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When talking about how muons reach earth even though their half-life is very short the explanation of time dilation is given. From earth's frame of reference, the Muon's clock is "slowed down" so is has longer to live and keep racing at it's speed, I think I understand this.

However, will the length between the Muon and the earth when seen from earth's reference also contract? If so, why is this never included in the calculation of amount of muons reaching the earth?

Also, from the muon's perspective the earth contracts which I understand but nobody talks about the time dilation for the earth. (Although this I understand better because it doesn't really matter how "fast" time flows for the earth relative to the muon, the earth will still take the same amount of time to reach the muon wether time dilation is included or not) Examples of places where they discuss these phenomena without including the length contraction (from earth's frame of reference):

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation


marked as duplicate by Kyle Kanos, Cosmas Zachos, ZeroTheHero, sammy gerbil, Jon Custer May 4 '18 at 2:17

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migrated from math.stackexchange.com Apr 30 '18 at 16:03

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You have to be consistent in each reference frame you consider.
1. In the earth frame the muon proper time runs slower, but the distance it covers is not length-contracted. The length contraction refers to the muon itself, in principle the muon dimension in the direction of motion is contracted, but that is irrelevant in this experiment.
2. In the muon frame the distance to the earth is contracted, that is why the muon life time is enough for it to reach the earth.

  • $\begingroup$ The line "the length contraction refers to the muon itself" cleared it up for me, thanks! $\endgroup$ – delivosa Apr 30 '18 at 16:43

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