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Imagine a big blob of jelly. Imagine poking it on a side. The whole thing
wiggles and distorts, as the force of your poke acts on the entire blob of
jelly. The mathematical mechanism that describes how your poke is distributed
is calle the stress tensor of the material. It tells how energy and
momentum are connected by the medium itself.
The same concept can be generalized to a four dimensional medium, where the
``jelly'' is space time itself. Let us now study what an electromagnetic
stress tensor is, and how it relates to electromagnetic ``pokes''. Recall
that

(17.76) 
is the canonical momentum corresponding to the variable in an arbitrary
Lagrangian. The Hamiltonian is given, in this case, by

(17.77) 
as usual. If
then one can show that
. For four dimensional fields we should probably have a
Lagrangian and Hamiltonian density whose 3integral are the usual Lagrangian
and Hamiltonians. The Hamiltonian is the energy of a particle or system, so
it should transform like the zeroth component of a four vector. Thus, since

(17.78) 
and
, then must transform like the time
component of a second rank tensor. If we define the Hamiltonian density
in terms of the Lagrangian density of a field,
then

(17.79) 
Well, great! The first factor in the sum is the conjugate momentum by
definition, and the second is the generalized ``velocity''. Since
must transform like the time component of a second rank tensor (and the time
derivative appears in this equation) it appears that the covariant
generalization of the Hamiltonian density is something that puts a covariant
derivative there, instead. We try

(17.80) 
This is called the canonical stress tensor, and is related to the stress
tensor defined and studied in Chapter 6. This tensor has the covariant
function of telling us how the energy and momentum carried by the
electromagnetic field transform.
What is this tensor? It is, in fact, highly nontrivial. The best we can do
is note that if we assume that only free fields are present and
that the free fields are localized in some finite region of space (neither
assumption is particularly physical), then we can show that

(17.81) 
and

(17.82) 
which are the ``usual'' expressions for the energy and momentum of the free
field. At least if I got the change to SI units right...
What, you might ask, is this good for? Well, aside from this correspondance
(which is full of holes, by the way), we can write the energymomentum
conservation law

(17.83) 
This is proven in Jackson, with a discussion of some of its shortcomings.
One of these is that it is not symmetric. This creates difficulties when we
consider the angular momentum carried by the field. Since the angular
momentum density is important when we go to create photons (which must have
quantized angular momenta), it is worthwhile to construct the symmetric
stress tensor

(17.84) 
in terms of which we can correctly construct a covariant generalization
of the energy momentum conservation law

(17.85) 
and the angular momentum tensor

(17.86) 
which is therefore conserved. This form of the stress tensor can also be
directly coupled to source terms, resulting in the covariant form of the work
energy theorem for the combined system of particles and fields.
This is about all we will say about this at this time. I realize that it is
unsatisfactory and apologize. If we had one more semester together, we could
do it properly, but we don't. Therefore, it is on to
Next: Covariant Green's Functions
Up: Relativistic Dynamics
Previous: Building a Relativistic Field
Contents
Robert G. Brown
20140819