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If there is a proton scattering, in which length contraction occurs, thus making a proton into a pancake shape in the lab frame, its scattering behavior differs from the scattering patterns of a spherical atoms at relatively lower speeds, as mentioned in the link below.

https://www.quora.com/Is-length-contraction-only-a-result-related-to-observation-or-a-real-contraction

Then if there is another proton traveling at the same speed, the proton will see its friend scattering as a sphere, while we observe it scatter as a pancake shaped projectile. But, by the time the the proton slows down after the scatter, the proton's frame and the lab frame will coincide and would have to match in position. However, before the frames end up coinciding, the proton has gone through spherical scattering in one frame and pancake scattering in the other thus would have to yield different scattering behavior and end position. So, how could this disparity be reconciled?

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  • $\begingroup$ Is a proton spherical? I hear particle physicists talk about cross section, but not thickness. $\endgroup$
    – Dale
    Commented Jan 20, 2020 at 4:26
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    $\begingroup$ @Dale , the proton is a composite particle made up by three valence quarks, and a sea of gluons quarks and antiquarks as an illustration. see profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/… . Sst the interactions are strong the spherical shape is a good approximation at rest. $\endgroup$
    – anna v
    Commented Jan 20, 2020 at 5:36
  • $\begingroup$ @anna v awesome link, thanks! $\endgroup$
    – Dale
    Commented Jan 20, 2020 at 17:31

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There is no contradiction.

The division you make into "spherical scattering" and "pancake scattering" is not quite there, because you fail to consider the other particle involved in the collision: If you collide two protons together at high relative speed, they will both be pancake-shaped in the centre-of-momentum frame (the lab frame). In the rest-frame of one of the protons, that one is spherical but the other one is even more pancake-shaped. (Furthermore, the relative speed will be slower because of time dilation.)

Relativity says that the two perspectives above describe the same situation, perhaps surprisingly. But note that in both cases, at least one of the protons is pancake-shaped. A collision between two spherical protons is not the same situation; here the relative speed must be small.

Hope that helps, but remember that there are many ways to discuss relativity in words, and they can all be confusing or even misleading outside the context of a proper mathematical picture.

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