If a hydrogen atom accelerates to near the speed of light, the total velocity of the system must not exceed that limit. Even if the electron has no net angular momentum around the proton, as it could be ejected from the atom by photoelectric effect, it must have a discrete velocity around the proton. If this velocity doesn't change then it is possible as the atom accelerates that the sum of this velocity and the velocity of the whole atom exceeds the speed of light. How does the system respond to this eventuality? By decreasing the angular velocity of the electron? But how?
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Wouldn't time dilation affect the atom? – Adrian Howard Feb 07 '20 at 15:14
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2A hydrogen atom in the lab is ultra-relativistic in some inertial frame. – JEB Feb 07 '20 at 15:20
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Does this answer your question? Do we know why there is a speed limit in our universe? – rghome Feb 07 '20 at 15:38
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2The orbiting electron is not an accurate model, but disregarding that: velocities are not additive in Special Relativity. Velocities cannot accumulate to exceed the speed of light. See the link above. – rghome Feb 07 '20 at 15:39
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@AdrianHoward Am I right? : For the proton time is dilated too but we still see it go same speed.. same for the electron.... A hypotethic spaceship going 0.99c is time dilated but the speed is still 0.99c, not less.... – Krešimir Bradvica Feb 07 '20 at 15:50
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@KrešimirBradvica In the spaceship's frame of reference the atom and c would seem normal, to an external frame, considering the spaceship at 0.99c, it would seem the atoms time frame dilated, giving the relative decrease in angular velocity of the electron. At least this is how I understand it. – Adrian Howard Feb 07 '20 at 16:15
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@rghome As that post has really a lot of long answers give me a day to review them. Thanks. – Krešimir Bradvica Feb 07 '20 at 20:44
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While your example with the electron has certain problems because of quantum mechanics, your general question has a simple answer: In special relativity, if $B$ is moving with velocity $u$ relative to $A$, and $C$ is moving with velocity $v$ relative to $B$, then the velocity of $C$ relative to $A$ is not simply $u + v$. Instead it is given by the relativistic formula for velocity addition $$ \frac{u + v}{1 + \frac{uv}{c^2}} $$ which you can show is always less than $c$.
Elias Riedel Gårding
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Does the angular velocity of the electron in the FoR of the proton decrease when the proton is accelerating? Thanks. – Krešimir Bradvica Feb 07 '20 at 16:29
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Because of quantum mechanics, the concept of "angular velocity" of the electron is not actually well-defined. However, if you were to think about a "classical electron" your question is fine. I'm not sure what would happen while the proton is accelerating, but after it has finished accelerating and is moving with a large constant speed relative to us, everything in the FoR of the proton would be the same. That is relativity: The proton can't know if it is moving or not. – Elias Riedel Gårding Feb 07 '20 at 23:00