Time dilation of Muons and a Paradox

[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

 

[Bob S]

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.

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.

 

[Mentz114]

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

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.

 

[universal_101]

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.

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.

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.

 

[Doc Al]

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.

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?

 

[universal_101]

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?

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.

 

[ghwellsjr]

Can you please show us your calculation?

 

[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.

 

[universal_101]

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.

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.

 

[DrGreg]

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

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

 

[Doc Al]

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.

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.)

 

[universal_101]

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

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 traveling 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.

 

[Vanadium 50]

Universal. you've been asked for your calculation multiple times. Please provide it in your next post, or this thread will be locked.

 

[universal_101]

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.)

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.

 

[universal_101]

Universal. you've been asked for your calculation multiple times. Please provide it in your next post, or this thread will be locked.

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.

 

[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.

 

[Dale]

This implies, that Time Dilation is same for every different observer, until and unless the Length Contraction is as real as Time Dilation itself.

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

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

Density is frame variant. What is hard to comprehend?

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.

Not if you use a relativistic version of Hookes law.

 

[universal_101]

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 any observer's frame that you wish to propose.

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.

 

[universal_101]

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.

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

 

[Mentz114]

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

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.

 

[universal_101]

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.

Thanks, and yes BCC stands for Body-Centered Cubic,

You very easily used the word "calculate" and "appear", but I'm trying to comprehend the actual/real increase in density since Time Dilation of Muons is as real/actual as anything, and Experiments confirms that.

But, it seems there is No role of Length contraction except when we need to explain the Time Dilation of Muons or other particles like that. This seems to be the reason why we don't have any Experiment that shows clearly the effects of increased density or any other property for that matter.

 

[Dale]

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

I don't know why you would think this.

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).

All that means is that the glow(density) function is not a law of nature. It is merely an approximation to the law of nature, which is valid only for v<<c.

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

I believe that bunch length contraction in a particle accelerator is an experimental confirmation of length contraction, but that is not a broadly held view.

 

[universal_101]

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).

All that means is that the glow(density) function is not a law of nature. It is merely an approximation to the law of nature, which is valid only for v<<c.

Well if glow-density function seems out of the world, you can always think of any property that can depend on the Length of an Object(as I included the lattice transition).

But what you are suggesting is, any law of nature will change relativistically so that one cannot have any net observation of Length Contraction, except when we consider the Time-Dilation of Muons.

Because if we can have different Length Contraction as Observed from different frames, it will immediately contradict, as different observers will not agree on the state of an object.

 

[bobc2]

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.

Here is one way of looking at length contraction and time dilation. In the description that follows, some physicists would accept this description as actually corresponding to physical reality whereas others will accept the description as a geometric picture that is consistent with special relativity but is not to be accepted as a literal description of the universe. That is, here I'll try to help visualize the sense in which space contracts and time dilates by resorting to a 4-dimensional spacetime universe--however, most physicists on the forum here would object to using this as a literal picture of physical reality, rather accepting it as a useful pedagogical tool.

In the spacetime sketch below I've tried to depict a red and blue rocket moving in opposite directions at the same speed with respect to some rest frame (the black frame with perpendicular coordinates). The rockets are 4-dimensional objects with fixed 4-dimensional geometry--their geometry does not change as 4-D objects, i.e., these objects do not shrink as 4-dimensional objects. However, in this universe model the observers have just limited 3-dimensional cross-section views of the 4-dimensional universe at any instant of time. And for some inexplicable reason (a fundamental feature of special relativity) these 3-D cross-section instantaneous views cut through the 4-D universe at different angles, depending on the slopes of the objects' worldlines (the slope of the worldline determines an object's speed--observers move along their worldlines at the speed of light). The red and blue rockets have the same slopes in the black reference frame, but they slope in opposite directions since they are moving in opposite directions.

Bottom line: Blue sees Red's rocket as being shorter than his own (Blue's). But the cross-section of the universe viewed by Red cuts through Blue's rocket in a direction that gives a cross-section length that is contracted by comparison to Red's. So, the dimensions of the rockets as 4-dimensional objects do not change at all--it's just the different cross-section views that give the appearance of contraction.

