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 functionWe 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 derivation of this result is virtually identical to what was presented in Sections 4.3.3 and 4.4.3. A direct consequence of the reciprocity relation is the statementThe 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
and it comes with the properties 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
relations