Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[yuiop]

Let's say we had a thin rod that expands significantly when heated and correspondingly shrinks significantly when cooled. I am sure you will agree such a thing exists. Now if the rod is heated to a high temperature and welded the rod in its hot state in-between two very strong pillars and then cooled the rod down rapidly, there could come a point when the rod would snap. Right up to the point before the rod snaps as it cools the length of the rod remains constant. Would you take that to mean that we had no theory to predict that the rod would snap in this case? If you agree that we can predict the rod will snap then can you agree that we can in principle predict things about stress, even when we can not measure a change in length?

 

[cfrogue]

Let's say we had a thin rod that expands significantly when heated and correspondingly shrinks significantly when cooled. I am sure you will agree such a thing exists. Now if the rod is heated to a high temperature and welded the rod in its hot state in-between two very strong pillars and then cooled the rod down rapidly, there could come a point when the rod would snap. Right up to the point before the rod snaps as it cools the length of the rod remains constant. Would you take that to mean that we had no theory to predict that the rod would snap in this case? If you agree that we can predict the rod will snap then can you agree that we can in principle predict things about stress, even when we can not measure a change in length?

I agree with all of you that the accelerating ships conclude the string will break.


The launch frame has available to it a collection of acceleration equations from SR.

These equations predict the distance between the ships does not change.

Should we ignore this scientific evidence of the launch frame?

 

[JesseM]

Does this argument appeal to QT to solve a problem in SR?

The argument about the equilibrium length of the string seems to have involved only classical electromagnetism, although of course we know that this is only an approximation for a more accurate quantum theory of electromagnetic bonds between atoms.

Should this be decidable in SR from the POV of the launch frame?

"SR" is not a theory of how materials behave under stress, it's just a general statement that all possible laws of physics (including whatever laws govern the behavior of materials) should be Lorentz-symmetric. So regardless of whether you use classical electromagnetism or a relativistic quantum theory of atomic bonds, you need to go outside the bare axioms of SR to say anything about how materials will behave, but whatever additional theory you use will still be consistent with SR as long as it's Lorentz-symmetric.

 

[atyy]

Does this argument appeal to QT to solve a problem in SR?

Should this be decidable in SR from the POV of the launch frame?

The full argument must appeal to quantum mechanics, because classical physics cannot give a microscopic explanation for the existence of rigid bodies, and this is precisely what is needed here.

However, I do believe Bell brings in enough classical arguments to show that the string will break when considered wholly from the point of view of the launch frame.

First he notes the electric field of a moving charge is not the same as that of a stationary charge. Thus the equilibrium state of a moving rod cannot be the same, and if the rod is stressed to start with, then the stress must either increase or decrease. It is not obvious (to me) which happens, but certainly the stress cannot stay the same.

To argue that the stress increases, Bell calculates (strictly wrongly, but I think correctly enough, and he discusses this in the text) the equilibrium radius of a negative charge orbiting a positive charge, and shows the equilibrium radius is smaller, which argues that the stress on the moving rod increases.

Bell, "How to teach special relativity" in http://books.google.com/books?id=FG...eakable+and+unspeakable&source=gbs_navlinks_s

Also useful is Fitzpatrick's "Fields due to a moving charge" http://farside.ph.utexas.edu/teaching/em/lectures/node125.html

 

[yuiop]

The launch frame has available to it a collection of acceleration equations from SR.

These equations predict the distance between the ships does not change.

Should we ignore this scientific evidence of the launch frame?

Did I not just demonstrate that I could snap a rod without changing the distance between the ends of the rod?

 

[cfrogue]

The argument about the equilibrium length of the string seems to have involved only classical electromagnetism, although of course we know that this is only an approximation for a more accurate quantum theory of bonds between atoms.

"SR" is not a theory of how materials behave under stress, it's just a general statement that all possible laws of physics (including whatever laws govern the behavior of materials) should be Lorentz-symmetric. So regardless of whether you use classical electromagnetism or a relativistic quantum theory of atomic bonds, you need to go outside the bare axioms of SR to say anything about how materials will behave, but whatever additional theory you use will still be consistent with SR as long as it's Lorentz-symmetric.

