In the famous muon experiment, time dilation is indeed a crucial factor that is taken into account when explaining the observed results. The experiment involves high-energy muons produced in the upper atmosphere, which have a short half-life. According to special relativity, when these muons travel at high speeds close to the speed of light, time dilation occurs, causing their internal clocks to appear to run slower from the perspective of an observer at rest.
From the frame of the muon itself, it would perceive its own internal clock as running normally, and it would see the Earth's clocks as running slower due to its high velocity. This is consistent with the predictions of special relativity.
However, the key point to understand is that the muons have a very short lifetime in their rest frame. This means that, regardless of time dilation, most of them would decay before reaching the Earth's surface. According to the Earth-based observer, who measures the muon's lifetime, it appears to be much longer due to time dilation.
The reason why the experiment is successful is that the high-energy muons travel at such high speeds that their time dilation effect allows a significant fraction of them to survive long enough to reach the Earth's surface, despite their short lifetime in their own frame. This is consistent with the predictions of special relativity.
Therefore, it's important to note that time dilation is indeed considered in the muon experiment, and it is precisely this effect that allows the muons to reach the Earth's surface before decaying, providing evidence for the validity of special relativity.