# Time Dilation Explained: Muon Decay & Reaching Earth

• I
• Quandry
In summary, muons have a half life of 2.2 microseconds and at the speed of light cannot reach the Earth. However, based on exponential decay, about 70 out of every billion muons will actually reach the Earth. It is important to note that the simple arrival of muons at the Earth's surface does not provide any information. Instead, tests at different altitudes have been conducted and the decay rates do not support a 2.2 microsecond half life when measured in our frame. This experimental evidence can be found in the "Experimental Basis for SR" sticky thread at the top of the forum. Further references and discussion can be found in the linked FAQ.

#### Quandry

As I understand it muons have a half life of 2.2 microseconds, thus, at the speed of light cannot get to the earth. But based on exponential decay, of every billion muons which head or way, about 70 will actually get here. Using time dilation the full billion will get here.
Do we need a billion to say we have muons, or is 70 enough?

You are correct that the simple arrival of muons at the Earth's surface doesn't tell you anything. However, if you assume that their half life is 2.2μs in our frame then you are making predictions about the numbers you would expect to see at different altitudes (and their production rate in the upper atmosphere, which would need a very good explanation), which can be tested. We have done tests at different altitudes (see the experimental basis for SR sticky thread at the top of this forum) and the decay rates don't support a 2.2μs half life when measured in our frame.

Ibix said:
You are correct that the simple arrival of muons at the Earth's surface doesn't tell you anything. However, if you assume that their half life is 2.2μs in our frame then you are making predictions about the numbers you would expect to see at different altitudes (and their production rate in the upper atmosphere, which would need a very good explanation), which can be tested. We have done tests at different altitudes (see the experimental basis for SR sticky thread at the top of this forum) and the decay rates don't support a 2.2μs half life when measured in our frame.
Many thanks. Your reference to test at altitude would be of great interest. I have found predictions based on time dilation but so far have not found actual tests. If you had a more specific reference for me that would be appreciated.

The experimental evidence thread is a single post with a link to an FAQ which is a list of formal references with some discussion.