Clocks may be used to measure time along the world lines. Notice that the different cross-section views of the 4-D universe result in Blue observing Red's clock at a position that is much earlier along Red's world line than the position of his own (Blue's) clock along his (Blue's) worldline. Thus, Blue thinks Red's clock is running slower than his own. In the example below, using Blue's reference frame, we see that Blue is at clock position #9 while Red is at clock position #8. But, from Red's point of view (using Red's frame), we see that Red is clock position #9 while Blue is at clock position #8. Thus, 4-dimensional clocks don't slow down or speed up at all--different observers just have different cross-section views of a 4-dimensional universe.

 

[Dale]

any law of nature will change relativistically so that one cannot have any net observation of Length Contraction

This is a non sequitor. The first part is true, but the second part is false and does not follow from the first. If it did, then you could say the same about time dilation.

 

[Mentz114]

but I'm trying to comprehend the actual/real increase in density since Time Dilation of Muons is as real/actual as anything, and Experiments confirms that.

I don't know what you mean by 'actual/real'. The relativistic effects are only apparent from another frame. The object itself has not changed.

But, it seems there is No role of Length contraction except when we need to explain the Time Dilation of Muons or other particles like that. This seems to be the reason why we don't have any Experiment that shows clearly the effects of increased density or any other property for that matter.

Length contraction is an effect of changing frames with the Lorentz transformation and is always present.

... you can always think of any property that can depend on the Length of an Object(as I included the lattice transition).

If the lattice transition does not happen in the rest frame, it won't be observed in any other frame. What other property can you think of that would be different when
observed from a moving frame ?

 

[Matterwave]

I'm just going to go ahead and say it. Bobc2, do you use that picture in every SR thread you reply in?

 

[Vanadium 50]

Universal. you've been asked for your calculation multiple times. Please provide it in your next post, or this thread will be locked.

As promised.

If you are ready to post a calculation, with numbers, PM me or another mentor and we can reopen this thread. But as it stands, we can't have a discussion based on "I have a calculation that shows otherwise, but I won't show it to you"

 

[universal_101]

As promised.

If you are ready to post a calculation, with numbers, PM me or another mentor and we can reopen this thread. But as it stands, we can't have a discussion based on "I have a calculation that shows otherwise, but I won't show it to you"

Alright, here is what I was able to get as calculation,

Consider the length if the accelerator to be L, initial number of Muons be x, rest frame Half-Life of Muons be \lambda, and for the simplicity of the calculations I would assume that Muons are traveling with speed v₀ in the Lab's Frame.

Now to calculate the number of Muons reaching the other End, from the Frame of the Lab's Frame. The time of flight of the Muons would be \frac{L}{v_0}, Now during this Time we have to include the Time Dilation and radioactivity law.

Since, \ N = x e^{-\lambda t},

Therefore, the number of Muons reaching the other End should be,

\ y = x e^{-\frac{\lambda}{\sqrt{1 - \frac{v_0}{c^2}}} (\frac{L}{v_0})},

Now, to transform these sets of Equations to another Frame, we should use Lorentz transformations, which says , all the Lengths along the direction of motion would be contracted and velocities need to be added relativistically.

Therefore let's assume a simple Frame which is moving along the length of the accelerator and whose speed w.r.t the Lab's Frame is 'v', and as viewed from the Lab's Frame this Frame is moving opposite to the direction of motion of Muons.

Now, speed of Muons w.r.t this Frame will be v_r = ({v + v_0})/({1 + \frac{v_0 v}{c^2}}) , the length of the accelerator would be, L_r = L (\sqrt{1 - v^2/c^2}) , where as the Time Dilation of half-lifr of Muons would be, \lambda_r = \lambda/\sqrt{1 - (v_r)^2/c^2}

Putting these values in the radioactivity law will give the calculation for the number of muons reaching the other end as observed by this new Frame.

That is, y_r = x e^{-(\lambda_r)({L_r}/{v_r})} ,

After this I just assumed that y_r = y will be the final result.

But since we were discussing the implications of Length contraction and the Experiments which can verify it, I would like to get a better insight on the real effect of Length Contraction.

 

[universal_101]

I don't know what you mean by 'actual/real'. The relativistic effects are only apparent from another frame. The object itself has not changed.

If this is the case, then how would you explain the breaking of a string in Bell's spaceship paradox. If the Length contraction is apparent and not real then according to you it should not break in any frame.

 

[universal_101]

This is a non sequitor. The first part is true, but the second part is false and does not follow from the first. If it did, then you could say the same about time dilation.

