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 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 [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 expansion; that is the quantum theory describing the interaction of the gravitational
field with
identical free matter fields. In the leading order, namely the limit in which
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
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
is characterized not only by its expectation value but also by its
-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.
To illustrate the relation between the semiclassical, stochastic and quantum theories, a simplified model of
scalar gravity interacting with scalar fields is considered here.
The large 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
matter fields, it shows that graviton loops are of higher order
than matter loops. To illustrate the large
expansion, let us first consider the following toy model of
gravity, which we will simplify as a scalar field
, interacting with a single scalar field
described by
the action
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
, 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
this diagram is of order
. The next diagram
contains two loops of matter, three gravitons and consequently it is of order
. 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
term in the action (20
). Since there are four
graviton propagators, which carry a
, but two vertices, which have
, this diagram is of
order
, 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.
|
Let us now consider the large expansion. We assume that the gravitational field is coupled with a
large number of identical fields
,
, which couple only with
. Next we rescale the
gravitational coupling in such a way that
is finite even when
goes to infinity. The action of
this system is:
|
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 and the vertices as
, thus this diagram is of
. Therefore
graviton loops contribute to a higher order in the
expansion than matter loops. Similarly
there are
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
of
them, their sum is of order
. To summarize, we have that when
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
.
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 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
expansion for the quantum theory of gravity interacting with
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.
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