Green's Function for the Wave Equation

This time we are interested in solving the inhomogeneous wave equation (IWE)

$$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2 }{\partial t... ...th$x$},t) = -\frac{\rho(\mbox{\boldmath$x$},t)}{\epsilon_0}$$ (11.52)

(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:

$$\left(\nabla^2 - \frac{1}{c^2} \frac{\partial^2 }{\partial ... ...a(\mbox{\boldmath$x$}- \mbox{\boldmath$x$}')\delta(t - t') .$$ (11.53)

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 dilemna, 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:

$$\delta(t - t_0) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-i\omega(t - t_0)} d\omega$$ (11.54)

to create a Fourier transform of the PDE for the Green's function:

$$\left(\nabla^2+ k^2\right) G(\mbox{\boldmath$x$},\mbox{\bol... ...ta(\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0)e^{i\omega t_0}$$ (11.55)

(where I'm indicating the explicit $\omega$ dependence for the moment).

From the previous section we already know these solutions:

$$G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0,\omega) = \frac{-\cos(k\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert)}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}e^{i\omega t_0}$$ (11.56)

$$G_+(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0,\omega) = \frac{-e^{+ik\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}e^{i\omega t_0}$$ (11.57)

$$G_-(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0,\omega) = \frac{-e^{-ik\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}e^{i\omega t_0}$$ (11.58)

At this point in time ^11.3 the only thing left to do is to Fourier transform back - to this point in time:

$$G_+(\mbox{\boldmath$x$},t,\mbox{\boldmath$x$}_0,t_0) = \frac{1}{2\pi}\int_{-\infty}^\infty \frac{-e^{+ik\vert\mbox{\bold... ...ert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}e^{-i\omega(t- t_0)} d\omega$$ (11.59)

$$= \frac{1}{2\pi} \frac{-1}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\bol... ...vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}e^{-i\omega(t-t_0)} d\omega$$ (11.60)

$$= \frac{-1}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert} \times$$

$$\left\{ \frac{1}{2\pi} \int_{-\infty}^\infty -\exp\left( - i \ome... ...x{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}{c}\right] \right) d\omega \right\}$$ (11.61)

$$= \frac{-\delta\left((t - t_0) - \frac{\vert\mbox{\boldmath$x$}- \m... ...$}_0\vert}{c}\right)}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}$$ (11.62)

so that:

$$G_\pm (\mbox{\boldmath$x$},t,\mbox{\boldmath$x$}_0,t_0) = \frac{-\delta\left((t - t_0) \mp \frac{\vert\mbox{\boldmath$x$}- ... ...$}_0\vert}{c}\right)}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}$$ (11.63)

$$G_0(\mbox{\boldmath$x$},t,\mbox{\boldmath$x$}_0,t_0) = \frac{1}{2}\left(G_+(\mbox{\boldmath$x$},t,\mbox{\boldmath$x$}_0,t_0) + G_-(\mbox{\boldmath$x$},t,\mbox{\boldmath$x$}_0,t_0)\right)$$ (11.64)

Note that when we set $k = \omega/c$, 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:

$$G^{(\pm)}(\mbox{\boldmath$x$},t;\mbox{\boldmath$x$}',t') = ... ...$}' \mid}{c} \right ]}{\mid vx - \mbox{\boldmath$x$}' \mid}$$ (11.65)

and are called the retarded (+) and advanced (-) Green's functions for the wave equation.

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 $(\mbox{\boldmath$x$},t)$ 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 $\mbox{\boldmath$x$}$ at the instant $t$ 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 $x$ at $t$ 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. $\mbox{\boldmath$A$}(\mbox{\boldmath$x$},t)$:

$$\mbox{\boldmath$A$}(\mbox{\boldmath$x$},t) = \chi_A(\mbox{\... ...$}',t) \mbox{\boldmath$J$}(\mbox{\boldmath$x$}',t') d^3x' dt'$$ (11.66)

where $\chi_A$ solves the HWE. This (with $\chi_A = 0$) is essentially equation (9.2), which is why I have reviewed this. Obviously we also have

$$\phi(\mbox{\boldmath$x$},t) = \chi_\phi(\mbox{\boldmath$x$}... ...box{\boldmath$x$}',t) \rho(\mbox{\boldmath$x$}',t') d^3x' dt'$$ (11.67)

for $\phi(\mbox{\boldmath$x$},t)$ (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 $k$ 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...