Question

Why is time dilation not applied in the muon’s frame in the muon experiment? That would then result in a wrong prediction by special relativity, that Earth’s clock is running slower than the muon’s clock?

Answer 1

Time dilation is indeed applied in the muon experiment when considering the perspective of the muon. The experiment is actually a classic example used to demonstrate the effects of time dilation in special relativity.

In the muon experiment, cosmic muons (high-energy subatomic particles) are created in the upper atmosphere when cosmic rays interact with the Earth's atmosphere. These muons are unstable and decay with a very short half-life. However, due to their high speed (close to the speed of light), they experience time dilation, which means their internal clocks appear to slow down from our perspective on Earth.

Here's how time dilation applies in the muon experiment:

1. From the Earth's frame of reference: We observe the muons created high in the atmosphere and measure their speed and lifetime as they travel toward the Earth's surface. Since muons move at a significant fraction of the speed of light, their clocks appear to be running slower from our perspective. As a result, they can cover greater distances than their short half-life would typically allow.

2. From the muon's frame of reference: In the frame of reference of the muon itself, it is at rest, and it is the Earth and the atmosphere that are moving at a high speed relative to it. In this frame, the muon's clock is running normally, but Earth's clocks (and everything else in the atmosphere) appear to be running slower due to time dilation. This means that the muon observes time passing more slowly on Earth than we do, which would seemingly result in a "wrong prediction" if we considered only the muon's perspective.

However, there's no contradiction here. Both observers, one on Earth and the other on the muon, would agree on the observed facts when they compare their measurements after the muon reaches Earth or decays. The key is understanding that time dilation is reciprocal. Each observer sees the other's clock running slowly, but both observations are consistent and do not violate the principles of special relativity.

The muon experiment is a compelling demonstration of time dilation and illustrates the counterintuitive yet fascinating aspects of Einstein's theory of special relativity.