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 speed of length contraction - explanation/contradiction 
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Post speed of length contraction - explanation/contradiction
In explaining relativity extremely high velocities are typically used so that relativistic effects occur with time and distance parameters we use in our everyday thinking. These same relativistic effects can be seen with common, everyday velocities if the distance between events becomes extremely large. Here is a simple problem. I don't understand how it is explained using the concepts of relativitity.
Let's say we have a very long rod of length L along the x-axis in a rest frame. At one end of the rod, we have an object A, one meter in length, sharing the same end point coordinate as one end of the rod. At the other end of the rod, we have an object B, one meter in length, sharing the same end point coordinate with the other end of rod. At time t0, as measured in the rest frame, we simultaneously accelerate all points of the rod, and all points of objects A and B along the x-axis. Let's say after the acceleration changes the velocity of the rod and objects A and B from zero meters / second to 1 meter / second and takes only a second or two. Let's make the length of the rod, L, a very long length such that we expect a length contraction of a few meters.
As measured in the rest frame, the distance between objects A and B hardly changes during this acceleration. Since both objects accelerated at the same time and in an identical manner, the only change in distance is related to how much a one meter object changes in length when its velocity changes from zero to one meter a second (which is generally negligible). But somewhere in this process, the rest frame must measure that the rod has changed its length by a few meters (because of its extremely long length). So we have object A and one end of the rod accelerate in an identical fashion, and at the other end of the rod we have object B accelerate in an identical fashion as that end of the rod. The distance between A and B after the acceleration is virtually identical to the distance between A and B before the acceleration. However the rest frame observers must measure a length contraction in the rod (per Einstein), so some time during or after the acceleration object A and object B can no longer be at the end points of the rod. But when did the length contraction occur?
Since the amount of length contraction depends on the length L of the rod, some sort of effect must have been transmitted along the rod. If the length contraction occurred instantly, then we can send information faster than c. For rods around a light-year in length, if info cannot be transmitted faster than c, then it would take at least a year to reach the contracted length. That would mean that moving frame observers (with V = one meter/second relative to the rest frame) would have to wait around a year before they notice any change in length of the rod. But the moving frame observers, per Einstein, notice a change in length of the rod immediately since one end of the rod starts accelerating before the other end starts (acceleration start was not simultaneous in the moving frame). They also notice that the distance between object A and object B changes immediately. Since the rest frame doesn't measure any significant change in the distance between A and B, and it might take a year or more for the change of a few meters in length of the rod to show up in the rest frame, how is possible to reconcile both views?
Isn't this contradictory, or how is this explained?
Thanks,
David Seppala
Bastrop TX


Mon Nov 14, 2011 8:19 pm
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Post Re: speed of length contraction - explanation/contradiction
Hi David,

Still trying to find ahole in SR, I see.
*********************


dseppala: In explaining relativity extremely high velocities are typically used so that relativistic effects occur with time and distance parameters we use in our everyday thinking. These same relativistic effects can be seen with common, everyday velocities if the distance between events becomes extremely large. Here is a simple problem. I don't understand how it is explained using the concepts of relativitity.
Let's say we have a very long rod of length L along the x-axis in a rest frame. At one end of the rod, we have an object A, one meter in length, sharing the same end point coordinate as one end of the rod. At the other end of the rod, we have an object B, one meter in length, sharing the same end point coordinate with the other end of rod. At time t0, as measured in the rest frame, we simultaneously accelerate all points of the rod, and all points of objects A and B along the x-axis. Let's say after the acceleration changes the velocity of the rod and objects A and B from zero meters / second to 1 meter / second and takes only a second or two. Let's make the length of the rod, L, a very long length such that we expect a length contraction of a few meters.
As measured in the rest frame, the distance between objects A and B hardly changes during this acceleration. Since both objects accelerated at the same time and in an identical manner, the only change in distance is related to how much a one meter object changes in length when its velocity changes from zero to one meter a second (which is generally negligible). But somewhere in this process, the rest frame must measure that the rod has changed its length by a few meters (because of its extremely long length). So we have object A and one end of the rod accelerate in an identical fashion, and at the other end of the rod we have object B accelerate in an identical fashion as that end of the rod. The distance between A and B after the acceleration is virtually identical to the distance between A and B before the acceleration. However the rest frame observers must measure a length contraction in the rod (per Einstein), so some time during or after the acceleration object A and object B can no longer be at the end points of the rod. But when did the length contraction occur?
Since the amount of length contraction depends on the length L of the rod, some sort of effect must have been transmitted along the rod. If the length contraction occurred instantly, then we can send information faster than c. For rods around a light-year in length, if info cannot be transmitted faster than c, then it would take at least a year to reach the contracted length. That would mean that moving frame observers (with V = one meter/second relative to the rest frame) would have to wait around a year before they notice any change in length of the rod. But the moving frame observers, per Einstein, notice a change in length of the rod immediately since one end of the rod starts accelerating before the other end starts (acceleration start was not simultaneous in the moving frame). They also notice that the distance between object A and object B changes immediately. Since the rest frame doesn't measure any significant change in the distance between A and B, and it might take a year or more for the change of a few meters in length of the rod to show up in the rest frame, how is possible to reconcile both views?
Isn't this contradictory, or how is this explained?

