Bell's Spaceship Paradox and Length Contraction

[Dale]

Just a nitpick. Lorentz contraction is a disagreement between different reference frames about the length of something at a particular time. It is NOT a change in length at two different points in time. In Bells paradox it is a misnomer to call the change in length before and after the acceleration "length contraction".

Before the acceleration different frames disagree about the length of the string, that is length contraction. After the acceleration different frames disagree about the length of the string, that is also length contraction. In a single frame the length of the string may be different before vs after acceleration, this is NOT length contraction.

 

[nosepot]

Lorentz contraction is a disagreement between different reference frames about the length of something at a particular time.

Yes, I understand what you are saying and agree. Once everybody is at their respective relative speeds, I think SR defines "length contraction" as what these apparent lengths are to each frame.

I think this is being confused (I'm to blame, I guess, for my choice of wording) with an actual length contraction. I should change my nomenclature to say "squishing" perhaps. This squishing causes the weak string to break, the strong string to pull the ships together. The squishing is required so that when everybody gets to their steady state speeds the speed of light in their frame appears to be the same in all directions... So what would appear to cause this squishing for my stationary observer?

Acceleration is a factor as it is required to get the object up to speed, but once the acceleration is removed, the object holds its squished state. Why?

 

[Dale]

Acceleration is a factor as it is required to get the object up to speed, but once the acceleration is removed, the object holds its squished state. Why?

The movement and shape of an object depends entirely on its initial state and the forces on each part of the object. Just like in Newtonian mechanics. It only maintains its "squished" state if that is how the forces act on it. If different forces are applied then it could just as well be stretched, or bent, or broken.

 

[nosepot]

The movement and shape of an object depends entirely on its initial state and the forces on each part of the object. Just like in Newtonian mechanics.

If Newtonian mechanics would explain the squishing of the object we would not be having this discussion. I don't think Newtonian mechanics would cause it to be squished into a single plane when it approaches the speed of light. We are asking the question of how materials are actually squished (and can break, as in the case of Bell's paradox), due to the change in the shape of their interatomic fields perhaps, when they increase their speed.

Given that the material is actually squishing along the direction of movement, then we would be forced to reconsider the null result of a Michelson-Morely-type experiment, as the poor guys didn't stand a chance if their inteferometer arms contracted. So sad.

 

[PeterDonis]

So agree the string is contracting.

In this *different* scenario, the string behaves differently. I'll defer the question of whether "contracting" is a good description of its behavior in this different scenario, because, as I said, I don't want to talk about that scenario because we haven't even gotten clear about your original scenario, which is simpler. All of my criticisms of the word "contracting" are directed at your original scenario, in which the distance between the ships, in your chosen frame, remains constant.

Oh dear. Now it doesn't contract.

Oh dear. You can't even keep your own scenarios straight. Please read the above and re-read my posts more carefully. You are mixing up different scenarios. No wonder you're confused.

Let's reconcile all the examples we've visited so far and set up three experiments side by side to demonstrate different aspects of what's happening; each experiment contains a pair of rockets; so there are now six rockets.

We haven't even got your original scenario straight, and now you want to add a *third* one to the second one you already added that I said I didn't want to talk about? Sorry, I'm not playing. If you don't want to stick to the original scenario, I'll just bow out of the thread.

If we can't agree that the string contracts then I guess the discussion is reaching an impass.

This is why I don't like multiplying scenarios; you can't even keep straight which one we are talking about, and you're the one who specified the scenarios. Once again, if you can't stick to your original scenario--two ships, keeping constant distance apart in your chosen frame, with a weak string between them that eventually snaps due to increasing tension--then I can't really add anything more to this discussion.

 

[PeterDonis]

Given that the material is actually squishing along the direction of movement

As I have pointed out repeatedly, it isn't.

Consider, once again, your original scenario: two ships maintaining a constant distance apart in your chosen frame, and a weak string between them that gets stretched more and more until it breaks.

In your chosen frame, the length of the string is constant. The lengths of the *ships* contract.

However, the string is the object that is subjected to increasing tension, until it breaks. The internal stresses of the ships remain constant.

So: the ships are undergoing length contraction in your chosen frame (btw, you also seem to be ignoring the fact that length contraction is frame-dependent), but they are certainly not being "squished": their internal stresses remain constant. The string is *not* undergoing length contraction in your chosen frame, but it *is* experiencing increasing internal stress--but the stress is *stretching*, not compression.

