Newton's 3rd Law & Acceleration Paradox

[AdranGarrison]

There are plenty of circumstances in special relativity where Newton's 3rd Law appears to be violated, usually with some simple (or not so simple) solution. But I've been thinking lately about a situation which appears extremely simple, but I can't seem to resolve:

Consider 2 points (A and B) separated by some large distance and not moving with respect to one another. Suppose one places a charged particle at A, and leaves it there for a sufficient length of time for observers at B to see the field. Once observers at point B are able to measure the field at B from A, they place a charged particle at B. This particle will, of course, accelerate under the influence of the field. Observers at point A have not yet received any information about the charge which was placed at point B, and remove the charge from point A before the field from point B can reach them.

Now, what has happened appears to be in violation of either form (strong or weak) of Newton's third law, and by extension, conservation of momentum. If the charge at A accelerated, then information would be carried from B to A faster than light (which we know is impossible). If the charge at A does not accelerate, momentum in our nonaccelerating reference frame has not been conserved (which we also know is impossible). Even if we speculate that the field from the charge at B carried some momentum away as B accelerated, we run afoul of energy conservation. I am sure there is an explanation, but over the last couple of days of thinking about it, I have yet to come up with one.

Oh, and as for the 'placing' and 'removing' of charges being nonphysical, they could just as easily be dipoles or a particle-antiparticle which we allow to annihilate at the appropriate times. I just kept them as single point charges for simplicity.

 

[Bill_K]

You forgot to take into account the momentum stored in the electromagnetic field. Every time you 'place' or 'remove' one of the charges, you must push against the existing field. This modifies the field, and that modification propagates at velocity c.

they could just as easily be dipoles or a particle-antiparticle which we allow to annihilate at the appropriate times. I just kept them as single point charges for simplicity.

Doesn't matter. Charge is also conserved, and if you change the charge at either point it has the same effect regardless of what it's made of.

 

[D H]

Now, what has happened appears to be in violation of either form (strong or weak) of Newton's third law, and by extension, conservation of momentum.

Newton's third law is not universally true. Failure of Newton's third law does not however mean that the conservation laws also fail. You are reasoning along the lines of P implies Q, but P is false, so therefore Q is false. This is an invalid line reasoning called "denying the antecedent". You are ignoring the possibility that the conservation laws can still be true even if Newton's third law isn't.

While it is true that the conservation laws can be derived from Newton's third law, the converse is also true with some additional assumptions. These additional assumptions are

1. Forces act instantaneously.
2. Forces (three-forces) are subject to the superposition principle.
3. Forces can be resolved to interactions between pairs of particles.
4. Forces are invariant with respect to (a) translation and (b) rotation.

Conservation of linear momentum plus assumptions 1, 2, 3, and 4a yield the weak form of Newton's third law; both conservation laws plus all of the assumption yields the strong form of Newton's third law.

Assumptions 4a and 4b are in a sense freebies; the conservation laws themselves result from these assumptions. The first three are anything but freebies. The first two assumptions fail in relativistic mechanics. Forces do not act instantaneously, and the three-force in Newtonian mechanics becomes the four-force in relativistic mechanics. The third assumption fails with some N-body interactions in quantum mechanics.

Here we do not have a problem with denying the antecedent. Fail these assumptions but maintain the conservation laws and Newton's third law is false. Without action at a distance (assumption #1), the field that mediates the force will store momentum. You can't have both Newton's third law and the conservation laws hold true if the field stores momentum. If forces always can be resolved to pairs then you would still see some force in those N-body interactions. You don't. There is no interaction if all of the objects are not present.

Conservation of linear and angular momentum are more fundamental than Newton's laws. Even more fundamental are the nature of space and time themselves. The conservation laws follow from the nature of space and time.

 

[AdranGarrison]

Thank you for the responses. I think I've resolved this to my own mind.

I think that my fundamental misunderstanding ultimately boiled down to trying to think of the field disturbance in the wrong way. The field (and its associated disturbance) "travel" in all directions equally, but the momentum of the field points exclusively antiparallel to the acceleration of the charge (and will, in fact, be absorbed by the charge at A if it is not removed). I think I've been trying really hard to think of a photon as traveling in a nice, well-defined direction, and somehow thinking of a 'field disturbance' as something else entirely (which is quite obviously false). Yes, this is ultimately basic stuff, but my background in optics has always been rather weak, I suppose, and I haven't had to confront this disperity directly until now.