I agree SR is not material science.

However, we have seen articles posted here that concluded the string contracts as the reason it will break.

Also, the latest mainstream publication shows the ships drift further apart and the string will break.

But, this is all from integrating within the context of the accelerating frames.

So, it is natural to ask what does the launch frame think. The accelerating frames did not have to appeal to an outside authority to decide the string will break. These papers concluded from the POV of the accelerating frame, the string will break strictly within SR.

Why do we need to go outside the theory to conclude the string will break from the launch frame perspective? Why can't this be concluded from within SR? Because all we have right now is undisputed evidence the distance between the ships does not change.

 

[cfrogue]

Did I not just demonstrate that I could snap a rod without changing the distance between the ends of the rod?

You did.

So, are you going to heat up the rod and cool it to make SR solve this problem?

Is this correct?

 

[matheinste]

I am OK with the fact that the string breaks.

I think that the problem gets sidetracked if it is necesary to explain the physics of the actual breaking. Couldn't we just propose an element between the ships that will break under any stress and then,if stress is the reason for the breaking, demonstrate that there is stress. After all, whether it breaks or it doesn't is the point, not the physics of the detailed behaviour of the molecular structure of the string under stress, interesting though that may be.

Matheinste.

 

[JesseM]

I agree SR is not material science.

However, we have seen articles posted here that concluded the string contracts as the reason it will break.

Also, the latest mainstream publication shows the ships drift further apart and the string will break.

But these publications don't rely purely on SR either--they have to make at least some implicit qualitative assumptions about the behavior of materials to conclude the string snaps, like the idea that if you keep increasing the length of a string in its rest frame it will eventually snap. There's no way to answer a question about a string snapping without at least some basic assumptions about materials science that go beyond the basic axioms of SR; it just so happens that your assumptions about materials science need to be a bit more detailed if you want to analyze things purely in the launch frame.

 

[yuiop]

You did.

So, are you going to heat up the rod and cool it to make SR solve this problem?

Is this correct?

I was making the point that you can induce stress in the rod (by cooling OR length contraction) and snap the rod, without changing the length of the rod. It follows that demonstrating that the length of the string does not change according to the launch frame does not prove that no stresses are induced. Now I am sure you will agree that calculating the intermolecular forces of the string is no easy matter, but myself, Jesse, Dr Greg and others have shown you many ways to logically deduce that the string will snap, even from the point of view of the launch frame.

 

[cfrogue]

But these publications don't rely purely on SR either--they have to make at least some implicit qualitative assumptions about the behavior of materials to conclude the string snaps, like the idea that if you keep increasing the length of a string in its rest frame it will eventually snap. There's no way to answer a question about a string snapping without at least some basic assumptions about materials science that go beyond the basic axioms of SR; it just so happens that your assumptions about materials science need to be a bit more detailed if you want to analyze things purely in the launch frame.

Yes, but none of these things apply to the launch frame do they?

In particular, one paper says the string continues to contact as the acceleration continues.

Another says the distance between the ships increases as the acceleration continues.

Eventually, breakage will occur in both cases.

But, the launch frame has no such limitations. In the launch frame, at least all of us agree, has no reason for the string to break without imposing standards from some other source of logic outside SR.

I wish someone had a mainstream paper on this.

 

[cfrogue]

I was making the point that you can induce stress in the rod (by cooling OR length contraction) and snap the rod, without changing the length of the rod. It follows that demonstrating that the length of the string does not change according to the launch frame does not prove that no stresses are induced. Now I am sure you will agree that calculating the intermolecular forces of the string is no easy matter, but myself, Jesse, Dr Greg and others have shown you many ways to logically deduce that the string will snap, even from the point of view of the launch frame.

I must have missed the argument from SR.

Could you explain it?

Please only use SR.

 

[DrGreg]

cfrogue

(a) Do you accept that a loose, straight piece of string, which is free at both ends, gets shorter when accelerated along its length, as measured in the launch frame? If yes, does that satisfy your stipulation that you must work things out in the launch frame only?