cinci: I'm not sure what teh 1 meter rods at the end of the long rod do so I won't comment on them.

Transmitting a force along he rod is probably limited to the speed of sound in the material so there's plenty of time for this change to occur. Earlier you suggested accelerating all parts of the rod simultaneously. After the rod has achieved the new velocity along its entire length, then it will be contracted by the factor (1 - (v/c)^2)^.5.

For your velocity of 1 m/s, this factor is 1. followed by at least 30 zeroes. So you're talking about a very long rod, impractically so.

Your problem is similar to one posed by JS Bell where he proposes tying a string between two rockets some distance apart and then accelerating them identically along the directin of the string. In this case, the string must break because identical acceleration of the rockets relative to the stationary frame will maintain their separation; but, the string will be accelerating relative to the stationary frame and, because of it's velocity, it will contract and break.
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Mon Nov 14, 2011 10:37 pm
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Post Re: speed of length contraction - explanation/contradiction
The one meter rods at each end of the very long rod are measurement reference points. As measured in the original rest frame, the distance between these two very short rods remains constant, before, during, and after the acceleration (or would if they were point sources). Since actual objects are not point sources, I just made their length very small (1 meter) so that any length contraction has negligible effect on the distance between points A and B.
The long rod and these objects accelerate in an identical manner. In the rest frame, the long rod is the same length as the distance between objects A and B before the acceleration. After the acceleration, due to length contraction, the long rod becomes shorter, as measured in the rest frame, then the distance between objects A and B.
When I looked at what happens between points A and B as viewed in the moving frame (that A and B accelerate into), it seems that after the acceleration A and B are closer together then the moving frame measured them to be before the acceleration. (I may have the signs wrong, so I'm looking for feedback on this). After the acceleration, the very long rod however as measured in this moving frame increases in length. So relative to points A and B, where does each frame say the end points of the very long rod are?
Thanks,
David


Tue Nov 15, 2011 8:43 pm
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Post Re: speed of length contraction - explanation/contradiction
Hmmm. I'd like to start with the problem of accelerating the rod properly, relative to the initial rest frame. I understand that you expect all points on the rod to become closer together, due to length contraction. By definition, this means the rear end must accelerate more than the front.

Also, you had declared that the reference objects should accelerate in the same way as the rod. Considering the different accelerations along the rod, I have to assume that you want the rear object to accelerate identically to the rear of the rod, and the front object to accelerate identically to the front of the rod. If this is the case, then it seems to me that the reference objects should remain aligned with the ends of the rod. However, you also declared that the rod should become smaller than the gap between the objects. So I think something must be wrongly defined somewhere.


Wed Nov 16, 2011 1:00 am
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Post Re: speed of length contraction - explanation/contradiction
The length contraction does not complete during the acceleration. In the problem, the acceleration only lasts for a few seconds. The length of the very long rod can be light-years. So if the length contraction was completed after a few seconds, then information about the length of the rod could be transmitted faster than c, contrary to Einstein's notion about exceeding c.

I thought Einstein's view was that the length contraction was a property of space/time and not some mechanical property of materials. It isn't clear that's the case in this simple example but I don't see how the length contraction concept works in this example.

