Here's a paradox about work-driven buildup of mass and conservation of
angular momentum. Give it a try.
In everyday mechanics, in order to redistribute mass we have to actually
move it from one place to another. However, the equivalency of mass and
energy complicates that issue. For example, we could convert the impact
energy U (I use U so E can be a field) of a falling mass m1 hitting the
floor into mass m2: m2 = m1gh/c^2. Hence potential energy can of course
be converted into mass, not just actual energy. Note that we could
violate conservation of angular momentum (sum of r cross p) if we could
just ****ft mass effortlessly (no forces, like "tele****tation") even if the
total did stay the same. That's because in a frame of ref. where the mass
is moving, its linear momentum vector would be ****fted sideways to itself.
(Thus changing the r cross p with no compensation.)
I am aware of various sorts of compensation etc. in apparently paradoxical
situations, but I imagined a thought experiment that I can't solve to
maintain CoAM. Have a line charge along "x." Have also two square
"solenoids" S1 and S2 with same sense of current and sides equal to Y.
All three lie in the same plane, with one solenoid centered at coordinate
y1 and the other at y2 = -y1. (Being lazy at constructing ASCII diagrams
has sharpened my verbal descriptions.) One one side of each solenoid ,
the current is being "pushed" in the direction of field E, and on the
other, the current is fighting against E. It helps the following if you
imagine not a literal current of electrons, but a mechanically driven belt
of little charged bodies: On the side of each where E is favorable to the
"current ", mass-energy builds up at a rate dm/dt = IEY/c^2. On the
unfavorable side, mass-energy is lost at a rate dm/dt = -IEY/c^2 (if abs.
vals used for the variables.) That already looks like a problem per the
previous discussion, but we usually consider such issues solved by the AM
etc. of the fields. (Note Feynman's paradox of the charged wheel, etc.)
Some say there's an "energy current" between the sides (see
Taylor/Wheeler, _Spacetime Physics_ etc.) , but how does that really
work?
However, the real test (?) of there being a problem is whether it is
reversible. Hence, let's move S1 and S2 respectively away from the line
charge at low velocity. Now, once they're accelerated, we have for the
rate of change of angular momentum L: dL/dt = rv dm/dt, using proper
signs in vector notation. If you check, you'll find that there's a net
change of L as S1 and S2 move to a distance from the line (same sign of
build ups at opposite sides as seen by observer looking at plane, times
opposite r and v, gives same dL/dt for S1 and S2.) I can't find an
influence on the wire from their motion that would compensate the right
amount. Then, at a distance, you can switch the direction of current in
the solenoids and again no net effect. Then, move S1 and S2 towards the
line charge, and reversed dm/dt and reversed v makes the same dL/dt as
before. Lather, rinse, repeat; I don't yet see how to foil it.
Give solving it a try, you may even get help from offbeat angles like
stress corrections in the solenoids etc.
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