You are right on the later part that I can say the same about Time Dilation.

This is exactly what I was thinking and even mentioned in my post# 12, which is quoted below,

That is, I think that it is the number of Muons reaching the other End which specifies the Time Dilation.

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 traveling 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.

 

[Mentz114]

If this is the case, then how would you explain the breaking of a string in Bell's spaceship paradox. If the Length contraction is apparent and not real then according to you it should not break in any frame.

You are confusing accelerating and inertial frames.

Whereas, it is the difference in the age which is called TIME DILATION.

No it is not. That is called differential ageing and is a physical and invariant effect. Time dilation is a coordinate effect.

That is, I think that it is the number of Muons reaching the other End which specifies the Time Dilation.

No. The number of detections is independent of time dilation.

 

[HallsofIvy]

I don't know what you mean by 'actual/real'. The relativistic effects are only apparent from another frame. The object itself has not changed.

If this is the case, then how would you explain the breaking of a string in Bell's spaceship paradox. If the Length contraction is apparent and not real then according to you it should not break in any frame.

Mentz114 did not say it was not "real"- that should have been clear from his first sentence. He said it depends upon the frame of reference. An observer moving with the object would not observe any change.

 

[universal_101]

You are confusing accelerating and inertial frames.

Are you suggesting that, the string in the Bell's spaceship paradox breaks because of acceleration, and not because of relative velocity !

No it is not. That is called differential ageing and is a physical and invariant effect. Time dilation is a coordinate effect.

If that is the case, then it means that younger twin returns with the same difference in his age w.r.t the other Twin, No matter with respect to whom it is moving at what speed.

That means, that only the relative speed of the traveling Twin w.r.t the staying Twin, is what accounts for the difference in the age.

No. The number of detections is independent of time dilation.

Yes they are, but I'm a bit confused about the definition of Time Dilation itself then.

Since you are suggesting that difference in age is frame invariant, don't you think it seems pretty preferential to the Frame w.r.t which the traveling Twin is moving !

 

[universal_101]

Mentz114 did not say it was not "real"- that should have been clear from his first sentence. He said it depends upon the frame of reference. An observer moving with the object would not observe any change.

Does that mean we can never Expect an Experiment confirming Length Contraction,

Since, any observer just by mere moving w.r.t the object cannot effect what happens to the Object, by that same analogy, how can one understand the breaking of the string in Bell's spaceship Paradox, if the string cannot be affected by the motion of the observer.

 

[Dead Boss]

Doesn't Michelson-Morley experiment confirm length contraction?

 

[universal_101]

Doesn't Michelson-Morley experiment confirm length contraction?

NO, at-least not in the real sense, as the Time Dilation of Muons.

But it confirms the validity of Lorentz transformations for Electromagnetic phenomenon, Yes.

Thanks.

 

[Mentz114]

Are you suggesting that, the string in the Bell's spaceship paradox breaks because of acceleration, and not because of relative velocity !

This is all about the strange fact that the spaceships must have unequal accelerations in order for the strings to stay whole. It has no relevance to your muon experiment.

No it is not. That is called differential ageing and is a physical and invariant effect. Time dilation is a coordinate effect.

If that is the case, then it means that younger twin returns with the same difference in his age w.r.t the other Twin, No matter with respect to whom it is moving at what speed.

The ages of the twins is equal to the proper time measured along their worldlines. This is a geometric invariant. All observers agree on those numbers. Time dilation is something that appears when the time in one reference frame is transformed to the time in a different frame.

That means, that only the relative speed of the traveling Twin w.r.t the staying Twin, is what accounts for the difference in the age.

It is the proper length of each worldline. It is not frame dependent.

... I'm a bit confused about the definition of Time Dilation itself then.

See above.

Since you are suggesting that difference in age is frame invariant, don't you think it seems pretty preferential to the Frame w.r.t which the traveling Twin is moving !

Each twin has his own worldline which gives the elapsed time irrespective of what the other one is doing.

 

[PeterDonis]

Since, any observer just by mere moving w.r.t the object cannot effect what happens to the Object, by that same analogy, how can one understand the breaking of the string in Bell's spaceship Paradox, if the string cannot be affected by the motion of the observer.

