for four complex values of
of the form
and
, where
is some complex value. This means that, in the
corresponding propagator, there are two poles on the upper half-plane of the complex
plane
and two poles in the lower half-plane. We will consider here the case in which
; a
detailed description of the situation for
can be found in Appendix A of [110
]. The two
zeros on the upper half of the complex plane correspond to solutions in spacetime coordinates,
which exponentially grow in time, whereas the two on the lower half correspond to solutions
exponentially decreasing in time. Strictly speaking, these solutions only exist in spacetime coordinates,
since their Fourier transform is not well-defined. They are commonly referred to as runaway
solutions and for
they grow exponentially in time scales comparable to the Planck
time.
Consequently, in addition to the solutions with , there are other solutions that in Fourier
space take the form
for some particular values of
, 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
terms in Equations (153
) and
(154
) come from two spacetime derivatives of the Einstein tensor, moreover, the
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
in Equations (153
) and (154
), we are left with only the solutions, which satisfy
. 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 (149
) reads:
A second possibility, proposed by Hawking et al. [161, 162
], is to impose boundary conditions, which
discard the runaway solutions that grow unbounded in time. These boundary conditions correspond to a
special prescription for the integration contour when Fourier transforming back to spacetime
coordinates. As we will discuss in more detail in Section 6.5.2, this prescription reduces here to
integrating along the real axis in the
complex plane. Following that procedure we get,
for example, that for a massless conformally-coupled matter field with
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 [162
].
Induced metric fluctuations are described by the second term in Equation (17). They are dependent on the
noise kernel that describes the stress-tensor fluctuations of the matter fields,
As we have seen in Section 6.4, following [259], 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 (121
) as
Following the steps after Equation (133), the Fourier transform of the two-point correlation for the
linearized Einstein tensor can be written in our case as,
We may also use the order-reduction prescription, which amounts in this case to neglecting terms in the
propagator, which are proportional to , corresponding to two spacetime derivatives of the Einstein
tensor. The propagator then becomes a constant, and we have
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
plane. This was, in fact, done in Section 6.4.3 and we may now write Equation (146
) as
To estimate the above integral let us follow Section 6.4.3 and consider spacelike separated points
and introduce the Planck length
. For space separations
we have that the
two-point correlation (161
) goes as
and for
we have that it goes as
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
.
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.
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
where, say, Semiclassical gravity is expected to provide reliable results as long as the characteristic length scales
under consideration, say , satisfy that
[110
]. 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
However, if we have a large number of matter fields, the estimates for the different terms change
in a remarkable way. This is interesting because the large
expansion seems, as we have
argued in Section 3.3.1, the best justification for semiclassical gravity. In fact, now the
vacuum-polarization terms involving loops of matter are of order
. For this reason, the
contribution of the graviton loops, which is just of order
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
. Now there is a regime, when
,
where the Einstein–Hilbert term is comparable to the vacuum polarization of matter fields,
, and yet the higher correction terms can be neglected because we still have
,
provided
. This is the kind of situation considered in trace anomaly driven inflationary
models [162
], such as that originally proposed by Starobinsky [339
], see also [355
], where exponential
inflation is driven by a large number of massless conformal fields. The 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
is considered, not the rescaled one,
, which is related to
by
.
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 , where
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 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, 162
], both the intrinsic and the induced
contributions to the quantum correlation function for the metric perturbations are found to be
stable [203
]. Thus, we conclude that Minkowski spacetime is a valid solution of semiclassical
gravity.
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