Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[matheinste]

Again, the original paradox relied on implicit knowledge about how materials behave in their own rest frame, but I think anyone would acknowledge that to make the proof 100% rigorous you need to actually calculate how the material would behave in a given frame, based on the internal forces holding the material together.

While the (forum) thread is fairly quiet and at the risk of my appearing ignorant, can you perhaps clear up a point that has long been bothering me. I have it in my head how, in principle, the stresses arise to break the thread using SR from either frame. What I am having a problem with from the point of view of an SR analysis are the words "the thread breaks through Lorentz contraction" when Lorentz contraction in SR is not stress inducing. I am of course assunming that Lorentz contraction in SR means the same as Lorentz transformation.

Matheinste.

 

[cfrogue]

What must overlap and cannot?

The distance between the two ships is within the same coords as the launch frame. The distance does not change and is the original distance as in the coords of the launch frame.

Yet, the string within that space is Minkowsky.

How do you resolve this?

 

[DrGreg]

While the (forum) thread is fairly quiet and at the risk of my appearing ignorant, can you perhaps clear up a point that has long been bothering me. I have it in my head how, in principle, the stresses arise to break the thread using SR from either frame. What I am having a problem with from the point of view of an SR analysis are the words "the thread breaks through Lorentz contraction" when Lorentz contraction in SR is not stress inducing. I am of course assunming that Lorentz contraction in SR means the same as Lorentz transformation.

The tension in the string is caused by the front rocket which is accelerating "too much" to keep the tension zero. For zero tension, which would be Born rigid acceleration, the front rocket would have to accelerate less than the back rocket. As their accelerations are actually equal, the front rocket is pulling too hard for equilibrium and the string breaks.

The phrase "the thread breaks through Lorentz contraction" isn't all that precise.

 

[matheinste]

The tension in the string is caused by the front rocket which is accelerating "too much" to keep the tension zero. For zero tension, which would be Born rigid acceleration, the front rocket would have to accelerate less than the back rocket. As their accelerations are actually equal, the front rocket is pulling too hard for equilibrium and the string breaks.

The phrase "the thread breaks through Lorentz contraction" isn't all that precise.

Thanks. I understand the mechanism. It was just the wording that bothered me. It does seem to be very common usage though.

Matheinste.

 

[DrGreg]

Thanks. I understand the mechanism. It was just the wording that bothered me. It does seem to be very common usage though.

Yes. I think it's related to the misuse of the phrase "Lorentz contraction" that I mentioned way back in post #38 of this thread.

 

[matheinste]

Yes. I think it's related to the misuse of the phrase "Lorentz contraction" that I mentioned way back in post #38 of this thread.

Yes I thought that was the problem. I suppose the context should make it obvious what meaning it has in any particular situation.

Thanks.

Matheinste.

 

[JesseM]

What I am having a problem with from the point of view of an SR analysis are the words "the thread breaks through Lorentz contraction" when Lorentz contraction in SR is not stress inducing.

I think that phrase is too ambiguous without additional explanation--the speaker could mean the ideal equilibrium length is contracting so the string "wants" to contract but can't, they could mean the atoms and electromagnetic fields surrounding them are contracting, or they could mean something else, it isn't clear.

I am of course assunming that Lorentz contraction in SR means the same as Lorentz transformation.

No, "Lorentz contraction" is just a synonym for length contraction, while the "Lorentz transformation" is a more general coordinate transformation for translating arbitrary coordinates from one frame to another; if frame B is moving along the x-axis of frame A at speed v, and the spatial origins of the two coordinate systems (x=0 in frame A and x'=0 in frame B) coincided at a time of 0 in each frame, then if a particular event is assigned coordinates x,t in frame A, the Lorentz transformation tells you what the coordinates of the same event would be in frame B:

x' = gamma*(x - vt)
t' = gamma*(t - vx)
where gamma = 1/sqrt(1 - v^2/c^2)

Note that length contraction (or Lorentz contraction), time dilation and the relativity of simultaneity can all be derived from the Lorentz transformation (which itself is derived from the two fundamental postulates of SR).

 

[JesseM]

The distance between the two ships is within the same coords as the launch frame. The distance does not change and is the original distance as in the coords of the launch frame.

Yet, the string within that space is Minkowsky.

