Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[cfrogue]

In the launch frame the string won't experience any change in length until it snaps. As I've said, the stress in the string will increase though. I think when people cite "Lorentz contraction" as an explanation for the string breaking, what they're getting at is that the string "wants" to contract but can't because it's attached to the ships...it may be easier to make sense of this if we think of a spring rather than a string, since you may remember from classical mechanics that springs have a "rest length" that they naturally assume when nothing is pulling or pushing on them (the rest length minimizing the stress in the spring), and that when they are pulled to a greater length than the rest length they pull back with greater and greater force, as if they are "trying" to return to that length (and obviously if you pull a spring far enough past its rest length, it'll snap). If you had two identical springs traveling alongside each other, one attached to the two ships and one with its ends free whose length was equal to its rest length, then the length of the free spring would grow shorter and shorter as seen by the launch frame as its velocity increased, which implies that the spring attached to the ships, whose length does not change in this frame, is being extended farther and farther past its own natural rest length.

Well, the SR acceleration equations indicate the distance between ther ships will not change.

From the POV of the rest observer, what is the math to indicate the space remains constant but a rod will contract if allowed between the two ships.

All these links show what happens from the POV of the accelerating ships.

I want to concentrate on the math from the rest/launch frame's POV.

Also, this paper seems to say something different.

4 Conclusion
We have seen that the physical length of an object is the rest frame length as
measured in the instantaneous rest frame of the object. 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

 

[JesseM]

Well, the SR acceleration equations indicate the distance between ther ships will not change.

From the POV of the rest observer, what is the math to indicate the space remains constant but a rod will contract if allowed between the two ships.

If the ends of the rod are connected to the ships then it can't contract, although it will eventually break. If it's not connected, then the math to indicate it contracts is just the fact that we expect the length of a free rod to stay constant in its own rest frame (assuming it behaves like a spring and has a natural 'rest length' it will return to after a small deformation due to acceleration), which means in the observer's frame it should contract according to the length contraction equation (if you want to calculate things without even referring to the rod's rest frame, I'm sure you could show why it contracts with a detailed analysis of the intermolecular forces in the rod at different velocities as defined in the observer's frame).

Also, this paper seems to say something different.

4 Conclusion
We have seen that the physical length of an object is the rest frame length as
measured in the instantaneous rest frame of the object. 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

I already addressed this paper (and pointed out that it definitely says that string will snap) in post #32, did you read that one? The paper certainly doesn't dispute the idea that in the frame of the observer the length of the string will be constant until it snaps, it just argues that defining "length" in terms of the coordinates of an outside observer is not very physical, and that it's better to use a quantity called "rest frame length" which is defined solely in the string's own rest frame.

 

[DrGreg]

I think it is perhaps worth pointing out that some people have a false impression about what Lorentz contraction is. They may think that "when something accelerates it gets shorter". Or to be a bit more precise, if Alice measures (=x) something at rest (relative to Alice) and then later measures (=y) the same thing in motion, the length contracts. There may then be some debate over whether or not the "things" this applies to are just solid objects, or gaps between objects, or "space itself".

The above description of Lorentz contraction is wrong.

In many circumstances, what I said above is true, but reason it is true is not simply Lorentz contraction alone; it is Lorentz contraction plus some other reason combined.

A more accurate description of Lorentz contraction is that when inertial observer Bob measures the length z between two things both at rest relative to Bob, and another inertial observer Alice in relative motion measures the length y between the same two things at the same time, Alice measures a shorter distance than Bob.

So, the situation I described in the first paragraph will arise if there is a reason why Alice's initial "rest distance" x between the two things beforehand is the same as the Bob's final "rest distance" z. For example if the the two things are the two ends of a rigid object that doesn't break into pieces as a result of the acceleration.

The attached illustration emphasises my point. The transformation of x to y is not Lorentz contraction. The transformation of z to y is Lorentz contraction. If there is a reason why x = z, then the transformation of x to y will be a contraction. But if there's no reason, then contraction need not occur.

 

[A.T.]

I want to concentrate on the math from the rest/launch frame's POV.