The string isn't broken by "motion" with respect to some arbitrary observer. The string is broken because the two ends of it are physically attached to spaceships that stretch the string until it breaks. If the string ends weren't attached to the spaceships then the string wouldn't break.

You can express this condition in a frame-invariant way by saying that the ratio of "actual length" to "unstressed length" for the string increases until it exceeds the string's tensile strength, at which point the string breaks.

It's true that how you "interpret" why the string breaks can depend on the motion of the "observer". With respect to the "lab" frame, the frame in which the spaceships are initially at rest and in which they follow identical "acceleration profiles", the string breaks because its "unstressed length" gets smaller and smaller due to "length contraction", while its "actual length" stays constant (because the ends of the string are physically attached to the spaceships, which remain a constant distance apart in this frame). With respect to the "spaceship frame", however, the string's "unstressed length" stays constant, while its "actual length" gets larger and larger because the spaceships are moving apart in this framem, and therefore the string ends are too. (Actually there isn't a single "spaceship frame", but we can use either the front spaceship or the rear spaceship's frame and get the same result.)

So whether or not "length contraction" is "real" depends on whether you are looking at motion relative to some arbitrary "observer", or whether you are looking at actual, physical constraints imposed by the actual, physical conditions of the problem (like the string ends being physically attached to the spaceships). I recommend sticking to the latter. Similar remarks would apply to "time dilation" as in the muon experiment.

 

[universal_101]

This is all about the strange fact that the spaceships must have unequal accelerations in order for the strings to stay whole. It has no relevance to your muon experiment.

So it does not have anything to do with the relative velocity,

First of all, I think that Bell's Spaceship paradox is only Theoretical, that is, there is No experimental confirmation on that.

If it does not have any relevance to the Time Dilation of Muons, then how come we use Length contraction to comprehend the breaking of the string and the Time Dilation of muons.

And, I don't think that, it is the acceleration which cause a relativistic effect of Special relativity, it is always the relative velocity.

The ages of the twins is equal to the proper time measured along their worldlines. This is a geometric invariant. All observers agree on those numbers. Time dilation is something that appears when the time in one reference frame is transformed to the time in a different frame.

Yes, this is what I learned till now, but all this suggests there should be a real length contraction in the sense, that we can measure it, just like Time Dilation of Muons.

It is the proper length of each worldline. It is not frame dependent.

Unfortunately, I am unable to understand physics in abstract form, please let's just stick to the length contraction, Time Dilation, velocity addition etc.

Each twin has his own worldline which gives the elapsed time irrespective of what the other one is doing.

In the end, it means that the younger Twin stays younger by the same amount, No matter which frame we observe his motion from. Please reply, yes or Mo, because I'm getting confused.

 

[Mentz114]

In the end, it means that the younger Twin stays younger by the same amount, No matter which frame we observe his motion from. Please reply, yes or no, because I'm getting confused.

Yes.

Unfortunately, I am unable to understand physics in abstract form, please let's just stick to the length contraction, Time Dilation, velocity addition etc.

That is a pity. The best thing about SR is the fact that all obervers agree about elapsed time on clocks. If that was not so, then 'time-bomb' paradoxes appear.

Yes, this is what I learned till now, but all this suggests there should be a real length contraction in the sense, that we can measure it, just like Time Dilation of Muons.

The distance that cosmic muons descend through the atmosphere is a test of length contraction and time dilation. In the muon frame the half-life is not affected but the distance is contracted. From the Earth frame the distance is the same but the half life is longer. Beautiful symmetry ? ( This was pointed out more than once in previous replies).

 

[Jonathan Scott]

Getting back to the original subject...

The way to calculate how many muons survive is to calculate what the time between the two events looks like from the point of view of an observer on the muon, which is the proper time that elapses along that path. The result is invariant, which means it is the same regardless of the frame of the reference of the observer. Time dilation and length contraction can be used to transform the description from one observer's frame to another, but the proper time is an invariant quantity.

Lorentz velocity transformations between space and time (known as "boosts") are in many ways similar to rotations in space (apart from a somewhat confusing minus sign, which relates to the fact that the rotation "angle" is imaginary). In space, one observer's (x,y,z) measurements between two events may be different from another, but the distance between the events is not affected by the direction of the axes. Similarly, in Lorentz transformations, different observers may measure different time and space displacements, but when they compute the magnitude of the total straight-line displacement between two events, or the length of a specific path connecting two events, everyone gets the same value.

