June 3, 2008

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 shift mass effortlessly (no forces, like "teleportation") 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 shifted 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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