We consider a massless scalar field in a curved spacetime with metric
. The field satisfies the wave equation
We let be the
retarded solution to Equation (268
), and
be the advanced solution; when viewed as functions
of
,
is nonzero in the causal
future of
, while
is nonzero
in its causal past. We assume that the retarded and advanced
Green’s functions exist as distributions and can be defined
globally in the entire spacetime.
Assuming throughout this section that is restricted to the normal convex neighbourhood of
, we make the ansatz
Before we substitute the Green’s functions of
Equation (269) into the differential
equation of Equation (268
), we proceed as in
Section 4.1.6 and shift
by the small positive quantity
. We shall therefore consider the distributions
and later recover the Green’s functions by taking
the limit . Differentiation of these objects is
straightforward, and in the following manipulations we will
repeatedly use the relation
satisfied by
the world function. We will also use the distributional identities
,
, and
. After a routine calculation we
obtain
According to Equation (268), the right-hand side
of Equation (270
) should be equal to
. This immediately gives us the
coincidence condition
Recall from Section 2.1.3 that is a vector at
that is tangent to the unique
geodesic
that connects
to
. This geodesic is affinely
parameterized by
and a displacement along
is described by
. The
first term of Equation (272
) therefore represents
the rate of change of
along
, and this can be expressed as
. For the second term we recall from Section 2.5.1 the differential equation
satisfied by
, the van Vleck determinant. This gives us
, and Equation (272
) becomes
It follows that is constant on
, and it must therefore be equal to its value at the
starting point
:
,
by virtue of Equation (271
) and the property
of the van Vleck determinant. Since this statement
must be true for all geodesics
that emanate from
, we have found that the unique solution to
Equations (271
) and (272
) is
We must still consider the remaining terms in
Equation (270). The
term can be eliminated by demanding that its
coefficient vanish when
. This, however, does not
constrain its value away from the light cone, and we thus obtain
information about
only. Denoting this by
- the restriction of
on the light
cone
- we have
Equations (97) and (273
) imply that near
coincidence,
admits the expansion
Equations (274) and (278
) give us a means to
construct
, the restriction of
on the null cone
. These
values can then be used as characteristic data for the wave
equation
To summarize: We have shown that with given by Equation (273
) and
determined uniquely by the wave equation of
Equation (279
) and the
characteristic data constructed with Equations (274
) and (278
), the retarded and
advanced Green’s functions of Equation (269
) do indeed satisfy
Equation (268
). It should be
emphasized that the construction provided in this section is
restricted to
, the normal convex neighbourhood of the
reference point
.
We shall now establish the following reciprocity relation between the (globally defined) retarded and advanced Green’s functions:
Before we get to the proof we observe that by virtue of Equation (280To prove the reciprocity relation we invoke the identities
and
and take their difference. On the left-hand side we have
Integrating both sides over a large
four-dimensional region that contains both
and
, we obtain
where is the boundary of
. Assuming that the Green’s functions fall off
sufficiently rapidly at infinity (in the limit
; this statement imposes some restriction on the
spacetime’s asymptotic structure), we have that the left-hand side
of the equation evaluates to zero in the limit. This gives us the
statement
, which is just Equation (280
) with
replacing
.
Suppose that the values for a scalar field and its normal derivative
are known on a spacelike hypersurface
. Suppose also that the scalar field satisfies the
homogeneous wave equation
To establish this result we start with the equations
in which and
refer to arbitrary points in spacetime. Taking their
difference gives
and this we integrate over a four-dimensional
region that is bounded in the past by the hypersurface
. We suppose that
contains
and we obtain
where is the outward-directed
surface element on the boundary
. Assuming that the
Green’s function falls off sufficiently rapidly into the future, we
have that the only contribution to the hypersurface integral is the
one that comes from
. Since the surface element on
points in the direction opposite to the
outward-directed surface element on
, we must change the
sign of the left-hand side to be consistent with the convention
adopted previously. With this change we have
which is the same as Equation (283) if we take into
account the reciprocity relation of Equation (280
).
