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I'm currently in high school so sorry if the answer to this question seems obvious but I’m only just learning about this stuff. I’ve been learning about special relativity, in particular length contraction and time dilation. I was wondering, if the Planck length is the smallest possible observable length, then what would an observer who is travelling at relativistic speeds measure the Planck length to be? Would it be the same or would he observe a smaller length?

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There are two aspects you have to consider for the answer:

1.The spacetime of special relativity does not consider lengths but only distances of worldlines.

Special relativity consists of the two postulates, the first one saying that in each frame of reference the same physical laws do apply and the second, that light is observed as to move with speed of light. As you can see, special relativity helps describing timelike and lightlike worldlines (of mass particles and lightlike phenomena), but it says absolutely nothing about the vacuum between particles which would permit to fill the space between particles in order to get a length.

In the same way, the most current example for the explanation of length contraction is regarding some rigid macroscopic object. This is a gross simplification because according to special relativity there are no perfectly rigid objects. According to special relativity, every rigid object consists of its particles with their interacting forces which may provide high, but not perfect rigidity.

In contrast, special relativity may consider traveled distances as far as they are referring to timelike or lightlike worldlines of particles.

2.According to your question and according to the Lorentz contraction of special relativity, such distances are observed as contracted by moving observers. This applies also to distance constants such as Planck length.

Lorentz contraction is a phenomenon which is specific to a frame of an observer. A distance traveled by a particle which is observed with a length of one light year from the point of view of the reference frame of a comoving observer is observed smaller (contracted) from the point of view of observers with a relativistic relative velocity with respect to the considered traveling particle. The distance may be subject to arbitrarily high contraction, for an observer which is moving very near to light speed with respect to the two events which are limiting the distance. But it is evident that the fact of the observation does not change the travel data of the particle. The particle does not travel the contracted distance which is observed by observers (otherwise you might ask: of which observer? with contradictory answers) but the distance before Lorentz contraction which is simply the longest distance that can be measured.

Thus, the answer to your question is: Yes, an observer traveling at relativistic relative speed with respect to the reference frame of a particle may observe any distance as contracted, even constants such as the Planck length.

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  • $\begingroup$ see also "doubly special relativity" for more about this. $\endgroup$ – niels nielsen Nov 18 '17 at 20:59
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    $\begingroup$ I guess you are saying that if one observer observes some particle traveling a distance numerically equal to the Planck length, then a different observer might observe the particle traveling a different distance? Or do you actually mean to say that the Planck length itself (the combination of $c$, $G$, and $\hbar$ with units of length) changes? $\endgroup$ – d_b Jul 13 '20 at 3:16
  • $\begingroup$ As I explained, Lorentz contraction is mere observation: An observer moving near light speed will measure a shorter distance (e.g. for the distance between Sun and Earth), but this does not mean that the distance between Sun and Earth has changed. The same is true for any distance, including Planck length. $\endgroup$ – Moonraker Jul 14 '20 at 8:15

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