Time dilation of Muons and a Paradox

[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.

 

[universal_101]

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.

Agreed, But it is the very use of Lorentz transformation, which is supposed to only transform the co-ordinates around a event, that makes the Length contraction a necessity, to which we don't have any experimental Evidence even after more than 100 years of it's introduction.

But since we, in our frame, use the Lorentz transformation to explain the Time dilation of unstable particles, which it was not supposed to be used because it is just a co-ordinate effect.

And It makes me think that, can the decay of these Time Dilating unstable particles be dependent on the Sun, since I already encountered several papers confirming the same.

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

I don't think a co-ordinate effect can be real, it cannot be by it's mere definition.

 

[Mentz114]

The physical situation in the muon scenario is simple - a muon is created at a certain place and time, travels to another point in space and time and decays. Special relativity predicts correctly what happens, which is support for the idea of time dilation/length contraction, but I don't believe those phenomena can be directly observed. I think you made this point earlier. They must belong to that class of things which are part of successful theories but cannot be directly observed. Like the wave function, or the vector potential.

Whether something which cannot be directly observed is 'real' may just be an argument over words.

 

[Dale]

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.

The Lorentz transform can be used to explain experimental observations because the laws of physics are invariant under Lorentz transforms.

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.

You can do a coordinate transform to any frame you like. It doesn't have to be the rest frame of the muons. I just picked that frame because it made the mistake very obvious.

 

[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.

I think you are right, that the expression for the $v_r$ is incorrect, since I did not include the relative speed of the accelerator in this new frame, when I calculated $v_r$,

But soon after, I realized that $v_r$, is the relative velocity of the Muons w.r.t the accelerator, and therefore a invariant, which implies, $v_r = v_0$ simply.

But even then I can't seem to get any closer to the understanding or the correct result.

 

[universal_101]

...Special relativity predicts correctly what happens, which is support for the idea of time dilation/length contraction, but I don't believe those phenomena can be directly observed. I think you made this point earlier.

First of all, Thanks for the insight on your part.

But what do you mean by, we cannot observe Time Dilation directly, since Experimentally we do observe the Time Dilation.

The problematic part is Length Contraction which we cannot observe because Everything else changes relativistic-ally.

Remembering that, it is the Explanation/support of Time Dilation of Muons(which is real/observable) from Special Relativity, that necessitated the introduction of real/observable Length contraction, to which we don't have any experimental Evidence,

And NOR do we Expect it in near Future, because according to some people here and me, any object does not change it's state just because a observer is moving relative to it. Even though, some how the strings breaking in Bell's spaceship paradox does exactly the same(Until and unless it is the acceleration which make the string to break, but then again, how?).

They must belong to that class of things which are part of successful theories but cannot be directly observed. Like the wave function, or the vector potential.

But Time Dilation of Muons is observable, unlike the wave function(which is extended interpretation concept).

Whether something which cannot be directly observed is 'real' may just be an argument over words.

Agreed, but Time Dilation of Muons can be directly observed.

 

[universal_101]

The Lorentz transform can be used to explain experimental observations because the laws of physics are invariant under Lorentz transforms.

If the Lorentz transformation keeps the laws of Physics invariant, then how were we able to comprehend the Time Dilation of Muons themselves, when we do a Lorentz transform to some observer who is moving w.r.t them.

But if we look at all this in the following way, Everything then makes sense,

That is if, Time Dilation of Muons, itself is a law of physics, which is invariant under Lorentz transform, we can make every other consequences like, real Length contraction etc. Go away. They can then just be only tools to transform from one frame to the other, around an event or state of an object.

You can do a coordinate transform to any frame you like. It doesn't have to be the rest frame of the muons. I just picked that frame because it made the mistake very obvious.

I corrected my equations, for the new frame,

Thanks.

 

[Mentz114]

Agreed, but Time Dilation of Muons can be directly observed.

I must have missed this. How ?

 

[universal_101]

Whether something which cannot be directly observed is 'real' may just be an argument over words.

Agreed, but Time Dilation of Muons can be directly observed.

I must have missed this. How ?

I am not sure how to respond to this. But I'm guessing you don't want to discuss Logic anymore.

 

[Dale]

But soon after, I realized that $v_r$, is the relative velocity of the Muons w.r.t the accelerator, and therefore a invariant, which implies, $v_r = v_0$ simply.

Actually, it is the approaching speed which is important.

 

[Samshorn]

It is the Explanation of time dilation of muons ... that necessitated the introduction of ... length contraction, to which we don't have any experimental evidence...

I assume you've been told many times that that's completely false, both historically and conceptually. Historically, length contraction was introduced by Fitzgerald and Lorentz in order to account for the results of the Michelson-Morley experiment, long before anyone ever dreamed of muons or even special relativity. And of course the MM experiment along with all other failed attempts to measure absolute velocity and all the experimental demonstrations of the invariance of light speed, and indeed the Lorentz invariance of all physical phenomena in terms of standard inertial coordinates, collectively are irrefutable experimental evidence of both length contraction and time dilation. Also, length contraction emerges from Lorentz's theorem of corresponding states based on the already-known laws of electrodynamics, and of course Lorentz always stressed the physical reality of (and necessity of) this contraction for active transformations.

An object does not change it's state just because a observer is moving relative to it.

You need to distinguish between intrinsic and extrinsic state variables, and between passive and active transformations.

 

[universal_101]

Actually, it is the approaching speed which is important.

Are you talking about the approaching speed of Muons w.r.t the other End ?

Is it different from the relative velocity of Muons w.r.t the the other End ?

And how important exactly !

 

[universal_101]

I assume you've been told many times that that's completely false,

Since you are assuming what have happened so far, can I safely assume that you did not read the previous discussions.

...Historically, length contraction was introduced by Fitzgerald and Lorentz in order to account for the results of the Michelson-Morley experiment, long before anyone ever dreamed of muons or even special relativity.

I agree, that Fitzgerald introduced/coined the term Length contraction to explain the results of MMX. But again, we are not discussing when was it first proposed.

And of course the MM experiment along with all other failed attempts to measure absolute velocity and all the experimental demonstrations of the invariance of light speed, and indeed the Lorentz invariance of all physical phenomena in terms of standard inertial coordinates, collectively are irrefutable experimental evidence of both length contraction and time dilation.

MMX was about the light and its properties, whereas, Time Dilation of Muons has nothing to do with Light and it's properties because if it does, there are contradictions of Logic. Again for the later part you need to read the previous posts.

Also, length contraction emerges from Lorentz's theorem of corresponding states based on the already-known laws of electrodynamics, and of course Lorentz always stressed the physical reality of (and necessity of) this contraction for active transformations.

Why don't then you just give me a real example of this physical reality, it will end all the conflicts.

You need to distinguish between intrinsic and extrinsic state variables, and between passive and active transformations.

Unfortunately, I don't know what all this mean. I suggest you read the previous discussions, if you think, there is an obvious mistake somewhere.

 

[Vanadium 50]

This thread is going around in circles. Enough.