In short, you appear to have a mistaken view of what is going on: you keep on describing things as "contraction" that don't appear to fit that term at all.

 

[Samshorn]

We are asking the question of how materials are actually squished ... due to the change in the shape of their interatomic fields perhaps, when they increase their speed.

I think your question was answered in post #27.

Lorentz contraction is a disagreement between different reference frames about the length of something at a particular time. It is NOT a change in length at two different points in time.

The first sentence is true, but not the second. Lorentz's theorem of corresponding states is, in a sense, the whole reason for the physical significance of systems of coordinates related by Lorentz transformations. The equilibrium configuration of a solid object, originally at rest in one standard system of inertial coordinates, when set into motion and allowed to reach equilibrium in another system of standard inertial coordinates, is found to be spatially contracted in terms of the original coordinates. Of course, in terms of the second system of coordinates the object was spatially contracted in its original state. The fact that the two things you mentioned BOTH represent length contraction is crucial for understanding special relativity.

 

[nosepot]

In short, you appear to have a mistaken view of what is going on: you keep on describing things as "contraction" that don't appear to fit that term at all.

I agree. I have been mixing terminologies. Length contraction as you know it is an apparent shortening in length when objects in different frames are moving relative to each other - transforming between frames on a spacetime diagram demonstrates this, but this is only an illusion because of the relativity of simultaneity. This is a consequence of attempting to measure the length.

We are now talking about "squishing", which is what happens to the object when it increases its speed. This is the inexplicable attraction of worldlines for an extended object on a spacetime diagram. This is not an illusion, as we know the string will break.

The string is *not* undergoing length contraction in your chosen frame, but it *is* experiencing increasing internal stress--but the stress is *stretching*, not compression.

The stress comes from the spaceships resisting the string's attempt to squish. It's squishing, for sure.

 

[nosepot]

Samshorn, you said the answer is in here somewhere:

It's because what you "know to be the laws of physics" is wrong. This was already known before special relativity came along. Lorentz had already shown by 1904 that the laws of physics (i.e., Maxwell's equations, the Lorentz force law, etc) imply that the equilibrium configuration of an object in motion is shortened in the direction of motion. In other words, if you have a "solid" object initially at rest and equilibrium in terms of one standard system of inertial coordinates, and then you impart some speed to the object (gently enough to avoid inducing any permanent plastic deformation) and allow it to reach equilibrium again at rest in some new standard system of inertial coordinates, it's spatial length in terms of the original system of coordinates is reduced. What Lorentz didn't clearly notice or articulate (but Einstein did a year later) is that the object in its original state was shorter by exactly the same amount in terms of the second system of coordinates.

I'm not sure I follow 'what you "know to be the laws of physics" is wrong'. Relativity expressly states that the laws of physics are to be the same for everyone in an inertial frame, but also that the speed of light is isotropic in that frame, everything else is a consequence of those postulates. If what we know to be the laws of physics (such as how atoms arrange themselves in a material included) is different, relativity is not doing its job.

You say, "Lorentz had already shown by 1904 that the laws of physics (i.e., Maxwell's equations, the Lorentz force law, etc) imply that the equilibrium configuration of an object in motion is shortened in the direction of motion.". I'm not sure your view would be accepted by many, although I would be inclined to agree with this, and this would make a satisfactory answer for me.

"What Lorentz didn't clearly notice or articulate (but Einstein did a year later) is that the object in its original state was shorter by exactly the same amount in terms of the second system of coordinates". True, but this is illusory, due to the relativity of simultaneity - this is very elegantly demonstrated on a spacetime diagram.

By the way, I've accepted several times above, that the SR explanation works, but it has holes. The one I'm poking my finger through at the moment is that it ignores the reality and mechanisms of how matter is arranged. It no better explaines beautful hyperbolic worldlines of the Rindler coordinates than to say, they must be so if the length must squish. When I look at that string squishing up, and in the case of Bell's Spaceship Paradox it breaks, it should be explainable by looking at relative speed of the string and the laws of physics holding the atoms of the string together in my frame.

