# Lorentz contraction

by matheinste
Tags: contraction, lorentz
P: 8,470
 Quote by cfrogue 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).
 Quote by cfrogue 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/...906.1919v2.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.
 Sci Advisor PF Gold P: 1,849 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. Attached Thumbnails
P: 4,212
 Quote by cfrogue 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.
P: 687
 Quote by JesseM 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.

 Quote by JesseM 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?
P: 4,212
 Quote by cfrogue Question, does the launch frame conclude based on observations that the string breaks?
Yes, see post #39
 Quote by cfrogue 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
P: 8,470
 Quote by cfrogue Question, does the launch frame conclude based on observations that the string breaks?
Yes.
 Quote by cfrogue 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.
 Quote by cfrogue 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.
P: 687
 Quote by A.T. 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?
P: 687
 Quote by JesseM 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?
P: 4,212
 Quote by cfrogue 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.
P: 8,470
 Quote by cfrogue 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.
P: 687
 Quote by JesseM 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.

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?
P: 8,470
 Quote by cfrogue 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?
P: 687
 Quote by JesseM 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.
P: 3,967
 Quote by cfrogue 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.
P: 687
 Quote by kev 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?
P: 8,470
 Quote by cfrogue 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.
P: 1,060
 Quote by cfrogue 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.
P: 3,967
 Quote by cfrogue 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.

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