This time we are interested in solving the *inhomogeneous wave
equation* (IWE)
(&nabla#nabla;^2 - 1c^2t)&phis#phi;(,t) =
-&rho#rho;(,t)&epsi#epsilon;_0
(for example) directly, without doing the Fourier transform(s) we did to
convert it into an IHE.

Proceeding as before, we seek a Green's function that satisfies: (&nabla#nabla;^2 - 1c^2 t) G(,t,_0,t_0) = &delta#delta;(- ')&delta#delta;(t - t') . The primary differences between this and the previous cases are a) the PDE is hyperbolic, not elliptical, if you have any clue as to what that means; b) it is now four dimensional - the ``point source'' is one that exists only at a single point in space for a single instant in time.

Of course this mathematical description leaves us with a bit of an
existential dilemma, as physicists. We generally have little trouble
with the idea of gradually restricting the support of a distribution to
a single point in space by a limiting process. We just squeeze it down,
mentally. However, in a supposedly conservative Universe, it is hard
for us to imagine one of those squeezed down distributions of charge
just ``popping into existence'' and then popping right out. We can't
even do it via a limiting process, as it is a bit bothersome to
create/destroy charge out of nothingness even gradually! We are left
with the uncomfortable feeling that this particular definition is
nonphysical in that it can describe *no* actual physical sources -
it is by far the most ``mathematical'' or ``formal'' of the constructs
we must use. It also leaves us with something to understand.

One way we can proceed is to view the Green's functions for the IHE as
being the *Fourier transform of the desired Green's function here!*
That is, we can exploit the fact that:
&delta#delta;(t - t_0) = 12&pi#pi;&int#int;_-&infin#infty;^&infin#infty;e^-i&omega#omega;(t -
t_0) d&omega#omega;
to create a Fourier transform of the PDE for the Green's function:
(+ k^2) G(,_0,&omega#omega;) = &delta#delta;(-
_0)e^i&omega#omega;t_0
(where I'm indicating the explicit
dependence for the moment).

From the previous section we already know these solutions:
G_0(,_0,&omega#omega;) & = &-(k|- _0|)4&pi#pi;|- _0|e^i&omega#omega;t_0

G_+(,_0,&omega#omega;) & = &-e^+ik|- _0|4&pi#pi;|- _0|e^i&omega#omega;t_0

G_-(,_0,&omega#omega;) & = &-e^-ik|- _0|4&pi#pi;|- _0|e^i&omega#omega;t_0
At this point in time ^{13.3} the only thing left
to do is to Fourier transform back - to this point in time:
G_+(,t,_0,t_0) & = & 12&pi#pi;&int#int;_-&infin#infty;^&infin#infty;
-e^+ik|- _0|4&pi#pi;|- _0|e^-i&omega#omega;(t-
t_0) d&omega#omega;

& = & 12&pi#pi; -14&pi#pi;|- _0|
&int#int;_-&infin#infty;^&infin#infty;
-e^+ i&omega#omega;c|- _0|e^-i&omega#omega;(t-t_0) d&omega#omega;

& = & -14&pi#pi;|- _0| ×

& & { 12&pi#pi;
&int#int;_-&infin#infty;^&infin#infty;-( - i &omega#omega;[(t - t_0)
- |- _0|c] ) d&omega#omega;}

& = & -&delta#delta;((t - t_0)
- |- _0|c)4&pi#pi;|- _0|
so that:
G_±(,t,_0,t_0) & = & -&delta#delta;((t - t_0)
&mnplus#mp;|- _0|c)4&pi#pi;|- _0|

G_0(,t,_0,t_0) & = & 12(G_+(,t,_0,t_0) +
G_-(,t,_0,t_0))
Note that when we set
, we basically asserted that
the solution is being defined without dispersion! If there *is*
dispersion, the Fourier transform will no longer neatly line up and
yield a delta function, because the different Fourier components will
not travel at the same speed. In that case one might still expect a
peaked distribution, but not an *infinitely sharp* peaked
distribution.

The first pair are generally rearranged (using the symmetry of the delta function) and presented as:

(13.6) |

and are called the

The second form is a very interesting beast. It is obviously a Green's
function by construction, but it is a symmetric combination of advanced
and retarded. Its use ``means'' that a field at any given point in
space-time
consists of two pieces - one half of it is due to
all the sources in space in the past such that the fields they emit are
contracting precisely to the point
at the instant
and the
other half is due to all of those *same* sources in space in the
*future* such that the fields currently emerging from the point
at
precisely arrive at them. According to this view, the field at
all points in space-time is as much due to the charges in the future as
it is those same charges in the past.

Again it is worthwhile to note that *any actual field configuration*
(solution to the wave equation) can be constructed from *any of
these Green's functions* augmented by the addition of an *arbitrary
bilinear solution* to the homogeneous wave equation (HWE) in primed and
unprimed coordinates. We usually select the retarded Green's function
as the ``causal'' one to simplify the way we think of an evaluate
solutions as ``initial value problems'', not because they are any more
or less causal than the others. Cause may precede effect in human
perception, but as far as the equations of classical electrodynamics are
concerned the concept of ``cause'' is better expressed as one of
interaction via a suitable propagator (Green's function) that may well
be time-symmetric or advanced.

A final note before moving on is that there are simply lovely papers
(that we hope to have time to study) by Dirac and by Wheeler and Feynman
that examine radiation reaction and the radiation field as constructed
by advanced and retarded Green's functions in considerable detail.
Dirac showed that the *difference* between the advanced and retarded
Green's functions *at* the position of a charge was an important
quantity, related to the *change* it made in the field presumably
created by all the *other* charges in the Universe at that point in
space and time. We have a lot to study here, in other words.

Using (say) the usual retarded Green's function, we could as usual write an integral equation for the solution to the general IWE above for e.g. : (,t) = &chi#chi;_A(,t) - &mu#mu;_0 &int#int;_V G_+(,t;',t) (',t') d^3x' dt' where solves the HWE. This (with ) is essentially equation (9.2), which is why I have reviewed this. Obviously we also have &phis#phi;(,t) = &chi#chi;_&phis#phi;(,t) - 1&epsi#epsilon;_0 &int#int;_V G_+(,t;',t) &rho#rho;(',t') d^3x' dt' for (the minus signs are in the differential equations with the sources, note). You should formally verify that these solutions ``work'' given the definition of the Green's function above and the ability to reverse the order of differentiation and integration (bringing the differential operators, applied from the left, in underneath the integral sign).

Jackson proceeds from these equations by fourier transforming back into
a
representation (eliminating time) and expanding the result to
get to multipolar radiation at any given frequency. However, because of
the way we proceeded above, we don't *have* to do this. We could
just as easily start by working with the IHE instead of the IWE and use
our HE Green's functions. Indeed, that's the plan, Stan...