(b) Everyone agrees the distance between the ships remains constant in the launch frame.

Now repeat the experiment with the string attached between the ships. (a) says the string contracts. (b) says it doesn't. There's only one way to resolve this apparent contradiction. The string must break.

I never mentioned any frame except the launch frame.

 

[cfrogue]

cfrogue

(a) Do you accept that a loose, straight piece of string, which is free at both ends, gets shorter when accelerated along its length, as measured in the launch frame? If yes, does that satisfy your stipulation that you must work things out in the launch frame only?

(b) Everyone agrees the distance between the ships remains constant in the launch frame.

Now repeat the experiment with the string attached between the ships. (a) says the string contracts. (b) says it doesn't. There's only one way to resolve this apparent contradiction. The string must break.

I never mentioned any frame except the launch frame.

(a) Two questions

1) Let a and b be two points at a distance d. Join them with a rod.

Now, move the unit in relative motion to another observer. Do the two points remain at a distance d and the rod contracts?

2) Do you have the integral from the launch frame to prove the rod contracts? We already know a and b remain at a distance d for the acceleration.

 

[DrGreg]

1) Let a and b be two points at a distance d. Join them with a rod.

Now, move the unit in relative motion to another observer. Do the two points remain at a distance d and the rod contracts?

If the rod is rigid, and a and b are firmly glued to the rod and not glued to anything else, the rod contracts and the points become less than d apart, relative to the original frame.

If the rod is rigid, and a and b are not glued to the rod and are firmly glued to something else, the rod contracts and the points will do whatever the "something elses" do.

If the rod is rigid, and a and b are firmly glued to the rod and are firmly glued to something else, the rod contracts and either the "something elses" will be pulled together by the force of the rod, or something will fall to pieces.

 

[cfrogue]

If the rod is rigid, and a and b are firmly glued to the rod and not glued to anything else, the rod contracts and the points become less than d apart, relative to the original frame.

If the rod is rigid, and a and b are not glued to the rod and are firmly glued to something else, the rod contracts and the points will do whatever the "something elses" do.

If the rod is rigid, and a and b are firmly glued to the rod and are firmly glued to something else, the rod contracts and either the "something elses" will be pulled together by the force of the rod, or something will fall to pieces.

Wow, this is very specific.

But, then you agree the points and rod contract together.

What about the perspective of the launch frame now?

 

[JesseM]

But these publications don't rely purely on SR either--they have to make at least some implicit qualitative assumptions about the behavior of materials to conclude the string snaps, like the idea that if you keep increasing the length of a string in its rest frame it will eventually snap. There's no way to answer a question about a string snapping without at least some basic assumptions about materials science that go beyond the basic axioms of SR; it just so happens that your assumptions about materials science need to be a bit more detailed if you want to analyze things purely in the launch frame.

Yes, but none of these things apply to the launch frame do they?

None of what things? I just said that you have to take into account assumptions about materials science in the string's rest frame, and the same is true in the launch frame.

In particular, one paper says the string continues to contact as the acceleration continues.

I don't think any of the papers say the string itself contracts, although they may say that the atoms and electromagnetic fields around them contract, or that the equilibrium length that the string would be if it weren't attached to the ship contracts.

Another says the distance between the ships increases as the acceleration continues.

It doesn't say the distance increases in the launch frame.

Eventually, breakage will occur in both cases.

But, the launch frame has no such limitations. In the launch frame, at least all of us agree, has no reason for the string to break without imposing standards from some other source of logic outside SR.

And exactly the same is true in every other frame, you always need at least some basic assumptions about materials that go beyond the basic axioms of SR! That was the whole point of my last post!

 

[Al68]

cfrogue

(a) Do you accept that a loose, straight piece of string, which is free at both ends, gets shorter when accelerated along its length, as measured in the launch frame? If yes, does that satisfy your stipulation that you must work things out in the launch frame only?

(b) Everyone agrees the distance between the ships remains constant in the launch frame.

