# Confused about muon evidence of relativity

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?

• The Earth experiences time dilation from the muon's perspective, not the muon. In the muon's rest frame the muon isn't moving at all. Commented Sep 13, 2021 at 14:23