What do you mean when you say the string "is" Minkowski? Note that even if we assumed a classical Newtonian universe, but also assumed that the laws of electromagnetism obeyed the familiar equations in one preferred frame (as in the old pre-relativistic aether theory), then an accelerating string with constant length in this frame would break too, and the analysis of the reason why would be exactly the same as it is in relativity (just applying the laws of electromagnetism in that frame to analyze a set of atoms with increasing velocity relative to the frame).

 

[matheinste]

I think that phrase is too ambiguous without additional explanation--the speaker could mean the ideal equilibrium length is contracting so the string "wants" to contract but can't, they could mean the atoms and electromagnetic fields surrounding them are contracting, or they could mean something else, it isn't clear.

No, "Lorentz contraction" is just a synonym for length contraction, while the "Lorentz transformation" is a more general coordinate transformation for translating arbitrary coordinates from one frame to another; if frame B is moving along the x-axis of frame A at speed v, and the spatial origins of the two coordinate systems coincided at a time of 0 in each frame, then if a particular event is assigned coordinates x,t in frame A, the Lorentz transformation tells you what the coordinates of the same event would be in frame B:

x' = gamma*(x - vt)
t' = gamma*(t - vx)
where gamma = 1/sqrt(1 - v^2/c^2)

Note that length contraction (or Lorentz contraction), time dilation and the relativity of simultaneity can all be derived from the Lorentz transformation (which itself is derived from the two fundamental postulates of SR).

Thanks. I'm OK with that.

Matheinste.

 

[matheinste]

I think that phrase is too ambiguous without additional explanation--the speaker could mean the ideal equilibrium length is contracting so the string "wants" to contract but can't, they could mean the atoms and electromagnetic fields surrounding them are contracting, or they could mean something else, it isn't clear.

No, "Lorentz contraction" is just a synonym for length contraction, while the "Lorentz transformation" is a more general coordinate transformation for translating arbitrary coordinates from one frame to another; if frame B is moving along the x-axis of frame A at speed v, and the spatial origins of the two coordinate systems (x=0 in frame A and x'=0 in frame B) coincided at a time of 0 in each frame, then if a particular event is assigned coordinates x,t in frame A, the Lorentz transformation tells you what the coordinates of the same event would be in frame B:

x' = gamma*(x - vt)
t' = gamma*(t - vx)
where gamma = 1/sqrt(1 - v^2/c^2)

Note that length contraction (or Lorentz contraction), time dilation and the relativity of simultaneity can all be derived from the Lorentz transformation (which itself is derived from the two fundamental postulates of SR).

I have had further thoughts about what you have said and it has now cleared up what was a very fundamental misunderstanding.

Matheinste.

 

[JesseM]

I have had further thoughts about what you have said and it has now cleared up what was a very fundamental misunderstanding.

Matheinste.

Glad it helped. By the way, I made a small error when I wrote out the Lorentz transformation equations: I said that t' = gamma*(t - vx), when actually it should be t' = gamma*(t - vx/c^2)

 

[matheinste]

Glad it helped. By the way, I made a small error when I wrote out the Lorentz transformation equations: I said that t' = gamma*(t - vx), when actually it should be t' = gamma*(t - vx/c^2)

I didn't actually read your mathematics because I have seen the transformation equations so often.

Thanks. Matheinste.

 

[cfrogue]

What do you mean when you say the string "is" Minkowski? Note that even if we assumed a classical Newtonian universe, but also assumed that the laws of electromagnetism obeyed the familiar equations in one preferred frame (as in the old pre-relativistic aether theory), then an accelerating string with constant length in this frame would break too, and the analysis of the reason why would be exactly the same as it is in relativity (just applying the laws of electromagnetism in that frame to analyze a set of atoms with increasing velocity relative to the frame).

The last article posted on this suggested that the ships drift further apart from the POV of the accelerating ships.

The launch frame sees the distance remaining the same.

These reasons are not the same for the string to break.
Do you agree?

 

[JesseM]

The last article posted on this suggested that the ships drift further apart from the POV of the accelerating ships.

The launch frame sees the distance remaining the same.

These reasons are not the same for the string to break.