Now repeat the experiment with the string attached between the ships. (a) says the string contracts. (b) says it doesn't. There's only one way to resolve this apparent contradiction. The string must break.

I never mentioned any frame except the launch frame.

Your second statement about (b) is false. In the launch frame, the string's length is contracted, and the distance between the ships is also contracted but stays constant, and the string's relaxed length becomes increasingly shorter than the distance between the ships.

Again, the word "contracted" in this context does not mean "shorter than before", it means shorter than its length in the co-moving frame of the string.

 

[matheinste]

Your second statement about (b) is false. In the launch frame, the string's length is contracted, and the distance between the ships is also contracted but stays constant, and the string's relaxed length becomes increasingly shorter than the distance between the ships.

Again, the word "contracted" in this context does not mean "shorter than before", it means shorter than its length in the co-moving frame of the string.

Yes, I am getting confused because sometimes the term contracted seems to be used in this discussion to mean Lorentz contraction and sometimes to mean the result of a Lorentz trnaformation withiut making it explicit which is meant. The context is not always clear.

Matheinste.

 

[A.T.]

I agree SR is not material science.

Good

Why do we need to go outside the theory to conclude the string will break from the launch frame perspective?

You need some material science in every frame to show that the string brakes. And applying SR to material science is not "going outside SR", but the opposite: it is using SR consistently on every level.

Why can't this be concluded from within SR?
...
I must have missed the argument from SR.
...
Please only use SR.

You sound like a broken record. See above.

 

[Fredrik]

If the rod is rigid, and a and b are firmly glued to the rod and not glued to anything else, the rod contracts and the points become less than d apart, relative to the original frame.

If the rod is rigid, and a and b are not glued to the rod and are firmly glued to something else, the rod contracts and the points will do whatever the "something elses" do.

If the rod is rigid, and a and b are firmly glued to the rod and are firmly glued to something else, the rod contracts and either the "something elses" will be pulled together by the force of the rod, or something will fall to pieces.

Wow, this is very specific.

But, then you agree the points and rod contract together.

What about the perspective of the launch frame now?

Why do you think that DrGreg's reply didn't completely answer all your concerns? It does. In the Bell's spaceship scenario, you're supposed to assume that the effect of the string on the spaceships is negligible, and that both rockets move in exactly the same way (except for their starting position). That last requirement implies that they stay a constant distance d apart in the launch frame. The Lorentz contraction of the string implies that its natural length in the launch frame at speed v is $d/\gamma<d$, so it must break. Note that the string can't change the motion of the rockets, because it's been specified in the problem that it doesn't.

 

[matheinste]

Why do you think that DrGreg's reply didn't completely answer all your concerns? It does. In the Bell's spaceship scenario, you're supposed to assume that the effect of the string on the spaceships is negligible, and that both rockets move in exactly the same way (except for their starting position). That last requirement implies that they stay a constant distance d apart in the launch frame. The Lorentz contraction of the string implies that its natural length in the launch frame at speed v is $d/\gamma<d$, so it must break. Note that the string can't change the motion of the rockets, because it's been specified in the problem that it doesn't.

So can we say by reasoning using SR and not LET and so avoiding Lorentz contraction, that to an observer in the launch frame, by the way the problem is posed, despite acceleration and thus change in relative velocity, the distance between the ships remains the same. As we would expect the result of aLorentz transformation into the launch frame from the ships frame(s) to result in smaller distances/lengths, this constant separation after Lorentz transformation, implies that the ships are actually moving apart in their own frame(s) as time progresses, thus breaking the string.

Matheinste

 

[Fredrik]

So can we say by reasoning using SR and not LET and so avoiding Lorentz contraction,

Huh?

...to an observer in the launch frame, by the way the problem is posed, despite acceleration and thus change in relative velocity, the distance between the ships remains the same.

Yes, that's what I said.

As we would expect the result of aLorentz transformation into the launch frame from the ships frame(s) to result in smaller distances/lengths,

I'm a bit uncomfortable with this argument since the two world lines aren't straight and parallel. You would need to supply more information to be convincing.

