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I've been recently digging into SR in my attempts to better understand electromagnetics. One thing that has been giving me trouble is the force transformation laws, which seem to give rise to some pretty gnarly paradoxes, one of which I've listed here:

Consider two objects, where object 2 is moving at a velocity v relative to object 1. There is a moment when both objects just touch, at which time object 1 exerts a force F (perhaps via an expertly-timed spring release) perpendicular to the direction of motion of object 2. Assuming Newton's 3rd law holds, then object 2 exerts a force -F on object 1.

Now consider the same scenario from the inertial frame of object 2, so that object 2 is now at rest, and object 1 moves with velocity -v relative to object 2. According to the force transformation laws, the force exerted by object 1 on object 2 is F/sqrt(1 - v^2/c^2), but the force exerted by object 2 on object 1 is F*sqrt(1 - v^2/c^2). The paradox is that Newton's 3rd law does not seem to be preserved by changes in the inertial frame.

Besides the lack of symmetry, the thing that most troubles me about this paradox is how can we have the conservation of momentum hold between inertial frames and yet Newton's 3rd law fail to hold?

Any answers would be greatly appreciated. But please answer only if you know what you're talking about.

Thanks,

Jeff

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# Another paradox from the force transformation laws?

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