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
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 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 
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
Those 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 relationsS2), and that its support excludes (property S3). We see also that the regular two-point function coincides with in (property R2), and that its support does not include (property R3).
Living Rev. Relativity 14, (2011), 7
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