...the ships are actually moving apart in their own frame(s) as time progresses, thus breaking the string.

That's the reason an observer at rest in one of the co-moving inertial frames (or in one of the accelerating frames) would use to explain why the string breaks.

 

[matheinste]

My words-----As we would expect the result of a Lorentz transformation into the launch frame from the ships frame(s) to result in smaller distances/lengths,-----

Your reply----I'm a bit uncomfortable with this argument since the two world lines aren't straight and parallel. You would need to supply more information to be convincing.---


I will have to give this more thought. I am quite happy to accept that my reasoning may be wrong. If you can see a specific problem with the quoted passage please let me know.

Part of my motivation is, as I said earlier, to remove references such as "the string contracts", or "Lorentz contraction" which I have found used ambiguously in the past. I would be happier viewing the problem purely in an SR context although I understand that the formulations always lead to the same results.

Matheinste.

 

[Fredrik]

I will have to give this more thought. I am quite happy to accept that my reasoning may be wrong. If you can see a specific problem with the quoted passage please let me know.

For starters, you didn't even say which event(s) you're applying the Lorentz transformation to.

Part of my motivation is, as I said earlier, to remove references such as "the string contracts", or "Lorentz contraction" which I have found used ambiguously in the past. I would be happier viewing the problem purely in an SR context although I understand that the formulations always lead to the same results.

What I said in #126 is a pure SR solution of this problem. (Hence the "Huh?" in my previous post).

 

[matheinste]

For starters, you didn't even say which event(s) you're applying the Lorentz transformation to.


What I said in #126 is a pure SR solution of this problem. (Hence the "Huh?" in my previous post).

I accept your reasonings completely. I am not here to argue, just to learn thoroughly off people who know more than me. I am afraid the term Lorentz contraction as far as I can see sometimes seems to be used as the physical contraction of the LET and sometimes as the result of a coordinate transformation in SR. Perhaps its a problem other people don't have so I'll just have to live it.

I will continue watching with interest.

Matheinste.

 

[Fredrik]

You need some material science in every frame to show that the string brakes.

It depends on what we consider the axioms of SR to be. One axiom must tell us how to measure lengths, and if we choose to say that length is measured using solid objects (i.e. objects that are doing Born rigid motion when they're accelerated gently), then it follows immediately from the axioms that the string must break.

Now that I think about it, I realize that that's the axiom we should use when we intend to add matter "manually" to the model (by simply associating properties like mass and charge with certain world lines). When we intend to introduce matter using a Lagrangian, we should use radar instead. Then it should be possible to prove the Born rigidity of a solid that's accelerated gently.

1. Physical events are represented by points in Minkowski space. (Note that this implies that the motion of a classical object can be represented by a set of curves in Minkowski space, and that this suggests that we define a classical particle to be "a physical system whose motion can be represented by exactly one such curve").

2. A clock measures the proper time of the curve in Minkowski space that represents its motion.

It's clear that we also need to postulate something about measurements of length, but this is more difficult because of Lorentz contraction. There might be more than one OK way to do it, and one of them seems to be to postulate that lengths are measured by rulers, and that a solid object (like a ruler) that's accelerated gently undergoes Born rigid motion (which guarantees that if we slowly change its velocity, its measurements before and after the acceleration will be consistent with the Lorentz contraction formula).

Unfortunately, this postulate is inappropriate when we formulate theories of matter in this framework (as described in my previous post). It must be possible to possible to use those theories to prove the Born rigidity of an accelerating solid. However, that result isn't going to be a testable prediction of the theory unless we use something other than solid objects to perform length measurements.

If we can't use solids, it seems natural to use light instead. We can use radar to define distance. If we emit a signal at time t₁ (as measured by a clock attached to the radar device) and detect the signal coming back at time t₂, then the distance to the reflection event can be defined as (t₂-t₁)/2. But of course this only works as well as we'd like if the radar isn't accelerating. So we can take the third postulate to be that "lengths are measured by radar devices that move on geodesics", or that "infinitesimal lengths are measured by radar devices".

