Please help me fill in the details of something I barely remember.
In my Physics for Poets class many, many years ago, I learned about some sort of short-lived particle that is formed in the upper atmosphere (I’m guessing by the interaction of cosmic rays and air molecules, but that doesn’t sound right).
Anyway, we know at what altitude these particles form, how long they last before they decay into something else (it’s measured in seconds), and what their speed is (a nontrivial fraction of c). Simple arithmetic tells us that all of these particles will have turned into something else before they reach (and I’m just making this number up now) 30 miles above Earth’s surface.
However, said particles can be detected at (making up, again) 10 miles above the surface. How can that be?
The answer was Lorenz contractions. To us, the distance between the area where the particles form and Earth’s surface is x miles. But because the particles are moving so quickly, in their frame of reference the distance between the area where they formed and Earth’s surface is x – 20 miles.
Does this sound familiar to anyone out there? I’d really love to find an article or something from which I could relearn what the name of the particle is and what the actual numbers involved are.
" Although their lifetime without relativistic effects would allow a half-survival distance of only about 0.66 km (660 meters) at most (as seen from Earth) the time dilation effect of special relativity (from the viewpoint of the Earth) allows cosmic ray secondary muons to survive the flight to the Earth’s surface, since in the Earth frame, the muons have a longer half life due to their velocity. From the viewpoint (inertial frame) of the muon, on the other hand, it is the length contraction effect of special relativity which allows this penetration, since in the muon frame, its life time is unaffected, but the length contraction causes distances through the atmosphere and Earth to be far shorter than these distances in the Earth rest-frame. Both effects are equally valid ways of explaining the fast muon’s unusual survival over distances."
There are actually two different ways you can look at this, depending on your frame of reference. From the frame of reference of the Earth, the thickness of the atmosphere is many miles, but the muon is subject to time dilation, and so it lasts for a longer time than it would if it were at rest. But what if we look at it from the muon’s point of view? In its own reference frame, it’s stationary, with the ground moving at it at high speed. Since it’s stationary, from its own point of view its clock is running perfectly normally, and so its lifespan is normal… But since the Earth is moving very fast relative to it, length contraction means that the thickness of the atmosphere is much less, and so the ground can still hit it before it decays.
Of course, either way of accounting for it gives the same answer, since time dilation and length contraction are really just two different aspects of the same effect.