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Let's imagine a boat on a lake. Observer A is sitting on the shore. Observer B is sitting in the boat on the bow. Observer B has a ball attached to the end of a string which he holds in his hand.

Observer A sees that the boat is not moving. Boat B has some mass. So does the ball. Let's ignore water/air friction.

Then observer B slowly moves towards stern. The boat moves appropriately in the opposite direction, as A can see. Then observer B stops, so does the boat.

Finally, the observer B goes back to the bow. Because of the law of conservation of momentum, in the end observer A sees the situation exactly as it was in the beginning.

Now let's repeat the situation, but before Observer B is coming back from the stern to the bow he starts spinning the ball on the string with sub-light speed.

Because of the speed, according to the special relativity, both observers should observe increased mass of the ball.

For observer A on the shore it would be like the observer in the boat carries a lightweight ball in one direction and then a heavy ball in the other direction.

In result, he would observe that the boat moved as compared to the initial position. This in turn would violate the momentum conservation law, so I conclude it is not going to happen.

Why? Isn't the increased mass of the spinning ball be perceived by observer A and B?

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  • $\begingroup$ @ACuriousMind I feel like your use of the term "conservation" here is misleading. An isolated system in SR conserves both energy and momentum separately in all inertial frames, which is equivalent to saying four-momentum is conserved. Perhaps you meant to say something about relativistic invariants instead of about conservation? $\endgroup$ – joshphysics Aug 11 '14 at 17:25
  • $\begingroup$ @joshphysics: Yeah. Reading that, I realize it is not at all clear what I wanted to say. I'm not that sure myself anymore. I'll delete that. $\endgroup$ – ACuriousMind Aug 11 '14 at 17:27
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The extra mass-energy that goes into the spinning ball has to come from somewhere. If it comes from observer B, then his mass decreases by the same amount that the mass of the ball+string increased.

Incidentally, observer A seems irrelevant here. Lorentz invariance means that you can pick a single reference frame and use that. You don't need more than one unless it somehow makes solving the problem simpler, or you're trying to show an inconsistency in special relativity by showing a difference between the frames, but there doesn't seem to be a difference here. The (time-averaged) mass of the ball increases in both frames, as you said.

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