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Originally Posted by Tim Thompson
We see the muons falling through the atmosphere, and we see "time dilation", the muon lives longer in our reference frame. But the muon does not see it that way. It's too easy to think of time dilation as "real", not in the sense that we see it, but that it somehow means more than what any other observer would see. I call it an illusion, because it is entirely dependent on the observer reference frame, which is not the case for proper time.
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That's true, but the muon does see the distance between upper atmosphere and ground as much shorter than we measure it, which is how it's able to travel from one to the other in the much shorter time measured by its own internal clock. Whether you think of the muon as travelling 10 km in 66 microseconds, or 0.33 km in 2.2 microseconds depends on whether you choose to measure the behavior in the "lab" frame or in the muon's rest frame, and both (as well as any others you might decide to use) are equally valid.
Time dilation and length contraction are thought of as "real" in the sense that measurements of time and space depend on your relative motion. But of course if you're moving along with a clock or a meter stick, they're at rest relative to you, so you'll never see them appear to do anything unusual regardless of how you (along with your clock and meter stick) might be moving relative to other things in the universe.