

6.4 Correlation functions for
gravitational perturbations
Here we solve the Einstein-Langevin equations (111) for the components
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 semiclassical 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 [137
] (see
also [91
]) to analyze the
stability of Minkowski spacetime in semiclassical gravity.
We first note that the tensors
and
can be written in terms of
as
where we have used that
. Therefore, the
Einstein-Langevin equation (111) can be seen as a
linear integro-differential stochastic equation for the components
. In order to find solutions to Equation (111), it is convenient to
Fourier transform. With the convention
for a given field
, one finds, from
Equation (112),
The Fourier transform of the Einstein-Langevin
Equation (111) now reads
where
with
In the Fourier transformed Einstein-Langevin Equation (114),
, the Fourier transform of
, is a
Gaussian stochastic source of zero average, and
where we have introduced the Fourier transform of the noise kernel.
The explicit expression for
is found
from Equations (95) and (94) to be
which in the massless case reduces to
6.4.1
Correlation
functions for the linearized Einstein tensor
In general, we can write
, where
is a solution to Equations (111) with zero average, or
Equation (114) in the Fourier
transformed version. The averages
must
be a solution of the linearized semiclassical Einstein equations
obtained by averaging Equations (111) or (114). Solutions to these
equations (specially in the massless case,
) have been studied by several authors [137
, 138, 141, 247, 248, 273, 274, 128
, 262, 182, 91
], particularly in
connection with the problem of the stability of the ground state of
semiclassical gravity. The two-point correlation functions for the
linearized Einstein tensor are defined by
Now we shall seek the family of solutions to the
Einstein-Langevin equation which can be written as a linear
functional of the stochastic source, and whose Fourier transform
depends locally on
. Each of
such solutions is a Gaussian stochastic field and, thus, it can be
completely characterized by the averages
and
the two-point correlation functions (120). For such a family of
solutions,
is the most general solution to
Equation (114) which is linear,
homogeneous, and local in
. It can be
written as
where
are the components of a Lorentz
invariant tensor field distribution in Minkowski spacetime, symmetric
under the interchanges
and
, which is the most general solution of
In addition, we must impose the conservation condition,
, where this zero must be understood as
a stochastic variable which behaves deterministically as a zero
vector field. We can write
, where
is a particular solution to
Equation (122) and
is the most general solution to the homogeneous
equation. Consequently, see Equation (121), we can write
. To find the particular solution, we
try an ansatz of the form
Substituting this ansatz into Equations (122), it is easy to see
that it solves these equations if we take
with
and where the notation
means that the zeros of the
denominators are regulated with appropriate prescriptions in such a
way that
and
are well defined
Lorentz invariant scalar distributions. This yields a particular
solution to the Einstein-Langevin equations,
which, since the stochastic source is conserved, satisfies the
conservation condition. Note that, in the case of a massless scalar
field (
), the above solution has a functional
form analogous to that of the solutions of linearized semiclassical
gravity found in the appendix of [91
]. Notice also that,
for a massless conformally coupled field (
and
), the second term on the
right-hand side of Equation (123) will not contribute
in the correlation functions (120), since in this case
the stochastic source is traceless.
A detailed analysis given in [209
] concludes that the
homogeneous solution
gives no contribution to the
correlation functions (120). Consequently
, where
is the
inverse Fourier transform of Equation (126), and the correlation
functions (120) are
It is easy to see from the above analysis that the prescriptions
in the factors
are irrelevant in the last
expression and, thus, they can be suppressed. Taking into account
that
, with
, we get from
Equations (123) and (124)
This last expression is well defined as a bi-distribution and can
be easily evaluated using Equation (118). The final explicit
result for the Fourier transformed correlation function for the
Einstein tensor is thus
To obtain the correlation functions in coordinate
space, Equation (120), we take the inverse
Fourier transform. The final result is
with
where
,
, are given
in Equations (116) and (125). Notice that, for a
massless field (
), we have
with
and
, and where
is the
Fourier transform of
given in Equation (110).
6.4.2
Correlation
functions for the metric perturbations
Starting from the solutions found for the
linearized Einstein tensor, which are characterized by the
two-point correlation functions (130) (or, in terms of
Fourier transforms, Equation (129)), we can now solve
the equations for the metric perturbations. Working in the harmonic
gauge,
(this zero must be understood in a
statistical sense) where
, the equations for the metric perturbations in terms
of the Einstein tensor are
or, in terms of Fourier transforms,
. Similarly to the analysis of the equation for the
Einstein tensor, we can write
, where
is a solution to these
equations with zero average, and the two-point correlation
functions are defined by
We can now seek solutions of the Fourier
transform of Equation (133) of the form
, where
is a Lorentz invariant scalar distribution in
Minkowski spacetime, which is the most general solution of
. Note that, since the linearized Einstein tensor is
conserved, solutions of this form automatically satisfy the
harmonic gauge condition. As in the previous subsection, we can
write
, where
is the most general solution to the associated
homogeneous equation and, correspondingly, we have
. However, since
has support on the set of points for which
, it is easy to see from Equation (129) (from the factor
) that
and, thus, the two-point correlation
functions (134) can be computed from
. From Equation (129) and due to the factor
, it is also easy to see that the
prescription
is irrelevant in this correlation
function, and we obtain
where
is given by Equation (129). The right-hand side
of this equation is a well defined bi-distribution, at least for
(the
function provides the suitable cutoff).
In the massless field case, since the noise kernel is obtained as
the limit
of the noise kernel for a massive
field, it seems that the natural prescription to avoid the
divergences on the lightcone
is a Hadamard
finite part (see [256, 302] for its definition).
Taking this prescription, we also get a well defined
bi-distribution for the massless limit of the last expression.
The final result for the two-point correlation
function for the field
is
where
and
, with
and
given by Equation (131). The two-point
correlation functions for the metric perturbations can be easily
obtained using
.
6.4.3
Conformally
coupled field
For a conformally coupled field, i.e., when
and
, the previous correlation
functions are greatly simplified and can be approximated explicitly
in terms of analytic functions. The detailed results are given
in [209
]; here we outline
the main features.
When
and
, we have
and
. Thus the two-point
correlations functions for the Einstein tensor is
where
(see Equation (110)).
To estimate this integral, let us consider
spacelike separated points
, and define
. We may now
formally change the momentum variable
by the
dimensionless vector
,
. Then
the previous integral denominator is
, where we have introduced the Planck length
. It is clear that we can consider two regimes: (a)
when
, and (b) when
. In case (a) the correlation function, for the
component, say, will be of the order
In case (b) when the denominator has zeros, a
detailed calculation carried out in [209
] shows that
which indicates an exponential decay at distances
around the Planck scale. Thus short scale fluctuations are strongly
suppressed.
For the two-point metric correlation the results
are similar. In this case we have
The integrand has the same behavior of the correlation function of
Equation (137), thus matter fields
tends to suppress the short scale metric perturbations. In this
case we find, as for the correlation of the Einstein tensor, that
for case (a) above we have
and for case (b) we have
It is interesting to write expression (138) in an alternative
way. If we use the dimensionless tensor
introduced in
Equation (100), which accounts for
the effect of the operator
, we can write
This expression allows a direct comparison with the graviton
propagator for linearized quantum gravity in the
expansion found by Tomboulis [277
]. One can see that
the imaginary part of the graviton propagator leads, in fact, to
Equation (139). In [255
] it is shown that
the two-point correlation functions for the metric perturbations
derived from the Einstein-Langevin equation are equivalent to the
symmetrized quantum two-point correlation functions for the metric
fluctuations in the large
expansion of quantum gravity
interacting with
matter fields.

