Imagine a closed electromagnetic system of charges interacting with one another at some time t. The field at every point can be written as the superposition of the Lienard-Wiechert field from every charge q_n at retarded time t_n = t - R_n/c.
Now consider a time reversed system of the above via: dt -> -dt, j -> -j, B -> -B. At the same time t, the retarded time t'' = t - R*_n/c. In particular, R_n and R*_n are generally different for each charge q_n since their trajectories for the twocases are generally different. But this now creates a problem for my original assumption that the system is time reversible!
A solution is to add another field that isn't coupled to the sources and hence is "free".
This then alters the trajectories of the charges to ensure the system is time reversible at all t,
but at the expense of no longer being coupled to the sources, becoming free radiation.
Any comments correcting or adding to the above would be appreciated.
On 20/06/30 1:41 AM, john mcandrew wrote:
Imagine a closed electromagnetic system of charges interacting with one another at some time t. The field at every point can be written as the superposition of the Lienard-Wiechert field from every charge q_n at retarded time t_n = t - R_n/c.
Your system is not "closed" unless you also give it a surrounding box,
or some other hull, to keep any radiation from escaping. If you don't do
that there will constantly be radiation escaping to infinity and also
you have to add mechanical energy to your radiating charges all the
time. I wouldn't call that a closed system.
cases are generally different. But this now creates a problem for my original assumption that the system is time reversible!Now consider a time reversed system of the above via: dt -> -dt, j -> -j, B -> -B. At the same time t, the retarded time t'' = t - R*_n/c. In particular, R_n and R*_n are generally different for each charge q_n since their trajectories for the two
Why? Logically the retarded time of the first system would become the *advanced* time for the reversed system. No reason why their reversed
times need to be related.
A solution is to add another field that isn't coupled to the sources and hence is "free".
How would that solve a problem with the defined retarded time? It just
is what it is: the particle's time where it crosses the retarded light
cone seen from the observer.
This then alters the trajectories of the charges to ensure the system is time reversible at all t,
But it already was time-reversable, you actually did reverse it! (Or
else your whole story about the "second" system that we get by time-
reversal makes no sense.. Anyway we know that EM is time reversible,
so that discussion seems irrelevant.)
but at the expense of no longer being coupled to the sources, becoming free radiation.
There *never was* a rule that EM radiation should be "coupled to
sources" in general. If there is such a coupling in a certain system
then that is a special case, but then don't be surprised if it is
not the case in the time-reversed system.
Any comments correcting or adding to the above would be appreciated.
My correction would be that the system simply *is* time-reversible,
that should not be the discussion to begin with. Apart from that, your conclusion seems basically valid: this system has sources for all its radiation when viewed in forward time, but does not have such a relation
in backward time. (But in backward time it has a "sink" for all its
occurring radiation!)
--
Jos
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