 

[nosepot]

There's a really nice explanation of a similar phenomenon when considering the barn-pole paradox, with the pole suddenly arrested inside the barn using a very sticky plank of wood:

http://arxiv.org/abs/0712.3891

The pole remains under immense squishing after being stopped in the frame of the barn. The author also explains how the relativity of simultaneity would explain the sequence of events from both frames, such that both disagree about how it happened, but both ultimately end up staring up at an exotically compressed pole, wondering why all those atoms managed to get so squished up when it was moving at speed.

 

[nosepot]

This is why I don't like multiplying scenarios; you can't even keep straight which one we are talking about, and you're the one who specified the scenarios.

The multiple scenarios are interjected to remove the urge to seek refuge in any of the usual places. They demonstrate that it's not resolved by a change of frame, nor by considering time dilation, nor relativity of simultaneity, nor by thinking that the space in between is squishing. After all are eliminated, we realize the string is really squishing and if not strong enough will break. It is a string made of atoms which are bound together. Moving at speed seems to make them bind together more closely. Relativity says, so it must be so the speed of light remains isotropic in any frame. The unexpected consequence is that real materials must somehow obey this prinicple. I'm asking how do they obey?

 

[Samshorn]

You say, "Lorentz had already shown by 1904 that the laws of physics (i.e., Maxwell's equations, the Lorentz force law, etc) imply that the equilibrium configuration of an object in motion is shortened in the direction of motion.". I'm not sure your view would be accepted by many, although I would be inclined to agree with this, and this would make a satisfactory answer for me.

Excellent. Then we're done! (If you encounter anyone who doesn't think that's what Lorentz's 1904 shows, then just refer them to the paper. It isn't a controversial point.)

"What Lorentz didn't clearly notice or articulate (but Einstein did a year later) is that the object in its original state was shorter by exactly the same amount in terms of the second system of coordinates".

True, but this is illusory, due to the relativity of simultaneity - this is very elegantly demonstrated on a spacetime diagram.

True but illusory? It is certainly true in the sense that it is a verifiable statement of fact with perfectly well defined operational meaning. One can choose to call any such empirical fact "illusory" if one is inclined to give precedence to some operationally baseless metaphysical notions, but that isn't what physics is about.

By the way, I've accepted several times above, that the SR explanation works, but it has holes.

Above you said the facts about equilibrium configuations under the laws of classical electrodynamics "make a satisfactory answer" for you... but now you say there are "holes". This seems contradictory. Also, I would say those "holes" are not in evidence. Nothing you've said gives any indication of any "holes" in how the explanations of special relativity work.

The one I'm poking my finger through at the moment is that it ignores the reality and mechanisms of how matter is arranged.

Huh? I just explained (and you agreed) that the classical laws of electrodynamics do this, as Lorentz was the first to show in detail. And this is entirely consistent with special relativity - indeed it is the foundation of special relativity. Why are you back-sliding now? Is it just because you can find people on this message board who don't understand this, and you enjoy trolling them?

 

[nosepot]

Excellent. Then we're done! (If you encounter anyone who doesn't think that's what Lorentz's 1904 shows, then just refer them to the paper. It isn't a controversial point.)

Sweet. I'm not sure it's as uncontroversial as you state. It certainly seems to be misunderstood. If you look at the Wiki page for Bell's Spaceship Paradox (no authority, I know) it's a dog's breakfast. The explanation for the string breaking by is not complete as it only considers events in the rest frame of the co-moving spaceships, the list of papers disputing whether the string even breaks at all is as long as your arm, and the "Talk" page looks like a battlefield.

Huh? I just explained (and you agreed) that the classical laws of electrodynamics do this, as Lorentz was the first to show in detail. And this is entirely consistent with special relativity - indeed it is the foundation of special relativity. Why are you back-sliding now?

Apologies. That point addresses what appears to be the view of other people, but not yours.

Is it just because you can find people on this message board who don't understand this, and you enjoy trolling them?

I'm not trying to troll (too much), and my original question was sincere. I'm trying understand why these paradoxes exist and persist (barn-pole and Ehrenfest paradoxes also rest on the dynamics effects within materials), and part of that is seeking a viewpoint from others who may have considered what I have not. The general consensus seems to be that there is no electrodynamic effect causing my length squishing, which you and I disagree with.

All that said, I'm still not clear why an ether theory was abandoned, just because it was considered to be immeasureable? It appears useful on a number of levels.

 

[jartsa]

jartsa: So you are saying it's related to the interatomic distances being compressed because of the acceleration?

No, I was saying it's related to the time schedule of delivery of kinetic energy to various parts of an object.

