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
in terms of a Green’s functionWe 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 section 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 Equation (316
) and then take the
limit, we obtain
Equation (319) can be integrated
along the unique geodesic
that links
to
. The initial conditions are provided by
Equation (318
), and if we set
, we find that these
equations reduce to Equations (272
) and (271
), respectively.
According to Equation (273
), then, we have
Similarly, Equation (320) 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 Equations (318
), (326
), and the additional
relation
. We arrive at
To summarize, the retarded and advanced
electromagnetic Green’s functions are given by Equation (317) with
given by Equation (322
) and
determined by Equation (321
) and the
characteristic data constructed with Equations (320
) and (327
). It should be
emphasized that the construction provided in this section 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
The derivation of Equation (328and
A direct consequence of the reciprocity relation is
the statement that the bitensorThe Kirchhoff representation for the
electromagnetic vector potential is formulated as follows. Suppose
that satisfies the homogeneous version of Equation (314
) 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
We shall now construct singular and radiative Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 4.3.5, and the reader is referred to that section for a more complete discussion.
We begin by introducing the bitensor with properties
Em.H1: satisfies the homogeneous wave
equation,
Em.H2: is symmetric in its indices and arguments,
Em.H3: agrees with the retarded Green’s
function if
is in the chronological future of
,
Em.H4: agrees with the advanced Green’s function if
is in the chronological past of
,
It is easy to prove that Property Em.H4 follows from
Property Em.H2,
Property Em.H3,
and the reciprocity relation (328) 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 4.3.5.
Equipped with the bitensor we define the singular Green’s function to be
Em.S1: satisfies the inhomogeneous wave
equation,
Em.S2: is symmetric in its indices and arguments,
Em.S3: vanishes if
is in the chronological past
or future of
,
These can be established as consequences of Properties Em.H1, Em.H2, Em.H3, and Em.H4, and the properties of the retarded and advanced Green’s functions.
The radiative Green’s function is then defined by
and it comes with the properties Em.R1: satisfies the homogeneous wave
equation,
Em.R2: agrees with the retarded Green’s function if
is in the chronological future of
Em.R3: vanishes if
is in the chronological past
of
,
Those follow immediately from Properties Em.S1, Em.S2, and Em.S3 and the properties of the retarded Green’s function.
When is restricted to the normal
convex neighbourhood of
, we have the explicit
relations