We are given a background spacetime for which the metric satisfies the Einstein field equations in
vacuum. We then perturb the metric from
to
The solution to the wave equation is written as
in terms of a Green’s function We will assume that the retarded Green’s function , which is nonzero if
is in the
causal future of
, and the advanced Green’s function
, which is nonzero if
is in the
causal past of
, exist as distributions and can be defined globally in the entire background
spacetime.
Assuming throughout this subsection that is in the normal convex neighbourhood of
, we make the
ansatz
To conveniently manipulate the Green’s functions we shift by a small positive quantity
. The
Green’s functions are then recovered by the taking the limit of
as . When we substitute this into the left-hand side of Eq. (16.6
) and then take the limit, we
obtain
Eq. (16.9) can be integrated along the unique geodesic
that links
to
. The initial conditions
are provided by Eq. (16.8
), and if we set
, we find that these
equations reduce to Eqs. (14.7
) and (14.6
), respectively. According to Eq. (14.8
), then, we have
Similarly, Eq. (16.10) can be integrated along each null geodesic that generates the null cone
. The initial values are obtained by taking the coincidence limit of this equation, using
Eqs. (16.8
), (16.16
), and the additional relation
. We arrive at
To summarize, the retarded and advanced gravitational Green’s functions are given by Eq. (16.7) with
given by Eq. (16.12
) and
determined by Eq. (16.11
) and the characteristic
data constructed with Eqs. (16.10
) and (16.17
). It should be emphasized that the construction provided in
this subsection is restricted to
, the normal convex neighbourhood of the reference point
.
The (globally defined) gravitational Green’s functions satisfy the reciprocity relation
The derivation of this result is virtually identical to what was presented in Sections 14.3 and 15.3. A direct consequence of the reciprocity relation is the statement The Kirchhoff representation for the trace-reversed gravitational perturbation is formulated as
follows. Suppose that
satisfies the homogeneous version of Eq. (16.4
) and that initial values
,
are specified on a spacelike hypersurface
. Then the value of the
perturbation field at a point
in the future of
is given by
In a spacetime that satisfies the Einstein field equations in vacuum, so that everywhere in the
spacetime, the (retarded and advanced) gravitational Green’s functions satisfy the identities [144
]
To prove Eq. (16.21) we differentiate Eq. (16.6
) covariantly with respect to
, use Eq. (13.3
) to
work on the right-hand side, and invoke Ricci’s identity to permute the ordering of the covariant derivatives
on the left-hand side. After simplification and involvement of the Ricci-flat condition (which,
together with the Bianchi identities, implies that
), we arrive at the equation
The identity of Eq. (16.22) follows simply from the fact that
and
satisfy the same
tensorial wave equation in a Ricci-flat spacetime.
We shall now construct singular and regular Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 14.5 and 15.5.
We begin by introducing the bitensor with properties
It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (16.18) satisfied by the
retarded and advanced Green’s functions. That such a bitensor exists can be argued along the same lines as
those presented in Section 14.5.
Equipped with we define the singular Green’s function to be
These can be established as consequences of H1 – H4 and the properties of the retarded and advanced Green’s functions.
The regular two-point function is then defined by
and it comes with the propertiesThose follow immediately from S1 – S3 and the properties of the retarded Green’s function.
When is restricted to the normal convex neighbourhood of
, we have the explicit relations
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Living Rev. Relativity 14, (2011), 7
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