We consider a massless scalar field in a curved spacetime with metric . The field satisfies the wave equation

where is the wave operator, the Ricci scalar, an arbitrary coupling constant, and is a prescribed source. We seek a Green’s function such that a solution to Eq. (14.1) can be expressed as where the integration is over the entire spacetime. The wave equation for the Green’s function is where is the invariant Dirac functional introduced in Section 13.1. It is easy to verify that the field defined by Eq. (14.2) is truly a solution to Eq. (14.1).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

where and are smooth biscalars; the fact that the spacetime is no longer homogeneous means that these functions cannot depend on alone.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

for the biscalar . To eliminate the term we make its coefficient vanish: As we shall now prove, these two equations determine uniquely.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

where we indicate that the right-hand side also must be restricted to the light cone. The first term of Eq. (14.9) can be expressed as and this equation can be integrated along any null geodesic that generates the null cone . For these integrations to be well posed, however, we must provide initial values at . As we shall now see, these can be inferred from Eq. (14.9) and the fact that must be smooth at coincidence.Eqs. (7.4) and (14.8) imply that near coincidence, admits the expansion

where is the Ricci tensor at and is the affine-parameter distance to (which can be either on or off the light cone). Differentiation of this relation gives and eventually, Using also , we find that the coincidence limit of Eq. (14.9) gives and this provides the initial values required for the integration of Eq. (14.9) on the null cone.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

which is obtained by elimination of the term in Eq. (14.5). While this certainly does not constitute a practical method to compute the biscalar , these considerations show that exists and is unique.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. (14.15), the biscalar must be symmetric in its arguments: To go from Eq. (14.15) to Eq. (14.16) we simply note that when and belongs to , then and .To 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

Then the value of the field at a point in the future of is given by Kirchhoff’s formula, where is the surface element on . If is the future-directed unit normal, then , with denoting the invariant volume element on ; notice that is past directed.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

- S1:
- satisfies the inhomogeneous scalar wave equation,
- S2:
- is symmetric in its arguments,
- S3:
- vanishes if is in the chronological past or future of ,

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

where is the retarded Green’s function. This comes with the properties- R1:
- satisfies the homogeneous wave equation,
- R2:
- agrees with the retarded Green’s function if is in the chronological future of ,
- R3:
- vanishes if is in the chronological past of ,

Property 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

- H1:
- satisfies the homogeneous wave equation,
- H2:
- is symmetric in its arguments,
- H3:
- agrees with the retarded Green’s function if is in the chronological future of ,
- 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

will satisfy all the properties listed previously: S1 comes as a consequence of H1 and the fact that both the advanced and the retarded Green’s functions are solutions to the inhomogeneous wave equation, S2 follows directly from H2 and the definition of Eq. (14.30), and S3 comes as a consequence of H3, H4 and the properties of the retarded and advanced Green’s functions.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

it should be stressed here that while is assumed to be defined globally in the entire spacetime, the existence of is limited to . With Eqs. (14.4) and (14.30) we find that the singular Green’s function is given explicitly by in the normal convex neighbourhood. Equation (14.32) shows very clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes , in which (property S3). From Eq. (14.22) we get an analogous expression for the regular two-point function: This reveals directly that the regular two-point function coincides with in , in which and (property R2), and that its support does not include , in which (property R3).

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

where is the scale factor expressed in terms of conformal time. For concreteness we take the universe to be matter dominated, so that , where is a constant. This spacetime is one of the very few for which Green’s functions can be explicitly constructed. The calculation presented here was first carried out by Burko, Harte, and Poisson [33]; it can be extended to other cosmologies [86].To solve Green’s equation we first introduce a reduced Green’s function defined by

Substitution yields where is a vector in three-dimensional flat space, and is the Laplacian operator in this space. We next expand in terms of plane-wave solutions to Laplace’s equation, and we substitute this back into Eq. (14.36). The result, after also Fourier transforming , is an ordinary differential equation for : where . To generate the retarded Green’s function we set in which we indicate that depends only on the modulus of the vector . To generate the advanced Green’s function we would set instead . The following manipulations will refer specifically to the retarded Green’s function; they are easily adapted to the case of the advanced Green’s function.Substitution of Eq. (14.39) into Eq. (14.38) reveals that must satisfy the homogeneous equation

together with the boundary conditions Inserting Eq. (14.39) into Eq. (14.37) and integrating over the angular variables associated with the vector yields where and .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

Substituting this into Eq. (14.42) and using the identity yieldsafter 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 distributions are solutions to the reduced Green’s equation of Eq. (14.36). The actual Green’s functions are obtained by substituting Eqs. (14.44) and (14.45) into Eq. (14.35). We note that the support of the retarded Green’s function is given by , while the support of the advanced Green’s function is given by .It 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) the singular Green’s function is given by and its support is limited to the interval . According to Eq. (14.22) the regular two-point function is given by its support is given by and for the regular two-point function agrees with the retarded Green’s function.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).

Living Rev. Relativity 14, (2011), 7
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