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energy etc. in the fields



Hugh Logan wrote:
When a baseball is thrown, it is
difficult for me to think of something like a fluid corresponding to the
baseball's energy flowing.

I replied:

That's a good point. That situation doesn't really correspond
to a flow of energy from region A to region B ... the energy is
just transformed in place, transformed from KE to PE and back
again. Since there's no spatial separation, flow images don't
really apply ... and I'm not sure what images should apply.
Maybe red fluid turning into blue fluid and back again?? I like
Bob's suggestion of pennies flipping heads <--> tails ... I
wonder if we could somehow refine it to give some control of
the PE/KE ratio.

Upon reflection, I think I ought to retract much of what I said,
and clarify the rest.

First, let's consider a slightly different question, namely a
charged ball moving under the influence of an electrostatic
field (instead of gravity). The simplest approach would be to
associate the electostatic potential energy with the charged
particle ... but a more satisfactory approach is to "locate"
the energy in the field, with an energy density proportional
to field^2.

If you extend this idea to include magnetostatics it gets a
bit more complicated, but it doesn't break. You can further
extend it to time-varying fields. You can even construct an
elegant Lagrangian density that behaves properly under Lorentz
transformations.

It is easy to construct situations where you are _required_
to attribute energy and momentum to the EM field ... for
example, the apparatus shown in Feynman volume II figure
17-5. (The resolution of the "paradox" is presented at the
end of chapter 27.)

So in the EM case, it seems 100% OK to visualize the KE <--> PE
exchange as a flow between the object (where the KE resides) and
the field (where the PE resides). In typical cases it probably
isn't worth visualizing the flow in super-fine detail, but the
general idea is sound, and you can make it work in detail if you
really care.

Now for a ball moving under the influence of a _gravitational_
field, things get a bit weird. We can use the analogy between
electrostatics and Newtonian gravity ... but beware that for
gravity, the field^2 has a _negative_ energy density, because
gravitation is attractive. Lots of references surrender at this
point, declaring a negative energy density to be unphysical, but
(pre general relativity), there is a gauge symmetry that tells
us that only energy _differences_ matter, and we can take the
difference of negative numbers just as easily as of positive
numbers.

So as far as I can tell, in an introductory course (Newtonian
gravity, not general relativity) you can get away with locating
the gravitational potential energy in the field.

However, this won't withstand very much scrutiny. People
have come up with many different schemes for attributing a
relativistically-correct energy density to the gravitational
field, but none (as far as I've heard) work very well except
in narrow special cases. To get some appreciation for the
real problem, consider that at any given point in space, there
will be plenty of freely-falling reference frames that measure
zero gravitational field at that point. So an energy density
proprtional to field^2 would depend not just on the velocity of
the observer (as might be described by a Lorentz transformation)
but would crucially depend on the acceleration of the observer.
It's a mess.

So ... for an introductory course, AFAICT it's best to just say
that the gravitational potential energy is (m g h) ... and
explicitly duck the question of _where_ this energy is stored.
It would be nice if I had a concrete picture of where the energy
is stored ... but I don't absolutely need one. The (m g h)
formula is all I really need. Hypotheses non fingo.

=========================

Additional pedagogical challenges arise when describing momentum
flow.

Each component of momentum obeys a strict local conservation law,
and therefore can be described in terms of flow. In particular,
the gravitational force is entirely equivalent to a flow of
z-momentum (pz) from the baseball to the earth.

The defining property of conservative flow is that whatever
disappeared from one region must have flowed across the boundary
into an adjoining region. Alas Newtonian gravitation (like
electrostatics) involves action-at-a-distance, so the pz seems
to leap across a possibly-vast distance, without in any very
obvious way flowing through the intervening space. Leaping to
a non-adjoining region is not a *local* conservation law. In
contrast, if the baseball were attracted to the earth by the
force of a rubber band, it would be easy to visualize the pz
flowing through the rubber band. This is *not* a problem with
the conservation law as expressed in terms of flow ... rather
it is a problem with Newtonian gravitation. Action-at-a-distance
is fundamentally unphysical, and can be tolerated only as an
approximation, valid in the limiting case where the timescales
(t) are not too short and the distances (x) are not too vast,
i.e. x/t << c.

=====

Also at one point I spoke of treating energy as an incompressible
indestructible fluid. The "incompressible" part is completely
wrong. Sorry.