# Time dilation of Muons and a Paradox

1. May 17, 2012

### universal_101

Hello Friends,

Consider a Linear accelerator, at the one end there is a Muon Generator, and it produces a certain amount of Mouns, let's say 'x'.

These Muons are accelerated to the other End of the accelerator, where the number of Muons reached are detected and displayed on a Digital display, let's say 'y',(where, x>y).

Now, there are several other inertial frames moving at different speeds and different directions w.r.t the Lab's Frame. Since, according to Special relativity, all these different inertial frames should see(observe,calculate or measure) different number of Muons reaching the other End.

Does that mean that the display shows different number for different frames, but it seems utterly impossible.

In case, your answer is, that all of them sees the same number of Muons reaching the other End, then how can one include this is Special relativity's domain.

Since, the above Experiment is exactly similar to the Experiment on cosmic Muons, one must conclude according to Special Relativity that different observers see different number of Muons reaching the other End of the accelerator. That is, different relative speed between observers and the Muons should make them Time Dilate differently for each different observer.

Or am I missing something very obvious ?

Thanks

2. May 17, 2012

### Bob S

The number of muons reaching the End is the same in all reference frames. This is even true if you include radioactive muon decay and time dilation.

3. May 17, 2012

### Mentz114

All observers will see the same number of detection events as the rest frame of the accelerator and detectors. If you think in terms of worldlines, then a detection event is when the worldline of a muon coincides with the worldline of a detector. This cannot be altered by the transformation between frames although the times and rates of detection may be different.

4. May 17, 2012

### universal_101

You missed the above consequence, entirely.

Well, let me put it this way, Since, the Time Dilation of Muons is calculated by the relative speed of Muons and the Lab Frame.

Why the same is not applicable for other different observing frames.

5. May 17, 2012

### Staff: Mentor

Why do you think it's not applicable in other frames? (You must include the effects of simultaneity and length contraction as well as time dilation.)

Have you done the calculation?

6. May 17, 2012

### universal_101

Well, I did the calculation(which is very simple for that matter) and found that,

If I use the relative speed of Muons and different observers to calculate Time Dilation of Muons,
then according to these observers(the calculations) the number of Muons reaching at the other End is different for different observers.

By the way, what does simultaneity has to do with anything? Since we are only considering relative speeds to calculate Time Dilation and/or length contraction, therefore we should not bother about which frame is located where w.r.t the Lab Frame.

7. May 17, 2012

### ghwellsjr

8. May 17, 2012

### DrGreg

• The half-life of a muon is relative to the frame in which it is at rest. Because of time-dilation, other observers will measure a longer half-life.
• The length of the accelerator is affected by length contraction, therefore different observers will disagree over the length of the accelerator.
• The velocity of the muon relative to the observer isn't obtained just by adding/subtracting the velocities of the muon and observer relative to the lab.
When you take all these effects into account, observers disagree over how much time it takes for a muon to travel along the accelerator, but they all agree over how many muons reach the end.

9. May 17, 2012

### universal_101

But it is the number of Muons reaching other end, which specify/tells us how much they(Muons) got Time Dilated.

And not how much time it takes to reach the other end, according to different observers.

It is analogous to, the younger twin is younger by the same amount irrespective of the different relative motion w.r.t different observers.

If this is the case then I think we will have more problems.

10. May 17, 2012

### DrGreg

But this isn't true. Explain why you think it is.

11. May 17, 2012

### Staff: Mentor

Please show your calculation. And how you concluded that different numbers of muons reach the end in different frames. (Realize that 'the end' is moving as seen from any frame but the lab frame.)

12. May 17, 2012

### universal_101

Very well, Consider all the observations from the Lab's Frame of reference.

First of all, there is No length contraction of the accelerator in this frame and we don't need to add any velocity.

So, there is only one thing left which is, Time Dilation due to motion Muons,

Which can be experimentally verified only by analyzing the number of Muons reached, produced and the half-life of Muons. That is, more Muons reached at the detector end says more time dilation and vice-verse.