The string is made up of atoms held together by electromagnetic forces. In the launch frame all these atoms and their electromagnetic fields are contracting and cannot fill the constant distance between the rockets anymore.

 

[cfrogue]

If the ends of the rod are connected to the ships then it can't contract, although it will eventually break. If it's not connected, then the math to indicate it contracts is just the fact that we expect the length of a free rod to stay constant in its own rest frame (assuming it behaves like a spring and has a natural 'rest length' it will return to after a small deformation due to acceleration), which means in the observer's frame it should contract according to the length contraction equation (if you want to calculate things without even referring to the rod's rest frame, I'm sure you could show why it contracts with a detailed analysis of the intermolecular forces in the rod at different velocities as defined in the observer's frame).

There are three frames, the launch frame, a theoretical instantaneous at rest frame and the accelerating frame.

In the theoretical instantaneous at rest frame, this is where the various papers prove one way or another the string snaps.

But, I want to focus on the launch frame. This frame is not seeing the distance change between the ships..

Question, does the launch frame conclude based on observations that the string breaks?

If so, what is the math from the launch frame to show this.

I already addressed this paper (and pointed out that it definitely says that string will snap) in post #32, did you read that one? The paper certainly doesn't dispute the idea that in the frame of the observer the length of the string will be constant until it snaps, it just argues that defining "length" in terms of the coordinates of an outside observer is not very physical, and that it's better to use a quantity called "rest frame length" which is defined solely in the string's own rest frame.

Yea, I am OK with that but, this author says the distance between them increases whereas before you mentioned the string wants to contract. Is this not a difference or am I misunderstanding you?

 

[A.T.]

Question, does the launch frame conclude based on observations that the string breaks?

Yes, see post #39

If so, what is the math from the launch frame to show this.

It is the same math that shows that the string breaks in its rest frame: The distances between the string atoms/molecules are to great for the bonding forces to hold them together. The only difference is:

- In the string rest frame the distances between the atoms/molecules are increased by stretching the string.
- In the launch frame the range of the bonding interactions is decreased as the atoms/molecules are contracted

 

[JesseM]

Question, does the launch frame conclude based on observations that the string breaks?

Yes.

If so, what is the math from the launch frame to show this.

As I've said before, if you wanted to do the calculation solely from the perspective of the launch frame I think you would need to actually do some detailed calculation of the inter-atomic forces in this frame. Even though the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame), as A.T. said the way the electromagnetic field between atoms varies as a function of distance would change, and from this you could presumably show that the stress in the string was increasing. The details of such a calculation are beyond me though.

Yea, I am OK with that but, this author says the distance between them increases whereas before you mentioned the string wants to contract. Is this not a difference or am I misunderstanding you?

You're misunderstanding. The author is talking about the actual length in the string's instantaneous rest frame, which does increase, while I was talking about the idea of a spring's "rest length" from classical mechanics (google 'spring' and 'rest length' to see that this is a common term) which has nothing to do with the spring's actual length in its rest frame, it just means the length the spring would naturally assume if it were relaxed and no forces were being applied to either end, which can of course be different from the spring's actual length if it is being stretched or compressed by outside forces.

 

[cfrogue]

Yes, see post #39

It is the same math that shows that the string breaks in its rest frame: The distances between the string atoms/molecules are to great for the bonding forces to hold them together. The only difference is:

- In the string rest frame the distances between the atoms/molecules are increased by stretching the string.
- In the launch frame the range of the interactions is decreased as the atoms/molecules are contracted

The integral for all of the solutions is calculated vs a theoretical instantaneous at rest frame not the launch frame.

Is this not correct?

 

[cfrogue]

Yes.

As I've said before, if you wanted to do the calculation solely from the perspective of the launch frame I think you would need to actually do some detailed calculation of the inter-atomic forces in this frame. Even though the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame), as A.T. said the way the electromagnetic field between atoms varies as a function of distance would change, and from this you could presumably show that the stress in the string was increasing. The details of such a calculation are beyond me though.

