Muons traveling to Earth, time dilation question

[Glenn G]

Hello learned people,
I've been looking at special relativity of muons formed in the upper atmosphere...
If I can summarise what I do understand (i think)...
A muon has 12km to travel to the Earth from the atmosphere at 0.994c. Alice records this as taking 40.2 micro seconds.
Now a muon has a half life of 2.2microseconds in the lab so how come a quarter of them reach Earth rather than next to none if relativity is not taken into account?
From the calculation above we find that tau = 4.4microseconds which is 2 half lives explaining why 1/4 remain as in the reference frame of the muons it has only taken 2 half lifes to reach Earth (hopefully so far so good?, time dilation)
Now at 0.994c it must also seem to the muons that they only travel 1.3km as opposed to 12km (length contraction?)

What then got me confused is that from special relativity there should be nothing that the muon can do that can tell it whether it is moving towards the Earth or the Earth is moving towards it...hence what if the muon can look at Alice's clock? surely he would think that she is the one that is affected by time dilation (otherwise if he does see Alice's clock as reading 40micro seconds and his at 4micro seconds if he knows a bit of relativity he could conclude that he is actually the one that is moving)

So, if the muon believes the journey take 4 microseconds, when he looks at Alice's clock he sees her clock running slow sees her clock running at 0.48 micro seconds).

Have I got this right? if I'm wrong would appreciate what the muon would see if (he as the moving one) were to look back at Alice's clock.

many thanks,
Glenn.

 

[Janus]

Relativity of simultaneity. If a muon is formed at an altitude of 12 km when, according to Alice, her clock reads zero, then according to the Muon, when it first comes into existence, 1.3 km from Alice and her Clock, Alice's clock already reads 39.52 microseconds, and then advances by 0.48 microseconds while the distance between them closes, to read 40 microseconds upon their meeting.

 

[Mister T]

What then got me confused is that from special relativity there should be nothing that the muon can do that can tell it whether it is moving towards the Earth or the Earth is moving towards it...hence what if the muon can look at Alice's clock?

Alice actually has two clocks. One where she is located and another 12 km above her. She synchronizes them. It's the only way for her to measure the time that elapses. In the rest frame of the muon those clocks are not synchronized.

It is best to think in terms of events. One event is the muon passing the 12 km altitude location and the other is the muon reaching Alice. Note that the muon is present at both events and therefore the time that elapses between the events in the muon rest frame is a proper time, by definition. Call it $\tau$. The dilated time is always larger than $\tau$.

 

[Glenn G]

Thanks folks. That’s fascinating!

 

[Glenn G]

Relativity of simultaneity. If a muon is formed at an altitude of 12 km when, according to Alice, her clock reads zero, then according to the Muon, when it first comes into existence, 1.3 km from Alice and her Clock, Alice's clock already reads 39.52 microseconds, and then advances by 0.48 microseconds while the distance between them closes, to read 40 microseconds upon their meeting.

Thanks Janus; but does the muon not record the time of passing the first marker and reaching Earth as 4.4 microseconds at a distance traveled of 1.3km (using his Instruments) but when he looks at Alice’s clock her clock time only shows up at +0.48 microseconds?

 

[Nugatory]

but when he looks at Alice’s clock her clock time only shows up at +0.48 microseconds?

It depends on when Alice's clock was zeroed.

If that clock was zeroed at the same time that the muon was created - using Alice's frame's definition of "at the same time" - then Alice's clock will read the distance in Alice's frame divided by the speed with which the muon approaches alice. If Alice's clock was zeroed at the same that the muon was created - using the muon's frame's definition of "at the same time" - then Alice's clock will read the distance in the muon frame divided by the speed with which Alice is moving towards the muon. Because of the relativity of simultaneity, these two procedures will yield different results; but no matter which you choose, Alice and the muon will agree about the reading on Alice's clock when the muon reaches it.

