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 section 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 Equation (351) and then take the limit, we obtain
Equation (354) can be integrated along the unique geodesic that links to . The initial conditions are provided by Equation (353), and if we set , we find that these equations reduce to Equations (272) and (271), respectively. According to Equation (273), then, we have
Similarly, Equation (355) 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 Equations (353), (361), and the additional relation . We arrive at
To summarize, the retarded and advanced gravitational Green’s functions are given by Equation (352) with given by Equation (357) and determined by Equation (356), and the characteristic data constructed with Equations (355) and (362). It should be emphasized that the construction provided in this section 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 Equation (349) 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
We shall now construct singular and radiative Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 4.3.5 and 4.4.4.
We begin by introducing the bitensor with properties
Gr.H1: satisfies the homogeneous wave equation,
Gr.H2: is symmetric in its indices and arguments,
Gr.H3: agrees with the retarded Green’s function if is in the chronological future of ,
Gr.H4: agrees with the advanced Green’s function if is in the chronological past of ,
It is easy to prove that Property Gr.H4 follows from Property Gr.H2, Property Gr.H3, and the reciprocity relation (363) 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 4.3.5.
Equipped with we define the singular Green’s function to be
Gr.S1: satisfies the inhomogeneous wave equation,
Gr.S2: is symmetric in its indices and arguments,
Gr.S3: vanishes if is in the chronological past or future of ,
These can be established as consequences of Properties Gr.H1, Gr.H2, Gr.H3, and Gr.H4, and the properties of the retarded and advanced Green’s functions.
The radiative Green’s function is then defined by
Gr.R1: satisfies the homogeneous wave equation,
Gr.R2: agrees with the retarded Green’s function if is in the chronological future of ,
Gr.R3: vanishes if is in the chronological past of ,
Those follow immediately from Properties Gr.S1, Gr.S2, and Gr.S3, and the properties of the retarded Green’s function.
When is restricted to the normal convex neighbourhood of , we have the explicit relationsGr.S2), and that its support excludes (Property Gr.S3). We see also that the radiative Green’s function coincides with in (Property Gr.R2), and that its support does not include (Property Gr.R3).
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