15 Electromagnetic Green’s functions

15.1 Equations of electromagnetism

The electromagnetic field tensor is expressed in terms of a vector potential . In the Lorenz gauge , the vector potential satisfies the wave equation

where is the wave operator, the Ricci tensor, and a prescribed current density. The wave equation enforces the condition , which expresses charge conservation.

The solution to the wave equation is written as

in terms of a Green’s function that satisfies
where is a parallel propagator and an invariant Dirac distribution. The parallel propagator is inserted on the right-hand side of Eq. (15.3) to keep the index structure of the equation consistent from side to side; because is distributionally equal to , it could have been replaced by either or . It is easy to check that by virtue of Eq. (15.3), the vector potential of Eq. (15.2) satisfies the wave equation of Eq. (15.1).

We 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 spacetime.

15.2 Hadamard construction of the Green’s functions

Assuming throughout this subsection that is in the normal convex neighbourhood of , we make the ansatz

where , are the light-cone distributions introduced in Section 13.2, and where , are smooth bitensors.

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 Eq. (15.3) and then take the limit, we obtain

after a routine computation similar to the one presented at the beginning of Section 14.2. Comparison with Eq. (15.3) returns: (i) the equations
and
that determine ; (ii) the equation
that determines , the restriction of on the light cone ; and (iii) the wave equation
that determines inside the light cone.

Eq. (15.6) can be integrated along the unique geodesic that links to . The initial conditions are provided by Eq. (15.5), and if we set , we find that these equations reduce to Eqs. (14.7) and (14.6), respectively. According to Eq. (14.8), then, we have

which reduces to
near coincidence, with denoting the affine-parameter distance between and . Differentiation of this relation gives
and eventually,

Similarly, Eq. (15.7) 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 Eqs. (15.5), (15.13), and the additional relation . We arrive at

With the characteristic data obtained by integrating Eq. (15.7), the wave equation of Eq. (15.8) admits a unique solution.

To summarize, the retarded and advanced electromagnetic Green’s functions are given by Eq. (15.4) with given by Eq. (15.9) and determined by Eq. (15.8) and the characteristic data constructed with Eqs. (15.7) and (15.14). It should be emphasized that the construction provided in this subsection is restricted to , the normal convex neighbourhood of the reference point .

15.3 Reciprocity and Kirchhoff representation

Like their scalar counterparts, the (globally defined) electromagnetic Green’s functions satisfy a reciprocity relation, the statement of which is

The derivation of Eq. (15.15) is virtually identical to what was presented in Section 14.3, and we shall not present the details. It suffices to mention that it is based on the identities

and

A direct consequence of the reciprocity relation is

the statement that the bitensor is symmetric in its indices and arguments.

The Kirchhoff representation for the electromagnetic vector potential is formulated as follows. Suppose that satisfies the homogeneous version of Eq. (15.1) and that initial values , are specified on a spacelike hypersurface . Then the value of the potential at a point in the future of is given by

where is a surface element on ; is the future-directed unit normal and is the invariant volume element on the hypersurface. The derivation of Eq. (15.17) is virtually identical to what was presented in Section 14.4.

15.4 Relation with scalar Green’s functions

In a spacetime that satisfies the Einstein field equations in vacuum, so that everywhere in the spacetime, the (retarded and advanced) electromagnetic Green’s functions satisfy the identities [54]

where are the corresponding scalar Green’s functions.

To prove this we differentiate Eq. (15.3) covariantly with respect to and use Eq. (13.3) to express the right-hand side as . After repeated use of Ricci’s identity to permute the ordering of the covariant derivatives on the left-hand side, we arrive at the equation

all terms involving the Riemann tensor disappear by virtue of the fact that the spacetime is Ricci-flat. Because Eq. (15.19) is also the differential equation satisfied by , and because the solutions are chosen to satisfy the same boundary conditions, we have established the validity of Eq. (15.18).

15.5 Singular and regular Green’s functions

We shall now construct singular and regular Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 14.5, and the reader is referred to that section for a more complete discussion.

We begin by introducing the bitensor with properties

H1:
satisfies the homogeneous wave equation,
H2:
is symmetric in its indices and 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 ,

It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (15.15) 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 14.5.

Equipped with the bitensor we define the singular Green’s function to be

This comes with the properties
S1:
satisfies the inhomogeneous wave equation,
S2:
is symmetric in its indices and arguments,
S3:
vanishes if is in the chronological past or future of ,

These can be established as consequences of H1 H4 and the properties of the retarded and advanced Green’s functions.

The regular two-point function is then defined by

and it 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 ,

Those follow immediately from S1 S3 and the properties of the retarded Green’s function.

When is restricted to the normal convex neighbourhood of , we have the explicit relations

From these we see clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes (property S3). We see also that the regular two-point function coincides with in (property R2), and that its support does not include (property R3).