How does muon time dilation explain the paradox of their journey to Earth?

[imsmooth]

when muons are created in the atmosphere and fall to Earth the Earth clock runs faster. Thus, the "slower" muon clock allows more of them to reach the Earth's surface. But from the muons' perspective, the Earth is moving towards the muon. Shouldn't the muon's clock run "faster" and there should be less muons when the Earth reaches them?

Both events should appear the same from the other' perspective. How does one resolve this?

 

[Nugatory]

when muons are created in the atmosphere and fall to Earth the Earth clock runs faster. Thus, the "slower" muon clock allows more of them to reach the Earth's surface. But from the muons' perspective, the Earth is moving towards the muon. Shouldn't the muon's clock run "faster" and there should be less muons when the Earth reaches them?

Both events should appear the same from the other' perspective. How does one resolve this?

There are two questions here. One is the surprising symmetry of time dilation: A is at rest while B is moving so B's clock should be running slow; but we could just as easily say that B is at rest while A is moving so A's clock is running slow. This is explained by the relativity of simultaneity; there are bunch of posts here that explain exactly how, such as this one: https://www.physicsforums.com/showpost.php?p=4720250&postcount=6

The second question is how the muons live long enough to make it to the ground when we're thinking in terms of the muons at rest (so no time dilation) while the Earth rushes towards them. The answer here is length contraction; the thickness of the atmosphere is less in this frame so the surface of the Earth reaches the muon more quickly, before it has had time to decay.

 

[PAllen]

1) The muons don't 'fall' to Earth in the sense that gravity plays no detectable role in their speed. In the hundred or so microseconds it takes for them to reach the ground, gravity would add essentially nothing to the original speed(energy) they inherit from a cosmic ray particle.

2) In the Earth frame, the muon decay clock runs slow, so almost all reach the ground though 50% should decay every 2 microseconds.

3) In the muon frame, the Earth clock does run slow, but this is irrelevant. The reason the muon reaches the ground, in the muon frame, is that the Earth and its atmosphere are extremely contracted in the direction of Earth motion (per the muon). Thus the atmosphere is is not 10 km, but only a few hundred meters thick per the muon.


To complete the account in both frames, we would also have to bring in the relativity of simultaneity (along with time dilation and length contraction). This is how the muon frame accounts for how the Earth frame considers the muon life to be long while the muon considers Earth clocks to be slow. Specifically, per the muon, the clock on the ground that the Earth considers synchronized with a clock reading the time of muon creation in the upper atmosphere is, instead, set way ahead. That is why (despite running slow) it reads a long time for the muon's life.

[edit: I see I cross posted with Nugatory.]