But according to your previous post that, "observers disagree over how much time it takes for a muon to travel along the accelerator", That is, It is the time taken by the Muons to reach the detector, in different frames, that produces different Time Dilation.

Does that mean, that the younger twin from Twin Paradox stays younger by the same amount, No matter from which inertial frame he is observed. Since the only effect of observation from different inertial frames is, the time taken by the travelling twin to return is different for different observers.

In short, returning back sooner or later, cannot/should not make the Twin aged different for different observers. Whereas, it is the difference in the age which is called TIME DILATION.

13. May 17, 2012

Staff Emeritus

14. May 17, 2012

### universal_101

I think, I did not realize that 'Length contraction' would have to be included(since, one can always assume frames moving at right angle to the linear accelerator). Apologies.

But then, it means the Length Contraction is as real as Time Dilation. That is, it can increase the density of any object if viewed from different frames.

Now again, does the density of a object is different for different frames ? Just like Time Dilation.

15. May 17, 2012

### universal_101

I finally got the calculations corrected, But even if the all the observers sees the same number of Muons reached the other End. This implies, that Time Dilation is same for every different observer, until and unless the Length Contraction is as real as Time Dilation itself.

But then, how can one comprehend different density(due to real length contraction) of the same object, observed from different inertial reference frames ?

Analogously, if we examine the Bell's spaceship paradox, and have many strings tied between the spaceships, which break at different tensions.

Then according to the length contraction(which must be applied in a real sense), there will be different number of broken strings for different observers.

16. May 17, 2012

### bobc2

universal 101, It has been pointed out for you that events that occur in spacetime exist in all frames of reference--it's just that the coordinate values are different in different frames. It was pointed out that you should consider the worldlines of the various objects, i.e., detector and the various muon particles. All particles showing up in the detector for one frame will correspond to the termination of the worldlines of the particles, and those worldlines and termination events will be present in all inertial frames.

The process for one muon is shown as a spacetime diagram below. I could easily add in the coordinate axes for all of the other observers moving at whatever velocities with respect to the reference frame that you wish. You cannot give me a frame (or any observer) for which I cannot have included in my spacetime diagram (the observers of course cannot move at the speed of light or greater). For any other observer you give me my spacetime diagram will show you his indertial frame--and that frame will definitely include the muon detection event.

The diagram would get very messy, but I could also include as many different muons as you wish--all having worldlines that terminate in the detector. And every event will show up in any observer's frame that you wish to propose. You tell me your observer and his velocity and his coordinate system may be added.

17. May 17, 2012

### Staff: Mentor

Of course length contraction is as "real" as time dilation. Both are frame variant consequences of the Lorentz transform.

Density is frame variant. What is hard to comprehend?

Not if you use a relativistic version of Hookes law.

Last edited: May 17, 2012
18. May 17, 2012

### universal_101

Thanks, and especially for the diagram.

But, I have another problem/complication as posted earlier, if we conclude that all the observing frames will see same number of muons reaching the other end. Since, there must be a real effective Length contraction so that different observing frame have different Time Dilation.

That is, the length contraction that we use to calculate the number of Muons reaching the other End, must be as real as Time Dilation itself. Which implies, there will be different Length Contraction w.r.t different observer.

How do we comprehend that, is there an experiment that shows that length contraction is real.

19. May 17, 2012

### universal_101

But,

it seems that Length contraction is real, but some how we cannot observe it's effects. Since, everything changes relativistic-ally.

That is, if I say, there is an element which glows at a critical density,(or let's say changes its lattice configuration from BCC to BCT), then I must edit my equations of the effects governing the change(i.e. BCC to BCT) relativistically, so that every observing frame agree on the change(BCC to BCT).

EDIT - Can it be the reason, why we don't have any experiment supporting Length Contraction

20. May 17, 2012

### Mentz114

Does BCC stand for 'Body-Centred Cubic' ?

In its own frame a crystal lattice will feel no changes due to another observers relative velocity. Another observer will calculate a higher density, but remember that the electric fields holding the crystal in equilibrium will also appear to have changed shape for this moving observer. If all the effects are taken into account, there's no difficulty.