In Section 5 of this review we will compute the retarded field of a moving scalar charge, and we will analyze its singularity structure near the world line; this will be part of our effort to understand the effect of the field on the particle’s motion. The retarded solution to the scalar wave equation is the physically relevant solution because it properly incorporates outgoing-wave boundary conditions at infinity - the advanced solution would come instead with incoming-wave boundary conditions. The retarded field is singular on the world line because a point particle produces a Coulomb field that diverges at the particle’s position. In view of this singular behaviour, it is a subtle matter to describe the field’s action on the particle, and to formulate meaningful equations of motion.
When facing this problem in flat spacetime
(recall the discussion of Section 1.3), it is convenient to decompose
the retarded Green’s function into a singular Green’s function
and a radiative Green’s function
. The singular Green’s
function takes its name from the fact that it produces a field with
the same singularity structure as the retarded solution: The
diverging field near the particle is insensitive to the boundary
conditions imposed at infinity. We note also that
satisfies the same wave equation as the retarded
Green’s function (with a Dirac functional as a source), and that by
virtue of the reciprocity relations, it is symmetric in its
arguments. The radiative Green’s function, on the other hand, takes
its name from the fact that it satisfies the homogeneous wave equation, without the Dirac
functional on the right-hand side; it produces a field that is
smooth on the world line of the moving scalar charge.
Because the singular Green’s function is
symmetric in its argument, it does not distinguish between past and
future, and it produces a field that contains equal amounts of
outgoing and incoming radiation - the singular solution describes
standing waves at infinity. Removing from the
retarded Green’s function will therefore have the effect of
removing the singular behaviour of the field without affecting the motion of the particle. The motion is not
affected because it is intimately tied to the boundary conditions:
If the waves are outgoing, the particle loses energy to the
radiation and its motion is affected; if the waves are incoming,
the particle gains energy from the radiation and its motion is
affected differently. With equal amounts of outgoing and incoming
radiation, the particle neither loses nor gains energy and its
interaction with the scalar field cannot affect its motion. Thus,
subtracting
from the retarded Green’s function
eliminates the singular part of the field without affecting the
motion of the scalar charge. The subtraction leaves behind the
radiative Green’s function, which produces a field that is smooth
on the world line; it is this field that will govern the motion of
the particle. The action of this field is well defined, and it
properly encodes the outgoing-wave boundary conditions: The
particle will lose energy to the radiation.
In this section we attempt a decomposition of the
curved-spacetime retarded Green’s function into singular and
radiative Green’s functions. The flat-spacetime relations will have
to be amended, however, because of the fact that in a curved
spacetime, the advanced Green’s function is generally nonzero when
is in the chronological future of
. This implies that the value of the advanced field at
depends on events
that will unfold
in the future; this dependence would
be inherited by the radiative field (which acts on the particle and
determines its motion) if the naive definition
were to be adopted.
We shall not adopt this definition. Instead, we
shall follow Detweiler and Whiting [23] and introduce a
singular Green’s function with the properties
Sc.S1: satisfies the inhomogeneous scalar wave
equation,
Sc.S2: is symmetric in its arguments,
Sc.S3: vanishes if
is in the chronological past
or future of
,
Properties Sc.S1
and Sc.S2 ensure that the singular Green’s
function will properly reproduce the singular behaviour of the
retarded solution without distinguishing between past and future;
and as we shall see, Property Sc.S3
ensures that the support of the radiative Green’s function will not
include the chronological future of .
The radiative Green’s function is then defined by
where Sc.R1: satisfies the homogeneous wave equation,
Sc.R2: agrees with the retarded Green’s function if
is in the chronological future of
,
Sc.R3: vanishes if
is in the chronological past
of
,
Property Sc.R1
follows directly from Equation (287) and Property Sc.S1 of the singular Green’s function.
