6.4 Correlation functions for gravitational perturbations
Here we solve the Einstein–Langevin equations (118) 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 [169
] (see also [110
]) 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
. Therefore, the Einstein–Langevin equation (118) can be seen as a linear
integro-differential stochastic equation for the components
. In order to find solutions to
Equation (118), it is convenient to Fourier transform it. With the convention
for a
given field
, one finds, from Equation (119),
The Fourier transform of the Einstein–Langevin Equation (118) now reads
where
with
In the Fourier transformed Einstein–Langevin Equation (121),
, 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 (101) and (102) 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 (118) with
zero average, or Equation (121) in the Fourier transformed version. The averages
must be a solution of the linearized semiclassical Einstein equations obtained by averaging
Equations (118) or (121). Solutions to these equations (especially in the massless case,
)
have been studied by several authors [110
, 152, 169
, 170, 174, 223, 309, 310, 330, 344, 345],
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 these such solutions is a Gaussian stochastic field and thus can be completely
characterized by the averages
and the two-point correlation functions (127). For such a family of
solutions,
is the most general solution to Equation (121), 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 (by “Lorentz-invariant” we mean invariant under transformations of the orthochronous Lorentz
subgroup; see [169
] for more details on the definition and properties of these tensor distributions). This
tensor is symmetric under the interchanges of
and
, and 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 (129) and
is the most general solution to the homogeneous equation. Consequently, see Equation (128),
we can write
. To find the particular solution, we try an ansatz of the
form
Substituting this ansatz into Equations (129), 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 [110
]. Notice also that, for a massless
conformally-coupled field (
and
), the second term on the right-hand side of
Equation (130) will not contribute in the correlation functions (127), since in this case the stochastic
source is traceless.
A detailed analysis given in [259
] concludes that the homogeneous solution
gives no
contribution to the correlation functions (127). Consequently
, where
is the inverse Fourier transform of Equation (133), and the correlation functions (127) are
It is easy to see from the above analysis that the prescriptions
in the factors
are irrelevant in
the last expression and thus can be suppressed. Taking into account that
, with
, we get from Equations (130) and (131)
This last expression is well-defined as a bi-distribution and can be easily evaluated using Equation (125).
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 (127), we take the inverse Fourier
transform. The final result is
with
where
,
, are given in Equations (123) and (132). Notice that, for a massless field
(
), we have
with
and
, and where
is the Fourier transform of
given in Equation (117).
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 (137) (or, in terms of Fourier transforms, Equation (136)), 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 (140) 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 Section 6.4.1 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 (136) (from the factor
)
that
and, thus, the two-point correlation functions (141) can be computed from
. From Equation (136), 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 (136). 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 divergences
on the lightcone
is a Hadamard finite part (see [322, 388] 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 (138). 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 [259
]; here we outline the main features.
When
and
, we have
and
. Thus the two-point
correlations functions for the Einstein tensor are written
where
; (see Equation (117)).
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 [259
] 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 as the correlation function of Equation (144), 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 (145) in an alternative way. If we use the dimensionless tensor
introduced in Equation (107), 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 [348
]. One can see that the imaginary
part of the graviton propagator leads, in fact, to Equation (146). In [317
] 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.