Hey, I learned something. Some of these things weren't perfectly clear to me before, in particular the reason why the postulate about length measurements should use radar instead of rulers.

 

[matheinste]

Fredrik may I offer this as a paraphrase of your description in #126, the SR resolution.

In the launch frame the distance between the ships is constant whereas it should be continually contracting in the launch frame due to increasing velocity of the ships relative to the launch frame . The thread occupies the distance between the ships and so too should appear contracted in the launch frame. It does not appear so and therefore must be increasingly stressed. Simple as that?

Matheinste

 

[atyy]

Fredrik may I offer this as a paraphrase of your description in #126, the SR resolution.

In the launch frame the distance between the ships is constant whereas it should be continually contracting in the launch frame due to increasing velocity of the ships relative to the launch frame . The thread occupies the distance between the ships and so too should appear contracted in the launch frame. It does not appear so and therefore must be increasingly stressed. Simple as that?

Matheinste

I guess this is what you meant by the confusing usage of "Lorentz contraction" you objected to above? If the distance should be continually contracting but remains constant, at this point we should say we have obtained a contradiction and the theory is wrong.

 

[yuiop]

I guess this is what you meant by the confusing usage of "Lorentz contraction" you objected to above? If the distance should be continually contracting but remains constant, at this point we should say we have obtained a contradiction and the theory is wrong.

Oh nooo.. we shouldn,t say that!

If we had a theory that a cooled metal rod contracts, and then anchored the two ends of the rod so they could not move and then cooled the rod, would that prove that the theory that a cooled rod contracts is wrong?

 

[atyy]

Oh nooo.. we shouldn,t say that!

If we had a theory that a cooled metal rod contracts, and then anchored the two ends of the rod so they could not move and then cooled the rod, would that prove that the theory that a cooled rod contracts is wrong?

Yes.

 

[yuiop]

Oh nooo.. we shouldn,t say that!

If we had a theory that a cooled metal rod contracts, and then anchored the two ends of the rod so they could not move and then cooled the rod, would that prove that the theory that a cooled rod contracts is wrong?

Yes.

LOL

OK, to satisfy you I would have to reformulate the cooled rod contracts to something like " An unstressed rod contracts when cooled if it remains unstressed" or maybe "A cooled rod contracts if it is allowed to do so". Similarly we could say something like"an accelerated unstressed rod/string length contracts if it remains unstressed". The fact that the string in the Bell's rocket paradox does not length contract when accelerated is proof that it does not remain unstressed.

 

[atyy]

LOL

OK, to satisfy you I would have to reformulate the cooled rod contracts to something like " An unstressed rod contracts when cooled if it remains unstressed" or maybe "A cooled rod contracts if it is allowed to do so". Similarly we could say something like"an accelerated unstressed rod/string length contracts if it remains unstressed". The fact that the string in the Bell's rocket paradox does not length contract when accelerated is proof that it does not remain unstressed.

Yes, something like that

 

[A.T.]

You need some material science in every frame to show that the string brakes.

It depends on what we consider the axioms of SR to be...

What I meant is: Even aside of SR, defining the condition for a material to break is already a form of material science.

The relevant part was: Combining SR & material science is just using SR consistently on every level. This is what cfrogue fails to acknowledge, by asking to use "only SR". SR as such doesn't tell you anything about breaking of some materials in any frame.

 

[cfrogue]

The relevant part was: Combining SR & material science is just using SR consistently on every level. This is what cfrogue fails to acknowledge, by asking to use "only SR". SR as such doesn't tell you anything about breaking of some materials in any frame.

OK, let's do this.

Let us abandon the logic of breakage.

One solution from the accelerating ships concludes a rod within the space will contract.

One solution from the accelerating ships concludes the ships get further apart.

Now, from the launch frame, the distance between the ships does not change and there is no prevision for rod contraction under the acceleration equations unless space and the rod length diverge and even with that there are no mainstream papers to support that either.

So, the theory concludes the rod experiences change from the accelerating frame and from the launch frame, no such conclusion can be drawn.

Is this a problem?