Let's consider an uniformly charged non-conducting human observer "falling" in a homogeneous electric field. The length axis of the observer is parallel to the electric field lines.

This observer feels his tendons are resisting some force that is trying to stretch him. And an "outside observer" says the person is contracting.

____________________________________________________________________Now let's consider an observer falling in a homogeneous gravity field. The length axis of the observer is vertical.

This observer does not feel any force that is trying to stretch him. But still an "outside observer" says the person is contracting.
Why the difference? Well let's see ... gravitational time dilation has an effect on the time schedule of the delivery of kinetic energy to various parts of a falling object.

 

[Dale]

If Newtonian mechanics would explain the squishing of the object we would not be having this discussion.

Sorry, I was unclear. The equation which relates force to acceleration is different in SR than in Newtonian mechanics, but the concept is the same:

In both, the shape and movement of any object is entirely determined by the initial position and velocity of each part and the forces on each part. That is what I meant.

We are asking the question of how materials are actually squished (and can break, as in the case of Bell's paradox), due to the change in the shape of their interatomic fields perhaps, when they increase their speed.

The answer is that in Bells paradox the string snaps because it is placed in tension. The formula for determining if something is placed in tension is the one PeterDonis linked to earlier. Conceptually, this is no different than in Newtonian mechanics, if you make a scenario in Newtonian mechanics where an inelastic string is placed in tension then it will snap.

The only difference is the formula which is used. Bells paradox merely hilights that the Newtonian equation must be wrong.

 

[Dale]

We are now talking about "squishing", which is what happens to the object when it increases its speed. This is the inexplicable attraction of worldlines for an extended object on a spacetime diagram.

It is not inexplicable. It comes directly out of the equation that relates force to acceleration, which is somewhat different from the Newtonian equation.

 

[nosepot]

Thank you all for your answers. It's been fun and educational. This is a great forum - keep it up.

For anybody still watching, there is a very interesting paper I found which describes very well what I've fumbled to express:

http://philsci-archive.pitt.edu/987/1/Michelson.pdf

 

[Dale]

Doesn't seem like the paper has anything to do with the topic of the thread. The paper is historical, not technical.

 

[ghwellsjr]

Thank you all for your answers. It's been fun and educational. This is a great forum - keep it up.

For anybody still watching, there is a very interesting paper I found which describes very well what I've fumbled to express:

http://philsci-archive.pitt.edu/987/1/Michelson.pdf

Thanks for providing the link to that paper. It was very interesting and informative.

However, since you claim that it describes very well what you've fumbled to express, then you should know that it describes very well what all the rest of us have clearly expressed, as you can read in the Final remarks, Einstein didn't err, he just provided a simpler approach than "the messier, less economical reasoning" that you want to take. And it's not that "the messier, less economical reasoning" cannot also be handled by Einstein's Special Relativity, it can, you just have to specify an incredibly more complex scenario. You can't just say that the string is inelastic or the spaceship is rigid, because, as I have stated earlier, the slightest acceleration of any part of those objects will instantly break or crush them. And until you are willing to recognize this as a problem with all the scenarios that you have presented, you are going to continue to think that you have an understanding that the rest of us are lacking, despite your ability to express your ideas.

 

[Samshorn]

For anybody still watching, there is a very interesting paper I found which describes very well what I've fumbled to express:
http://philsci-archive.pitt.edu/987/1/Michelson.pdf

Harvey Brown has made a career out of writing articles for philosophical magazines advocating the Lorentzian interpretation of special relativity. In the end, his arguments are all based on a misunderstanding of special relativity - a misunderstanding that was unfortunately encouraged by some misleading features of Einstein's 1905 paper (such as mislabeling the first section "The Kinematic Part" when the very first sentence places it squarely in the realm of dynamics). But Harvey's biggest problem (coincidentally the same as nosepot's) is that he mistakenly thinks if the laws of physics, expressed in terms of one system of coordinates S1, predict that physical phenomena will behave in a way (contracting, slowing, etc) that ensures they will satisfy the same formal laws in terms of a relatively moving system of coordinates S2 with a different simultaneity, then (so Harvey and nosepot contend) this proves that the S1 coordinates are the true coordinates and S2 are just mathematical artifacts. The obvious flaw in this reasoning is that it applies equally well to S2 as the true coordinates and S1 as mathematical artifacts. Lorentz himself credited Einstein with pointing out this "remarkable reciprocity", which reveals Lorentz invariance as a fundamental symmetry of nature, and makes it meaningless to argue for the primacy of S1 or S2 - at least in terms of the local physics. Neo-Lorentzians habitually conflate the possibility of a Lorentzian interpretation with its necessity or physical meaningfulness.

