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 Eq. (14.3
), and
is 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 subsection that is restricted to the normal convex neighbourhood of
, we
make the ansatz
Before we substitute the Green’s functions of Eq. (14.4) into the differential equation of Eq. (14.3
) we
proceed as in Section 12.6 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 Eq. (14.3), the right-hand side of Eq. (14.5
) should be equal to
. This
immediately gives us the coincidence condition
Recall from Section 3.3 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 Eq. (14.7
) therefore represents the logarithmic rate of
change of
along
, and this can be expressed as
. For the second term we recall
from Section 7.1 the differential equation
satisfied by
, the van
Vleck determinant. This gives us
, and Eq. (14.7
)
becomes
It follows that is constant on
, and this must therefore be equal to its value at the starting
point
:
, by virtue of Eq. (14.6
) and the property
of the van Vleck
determinant. Because this statement must be true for all geodesics
that emanate from
, we have
found that the unique solution to Eqs. (14.6
) and (14.7
) is
We must still consider the remaining terms in Eq. (14.5). 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
Eqs. (7.4) and (14.8
) imply that near coincidence,
admits the expansion
Eqs. (14.9) and (14.13
) 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 Eq. (14.8
) and
determined
uniquely by the wave equation of Eq. (14.14
) and the characteristic data constructed with Eqs. (14.9
) and
(14.13
), the retarded and advanced Green’s functions of Eq. (14.4
) do indeed satisfy Eq. (14.3
). It should
be emphasized that the construction provided in this subsection 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 Eq. (14.15To prove the reciprocity relation we invoke the identities
and
and take their difference. On the left-hand side we have
while the right-hand side gives
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 Eq. (14.15
) 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 statement as Eq. (14.18) if we take into account the reciprocity relation of
Eq. (14.15
).
In Part IV 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 regular two-point 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 regular two-point 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 regular on the world line of the moving scalar charge. (We reserve
the term “Green’s function” to a two-point function that satisfies the wave equation with a Dirac
distribution on the right-hand side; when the source term is absent, the object is called a “two-point
function”.)
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 a standing wave at infinity. Removing from the retarded Green’s
function will 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
regular two-point function, which produces a field that is regular 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 subsection we attempt a decomposition of the curved-spacetime retarded Green’s function into
singular and regular pieces. 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 regular 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 [53] and introduce a
singular Green’s function with the properties
Properties S1 and 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 S3
ensures that the support of the regular two-point function will not include the chronological future of
.
The regular two-point function is then defined by
whereProperty R1 follows directly from Eq. (14.22) and property S1 of the singular Green’s function. Properties R2
and R3 follow from S3 and the fact that the retarded Green’s function vanishes if
is in past of
.
The properties of the regular two-point function ensure that the corresponding regular field
will be nonsingular at the world line, and will depend only on the past history of the scalar
charge.
We must still show that such singular and regular Green’s functions can be constructed. This relies on
the existence of a two-point function that would possess the properties
With a biscalar satisfying these relations, a singular Green’s function defined by
The question is now: does such a function exist? We 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 14.6 we will see that there
exist particular spacetimes for which
can be defined globally.
To satisfy all of H1 – H4 might seem a tall order, but it should be possible. We first note that property
H4 is not independent from the rest: it follows from H2, H3, and the reciprocity relation (14.15) satisfied by
the retarded and advanced Green’s functions. Let
, so that
. Then
by H2, and by H3 this is equal to
. But by the reciprocity
relation this is also equal to
, and we have obtained H4. Alternatively, and this
shall be our point of view in the next paragraph, we can think of H3 as following from H2 and
H4.
Because satisfies the homogeneous wave equation (property H1), it can be given the Kirkhoff
representation of Eq. (14.18
): 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 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 the entire 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 H2 is enforced and we conclude that the properties H1, H2, H3, and H4 can all be
satisfied.
When is restricted to the normal convex neighbourhood of
, properties H1 – H4 imply that
To illustrate the general theory outlined in the previous subsections 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 Eq. (14.39) into Eq. (14.38
) reveals that
must satisfy the homogeneous equation
Eq. (14.40) has
and
as linearly
independent solutions, and
must be given by a linear superposition. The coefficients can be
functions of
, and after imposing Eqs. (14.41
) 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 H1 – H4 listed in Section 14.5; it may thus be used to define singular and regular Green’s functions. According to Eq. (14.30 As 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 Eqs. (14.44
) and (14.45
).
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Living Rev. Relativity 14, (2011), 7
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