6.1 Perturbations around Minkowski spacetime
The Minkowski metric
in a manifold
, which is topologically
, together with
the usual Minkowski vacuum, denoted as
, is the simplest solution to the semiclassical
Einstein equation (8), the so-called trivial solution of semiclassical gravity [110
]. It constitutes the
ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in
which the renormalized expectation value
. Thus, the Minkowski spacetime
plus the vacuum state
is a solution to the semiclassical Einstein equation with
renormalized cosmological constant
. The fact that the vacuum expectation value of the
renormalized stress-energy operator in Minkowski spacetime should vanish was originally proposed
by Wald [359], and it may be understood as a renormalization convention [121, 135]. Note
that other possible solutions of semiclassical gravity with zero vacuum expectation value of
the stress-energy tensor are the exact gravitational plane waves, since they are known to be
vacuum solutions of Einstein equations, which induce neither particle creation nor vacuum
polarization [91, 125, 128].
As we have already mentioned, the vacuum
is an eigenstate of the total four-momentum operator
in Minkowski spacetime, but not an eigenstate of
. Hence, even in the Minkowski background
there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel
does not vanish. This fact leads us to consider stochastic corrections to this class of trivial
solutions of semiclassical gravity. Since in this case the Wightman and Feynman functions (44),
their values in the two-point coincidence limit and the products of derivatives of two of such
functions appearing in expressions (45) and (46) are known in dimensional regularization,
we can compute the Einstein–Langevin equation using the methods outlined in Sections 3
and 4.
To perform explicit calculations it is convenient to work in a global inertial coordinate system
and in
the associated basis, in which the components of the flat metric are simply
.
In Minkowski spacetime, the components of the classical stress-energy tensor (4) reduce to
where
and the formal expression for the components of the corresponding “operator” in
dimensional regularization, see Equation (5), is
where
is the differential operator (6), with
,
, and
. The field
is the field operator in the Heisenberg representation in an
-dimensional Minkowski spacetime,
which satisfies the Klein–Gordon equation (2). We use here a stress-energy tensor, which differs from
the canonical one that corresponds to
; both tensors, however, define the same total
momentum.
The Wightman and Feynman functions (44) for
are well known:
with
where
and
. Note that the derivatives of these functions satisfy
and
, and similarly for the Feynman
propagator
.
To write down the semiclassical Einstein equation (8) in
dimensions for this case, we need to
compute the vacuum expectation value of the stress-energy operator components (86). Since, from (87), we
have that
, which is a constant (independent of
), we have simply
where the integrals in dimensional regularization have been computed in the standard way [259
], and where
is Euler’s gamma function. The semiclassical Einstein equation (8) in
dimensions before
renormalization reduces now to
Thus, this equation simply sets the value of the bare coupling constant
. Note from Equation (89)
that in order to have
, the renormalized and regularized stress-energy tensor
“operator” for a scalar field in Minkowski spacetime, see Equation (7), has to be defined as
which corresponds to a renormalization of the cosmological constant
where
with
being Euler’s constant. In the case of a massless scalar field,
, one simply has
. Introducing this renormalized coupling constant into Equation (90), we can take the limit
. We find that, for
to satisfy the semiclassical Einstein equation, we must take
.
We can now write down the Einstein–Langevin equations for the components
of the stochastic
metric perturbation in dimensional regularization. In our case, using
and the
explicit expression of Equation (41), we obtain
The indices in
are raised with the Minkowski metric and
; here a superindex
denotes
the components of a tensor linearized around the flat metric. Note that in
dimensions the two-point
correlation functions for the field
is written as
Explicit expressions for
and
are given by
with the differential operators
and
.