Sure, it is common for different frames to have different explanations for the same events. For example, see this page on electromagnetism in relativity--the cause of a charge's movement may purely involve the electric force in one frame, but involve both the electric and magnetic force in another.

 

[A.T.]

These reasons are not the same for the string to break.
Do you agree?

Depends how you formulate the 'reasons'. This 'reason' applies in every frame:

The string breaks, because the total length that can be spanned by its elements is less than the distance between it's ends

Reasons are just qualitative explanations, and can be given at different levels of abstraction. Talk to a kid in his why-phase and you will see how many different reasons you can give for the same effect. In some counter-intuitive cases the cause-effect-reasoning doesn't work at all, like for feedback-loops:
http://karmatics.com/dwfttw

What matters is that the frames agree on the result.

 

[Dale]

Depends how you formulate the 'reasons'. This 'reason' applies in every frame:

The string breaks, because the total length that can be spanned by its elements is less than the distance between it's ends

Or "the proper distance between the spaceships is greater than the proper length of the unstressed string". Since proper distances are frame invariant this is a coordinate-independent reason.

What matters is that the frames agree on the result.

That is a very succinct way to express one of the core ideas of the first postulate.

 

[cfrogue]

Depends how you formulate the 'reasons'. This 'reason' applies in every frame:

The string breaks, because the total length that can be spanned by its elements is less than the distance between it's ends

Reasons are just qualitative explanations, and can be given at different levels of abstraction. Talk to a kid in his why-phase and you will see how many different reasons you can give for the same effect. In some counter-intuitive cases the cause-effect-reasoning doesn't work at all, like for feedback-loops:
http://karmatics.com/dwfttw

What matters is that the frames agree on the result.

The result is that the accelerating frame sees the distance increasing between the ships and the launch frame sees a constant distance between the ships.

That is really the issue.

The string is incidental.

 

[Dale]

That doesn't make any sense at all. How can you possibly think that the string is incidental to the question of whether or not the string breaks?

 

[A.T.]

The result is that the accelerating frame sees the distance increasing between the ships and the launch frame sees a constant distance between the ships. That is really the issue.

Big news! Lengths are frame dependent in SR.

The string is incidental.

The frames must agree on the breaking of string, but not on the length of the string when it breaks. The Lorentz transformation can change the length of the string, but it cannot fix a broken string.

 

[cfrogue]

That doesn't make any sense at all. How can you possibly think that the string is incidental to the question of whether or not the string breaks?

The latest paper shows the distance between the ships increases in the accelerating frame. Thus, the "cause of action" for the string breaking is the increasing distance between the ships.

Thus, the question is why does this string break. Is this not a scientific question?

Yet, in the launch frame the distance does not increase and remains constant.

For two spaceships having equal accelerations, as in Bell’s spaceship example, the distance between the moving ships appears to be constant, but the rest frame distance between them continually increases. This means that a cable between the two ships must eventually break if the acceleration continues.

http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

 

[cfrogue]

Big news! Lengths are frame dependent in SR.

The frames must agree on the breaking of string, but not on the length of the string when it breaks. The Lorentz transformation can change the length of the string, but it cannot fix a broken string.

Agreed lengths are frame dependent.

But, given that the distance between the ships remains constant in the launch frame and increases in the instantaneous rest frame of the accelerating ships, we have one frame, the "accelerating frame" claiming the distance between the ships increases. That is not a frame dependent length issue because it is within the same instantaneous frame of the ships.

 

[JesseM]

The latest paper shows the distance between the ships increases in the accelerating frame. Thus, the "cause of action" for the string breaking is the increasing distance between the ships.

The increasing distance in this frame alone does not prove the string breaks, you have to make some additional assumptions about how materials behave in this frame to show it breaks. Do you disagree?

 

[A.T.]

Thus, the "cause of action" for the string breaking is the increasing distance between the ships.

I already explained you the fallacy of cause-effect-reasoning: it is pretty arbitrary what you call the 'cause'.

Thus, the question is why does this string break. Is this not a scientific question?

As I already told you, even kids can create an infinite chain of why-questions. Physics doesn't finally tell you why. But it can predict if & when the string breaks. The closest to "why" is: The string breaks, because the total length that can be spanned by its elements is less than the distance between it's ends

But, given that the distance between the ships remains constant in the launch frame and increases in the instantaneous rest frame of the accelerating ships, we have one frame, the "accelerating frame" claiming the distance between the ships increases. That is not a frame dependent length issue because it is within the same instantaneous frame of the ships.