Properties Sc.R2
and Sc.R3 follow from Property Sc.S3 and the fact that the retarded
Green’s function vanishes if
is in past of
. The properties of the radiative Green’s function
ensure that the corresponding radiative field will be smooth at the
world line, and will depend only on the past history of the scalar
charge.
We must still show that such singular and
radiative Green’s functions can be constructed. This relies on the
existence of a two-point function that would
possess the properties
Sc.H1: satisfies the homogeneous wave equation,
Sc.H2: is symmetric in its arguments,
Sc.H3: agrees with the retarded Green’s function if
is in the chronological future of
,
Sc.H4: agrees with the advanced Green’s function if
is in the chronological past of
,
With a biscalar satisfying
these relations, a singular Green’s function defined by
The question is now: Does such a function exist? I will present a plausibility argument for an
affirmative answer. Later in this section we will see that
is guaranteed to exist in the local convex
neighbourhood of
, where it is equal to
. And in Section 4.3.6 we will see that there exist
particular spacetimes for which
can be
defined globally.
To satisfy all of Properties Sc.H4, Sc.H2, Sc.H3, and Sc.H4 might seem a tall order, but it
should be possible. We first note that Property Sc.H4 is not independent from the rest:
It follows from Property Sc.H2,
Property Sc.H3, and
the reciprocity relation (280) satisfied by the
retarded and advanced Green’s functions. Let
, so that
. Then
by Property Sc.H2, and by Property Sc.H3 this is equal to
. But by the reciprocity relation this is also equal
to
, and we have obtained Property Sc.H4. Alternatively, and this shall be
our point of view in the next paragraph, we can think of
Property Sc.H3 as
following from Properties Sc.H2 and
Sc.H4.
Because satisfies the
homogeneous wave equation (Property Sc.H1), it can be given the Kirkhoff
representation of Equation (283
): If
is a spacelike hypersurface in the past of both
and
, then
where is a surface element on
. The hypersurface can be partitioned into two
segments,
and
, with
denoting the intersection of
with
. To enforce Property
Sc.H4 it suffices to choose for
initial data on
that agree
with the initial data for the advanced Green’s function; because
both functions satisfy the homogeneous wave equation in
, the agreement will be preserved in all of the
domain of dependence of
. The data on
is still free, and it should be possible to choose
it so as to make
symmetric. Assuming that
this can be done, we see that Property Sc.H2 is enforced and we conclude that
the Properties Sc.H1, Sc.H2, Sc.H3,
and Sc.H4 can all be satisfied.
When is restricted to the normal
convex neighbourhood of
, Properties Sc.H1, Sc.H2, Sc.H3, and Sc.H4 imply that
To illustrate the general theory outlined in
the previous Sections 4.3.1, 4.3.2,
4.3.3, 4.3.4,
and 4.3.5, we consider here the
specific case of a minimally-coupled () scalar field in
a cosmological spacetime with metric
To solve Green’s equation we first introduce a reduced Green’s
function
defined by
Substitution of Equation (304) into
Equation (303
) reveals that
must satisfy the homogeneous equation
Equation (305) has
and
as linearly independent solutions, and
must be given by a linear superposition. The
coefficients can be functions of
, and after
imposing Equations (306
) we find that the
appropriate combination is
after integration by parts. The integral evaluates
to .
We have arrived at
for our final expression for the retarded Green’s function. The advanced Green’s function is given instead by The distributionsIt may be verified that the symmetric two-point function
satisfies all of the Properties Sc.H1, Sc.H2, Sc.H3, and Sc.H4 listed in Section 4.3.5; it may thus be used to define singular and radiative Green’s functions. According to Equation (295As a final observation we note that for this
cosmological spacetime, the normal convex neighbourhood of any
point consists of the whole spacetime manifold (which
excludes the cosmological singularity at
). The Hadamard construction of the Green’s functions
is therefore valid globally, a fact that is immediately revealed by
Equations (309
) and (310
).