Muons: Why time dilation takes precedence?

[Raze]

I was browsing through old threads and a user named universal_101 kept asking about why we can use muons as a direct measurement of time dilation but only an indirect measurement of length contraction. It was pointed out that the two go together and cannot be separated, but it got me thinking that there might be something to his/her confusion. What do you think of the following?


Since the proper time clock rides with the muon, obviously it is the muon that measures length contraction. The decay time is normal from the muon's frame of reference, but the distance the Earth moves toward it is contracted to the point that the muon decays at the surface.

Conversely on earth, we observe the length the muon travels to be maximized, which means we must measure the muon's travel to include time dilation enough to allow the muon to not decay until it reaches the surface.

The two are symmetrical in the same way there is a symmetry in the Moving Conductor Problem, it seems.


But what I am getting at is that the reason we measure time dilation instead of length contraction with this scenario is because in OUR reference frame time dilation is what we see. The only way we could directly measure length contraction is if we were in the reference frame in which the Earth is moving toward us (that is, the muon's reference frame).


Does this make any sense at all? Thanks.

 

[Nugatory]

But what I am getting at is that the reason we measure time dilation instead of length contraction with this scenario is because in OUR reference frame time dilation is what we see. The only way we could directly measure length contraction is if we were in the reference frame in which the Earth is moving toward us (that is, the muon's reference frame).

That's pretty much it. The distance in question is the thickness of the Earth's atmosphere, and because the Earth's atmosphere is not moving relative to us, we measure its rest length. The Earth and the Earth's atmosphere is moving relative to the muon, so the muon frame would observe length contraction. The muon is moving relative to us, so a clock on at rest relative to the muon will be time-dilated relative to our clock, and that's what extends the muon's life.

If the muon had some measurable length instead of being a point particle, then in principle we could observe length contraction of the muon itself... but for all practical purposes the muon is a point particle with zero length, and a length-contracted zero is still zero so there's nothing to see here.