Sorry, you lost me here. What is "within the same instantaneous frame" ?

Is your problem that the ships have equal speed in one frame, but different speeds in a second frame? That is relativity of simultaneity:
http://en.wikipedia.org/wiki/Relativity_of_simultaneity
Since the acceleration of the rockets is synchronized in the launch frame, it cannot be synchronous in the rocket's frame.

 

[atyy]

Yet, in the launch frame the distance does not increase and remains constant.

In the launch frame, if you know you have Lorentz symmetry, you can always do a transformation to another frame. This is elegant.

But let's say one doesn't know one has Lorentz symmetry in the launch frame, can one do a stupid masochistic brute force calculation that shows the same as the elegant calculation? Yes, I think this is interesting, and Bell supplies a handwavy calculation. However, I do think maybe it is a bit too handwavy, but I haven't been able to come up with anything better - perhaps in a week or so.

 

[Dale]

Thus, the question is why does this string break. Is this not a scientific question?

Sure, and it has been answered in posts 226, 225, and many other times throughout this long thread.

 

[cfrogue]

Sure, and it has been answered in posts 226, 225, and many other times throughout this long thread.

Yes it has.

But, what has not been answered is how the space between the ships can be both expanding and remaining constant at the same time.
Since there exist an instantaneous rest frame for the acceleration calculations, the launch frame is not at all involved in the calculation of the expanding distance between the ships.

Thus, two different methods arrive at different conclusions about the distance between the ships.

Certainly, SR says a moving rod contracts whereas the stationary system metrics are not altered for the rod.

Also, this may be simply a natural consequence of accelerating where the lanuch frame "sees" a constant metric and the accelerating frame sees and expanding metric.

It seems to be simply the reverse from what one would expect for motion, but that does not necessarily mean anything.

Then after the acceleration, I guess the launch frame see length contraction from the original distance.

I wonder what the accelerating frame thinks about its metrics after the acceleration. If it does not snap back to it original distance, then LT would not work correctly from the length contraction calculation of the original launch frame.

 

[cfrogue]

In the launch frame, if you know you have Lorentz symmetry, you can always do a transformation to another frame. This is elegant.

But let's say one doesn't know one has Lorentz symmetry in the launch frame, can one do a stupid masochistic brute force calculation that shows the same as the elegant calculation? Yes, I think this is interesting, and Bell supplies a handwavy calculation. However, I do think maybe it is a bit too handwavy, but I haven't been able to come up with anything better - perhaps in a week or so.

I would be interested in your calculation.

This seems to be an area that has been left off from a rigorous analysis.

In the prior post, there are also some interesting possibilities after the acceleration is done.

 

[cfrogue]

In the launch frame, if you know you have Lorentz symmetry, you can always do a transformation to another frame. This is elegant.

But let's say one doesn't know one has Lorentz symmetry in the launch frame, can one do a stupid masochistic brute force calculation that shows the same as the elegant calculation? Yes, I think this is interesting, and Bell supplies a handwavy calculation. However, I do think maybe it is a bit too handwavy, but I haven't been able to come up with anything better - perhaps in a week or so.

OK, there is no paradox.
Equation 4 shows the length will adjust back to its original d when the acceleration stops.
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

Then, the launch frame will apply length contraction once the acceleration stops and all is correct.

This paper is just showing a natural behavior of SR for acceleration. The ships grow further apart as the acceleration continues, but will snap back to their original distance once the acceleration stops based on equation 4.

It is kind of an interesting behavior that the metrics of an accelerating frame expand in the direction of acceleration until the acceleration stops. Then, they instantly restore to the original pre-acceleration metrics.

That is at least what I read.

 

[atyy]

OK, there is no paradox.
Equation 4 shows the length will adjust back to its original d when the acceleration stops.
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

Then, the launch frame will apply length contraction once the acceleration stops and all is correct.

This paper is just showing a natural behavior of SR for acceleration. The ships grow further apart as the acceleration continues, but will snap back to their original distance once the acceleration stops based on equation 4.