 

[PeterDonis]

Length contraction as you know it is an apparent shortening in length when objects in different frames are moving relative to each other - transforming between frames on a spacetime diagram demonstrates this, but this is only an illusion because of the relativity of simultaneity.

And none of this is what I've been talking about. I've been talking about invariants, like the observed internal stresses in an object.

We are now talking about "squishing", which is what happens to the object when it increases its speed.

Only it doesn't. As I've repeatedly said, the spaceships themselves have unchanging internal stresses, even though their measured length shortens in your chosen frame. So whatever is happening to them, they are not being "squished". If they were, they would show increasing internal stresses, and they don't.

The stress comes from the spaceships resisting the string's attempt to squish.

No, the stress comes from the spaceships pulling on the string. If there were "squishing" going on, as I have repeatedly said, you would see evidence of it in the spaceships themselves, since there is nothing preventing them from squishing.

The multiple scenarios are interjected to remove the urge to seek refuge in any of the usual places.

None of which I have attempted to take refuge in. You are not responding to what I'm saying. What I'm saying is that your model of what's going on doesn't explain what you claim it explains. I have repeatedly said why: if your model was really the explanation, you would expect the spaceships to experience increasing internal stresses. They don't. Can you address that point?

 

[PeterDonis]

You say, "Lorentz had already shown by 1904 that the laws of physics (i.e., Maxwell's equations, the Lorentz force law, etc) imply that the equilibrium configuration of an object in motion is shortened in the direction of motion.".

You left out a key qualifier: shortened in the direction of motion with reference to the original coordinates. The phenomenon you are talking about is frame-dependent, and Lorentz never claimed otherwise.

I'm not sure your view would be accepted by many

I accept it; I just don't agree that it means what you think it means. You are taking a frame-dependent phenomenon, the length of an object, and trying to relate it to a frame-independent, invariant phenomenon, the observed internal stresses of an object. That's not a fruitful procedure. If you want to understand the frame-independent internal stresses of an object, you need to relate them to a frame-independent feature of their motion: in this case, the expansion scalar.

 

[nosepot]

PeterDonis: The entire point of this exercise was to understand how things would look to the original reference frame, and not from a frame independent perspective, and only Samshorn and yossell could confirm my intuition that the material would actually appear to contract, and Samshorn the only to state it was due to the electrodynamics of the atomic bonds.

I must admit, once I separated in my mind that "length contraction" is two different phenomenon (the squishing caused by reshaping of moving atomic fields as the object accelerates, and the apparent shortening of moving objects due to relativity of simultaneity) the whole thing becomes much clearer. And I agree, the symmetry is absurdly strange.

When I was schooled in SR, we only dealt with two frames of reference moving at a steady state relative to each other; which means we only touched on relativity of simultaneity. No mention of how why materials would contract when accelerated was ever made to us. As Samshorn says, an understanding of both is needed. I'm not sure many textbooks deal with both? Or maybe I had a garbagety physics lecturer! Perhaps that's why the ether is so hard to let go.

For me there is something easier to grasp when explaining these concepts with an ether, and there will always be the question of "what's doing the waving?", which would also add to the reluctance of an neo-Lorentizan to down tools.

 

[Dale]

You are taking a frame-dependent phenomenon, the length of an object, and trying to relate it to a frame-independent, invariant phenomenon, the observed internal stresses of an object. That's not a fruitful procedure. If you want to understand the frame-independent internal stresses of an object, you need to relate them to a frame-independent feature of their motion: in this case, the expansion scalar.

That is, in my opinion, the most important message in the whole thread.

 

[Dale]

For me there is something easier to grasp when explaining these concepts with an ether, and there will always be the question of "what's doing the waving?", which would also add to the reluctance of an neo-Lorentizan to down tools.

There is no need to "down tools" as long as you realize that your tools are not necessary and that there are other tools in the toolbox. I personally prefer the "block universe tool" because it makes identifying the invariants much easier as well as making the leap to GR easier. However, I still break out my "aether tool" to do Doppler problems. I think it is best to learn all of the tools and use the best one for the job at hand.

