In the scenario you described, where a muon is approaching Earth at a high velocity, both the muon and Earth can consider themselves at rest relative to their own frames of reference. However, when we compare the two frames of reference, we observe a phenomenon known as time dilation.
From the perspective of an observer on Earth, the muon's clock appears to be running slower as it travels at high speed. This is because, according to special relativity, time dilation occurs when there is relative motion between two frames of reference. The faster an object moves relative to an observer, the slower time appears to pass for that object.
On the other hand, from the muon's perspective, it considers itself at rest and observes Earth approaching it at a high velocity. The muon would perceive Earth's clock to be running slower due to the relative motion between them.
The key point to understand here is that time dilation is a relative effect. Each observer in their own frame of reference measures time passing normally, but when they compare their measurements to the other observer's measurements, they observe a difference.
In the scenario of the muon approaching Earth, both the muon and Earth experience time dilation as observed from the other's frame of reference. This apparent paradox is resolved because each observer perceives the other's clock to be running slower due to their relative motion.
It's worth noting that the muon's high velocity and relativistic effects allow it to survive longer in Earth's atmosphere than would be expected based on its relatively short half-life. This phenomenon is a consequence of time dilation and has been experimentally confirmed.