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Wed Nov 16, 2011 9:15 am
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Post Re: speed of length contraction - explanation/contradiction
dseppalla: The length contraction does not complete during the acceleration. In the problem, the acceleration only lasts for a few seconds. The length of the very long rod can be light-years. So if the length contraction was completed after a few seconds, then information about the length of the rod could be transmitted faster than c, contrary to Einstein's notion about exceeding c.

cinci: Pushing on the end: In this case, the force will only travel along teh length of the rod at the speed of sound in the rod. At the instnat you push on one end, the other end cannot know this is happening for at least the time it takes sound to reach the other end.
******************************


dsepalla: I thought Einstein's view was that the length contraction was a property of space/time and not some mechanical property of materials. It isn't clear that's the case in this simple example but I don't see how the length contraction concept works in this example.

cinci: The phenomena of relativity and normal mechanics of materials arent' mutually exclusive. You get both.
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Wed Nov 16, 2011 3:34 pm
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Post Re: speed of length contraction - explanation/contradiction
Here's something that might help. I'm going to change the experiment quite a bit. I'm going to try to eliminate accelerations entirely. The very long rod and the reference objects will remain inertial throughout the entire experiment.

Let's define two inertial frames. The first one we will call K, and that is the inertial frame in which the very long rod and reference objects remain stationary. The other one will be called K' and that is an inertial frame that is moving along the length of the rod at some constant speed.

First we will ask frame K to measure the length of the rod. All of the clocks in that frame have been synchronised in advance, and they are all programmed to take a photo at a pre-arranged time. At that time, the information regarding the location of the endpoints of the rod becomes known to the clocks at each end of the rod. However, the length of the rod will not be known at both ends until the information from one end is sent to the other, and that information will have to travel at (c). So, the time required to measure the rod is not really instantaneous, but rather requires a finite amount of time (dt=L/c) where (L) is the length of the rod, and (c) is the speed of light.

Next, we will ask frame K' to measure the length of the rod. The procedure is the same as before, except the length of the rod is now (L') instead of (L). So, the time required to measure the rod requires a finite amount of time (dt'=L'/c).

So, if we imagine an accelerating frame which starts off identical to K, and ends up identical to K', I think there would still be the same time lag in determining the length of the rod. I'm not sure if this really answers any question in the OP, but I thought it might help.


Wed Nov 16, 2011 5:56 pm
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Post Re: speed of length contraction - explanation/contradiction
If the long rod is a lightyear long, and you accelerate it by giving it a push on the rear end, then it'll be some time before the front end starts moving (at least a year, possibly more, depending on the speed of sound in the material). When the front end does start moving, there will be some oscillation before it settles at its contracted length; but in that first year, before the front end starts moving, the front rod will move quite a bit ahead of it, and the front of the long rod will end up a bit behind the front short rod.

That's if it is accelerated from the rear. Different considerations apply depending on where exactly the acceleration is applied.

JammyTown wrote:
So, the time required to measure the rod is not really instantaneous, but rather requires a finite amount of time (dt=L/c) where (L) is the length of the rod, and (c) is the speed of light.


I should note that if there is someone at the midpoint of the rod, he can find the length of the rod in half the time by getting a signal from both ends.


Fri Nov 18, 2011 2:15 pm
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Post Re: speed of length contraction - explanation/contradiction
In the problem I posted, every point of the long rod was simultaneously accelerated at an identical rate. What happens in that case?
David


Fri Nov 18, 2011 6:39 pm
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Post Re: speed of length contraction - explanation/contradiction
David: In the problem I posted, every point of the long rod was simultaneously accelerated at an identical rate. What happens in that case?

cinci: Sounds like a simple question but it isn't. This can be read at least two ways:

1. If you can arrange it so every point accelerates indentically in the rest frame, then they rod will pull itself apart because it wants to contract.

2. If you measure the acceleration in the frame of the rod, I think it proceeds without damaging the rod and it simply contracts relative to the rest frame because of it's relative velocity to the rest frame. But there might be some complicatins there also becasue what is simultaneous in the rod frame will not be simultaneous in the rest frame.
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Fri Nov 18, 2011 9:01 pm
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Post Re: speed of length contraction - explanation/contradiction
cincirob wrote:
2. If you measure the acceleration in the frame of the rod, I think it proceeds without damaging the rod and it simply contracts relative to the rest frame because of it's relative velocity to the rest frame. But there might be some complicatins there also becasue what is simultaneous in the rod frame will not be simultaneous in the rest frame.

Just to amplify a little ... Indeed, if the equal parallel accelerations are per the rod frame, or if a single point of acceleration is applied to the rod (or, say an aft thruster burn), then the accelerations do not act to stretch the rod into dismemberment. The rod simply contracts during acceleration per the rest frame without dismemberment.

EDIT: The assumption, in any case, is that the rod becomes deformed by dismemberment (or crushing) only if the force of proper acceleration exceeds the internal molecular forces of the rod. Otherwise, the rod always maintains its own proportion per those at rest with the rod, and length contracts per those in motion wrt the rod (eg the rest frame POV).