You're misunderstanding. The author is talking about the actual length in the string's instantaneous rest frame, which does increase, while I was talking about the idea of a spring's "rest length" from classical mechanics (google 'spring' and 'rest length' to see that this is a common term) which has nothing to do with the spring's actual length in its rest frame, it just means the length the spring would naturally assume if it were relaxed and no forces were being applied to either end, which can of course be different from the spring's actual length if it is being stretched or compressed by outside forces.

OK, I have not seen any mainstream articles that calculate the integral and prove the string breaks from strictly the POV of the launch frame. All I have seen use an instantaneous at rest frame within the context of the accelerating frame.

Do you have such calculations or mainstream articles strictly from the launch frame?

 

[A.T.]

The integral for all of the solutions is calculated vs a theoretical instantaneous at rest frame not the launch frame.

Is this not correct?

Not sure what you mean here. You can use both frames, but I guess the rest frame of the string is easier.

EDIT: Oh I see what you mean. No you are not correct. You don't need the rest frame of the string to conclude that the string will snap. In the launch frame you observe constant atom distances, but decreasing range of bonding forces.

 

[JesseM]

OK, I have not seen any mainstream articles that calculate the integral and prove the string breaks from strictly the POV of the launch frame. All I have seen use an instantaneous at rest frame within the context of the accelerating frame.

Do you have such calculations or mainstream articles strictly from the launch frame?

No, I don't know of any. It seems like it'd be a needlessly complicated approach, since it's easier to understand why it breaks by looking at the string's rest frame, and we know that in relativity all frames always agree about the answers to local physical questions like whether a string breaks.

 

[cfrogue]

No, I don't know of any. It seems like it'd be a needlessly complicated approach, since it's easier to understand why it breaks by looking at the string's rest frame, and we know that in relativity all frames always agree about the answers to local physical questions like whether a string breaks.

I have not seen any either, but that does not mean they do no exist.

Let me ask you this.

If you have two rockets at a distance d with a string of length d between them and the rockets at in the same frame moving relative v to a stationary observer, would the string break?

 

[JesseM]

Let me ask you this.

If you have two rockets at a distance d with a string of length d between them and the rockets at in the same frame moving relative v to a stationary observer, would the string break?

Are the distance d between rockets and the length d of the string measured in the rocket/string rest frame or the observer's frame? And all questions about whether a string would break depend on the elasticity of the string...if an identical string were placed at rest relative to the observer and gradually both ends were pulled apart, at what length would the string stretch to in the observer's frame before it snapped?

 

[cfrogue]

Are the distance d between rockets and the length d of the string measured in the rocket/string rest frame or the observer's frame? And all questions about whether a string would break depend on the elasticity of the string...if an identical string were placed at rest relative to the observer and gradually both ends were pulled apart, at what length would the string stretch to in the observer's frame before it snapped?

Oh, the d's are measured in the moving frame and are initially known in the rest frame.

Say that the string is very weak and brittle.

 

[yuiop]

OK, would the rest frame/launch frame conclude the string will break given the distance does not change between the ships from the POV of the rest frame?

In other words, does the launch frame conclude the distance does not change yet the string contracts?

Here is another way of looking at it. The two ships accelerate as per Bells's paradox, but this time the string is only connected to the front ship. The gap between the two ships stays constant according to the launch frame, but the string is length contracting. When the sting has contracted to say one hundredth of its original length, any attempt to force the string to connect the two ships, without bringing the two ships closer together (as measured in the launch frame) will snap the string. Of course, if the string is very flexible and stretching one hundred times is not sufficient to snap it, then we only have to run the experiment for a little longer until a point is reached where the string does snap, assuming that is impossible to have a string with infinite elasticity.

[EDIT] I have just noticed noticed that what I said is basically what Dr Greg said in post #33. Sorry about that. The posts in this thread are coming so fast, I missed a few.