It is kind of an interesting behavior that the metrics of an accelerating frame expand in the direction of acceleration until the acceleration stops. Then, they instantly restore to the original pre-acceleration metrics.

That is at least what I read.

Yes, there is no paradox - the elegant calculation is enough, the stupid masochistic brute force calculation is just for fun. However, once the acceleration stops, a ship will continue moving with constant non-zero velocity.

 

[JesseM]

But, what has not been answered is how the space between the ships can be both expanding and remaining constant at the same time.

What does "expanding space" mean? Is this just a vague poetic description, or something that can be defined in a precise mathematical way?

Also, this may be simply a natural consequence of accelerating where the lanuch frame "sees" a constant metric and the accelerating frame sees and expanding metric.

What does it mean to see a constant metric? Certainly the spacetime metric is the same in all frames, and in relativity the spacetime metric is the fundamental one. I suppose you can talk about a spatial metric in any given coordinate system, but what does it mean to say it's expanding? Are you just saying that the distance between the rockets expands? But of course even in classical Newtonian mechanics the distance between rockets will expand if they have different coordinate accelerations, yet in this situation I doubt you would talk about an "expanding metric". Do you have any exact definition for this phrase?

 

[cfrogue]

What does "expanding space" mean? Is this just a vague poetic description, or something that can be defined in a precise mathematical way?

You are good.

I mean that the posted paper shows the nature of SR's acceleration that the accelerating frame sees an expanding metric in the direction of acceleration when compared to the instantaneous rest frame. It did not matter that there were two ships. Equation 4 shows the metric expansion within the frame. Then, once this v compare to the instantaneous subsides, ie the acceleration stops, the metric reduces to just d since the v is 0 in equation 4.

Furthermore, the launch frame sees a constant distance during the acceleration. However, once the acceleration stops, LT length contraction applies since the SR acceleration equations for the launch frame no longer apply once acceleration stops.

What does it mean to see a constant metric? Certainly the spacetime metric is the same in all frames, and in relativity the spacetime metric is the fundamental one. I suppose you can talk about a spatial metric in any given coordinate system, but what does it mean to say it's expanding? Are you just saying that the distance between the rockets expands? But of course even in classical Newtonian mechanics the distance between rockets will expand if they have different coordinate accelerations, yet in this situation I doubt you would talk about an "expanding metric". Do you have any exact definition for this phrase?

The metric I describe is only the one in the frame.

I think this is now a required term since the accelerating frame sees its meter stick expand since the speed of light is constant.

What other choice do you have?

Have you seen anything in the literature to describe this phenomena?

 

[Dale]

what has not been answered is how the space between the ships can be both expanding and remaining constant at the same time.

That, at least, is very easy to answer. The answer is simply that distance is a coordinate-dependent (a.k.a. "Relative")quantity, so by definition different reference frames will disagree. Now, if you want a coordinate-independent explanation of the string breaking then I refer you to my earlier post.

 

[atyy]

Equation 4 shows the metric expansion within the frame. Then, once this v compare to the instantaneous subsides, ie the acceleration stops, the metric reduces to just d since the v is 0 in equation 4.

v does not become 0 in the launch frame unless the ships decelerate.

 

[cfrogue]

v does not become 0 in the launch frame unless the ships decelerate.

Agreed, but this v is relative to the instantaneous rest frame which is not the launch frame.
This instantaneous rest frame is auxiliary in order to solve the problem.

The paper calls the frame S'.

 

[cfrogue]

That, at least, is very easy to answer. The answer is simply that distance is a coordinate-dependent (a.k.a. "Relative")quantity, so by definition different reference frames will disagree. Now, if you want a coordinate-independent explanation of the string breaking then I refer you to my earlier post.

Universal generalization of "distance is a coordinate-dependent" is not logically true.

It is existentially quantified by the following.

1) The distance between objects in an inertial frame is constant and it is not the case that distance is a coordinate-dependent.

2) If there exists relative motion, then the stationary frame will calculate length contraction for the moving frame metrics.

3) If there is an accelerating frame, then the frame will experience metric expansion in the direction of acceleration. However, the launch frame will calculate a constant distance for objects in the accelerating frame.

These are all different behaviors based on the particular SR motion and thus the phrase "distance is a coordinate-dependent" is not a universally descriptive phrase.