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In Special Relativity, when length contracts inside an atomic Nucleus according to an independent observer, it is my understanding that the contraction does not affect the velocity of an electron. Yet, it seems that potential energy is gained by the electron by virtue of it getting closer to the nucleus, how does this not violate the conservation of energy? Where is the misconception/flaw in this logic?

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  • $\begingroup$ Does this really need to be atomic scale or does the question make sense at macroscopic scale? If it can be posed at macroscopic scale then it probably should, so that you can avoid quantum mechanical issues. (In particular, quantum mechanics stabilizes the "orbit" of the electron, which would classically need to emit radiation by virtue of its acceleration.) $\endgroup$ – Ian Sep 25 '17 at 0:31
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In Special Relativity, when length contracts inside an atomic Nucleus according to an independent observer,

This is a misunderstanding.

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame

The length does not contract as far as the atom is conserned in its center of mass. The observer sees a contraction because of the relative motion.

it is my understanding that the contraction does not affect the velocity of an electron.

The electron does not have a measurable velocity because the atom is quantum mechanical. Let us assume the (wrong) Bohr model . In its rest system the atom does not know it is moving so the potential solutions are the same.

Yet, it seems that potential energy is gained by the electron by virtue of it getting closer to the nucleus,

If you wanted to solve for the Bohr atom in the system where the observer is at rest, you would need a lot of lorenz tranformed quanties, including that of the potential , you will find that the potential also changes, but it is not a very wise decision in terms of computing time, as Lorenz transformations tell us that still, at the rest system the solutions for velocity are invariant.

how does this not violate the conservation of energy? Where is the misconception/flaw in this logic?

Conservation of energy will not be violated, due to the Lorenz covariance of the systems, conservation of "simplicity in calculations" will be.

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When an accelerating spaceship is towing another spaceship, the towing rope has potential energy as it is under stress. The rope is getting shorter, so it is losing potential energy.

The speed of the towed spaceship is the speed of towing spaceship + contraction speed of the rope.

Now we have two problems: 1: Were does the potential energy of the rope go? 2: Were does the extra kinetic energy of the towed spaceship come from?

The solution to those two problems is obvious.

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