 

[PeterDonis]

the squishing caused by reshaping of moving atomic fields

I don't disagree with much of what you say, but once again, I do not think "squishing" is an apt description of what you are talking about here. Once again, consider the spaceships: they undergo Lorentz contraction, with reference to your chosen frame, but their internal stresses do not change. So whatever Lorentz contraction is, it isn't "squishing"; if the spaceships were being squished, they would be subjected to increasing internal stresses, and they aren't.

The entire point of this exercise was to understand how things would look to the original reference frame

And, again, I don't think "squishing" is a good way to describe that, for the reasons given. Basically, you are trying to make an analogy between what happens to the spaceships as they accelerate and Lorentz contract more and more, and what would happen if you put them inside a big hydraulic press and gradually squeezed them. That's not a good analogy, because in the latter case, the ships (or any objects) would be subjected to increasing internal stresses, and in the former case, they aren't. So again, whatever Lorentz contraction is doing, whatever is happening with the internal forces between the atoms, as viewed from your chosen frame, it doesn't seem like it can be fruitfully understood as "squishing".

 

[nitsuj]

"length contraction" is two different phenomenon (the squishing caused by reshaping of moving atomic fields as the object accelerates, and the apparent shortening of moving objects due to relativity of simultaneity). And I agree, the symmetry is absurdly strange.

I find when SR - time length are reduced physically what's left is the concept of causality.

From that I learned to appreciate that length contraction is not merely an apparent shorting, but one with physical consequences. What can be attributed to appearance is calling it length, as has been pointed out that is a frame dependent value. Proper length, like proper time is what's familiar.

So when using the term length contraction it really isn't the traditional length, or proper length that is a meter stick in your hands. It's length from the perspective of causality, or maybe better said from the perspective of a spacetime interval.

A la either the barn doors don't close simultaneously and you measure proper length the whole time, or you see the doors close simultaneously and can't measure "proper length" for anything greater than an instant (if you catch what I am trying to say it that can "make sense", strictly speaking though it doesn't )

So at those relativistic speeds* length is not proper length like a meter stick in your hands, but is a length from a "causal system" perspective, like the continuum we live in. Specifically between you and the ladder / barn there is motion or a speed. the continuum has a speed constant of c (length/time). So got to "swap one for the other" to maintain causality. something has to "give", ie the comparable measures of time & length. Or in this case the definition of what length is when it's in motion. It's shorter by the same amount there is longer time That equates to the same continuum speed constant c.

So long and short of it proper length is easily understood and defined visually, length in motion is a "nominal" term/value/measurement compared to good ol' familiar proper length

*any speed of course, proper length is "at rest"

 

[StarsRuler]

Posts: 23 Thanks, Nugatory and ghwellsjr. I agree with both of your interpretations, but it doesn't resolve the problem. Saying the ships stay the same distance and the string contracts in the orginal frame, is the same as saying the distances increase and the string stays the same length in the frame of either ship. That's not the issue.

No, it is not the same. Suppose there are no ships, but another objects very more small, and his distance is Planck length in the "system" with increased distances and the string with the same length. ¡¡¡ There is no contractional invariance in physics !

 

[nosepot]

This has been enlightening.

One last question. A bit off topic...

Would it make sense to invoke length contraction to resolve the twin paradox, since in the traveling twin's frame both the origin and destination (and the observable universe) suddenly accelerate and appear to contract, making the journey there and back a much shorter distance for them?

I guess during the traveller's acceleration phase the destination would appear to rush towards them dramatically to shorten the trip.

 

[nosepot]

There is a good chapter here which probably would have answered my original question had I seen it first:

http://books.google.com.au/books?id...a=X&ei=sGS4UaC4G8uikgWvn4H4CA&ved=0CC8Q6AEwAQ

 

[ghwellsjr]

There is a good chapter here which probably would have answered my original question had I seen it first:

http://books.google.com.au/books?id...a=X&ei=sGS4UaC4G8uikgWvn4H4CA&ved=0CC8Q6AEwAQ

Another very interesting paper, thanks for finding it.

However, it adds absolutely nothing to the discussion except to reinforce what we have all been saying all along. I wonder why you find it more satisfying than our answers to you.