 

[universal_101]

The string isn't broken by "motion" with respect to some arbitrary observer. The string is broken because the two ends of it are physically attached to spaceships that stretch the string until it breaks. If the string ends weren't attached to the spaceships then the string wouldn't break.

You can express this condition in a frame-invariant way by saying that the ratio of "actual length" to "unstressed length" for the string increases until it exceeds the string's tensile strength, at which point the string breaks.

It's true that how you "interpret" why the string breaks can depend on the motion of the "observer". With respect to the "lab" frame, the frame in which the spaceships are initially at rest and in which they follow identical "acceleration profiles", the string breaks because its "unstressed length" gets smaller and smaller due to "length contraction", while its "actual length" stays constant (because the ends of the string are physically attached to the spaceships, which remain a constant distance apart in this frame). With respect to the "spaceship frame", however, the string's "unstressed length" stays constant, while its "actual length" gets larger and larger because the spaceships are moving apart in this framem, and therefore the string ends are too. (Actually there isn't a single "spaceship frame", but we can use either the front spaceship or the rear spaceship's frame and get the same result.)

So whether or not "length contraction" is "real" depends on whether you are looking at motion relative to some arbitrary "observer", or whether you are looking at actual, physical constraints imposed by the actual, physical conditions of the problem (like the string ends being physically attached to the spaceships). I recommend sticking to the latter. Similar remarks would apply to "time dilation" as in the muon experiment.

Does the same applicable if instead of the spaceships, it is the observer which is accelerating ?

Since, you are suggesting that, observing from any frame, it is the acceleration which makes the two spaceships moving apart, even though they are treated identically,

But what is it that is moving that apart, is it the acceleration, or the space itself is expanding. Please, can you provide a reason what makes these spaceships move away from each other, despite they are treated identically.

 

[Mentz114]

Suppose a muon travels distance L measured in the Earth frame. If the lifetime of the muon is T, measured in the muon frame, then in the muon frame

L/λ = vT ( sees length contraction )

and in the Earth frame

L = v(γT) (sees time dilation )

 

[universal_101]

Yes.

Thanks,

But according to the Time Dilation Equation from Lorentz transformation, what matters is the relative velocity.

And if the differential age is invariant then don't you think that differential age is preferential w.r.t whom the traveler started his journey, and the motion of other observers w.r.t the traveler is irrelevant.

That is a pity.

Yes I know, but I cannot help it.

The distance that cosmic muons descend through the atmosphere is a test of length contraction and time dilation. In the muon frame the half-life is not affected but the distance is contracted. From the Earth frame the distance is the same but the half life is longer. Beautiful symmetry ? ( This was pointed out more than once in previous replies).

So, does that mean we have an experimental confirmation of Length contraction, by the Time Dilation of muons,

But with all due respect, it is the Explanation of real Time Dilation of Muons w.r.t every observer, which necessitated the introduction of real Length contraction.

By the way, you never responded on the apparent or real length contraction, which I'm struggling with.

 

[Dale]

Alright, here is what I was able to get as calculation,

Consider the length if the accelerator to be L, initial number of Muons be x, rest frame Half-Life of Muons be \lambda, and for the simplicity of the calculations I would assume that Muons are traveling with speed v₀ in the Lab's Frame.

Now to calculate the number of Muons reaching the other End, from the Frame of the Lab's Frame. The time of flight of the Muons would be \frac{L}{v_0}, Now during this Time we have to include the Time Dilation and radioactivity law.

Since, \ N = x e^{-\lambda t},

The easiest way to approach this problem is to write the radioactivity law in a relativistically invariant form. Specifically:
\ N = n e^{-\lambda \tau} where there are n muons at the starting event and τ is the proper time from the starting event. Since τ is a relativistic invariant then it is immediately clear that all frames agree on N.

Therefore, the number of Muons reaching the other End should be,

\ y = x e^{-\frac{\lambda}{\sqrt{1 - \frac{v_0}{c^2}}} (\frac{L}{v_0})},

Now, to transform these sets of Equations to another Frame, we should use Lorentz transformations, which says , all the Lengths along the direction of motion would be contracted and velocities need to be added relativistically.

