Natural Science Forum / Physics / Research / October 2007

On Oct 2, 12:40 pm, lands...@troi.cc.rochester.edu (Steven E.
Landsburg) wrote:

The equations are linear and are therefore determined up to a +/-
solution to the homogeneous equations. That's the 'radiation' part of
the field. However, there is no unambiguous separation of "Coulomb"
part from "radiation" part, since the distinction between the two
varies with a change in frame of reference.

The question, itself, boils down to that of what types of boundary
data suffice to parametrize the solution space of a quasi-linear 2nd
order hyperbolic partial differential equation.

If you take a compact spacetime region and are able to parametrize it
into a series of layers, each layer identified by a value of a
parameter t, with increasing values of t giving you a generally future-
like direction in the spacetime region, then you might reduce the
question to that of what data to take from the slices corresponding to
the extremal values of t, which I'll call t+ and t-; as well as from
the boundaries of the successive snapshots for each t.

Call the regions S_t, where t ranges over [t-, t+]. Their union (union
S_t: t in [t-,t+]) = S is the spacetime region in question. The
boundary dS is the 3-chain S_{t+} - S_{t-} + sum (dS_t: t in [t-,t+]),
the last component giving you all the "sideways" boundary of the
region S.

On each boundary segment, you can configuration data for the field,
and the 1st derivatives with respect to t. If the boundary surfaces
are not characteristic surfaces of the hyperbolic system (i.e. they
are not null surfaces, here, in the context where one is talking about
the light wave equation) then you pick half of the total set of data
from the surfaces S_{t+} and S_{t-}. I don't know exactly which
combinations will yield well-posed problems, but the most common
choices are either (a) to pick all configuration data, only, from both
surfaces; or (b) to pick configuration and velocity data from one of
the two surfaces.

I'm not entirely sure how you deal with the sideways boundary. One way
to remove it from consideration is to restrict attention to regions S
that are saucer-shaped, so that all the snapshot surfaces S_t have a
common boundary dS_t = H (the "horizon"). Then it cancels out from the
equation for dS and you have just: dS = S_{t+} - S_{t-}.

The extreme case of a saucer-shaped region is an Alexandroff interval
-- where both S_{t+} and S_{t-} are, in fact, null surfaces. I'm not
sure how the problem reduces in this case. I think it might actually
be true that you only need the configuration data from just ONE of the
surfaces! So you may be able to get away with just half of what you'd
ordinarily need. However, I'm pretty sure this size reduction won't
work well with the usual quantization procedures. You need two
complementary sets of data to generate the Heisenberg relations.

You need boundary conditions.  If you're dealing with all of space,
and your sources are localized, then it's sensible to demand
that E and B go to zero at spatial infinity.  For static fields,
that's sufficient to give you uniqueness.

If you're not assuming things are static, then it's not true in any
sense that the sources determine the fields.  After all, you can add
to any solution a localized electromagnetic wave packet zooming any
way you like at any location you like.

-Ted

On Oct 2, 1:40 pm, lands...@troi.cc.rochester.edu (Steven E.
Landsburg) wrote:

Get a copy of Panofsky and Phillips and study chapter 1 until it is in
your bones. They prove the following theorem:

Give any vector field V (satisfying natural differentiability
conditions) and let

D = div V
C = curl V

Define

phi(x,y,z) = (1/4pi) integral over all space ( D(X,Y,Z) / R ) dX dY dZ
A(x,y,z) = (1/4pi) integral over all space ( C(X,Y,Z) / R ) dX dY dZ

where R = sqrt ( (x - X)^2 + (y - Y)^2 + (z - Z)^2 ) and the
integration is over X,Y,Z. Then

V = -grad phi + curl A

If s -> 0 and C -> 0 at infinity, then this resolution is unique. The
import of this theorem is - if there are no sources at infinity, then
a vector field is uniquely determined by its divergence and curl. One
cannot overstate the importance of this result for a proper
understanding of EM. Corollary - any non-uniqueness in the resolution
of a vector field in terms of its sources must come from additional
sources at infinity.

-drl

As others have pointed out, the fields are determined by boundary conditions
in space and time. The physically valid boundary conditions add the second
law of thermodynamics, which makes radiation an irreversible process. That
is, a moving charge will always radiate energy. Maxwell's equations would
allow a moving charge to absorb energy in the time-reversed process, but
this would violate the second law.

drl wrote:

My question:

Does the theorem above apply to fields that depend on time or only to
static fields?

What evidence do you have that this is correct?? To me this is quite false
because it is E and B that are measureable quantities and are thus unique.
Only the scalar and vector potentials are not unique. The math for a gauge
transformation is worked out in one of my web pages. If you follow this
you'll see what I mean

http://www.geocities.com/physics_world/em/gauge_transformation.htm

The uniqueness of the fields is proven by the Helmholtz theorem which I made
another web page out of. See
http://www.geocities.com/physics_world/em/helmholtz_theorem.htm

I hope this helps. Let me know if there is anything you object to in my
derivations or if you have any comments for me to respond to. Good luck.

Best regards

Pete