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:
To prove the reciprocity relation we invoke the identities
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  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
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 bySc.S1 comes as a consequence of Property Sc.H1 and the fact that both the advanced and the retarded Green’s functions are solutions to the inhomogeneous wave equation, Property Sc.S2 follows directly from Property Sc.H2 and the definition of Equation (295), and Property Sc.S3 comes as a consequence of Properties Sc.H3, Property Sc.H4 and the properties of the retarded and advanced Green’s functions.
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 thatSc.S2), and that its support excludes , in which (Property Sc.S3). From Equation (287) we get an analogous expression for the radiative Green’s function: Sc.R2), and that its support does not include , in which (Property Sc.R3).
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; it can be extended to other cosmologies.
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
It may be verified that the symmetric two-point functionSc.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 (295) the singular Green’s function is given by
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 Equations (309) and (310).
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