### 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.