The electromagnetic field tensor is expressed in terms of a vector potential . In the Lorenz gauge , the vector potential satisfies the wave equation
The solution to the wave equation is written as
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.
Assuming throughout this subsection that is in the normal convex neighbourhood of , we make the ansatz
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
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
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
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 .
Like their scalar counterparts, the (globally defined) electromagnetic Green’s functions satisfy a reciprocity relation, the statement of which is
A direct consequence of the reciprocity relation is
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
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 
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
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
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
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
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 relationsS2), 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).
Living Rev. Relativity 14, (2011), 7
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