 

[cfrogue]

Here is another way of looking at it. The two ships accelerate as per Bells's paradox, but this time the string is only connected to the front ship. The gap between the two ships stays constant according to the launch frame, but the string is length contracting. When the sting has contracted to say one hundredth of its original length, any attempt to force the string to connect the two ships, without bringing the two ships closer together (as measured in the launch frame) will snap the string. Of course, if the string is very lexible and stretching one hundred times is not sufficient to snap it, then we only have to run the experiment for a little longer until a point is reached where the string does snap, assuming that is impossible to have a string with infinite elasticity.

OK, does this imply space does not contract only rods?

Next, at any instant t in the two rocket and string frame, all three are at rest?

 

[JesseM]

Oh, the d's are measured in the moving frame and are initially known in the rest frame.

Say that the string is very weak and brittle.

Yes, but even a brittle string might have a relaxed length much greater than d...do you want to say that if we had laid out the string at rest relative to the observer with nothing pulling on either end, the distance in the observer's frame would be d? In that case, if the two ships are moving relative to the observer and the distance between them in the observer's frame is d, then since the distance between the ships in their own rest frame is greater than d, you couldn't stretch the string between the ships without breaking it.

 

[matheinste]

OK, does this imply space does not contract only rods?

Next, at any instant t in the two rocket and string frame, all three are at rest?

I think a direct answer to this in the context of standard SR would help to clarify the explanations.

Matheinste.

 

[yuiop]

Next, at any instant t in the two rocket and string frame, all three are at rest?

Nope, to one of the rocket observers the gap between the rockets is getting larger and the other rocket is getting further away, so the two rockets do not regard themselves as being at rest with respect to each other.

 

[cfrogue]

Nope, to one of the rocket observers the gap between the rockets is getting larger and the other rocket is getting further away, so the two rockets do not regard themselves as being at rest with respect to each other.

OK, so the rockets see themselves as getting further apart.

Yet, the launch frame does not see it this way. It sees the distance as constant.

How is this so?

 

[yuiop]

OK, so the rockets see themselves as getting further apart.

Yet, the launch frame does not see it this way. It sees the distance as constant.

How is this so?

The launch frame is using rulers that are not length contracted, while the rocket observers are measuring the gap using rulers that are gettting progressively shorter so to them the gap appears to be expanding.

 

[cfrogue]

The launch frame is using rulers that are not length contracted, while the rocket observers are measuring the gap using rulers that are gettting progressively shorter so to them the gap appears to be expanding.

Is there evidence that length actually contracts within a frame, I mean within the internals of a frame?

Isn't length contraction a phenomena of the "at rest" frame when viewing the moving frame?

 

[yuiop]

OK, does this imply space does not contract only rods?

Let's try a slightly modified experiment, to try and shed light on your question.

We have four rockets all with identical solid fuel propellants that burn at a fixed rate for a fixed length of time. Rocket A and B are joined by a tough cable of length d and rockets C and D are separated by a distance of d but not physically connected. All 4 rockets launch simultaneously in the launch frame. When they have exhausted their fuel rockets C and D are still a distance d apart, but rockets A and B are less than d apart. Rockets A and B have been physically pulled closer together by the length contraction of the cable.

If a fifth rocket and observer was introduced and this time only the fifth observer accelerated, then distances between the unconnected and connected rocket pairs would appear to length contract equally, but the apparent length contraction of the gap between the unconnected rockets is not physical. The changes in the clock rates and ruler lengths of the fifth observer makes the gap appear to contract.

 

[yuiop]

Is there evidence that length actually contracts within a frame, I mean within the internals of a frame?

Isn't length contraction a phenomena of the "at rest" frame when viewing the moving frame?

You can not actually observe length contraction if you are in the frame moving with the object. In the case of Bell's rocket observers they do not see the string as shrinking, but rather see it being stretched across a larger gap.

 

[cfrogue]

You can not actually observe length contraction if you are in the frame moving with the object. In the case of Bell's rocket observers they do not see the string as shrinking, but rather see it being stretched across a larger gap.

But why?

The rest frame does not see the gap getting wider.

 

[cfrogue]

Let's try a slightly modified experiment, to try and shed light on your question.

