Covering chapter 8 material from the text .
Covering lecture notes pp. 136-146: continued reminder of electrostatic Greens function (136); the retarded Greens function of the d’Alembert operator: derivation and properties (137-140); the solution of the d’Alembert equation with a source: retarded potentials (141-142)
Solving the forced wave equation.
See the notes for a complex variables and Fourier transform method of deriving the Green’s function. In class, we’ll just pull it out of a magic hat. We wish to solve
(with a gauge choice).
Our Green’s method utilizes
If we know such a function, our solution is simple to obtain
This is the retarded Green’s function of the operator , where
Proof of the d’Alembertian Green’s function
Our Prof is excellent at motivating any results that he pulls out of magic hats. He’s said that he’s included a derivation using Fourier transforms and tricky contour integration arguments in the class notes for anybody who is interested (and for those who also know how to do contour integration). For those who don’t know contour integration yet (some people are taking it concurrently), one can actually prove this by simply applying the wave equation operator to this function. This treats the delta function as a normal function that one can take the derivatives of, something that can be well defined in the context of generalized functions. Chugging ahead with this approach we have
This starts things off and now things get a bit hairy. It’s helpful to consider a chain rule expansion of the Laplacian
In vector form this is
Applying this to the Laplacian portion of 2.6 we have
Here we make the identification
This could be considered a given from our knowledge of electrostatics, but it’s not too much work to just do so.
An aside. Proving the Laplacian Green’s function.
If is a Green’s function for the Laplacian, then the Laplacian of the convolution of this with a test function should recover that test function
We can directly evaluate the LHS of this equation, following the approach in . First note that the Laplacian can be pulled into the integral and operates only on the presumed Green’s function. For that operation we have
It will be helpful to compute the gradient of various powers of
In particular we have, when , this gives us
For the Laplacian of , at the points where this is well defined we have
So we have a zero. This means that the Laplacian operation
can only have a value in a neighborhood of point . Writing we have
Observing that we can put this in a form that allows for use of Stokes theorem so that we can convert this to a surface integral
where we use as the outwards normal for a sphere centered at of radius . This integral is just , so we have
The convolution of with produces , allowing an identification of this function with a delta function, since the two have the same operational effect
Returning to the d’Alembertian Green’s function.
We need two additional computations to finish the job. The first is the gradient of the delta function
Consider . This is
so we have
The Laplacian is similar
so we have
With , we’ll need the Laplacian of this vector magnitude
So that we have
Now we have all the bits and pieces of 2.8 ready to assemble
Since we also have
The terms cancel out in the d’Alembertian, leaving just
Noting that the spatial delta function is non-zero only when , which means in this product, and we finally have
Elaborating on the wave equation Green’s function
The Green’s function 2.26 is a distribution that is non-zero only on the future lightcone. Observe that for we have
We say that is supported only on the future light cone. At , only the contributions for matter. Note that in the “old days”, Green’s functions used to be called influence functions, a name that works particularly well in this case. We have other Green’s functions for the d’Alembertian. The one above is called the retarded Green’s functions and we also have an advanced Green’s function. Writing for advanced and for retarded these are
There are also causal and non-causal variations that won’t be of interest for this course.
This arms us now to solve any problem in the Lorentz gauge
The additional EM waves are the possible contributions from the homogeneous equation.
Since is non-zero only when , the non-homogeneous parts of 3.28 reduce to
Our potentials at time and spatial position are completely specified in terms of the sums of the currents acting at the retarded time . The field can only depend on the charge and current distribution in the past. Specifically, it can only depend on the charge and current distribution on the past light cone of the spacetime point at which we measure the field.
Example of the Green’s function. Consider a charged particle moving on a worldline
( for classical)
For this particle
PICTURE: light cones, and curved worldline. Pick an arbitrary point , and draw the past light cone, looking at where this intersects with the trajectory
For the arbitrary point we see that this point and the retarded time obey the relation
This retarded time is unique. There is only one such intersection.
Our job is to calculate
 L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
 M. Schwartz. Principles of Electrodynamics. Dover Publications, 1987.