You haven't written your equation in terms of coordinates, so you really can't use the Lorentz transform since the Lorentz transform transforms between different coordinate systems. That said, this is a correct expression in the accelerator's frame except that you are missing a square on v0.

Therefore let's assume a simple Frame which is moving along the length of the accelerator and whose speed w.r.t the Lab's Frame is 'v', and as viewed from the Lab's Frame this Frame is moving opposite to the direction of motion of Muons.

Now, speed of Muons w.r.t this Frame will be v_r = ({v + v_0})/({1 + \frac{v_0 v}{c^2}}) , the length of the accelerator would be, L_r = L (\sqrt{1 - v^2/c^2}) , where as the Time Dilation of half-lifr of Muons would be, \lambda_r = \lambda/\sqrt{1 - (v_r)^2/c^2}

Putting these values in the radioactivity law will give the calculation for the number of muons reaching the other end as observed by this new Frame.

That is, y_r = x e^{-(\lambda_r)({L_r}/{v_r})} ,

This expression is incorrect because the time that it takes for the muons to reach the detector is not equal to L_r/v_r in any frame except the accelerator frame. For example consider v=-v0, i.e. v_r=0 or the muon's rest frame. In this frame, since the muons are at rest, the formula L_r/v_r predicts that the time to reach the detector is infinite. However, the detector is moving towards the muons and therefore the muons reach the detector in a finite amount of time (specifically L_r/v0).

I would recommend using the invariant form of the equation. It solves all of the hassles immediately.

 

[Dale]

That is, I think that it is the number of Muons reaching the other End which specifies the Time Dilation.

The number of muons depends on the decay rate and the decay time, both of which vary relativistically. The net result is that all frames agree on the number of muons that arrive, although they may disagree about the rate that they are decaying and how long it takes them to reach the detector.

 

[universal_101]

Getting back to the original subject...

The way to calculate how many muons survive is to calculate what the time between the two events looks like from the point of view of an observer on the muon, which is the proper time that elapses along that path. The result is invariant, which means it is the same regardless of the frame of the reference of the observer. Time dilation and length contraction can be used to transform the description from one observer's frame to another, but the proper time is an invariant quantity.

I guess you are right, that observing the experiment from the reference frame of the Muons produces the same number of Muons surviving to the other End.

Does that mean, my calculations are incorrect because I am using a frame other than that of Muons, or there is very obvious trivial mistake in my calculations.

Lorentz velocity transformations between space and time (known as "boosts") are in many ways similar to rotations in space (apart from a somewhat confusing minus sign, which relates to the fact that the rotation "angle" is imaginary). In space, one observer's (x,y,z) measurements between two events may be different from another, but the distance between the events is not affected by the direction of the axes. Similarly, in Lorentz transformations, different observers may measure different time and space displacements, but when they compute the magnitude of the total straight-line displacement between two events, or the length of a specific path connecting two events, everyone gets the same value.

Agreed...I learned about these properties of co-ordinate transformations from this thread only.

Thanks.

 

[universal_101]

This expression is incorrect because the time that it takes for the muons to reach the detector is not equal to L_r/v_r in any frame except the accelerator frame. For example consider v=-v0, i.e. v_r=0 or the muon's rest frame. In this frame, since the muons are at rest, the formula L_r/v_r predicts that the time to reach the detector is infinite. However, the detector is moving towards the muons and therefore the muons reach the detector in a finite amount of time (specifically L_r/v0).

I would recommend using the invariant form of the equation. It solves all of the hassles immediately.

Thanks,

Of-course, changing the co-ordinates around an event does not change the event, this is what I myself was suggesting ever since.

But if Lorentz transformation is just a co-ordinate transform, then how can it support/explain/justify the real events like Time Dilation of Muons and a real Length contraction in order to justify the Time Dilation of Muons themselves. To me it seems very preferential to the two observers, the Muons and the Lab Frame.

Besides, I included every thing you asked me to include in my calculations, but in the end what you are suggesting is, just do a co-ordinate transform for the reference frame of Muons.

 

[Mentz114]

So, does that mean we have an experimental confirmation of Length contraction, by the Time Dilation of muons,

Unless the Earth frame sees a time dilation, and the muon frame sees length contraction there will be a paradox. So if time dilation is 'real', so is length contraction.

By the way, you never responded on the apparent or real length contraction, which I'm struggling with.

It's a coordinate effect. I don't know if that makes it real.