The really salient comment is at the bottom of page 7 where Bell says:

We do not need to get involved in these details if we assume with Lorentz that the complete theory is Lorentz invariant...

He should have said "since" instead of "if" because that is fundamental to Einstein's theory but, of course, then he would be admitting that the import of his whole paper is unnecessary, which it is.

I would also like to point out that he is very sloppy and fuzzy in his analysis of his paradox. He uses undefined and unspecified words and phrases like "accelerate gently", "sufficiently high velocity", "set brutally in motion", "moved smoothly", "jerked", "sufficiently strong thread", "fragile thread", etc.

Look at these last two. The difference between those two threads is that one breaks and the other one keeps the rockets from following the same acceleration profile. He hasn't set up a scenario where he describes the actual physical characteristics of the thread nor of the physical characteristics of the rockets (and neither have you). If he (and you) had done so in enough detail (oh that ugly word), then you could show in your one frame that the thread would or would not break.

But your only specified details were that the thread was inelastic and that the rockets caused identical accelerations. And that means that your thread would break at the first instant the rockets were turned on right at the point of attachment to the leading rocket. Bell points this out in his note 3 where he says (sloppily, I might add), "Violent acceleration could break the thread just because of its own inertial while velocities are still small." Yes, but I would add that any acceleration is violent for an inelastic thread even without another rocket ship attached at the other end.

 

[Dale]

Would it make sense to invoke length contraction to resolve the twin paradox

I don't think so. The point of the paradox is a misunderstanding of the symmetries involved, and length contraction is symmetrical also. You have to resolve it by pointing out the asymmetry in the scenario.

 

[nosepot]

However, it adds absolutely nothing to the discussion except to reinforce what we have all been saying all along. I wonder why you find it more satisfying than our answers to you.

It may not add much for you, as you have a clearer understanding. My original question showed that I misunderstood that length contraction also involve an apparent dynamical shortening of objects when they are seen to accelerate. Bell's chapter was a sensible step by step argument towards why SR works. I've put it here for the benefit of anyone else who might be struggling with the same misconceptions I am. I'm for the most part converted - you should be pleased! :P

And that means that your thread would break at the first instant the rockets were turned on right at the point of attachment to the leading rocket. Bell points this out in his note 3 where he says (sloppily, I might add), "Violent acceleration could break the thread just because of its own inertial while velocities are still small." Yes, but I would add that any acceleration is violent for an inelastic thread even without another rocket ship attached at the other end.

If you are talking about inertia of the string, then in retrospect, yes I agree - I wasn't detailed enough. That wasn't really what the paradox is about, but your strictly correct. I suppose we should say, a not particularly elastic string, and that the rockets acceleration is never so great that it exerts a force on the string that might exceed its tensile strength. We should also say not that the rockets accelerate identically, but that their engines fire in an identical way that would cause them to accelerate identically if not connected by a string. Then after all that, we would see that the string is appearing to contract and draw the rockets closer.

 

[nosepot]

I don't think so. The point of the paradox is a misunderstanding of the symmetries involved, and length contraction is symmetrical also. You have to resolve it by pointing out the asymmetry in the scenario.

Aw, man. And there I thought I was getting it. It seems the only source of asymmetry beside the acceleration, which can be obviated by swapping clocks with a rocket going in the other direction, is length contraction. The length contraction is symmetric in that the rocket is seen to contract from the Earth frame, and everything else seems to contract from the rocket frame. [I edited this pargraph for clarity.]

The wiki page (argh!) gives a sensible resolution using length contraction, but then the page disintegrates into a mess of Doppler shifts and the like, mostly viewed from the Earth frame only. I searched the page and only found the word "contraction" listed twice. It's no wonder people are confused:

http://en.wikipedia.org/wiki/Twin_paradox#Specific_example

If the Talk page for Bell's Spaceship Paradox wiki page was a battlefield, the Talk page for the Twin Paradox is a holocaust. Obviously wiki is not a good source to go to for commonly misunderstood topics, but someone should be able to clean it up and put a sensible resolution there.

 

[Nugatory]

If the Talk page for Bell's Spaceship Paradox wiki page was a battlefield, the Talk page for the Twin Paradox is a holocaust. Obviously wiki is not a good source to go to for commonly misunderstood topics, but someone should be able to clean it up and put a sensible resolution there.

Cleaning it up and putting a sensible resolution there isn't hard... But keeping the sensible resolution there is a lost cause.