Last edited by Optimist5 on Sun Nov 20, 2011 11:36 pm, edited 1 time in total.



Sun Nov 20, 2011 2:41 am
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Post Re: speed of length contraction - explanation/contradiction
Yes, I thnk so.


Sun Nov 20, 2011 5:03 am
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Post Re: speed of length contraction - explanation/contradiction
Will we get a different result if we accelerate the observer instead of the rod?

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Post Re: speed of length contraction - explanation/contradiction
dseppala wrote:
In the problem I posted, every point of the long rod was simultaneously accelerated at an identical rate. What happens in that case?
David


As the rod accelerates, each molecule feels a force pulling it towards the molecules on either side (except the ones at the end, which have molecules on only one side to be pulled towards). When released, they'll spring together like an elastic that has been slightly stretched.

mhyworld wrote:
Will we get a different result if we accelerate the observer instead of the rod?


Yes. In this case, no forces are felt by any molecules in the rod at all. The observer will measure (after acceleration) that the rod is shorter than it was, due to their simultaneities now being different.


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Post Re: speed of length contraction - explanation/contradiction
Hi David,

Still trying to find ahole in SR, I see.
*********************

dseppala: In explaining relativity extremely high velocities are typically used so that relativistic effects occur with time and distance parameters we use in our everyday thinking. These same relativistic effects can be seen with common, everyday velocities if the distance between events becomes extremely large. Here is a simple problem. I don't understand how it is explained using the concepts of relativitity.
Let's say we have a very long rod of length L along the x-axis in a rest frame. At one end of the rod, we have an object A, one meter in length, sharing the same end point coordinate as one end of the rod. At the other end of the rod, we have an object B, one meter in length, sharing the same end point coordinate with the other end of rod. At time t0, as measured in the rest frame, we simultaneously accelerate all points of the rod, and all points of objects A and B along the x-axis. Let's say after the acceleration changes the velocity of the rod and objects A and B from zero meters / second to 1 meter / second and takes only a second or two. Let's make the length of the rod, L, a very long length such that we expect a length contraction of a few meters.
As measured in the rest frame, the distance between objects A and B hardly changes during this acceleration. Since both objects accelerated at the same time and in an identical manner, the only change in distance is related to how much a one meter object changes in length when its velocity changes from zero to one meter a second (which is generally negligible). But somewhere in this process, the rest frame must measure that the rod has changed its length by a few meters (because of its extremely long length). So we have object A and one end of the rod accelerate in an identical fashion, and at the other end of the rod we have object B accelerate in an identical fashion as that end of the rod. The distance between A and B after the acceleration is virtually identical to the distance between A and B before the acceleration. However the rest frame observers must measure a length contraction in the rod (per Einstein), so some time during or after the acceleration object A and object B can no longer be at the end points of the rod. But when did the length contraction occur?
Since the amount of length contraction depends on the length L of the rod, some sort of effect must have been transmitted along the rod. If the length contraction occurred instantly, then we can send information faster than c. For rods around a light-year in length, if info cannot be transmitted faster than c, then it would take at least a year to reach the contracted length. That would mean that moving frame observers (with V = one meter/second relative to the rest frame) would have to wait around a year before they notice any change in length of the rod. But the moving frame observers, per Einstein, notice a change in length of the rod immediately since one end of the rod starts accelerating before the other end starts (acceleration start was not simultaneous in the moving frame). They also notice that the distance between object A and object B changes immediately. Since the rest frame doesn't measure any significant change in the distance between A and B, and it might take a year or more for the change of a few meters in length of the rod to show up in the rest frame, how is possible to reconcile both views?
Isn't this contradictory, or how is this explained?

cinci: I'm not sure what teh 1 meter rods at the end of the long rod do so I won't comment on them.

Transmitting a force along he rod is probably limited to the speed of sound in the material so there's plenty of time for this change to occur. Earlier you suggested accelerating all parts of the rod simultaneously. After the rod has achieved the new velocity along its entire length, then it will be contracted by the factor (1 - (v/c)^2)^.5.

For your velocity of 1 m/s, this factor is 1. followed by at least 30 zeroes. So you're talking about a very long rod, impractically so.

Your problem is similar to one posed by JS Bell where he proposes tying a string between two rockets some distance apart and then accelerating them identically along the directin of the string. In this case, the string must break because identical acceleration of the rockets relative to the stationary frame will maintain their separation; but, the string will be accelerating relative to the stationary frame and, because of it's velocity, it will contract and break.

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