Warning: extreme science nerdiness follows.
One of the proofs of relativity is what happens to a particle called a muon. Muons are created in our atmosphere by cosmic rays. The half-life of a muon is only 2.2 1.56 microseconds (which means if you have a certain number of muons, every 2.2 1.56 microseconds you will have only half as many.) Even though the muons are traveling at near the speed of light, without relativity, very few of the muons generated high in the atmosphere are able to travel far enough to reach the earth’s surface.
However, according to the theory of relativity, when an object travels at close to the speed of light, a stationary observer sees time slow down for that object–that’s known as time dilation. Also, from the point of view of the object, distances along its path of travel get shorter–that’s known as length contraction.
From the point of view of someone on Earth, time slows down for a muon, which makes its half-life longer. That means more muons reach the surface of the earth.
From the point of view of the muon, its half-life is still the same, but the distance to the earth’s surface is shorter, so more of them are able to make it.
A web page that shows all the calculations can be found here (although, I’m not sure why, but it uses 1.56 microseconds as the half-life of a muon): http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html
So far, so good: if you work things out from the point of view of a muon traveling at 98% of the speed of light, about 49,000 out of a million muons can travel 10 kilometers to the earth; and you get the same result if you work things out from the point of view of an observer on the surface of the earth.
Unfortunately, I’ve run into a problem that I don’t know the answer to.
First, let’s assume we have muons that are only moving at 49% of the speed of light, instead of 98%. That actually makes a big difference in the rate of time dilation–it’s only 1.15 instead of 5.02. That means the half-life is only 15% longer than normal, rather than five times longer.
Naturally, that means much fewer muons will manage to travel 10 kilometers. Instead of 49,000 out of a million, only 0.0000035 out of a million will make it. That’s a huge difference–14 billion times difference, about ten orders of magnitude.
Now, a little thought experiment:
(UPDATED to account for the non-linear addition of relativistic velocities.)
Assume we have a muon detector that is moving toward the earth’s surface at 49% 82% of the speed of light. Relative to the frame of reference of that detector, the muons are moving at only 49% 82% of the speed of light. (This is not a coincidence: I specifically chose 82% because that’s the number at which the relative velocities of the muons and the earth are of the same absolute value relative to the frame of reference of the detector. It’s actually somewhere between 0.8169 and 0.8178, but I rounded it to 0.82.)
Let’s say the detector checks the number of muons at a height of 10km and finds one million. Then, just before it hits the surface of the earth, it checks again. What does it find?
Due to length contraction, from the detector’s frame of reference, the 10km is only 8.7km 5.7km, so from its point of view, the muons only have to travel 8.7km 5.7km. However, from its point of view, the muons are only traveling at 49% 82% of the speed of light, so their time dilation factor is only 1.15 1.75. Using the calculator, we find that only 0.0001 about 2700 out of a million muons will be able to travel 8.7km 5.7km at 49% 82% of the speed of light.
So there’s only a 1 in 10,000 chance our detector will even be able to detect even one muon at the surface of the earth. So our detector should detect about 2700 muons.
Yet the stationary detector on the surface of the earth detects 49,000 muons, just as our detector crashes into it.
So, here’s my problem: According to an observer on the surface of the Earth, 49,000 muons out of a million should reach the surface. According to the observer in our thought experiment, 0.0001 about 2700 muons out of a million should reach the surface. Since experimental evidence confirms the first result, the result of the thought experiment must be wrong. But as far as I can tell, the thought experiment is an accurate application of the theory of relativity.
Can someone explain to me how the thought experiment is wrong?
UPDATE: Over on my livejournal, dvandom reminds me that relativistic velocities don’t add in a linear fashion. I’m looking into those equations now.
UPDATE 2: I’ve updated the example to take into account the non-linear addition of relativistic velocities. It’s no longer off by as much, but it’s still off.
UPDATE 3: I’ve updated the half-life of the muon. The 2.2 figure was the mean life. It did not affect the calculations because the online calculator was already using the correct figure.