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My knowledge of atomic dynamics is a little superficial. But to my understanding an electron travels an orbital path around a nucleus of an atom. "correct?" So let's say that if a hydrogen atom were to travel "extremely close to" the speed of light (relative to some laboratory system). With the electron circling the proton approximately already traveling "extremely close to" the speed of light, how is it that the electron can circle the proton without exceeding light speed? (or does it continue to orbit at all?)

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In the rest frame of the atom there are of course no changes, so I assume you're asking what the atom will look like to the stationary observer watching the moving atom.

First note that electrons don't orbit the atom like planets orbiting a star. The electrons in atoms exist as a delocalised probability distribution. This distribution can have a non-zero angular momentum, but that can't be simply interpreted as the electrons circling the nucleus.

Can I answer what I think is the spirit of your question and consider a solar system moving at nearly the speed of light relative to us. That is a lot clearer because (viewed in the plane of the ecliptic) the planets are unambiguously moving towards and away from us as they orbit. The answer is that at relativistic speeds velocities don't just add. So if the solar system is moving at $0.999c$ and the orbital velocity of the planet is $0.002c$ the planet is not moving away from us at $1.001c$. The relativistic formula for adding velocities $u$ and $v$ is:

$$ V = \frac{u + v}{1 + uv/c^2} $$

In limit of $u$ and $v$ much less than $c$ this simplifies to the usual $V \approx u + v$, but if you do the sums for very high velocities you'll find the combined velocity can never exceed $c$.

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To emphasize John's first point: What the difference between “orbital” and “orbit”?. –  dmckee Feb 12 at 14:49
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The situation you described gives evidence that, if we really believe nothing can travel faster than the speed of light, the usual view of an electron actually orbiting the nucleus (like a planet orbits a star) can't be correct.

The model of the atom you described, with electrons moving around the nucleus, also has other problems. Physicists noticed these problems and responded by creating new theories to address them. They did this in stages.

First came non-relativistic quantum mechanics (usually just called quantum mechanics). In this model, the electron doesn't orbit the atom in a definite path like the planets do. The electrons's location can only be predicted probabilistically. One common view is a cloud; where the cloud is thickest, the electron is more likely to be found.

Next came, quantum electrodynamics, which is more applicable than just plain quantum mechanics because it takes special relativity into account. This is what physicists use today when dealing with electrical interactions on the particle scale. It's similar to Q.M. in that electrons have a probabilistic location. Beyond that I won't go into detail, mainly because I don't know it well. But I think it's safe to say that the "shape" of the probability cloud changes when the atom moves faster.

While I didn't answer your question directly, the biggest thing to take away is that the view of the electron orbiting a nucleus leads to problems like the one you thought of, so that view is no longer thought to be an accurate description of an atom. Instead, physicists invented a new model that doesn't have the problem you described.

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There's nothing logically inconsistent about a universe with Bohr-style atoms obeying relativity. The OP's conceptual problem lies in Galilean vs. Lorentzian shifting of reference frames. –  Chris White Feb 13 at 2:30
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