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We have defined the retarded time

(522) 
as the latest time at which a light signal emitted from position would
reach position before time . We have also shown that a solution to Maxwell's equations
can be written in terms of retarded potentials:

(523) 
etc.
But, is this the most general solution? Suppose that we define the advanced time.

(524) 
This is the time a light signal emitted at time from position would reach position
. It turns out that we can also write a solution to Maxwell's equations in
terms of advanced potentials:

(525) 
etc. In fact, this is just as good a solution to Maxwell's equation as the one involving retarded
potentials. To get some idea what is going on, let us examine the Green's function corresponding
to our retarded potential solution:

(526) 
with a similar equation for the vector potential. This says that the charge density present
at position and time emits a spherical wave in the scalar potential which propagates forwards in time. The Green's function corresponding to our advanced potential
solution is

(527) 
This says that the charge density present
at position and time emits a spherical wave in the scalar potential which propagates backwards in time. ``But, hang on a minute,'' you might say, ``everybody knows that electromagnetic
waves can't travel backwards in time. If they did then
causality would be violated.'' Well, you know that electromagnetic waves do not
propagate backwards in time, I know that electromagnetic waves do not propagate
backwards in time, but the question is do Maxwell's equations know this? Consider the wave equation
for the scalar potential:

(528) 
This equation is manifestly symmetric in time (i.e., it is invariant under the transformation
). Thus, backward traveling waves are just as good a solution to this
equation as forward traveling waves. The equation is also symmetric in space (i.e., it
is invariant under the transformation
). So, why do we adopt the Green's
function (526) which is symmetric in space (i.e., it is invariant under
)
but asymmetric in time (i.e., it is not invariant under
)? Would it not
be better to use the completely symmetric Green's function

(529) 
In other words, a charge emits half of its waves running forwards in time (i.e.,
retarded waves), and the other half running backwards in time (i.e., advanced waves).
This sounds completely crazy! However, in the 1940's Richard P. Feynman and John A. Wheeler
pointed out that under certain circumstances this prescription gives the right answer. Consider
a charge interacting with ``the rest of the Universe,'' where the ``rest of the Universe''
denotes all of the distant charges in the Universe, and is, by implication, an awful long way away
from our original charge. Suppose that the ``rest of the Universe'' is a perfect reflector of
advanced waves and a perfect absorbed of retarded waves. The waves emitted by the charge can
be written schematically as

(530) 
The response of the rest of the universe is written

(531) 
This is illustrated in the spacetime diagram Fig. 40.
Here, and denote the advanced and retarded waves emitted by the charge, respectively.
The advanced wave travels to ``the rest of the Universe'' and is reflected: i.e., the
distant charges oscillate in response to the advanced wave and emit a retarded wave ,
as shown. The retarded wave is spherical wave which
converges on the original charge, passes through the charge, and then diverges again. The
divergent wave is denoted . Note that looks like a negative advanced wave emitted by
the charge, whereas looks like a positive retarded wave emitted by the charge. This is
essentially what Eq. (531) says. The retarded waves and are absorbed by
``the rest of the Universe.''
Figure 40:

If we add the waves emitted by the charge to the response of ``the rest of the Universe''
we obtain

(532) 
Thus, charges appear to emit only retarded waves, which agrees with our everyday experience.
Clearly, in this model we have sidestepped the problem of a time asymmetric Green's function
by adopting time asymmetric boundary conditions to the Universe: i.e., the distant charges in the
Universe absorb retarded waves and reflect advanced waves. This is possible because the
absorption takes place at the end of the Universe
(i.e., at the ``big crunch,'' or whatever) and the reflection takes place at
the beginning of the Universe (i.e., at the ``big bang''). It is quite plausible that the
state of the Universe (and, hence, its interaction with electromagnetic
waves) is completely different at these two times. It should be pointed out that the FeynmanWheeler
model runs into trouble when one tries to combine electromagnetism with quantum mechanics.
These difficulties have yet to be resolved, so at present the status of this model
is that it is ``an interesting
idea,'' but it is still not fully accepted into the canon of physics.
Next: Retarded fields
Up: Timedependent Maxwell's equations
Previous: Retarded potentials
Richard Fitzpatrick
20060202