Time dilation and length contraction are not illusions of special relativity (SR); rather, they are real physical effects that arise due to the principles of SR. The muon experiment provides a good example to understand these effects.
In the context of the muon experiment, muons are high-energy subatomic particles that are created in the upper atmosphere and travel towards the Earth's surface. These muons have a relatively short lifetime as measured in their rest frame. However, due to their high speeds close to the speed of light, they experience time dilation as observed from a stationary frame of reference.
From the perspective of an observer on Earth, the muons appear to have an extended lifetime because time is dilated for them. This phenomenon occurs because the relative motion between the muons and the observer causes time to flow differently for each. As a result, the muons can travel farther distances within their dilated timeframe than what would be expected if time was not dilated. This explains why the muons can reach the Earth's surface even though their rest frame lifetime is relatively short.
Regarding length contraction, it is essential to understand that the contraction is observed in the direction of relative motion. In the frame of reference of the muons, they observe the length of the distance between the upper atmosphere and the Earth's surface to be contracted. This contraction is a consequence of the relative motion between the muons and the Earth, as observed from their frame.
However, it is important to note that the effects of time dilation and length contraction are not symmetrical between the observers. The observer on Earth sees the muons' clocks ticking slower and measures the distance between the upper atmosphere and the Earth's surface as it appears in their rest frame. In contrast, the muons themselves perceive their clocks to be ticking at a normal rate and observe the distance contracted as it appears in their frame.
These effects are not illusions but are well-established consequences of special relativity. They have been verified through numerous experiments and are essential components of our understanding of the behavior of particles moving at relativistic speeds. The muon experiment is just one example that demonstrates the validity of these effects and how they can lead to seemingly contradictory observations between different frames of reference.