If a proton travels exactly at 99% of speed of light are the individual velocities of the quarks inside it also exactly 99% of $c$? So can we think that they all travel linearly as the center of mass of the proton or they make some extra path due to rotation inside the proton? In that case could the quarks trajectory be approximated with the form of a coil, spring?
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2 Answers
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The quarks inside a proton do not have well-defined trajectories. The reason is basically the same set of problems as the “planetary orbits” of electrons in the Bohr model, only more so because the energies associated with the quarks make them highly relativistic.
All protons are indistinguishable in their rest frames. A moving proton is entirely the same particle as a stationary one.
A circular path, viewed from a moving frame, turns into a helix if the motion is parallel to the axis of the circle. If the axis and the motion are perpendicular, you get one of the points on the “relativistic wheel,” which has a surprisingly complicated shape.
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I suppose the answer to this question is "No" because the quarks certainly don't travel in a spiral. The problem with answering "No" is that gives the impression that the question makes sense in the first place.
It is a very classical question, and there is nothing classical about a proton. @rob's answer invokes the Bohr model of hydrogen, which is a good start for considering a quantum system. Looking at a hydrogen atom moving at 99% $c$, the electron does not orbit, rather it has its wave function distributed with spherical symmetry around the proton.
In the frame where it is moving, the $S$ orbital is Lorentz contracted into an oblate spheroid, and time dilated too. Whatever that traces out, it is not a spiral.
Note also: to use the language of old fashioned perturbation theory, about $\alpha^2\approx 0.01$% of the time, there are 2 electrons and a position in the atom, thanks to QED. Which electron are you tracking? idk, maybe you can tell them apart.
Now in the proton, there are no orbitals nor wave functions; rather, there are parton distribution in which quarks (distinguished by flavor) carry a fraction of the proton momentum. That causes problem for the spiral question: 1) you can't always tell the two valence up quarks apart so how do you know which one to track? 2) there are also an infinite number of sea quarks in there, and they can exchange identity with a valence quark too.
What mean by that is: in the effective field theory picture (sometimes called Quantum Hadro Dynamics), the binding force is due to meson-exchange, so a proton does this (virtually):
$$ p + \pi^- \rightarrow n + \pi^0 \rightarrow p + \pi^- $$
There's no reason the valence up quark you are tracking can't flow over to the $\pi^0$ for a while. Or not.
tl;dr A proton is a strongly coupled collection of 3 valence quarks ($uud$) and an infinite number of sea quarks (and gluons, and antiquarks) that are quantum vacuum fluctuations. At least 1/2 the proton's momentum is carried by sea-quarks and gluons. Regarding the spatial positions of the valence quarks: we don't really talk about that (caveat: EDM of the neutron, but that's different).