 

[Ibix]

Here's a couple of Minkowski diagrams that may explain things a bit. If you haven't come across these before, they're just displacement-time graphs, with the time axis drawn vertically and increasing up the page. Here's the situation in the Earth frame:

Apologies for the lack of labelling - I haven't finished writing the program to draw these. The left hand side of the diagram is the top of the atmosphere. A sequence of muons gets created at the events marked with red crosses (all at different times but the same altitude); their worldlines (position as a function of time) are shown in red - they travel down to a point on the Earth's surface, the worldline of which is drawn in blue. I've also added a grey dashed line which shows "time zero" according to the Earth. A muon happens to have been created at time zero. Note that the muons in this diagram are only doing 0.8c here, for reasons I'll explain in a minute.

Hopefully the above is fairly straightforward. But what does it look like in the rest frame of the muons? You can just take the Lorentz transform of the coordinates of every event shown on the graph and plot another diagram:

In this frame, the muons' worldlines are vertical because their x-position (or x'-position, I suppose) isn't changing. The lines are also a lot shorter because the muons aren't time-dilated in this frame so they don't have to live very long (and this is why I slowed the muons to 0.8c - at 0.9974c the lines were so short the diagram was illegible). In this frame the top of the atmosphere is moving to the left, so the muon creation events form a line sloped to the left. Similarly the Earth's surface is moving to the left so its worldline is also sloped. But the really interesting thing is what's happened to the grey dashed "time zero" line. It connects all events that the Earth frame says happened at t=0. But in this frame, the line is not horizontal - events that the Earth frame says happened at the same time do not happen at the same time in the muon frame (in fact, the inverse Lorentz transform tells you that $t=\gamma(t'+vx'/c^2)=0$, which means that this line satisfies $t'=-vx'/c^2$).

So this is how the muon frame explains that Alice's clock has such a high reading when the first muon reaches the ground - in the muon frame, Alice started her clock waaaaay early.

Hope that makes sense.

 

[Janus]

Thanks Janus; but does the muon not record the time of passing the first marker and reaching Earth as 4.4 microseconds at a distance traveled of 1.3km (using his Instruments) but when he looks at Alice’s clock her clock time only shows up at +0.48 microseconds?

Let's consider the two clock scenario and mentioned by Mister T.
A clock at Alice's position and one where the muon is created. According to Alice, both clocks are in sync and read 0 at the moment of the Muon's creation.
However, for the muon, the clock's are not in sync and the while the clock where it is created does read zero at the moment of its creation, Alice's clock already reads 39.52 microseconds.
Consider what the muon and the creation point clock would actually visually "see" on Alice's clock.
According to the clock, it took light ~40 microseconds to reach it from Alice's clock. Thus when it itself reads zero, it would "see" Alice's clock as reading 40 microseconds before zero. In the ~40 microseconds it took the light to reach it, Alice's clock has advanced 40 microseconds and reads zero at the moment the creation point clock "sees" it read 40 microseconds before 0.
The muon. upon its creation, sees this very same light and thus "sees" Alice's clock as also reading ~40 microseconds before zero. However, it cannot assume that it took that light 40 microseconds to reach it from Alice's clock. For one, at that moment Alice's clock is only 1.3 km distant, and for the other, that distance has not been constant. It has been decreasing ever since the light left Alice and her clock. In other words, Alice and her clock had to be further away when the light left than it is now.
If the light from the clock is traveling at c, and Alice is approaching at 0.994c, then it was ~216 2/3 km away when the light left. It thus took ~722.22 microseconds for the light to reach the point of muon creation( as would have been measured by the muon). During which time, it ticked off ~ 79.52 microseconds due to time dilation. Thus, if the muon "sees" a time of 40 microseconds before zero on Alice's clock, and 79.52 microseconds have ticked off on Alice's clock since it actually read that value, then according to the muon, Alice's clock reads 39.52 microseconds at the moment of it's creation.
The clock at the creation point, at rest with respect to Alice, and the muon. moving with respect to Alice, reach different conclusions as to what time it is at Alice's clock, even though they both visually read the same time on Alice's clock when the muon and clock when they are at the same position.