Muon Time Dilation: Earth vs Muon Perspective

[Allison]

From the reference frame of the earth, the distance between the surface of the Earth and the muon is longer, but the muon survives because time for the muon is slowed down.

From the reference frame of the muon, the time experienced by the muon is not slowed down but the muon survives because the distance between the surface of the Earth and the muon is contracted. However, the time experienced by the Earth is slowed down.

Please tell me if my logic is correct. I am not sure as to what time the muon's clock and the Earth's clock would be showing when the muon and the Earth's surface meet (supposing we could attach clocks to the two objects), since each object observes the other's clock slowed down from its own frame of reference.

Please forgive me if I seem ignorant!

 

[Nugatory]

I am not sure as to what time the muon's clock and the Earth's clock would be showing when the muon and the Earth's surface meet (supposing we could attach clocks to the two objects), since each object observes the other's clock slowed down from its own frame of reference.

This is a surprisingly tricky question. We'd like to set both clocks to zero at the moment that the muon is created, then compare them when the the muon reaches the surface. But because of the relativity of simultaneity (if you're not familiar with that, google for "Einstein train simultaneity" - relativity of simultaneity is essential to understanding relativity) an observer moving with the muon and an observer on the surface of the Earth will disagree about what the moment the muon is created is. Thus, if the clocks are properly zeroed according to the Earth observer, they won't be properly zeroed according to the muon observer.

However, we can ask a different question: the Earth observer sets his clock to zero at the moment that the muon is created, according to the frame in which the Earth is at rest. What value should the muon observer set on his clock so that the two clocks will read the same when the muon reaches the earth? This works because relativity of simultaneity won't get in the way when the clocks are both in the same place on the surface of the earth. The answer is that the muon observer must set his clock to some value greater than zero, by exactly the amount that the time dilation formula would require, to allow for the fact that less time passes on the muon's clock than on the Earth clock.

 

[Ibix]

Sounds about right to me.

I find the idea of "time experienced by the Earth is slowed down" rather uncomfortable. Better to put it as you did in your last paragraph: "each object observes the other's clock slowed down from its own frame of reference". You never experience time slowed down. You might measure someone else's clocks ticking slow, however.

With regard to your question about the time on the clocks, the solution is the relativity of simultaneity (it doesn't get as much press as length contraction and time dilation but is arguably more fundamental). This means that, for things not in the same place, "at the same time" means different things in different frames. So in the muon case you can stop the clocks when the muon reaches the ground. But to get a meaningful measure you need to have started the muon and Earth clocks at the same time - and the frames don't agree what that is. The difference of opinion accounts for the apparent paradox you spotted.

You may wish to look up the Lorentz transforms, which will give you the mathematical details.

 

[Mister T]

I am not sure as to what time the muon's clock and the Earth's clock would be showing when the muon and the Earth's surface meet (supposing we could attach clocks to the two objects), since each object observes the other's clock slowed down from its own frame of reference.

Note that with just one clock at rest relative to Earth and one clock at rest relative to the muon, you can't determine anything about how the clocks are ticking relative to each other. This is because the two clocks move relative to each other and will therefore be in the same place only once. You need another event so you can measure the time that elapses between the two events. You therefore need a third clock (as I discuss in the example below).

When that third clock is also at rest relative to Earth, it's that pair of Earth clocks that run slow relative to the single muon clock.

But when that third clock is instead at rest relative to the muon, it's that pair of muon clocks that run slow relative to the single Earth clock.

I agree that the situation is tricky and the questions you're asking are often avoided by authors introducing relativity to the reader in an effort to make the presentation less confusing.

So let's look at that example. In this first example there are two Earth clocks and one muon clock. Suppose the muon (carrying its clock with it) passes by a clock resting on top of a mountain, and then another resting at the bottom of a valley. The time that elapses on the muon's clock is, let's say, 2.20 μs between these two events. We call that a proper time. The time that elapses on those two earth-based clocks is a dilated time of, let's say, 22.00 μs. But to really understand how this is possible you have to know, at the location of the valley clock, when it is that the muon passes the mountain clock. This is crucial because the muon and the Earth will disagree as to when that happens! So you have to specify which one it is that you're talking about.

So let's first look at the simpler case, from Earth's rest frame. Suppose the two Earth clocks are synchronized in Earth's frame. Thus if all three clocks are set to zero as the muon passes the mountain top, the muon clock will read 2.20 μs and the Earth clocks will read 22.00 μs when the muon is at the bottom of the valley.

If instead you synchronize the clocks in the muon's rest frame, and moreover set all three to read zero when the muon passes the mountain clock, the muon's clock will still read 2.20 μs when it passes the valley clock, but the valley clock will read only 0.22 μs. You can't correctly conclude, though, that only 0.22 μs elapsed between events in Earth's rest frame, because in that frame the mountain clock and the valley clock are not synchronized!

This is what we mean by tricky. And we're only halfway through a more complete exploration of the situation.

To finish, if you want, we can have a second example where we switch things around and have a single Earth clock and two muon clocks. Let's say the one Earth clock is the one at the mountain top. It is now the one measuring the proper time that elapses between two events but, and this is crucial, it's not the same two events as before. To see that imagine a tether following the muon, with a clock at the end of the tether, giving us two muon clocks. The second event will be the passing of the tethered clock by the mountain clock. We would then look at the two cases as we did above, one where the two muon clocks are synchronized in their own rest frame, and one where they are synchronized in Earth's rest frame.

I recommend you attempt to carry out the rest of this exercise.