

1.3 Green’s functions in flat
spacetime
To see how Equation (5) can eventually be
generalized to curved spacetimes, I introduce a new layer of
mathematical formalism and show that the decomposition of the
retarded potential into symmetric-singular and regular-radiative
pieces can be performed at the level of the Green’s functions
associated with Equation (1). The retarded
solution to the wave equation can be expressed as
in terms of the retarded Green’s function
. Here
is an arbitrary field point,
is a source point, and
; tensors
at
are identified with unprimed indices, while primed
indices refer to tensors at
. Similarly, the advanced
solution can be expressed as
in terms of the advanced Green’s function
. The retarded Green’s function is zero
whenever
lies outside of the future light cone of
, and
is infinite at these points.
On the other hand, the advanced Green’s function is zero whenever
lies outside of the past light cone of
, and
is infinite at these points.
The retarded and advanced Green’s functions satisfy the reciprocity
relation
this states that the retarded Green’s function becomes the advanced
Green’s function (and vice versa) when
and
are interchanged.
From the retarded and advanced Green’s functions
we can define a singular Green’s function by
and a radiative Green’s function by
By virtue of Equation (8) the singular Green’s
function is symmetric in its indices and arguments:
. The radiative Green’s
function, on the other hand, is antisymmetric. The potential
satisfies the wave equation of Equation (1) and is singular on
the world line, while
satisfies the homogeneous equation
and is well
behaved on the world line.
Equation (6) implies that the
retarded potential at
is generated by a single event in
spacetime: the intersection of the world line and the past light
cone of
’ (see Figure 1). I shall call this
the retarded point associated with
and denote it
;
is the retarded time, the value of the proper-time
parameter at the retarded point. Similarly we find that the
advanced potential of Equation (7) is generated by the
intersection of the world line and the future light cone of the
field point
. I shall call this the advanced point associated with
and denote it
;
is the advanced time, the value of the proper-time
parameter at the advanced point.

