Confused about muon evidence of relativity

Question

As I understand it: muons are created $20$km high in the troposphere and since they only have a half life of $\tau = 1.56 \mu s$, they should only be able to travel $660m$ before half are lost, so we should expect to detect far fewer of them on Earth than we actually do. The inflated amount that we detect can be explained by Special Relativity so acts as evidence of it.

From our IRF, the situation is explained by time dilation: $$u \gamma(u) \tau =l > 20km$$

where $u$ is the speed of the muons and $l$ is the length they can travel before their half life

From their IRF, the situation is explained by length contraction:

$$ L = \frac{L_0}{\gamma(u)} \approx 630km$$

My question is, doesn't length contraction and time dilation occur from the muon perspective? In that case, isn't the advantage offered by time dilation compounded by the compression of the atmosphere so that from the muon's reference frame, it can travel further than in our reference frame, which would be an obvious contradiction?

Answer 1

In the muon's rest frame, the muon's lifetime takes its ordinary value. The distance between the atmosphere and the surface of the Earth is length contracted, so it can travel more of the distance between atmosphere and surface than it would be able to if the distance between the atmosphere and surface took its "at rest" value in the muon's rest frame. Additionally, the observer's clocks are time dilated; I'm only pointing this out so you see where time dilation occurs in this frame, but it's hard to draw simple conclusions from this statement. To follow what the observer sees, you need to compare two events: event A is when the muon is created in the atmosphere, and event B is when it is detected. To properly follow this, you need to account for both length contraction of the atmosphere-surface distance and the observer clock time-dilation; if done correctly, you'll show that the observer's clock ticks the same amount of time between events A and B, as it does in the observer rest frame, which is easier because you only need to account for the time-dilation of the muon lifetime.

In the observer frame, the muon's lifetime takes a time-dilated value. The distance between the atmosphere and surface takes its "normal" value, and the observer's clock runs at a "normal" rate. The muon is able to travel further in this frame than it would if the muon's lifetime were not time-dilated.

Of course, there are no contradictions if you do out all the math.