We have four rockets all with identical solid fuel propellants that burn at a fixed rate for a fixed length of time. Rocket A and B are joined by a tough cable of length d and rockets C and D are separated by a distance of d but not physically connected. All 4 rockets launch simultaneously in the launch frame. When they have exhausted their fuel rockets C and D are still a distance d apart, but rockets A and B are less than d apart. Rockets A and B have been physically pulled closer together by the length contraction of the cable.

If a fifth rocket and observer was introduced and this time only the fifth observer accelerated, then distances between the unconnected and connected rocket pairs would appear to length contract equally, but the apparent length contraction of the gap between the unconnected rockets is not physical, but brought about by changes in the clock rates and ruler lengths of the fifth observer who has undergone acceleration.
.

This thought experiment changes the game.

It should be solvable in the context we were in.

If the string contracts from the rest observer and the distance does not change, does this imply space does not contract but rods do?

 

[DrGreg]

If the string contracts from the rest observer and the distance does not change, does this imply space does not contract but rods do?

With the rapid pace of this thread, I think my post #38 may have been overlooked. I think it might be relevant to the difficulty you are having.

 

[cfrogue]

With the rapid pace of this thread, I think my post #38 may have been overlooked. I think it might be relevant to the difficulty you are having.

I read this and thought to ask you how you did those perfect graphics. I really mean this.

Assuming your post though, how do you explain this?

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context.

How do you reconcile this?

 

[DrGreg]

I read this and thought to ask you how you did those perfect graphics. I really mean this.

I used Microsoft Powerpoint to draw the pictures. The latest version has an option to save as a PNG file.

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context.

How do you reconcile this?

Using the notation of my diagram. "Alice" is the launch frame. "Bob" is a frame in which one of the rockets is momentarily at rest (some time later). P & Q are the two rockets.

We know y < z. That is Lorentz contraction.

We also know x = y. ("We have the rest frame not seeing any distance differentials. ")

Therefore z > x. ("We have the accelerating frames getting further apart in their context.")

 

[cfrogue]

I used Microsoft Powerpoint to draw the pictures. The latest version has an option to save as a PNG file.

Using the notation of my diagram. "Alice" is the launch frame. "Bob" is a frame in which one of the rockets is momentarily at rest (some time later). P & Q are the two rockets.

We know y < z. That is Lorentz contraction.

We also know x = y. ("We have the rest frame not seeing any distance differentials. ")

Therefore z > x. ("We have the accelerating frames getting further apart in their context.")

In order to compare these like this, you must have a uniform space.

You are depending on the trichotomy of the real numbers but the spaces are not the same in the frame to frame analysis.


Do you compare these another way I am not seeing?

 

[A.T.]

You are depending on the trichotomy of the real numbers

x,y,z are just real numbers here.

but the spaces are not the same in the frame to frame analysis.

The frame to frame part is handled by:

We know y < z. That is Lorentz contraction.

 

[A.T.]

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context. How do you reconcile this?

Actually you answer it yourself:

but the spaces are not the same in the frame to frame analysis.

 

[DrGreg]

In order to compare these like this, you must have a uniform space.

You are depending on the trichotomy of the real numbers but the spaces are not the same in the frame to frame analysis.

I've no idea what any of that means.

 

[cfrogue]

I've no idea what any of that means.

OK, sorry, when you have some time, I am not seeing your explanation.

1) One solution suggests there exists length contraction for the string.
2) One solution suggests the ships get further apart.
3) The rest frame concludes the distance remains constant between the ships and the v and any time t is the same.

Actually, if you look from the rest frame, a reaction may be that as v increases, length contraction for the string should increase.

Yet, the SR acceleration equations do not predict this and predict a constant distance between the ships.

How is this worked out?

 

[JesseM]

Actually, if you look from the rest frame, a reaction may be that as v increases, length contraction for the string should increase.

That would be wrong, you can only use the length contraction equation for an object with a constant length in its rest frame, but the string's length in its rest frame is changing because its ends are attached to the ships.

Yet, the SR acceleration equations do not predict this and predict a constant distance between the ships.

Well, only in the launch frame, not in other frames.