Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[yuiop]

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

I do not think that would be a correct solution. If the rod is attached to just one ship the observers on that ship will not consider the rod to be contracting. If the rod is connected to both ships they will consider the rod as being stretched.

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

That's an OK solution. The ships are getting further apart and anything connecting them is being stretched from the POV of the ship observers.

In the lauch frame the observers calculate the relaxed length of the accelerating rod to be d' which is less than the relaxed length of the rod when it was at rest in the launch frame. Since they observe the rod is still spanning a distance of d when it moving relative to them the launch frame observers consider the rod to stretched or under stress and will probably break if the stress continues to increase unless the rod has infinite strength which is unlikely.

 

[cfrogue]

I do not think that would be a correct solution. If the rod is attached to just one ship the observers on that ship will not consider the rod to be contracting. If the rod is connected to both ships they will consider the rod as being stretched.

This is the latest mainstream paper.

Although Bell’s name has been attached to the paradox, the
thought experiment involved was first considered by Dewan and Beran[5] as
a demonstration “that relativistic contraction can introduce stress effects in a
moving body.” We have disputed this contention in the previous section.

http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

That's an OK solution. The ships are getting further apart and anything connecting them is being stretched from the POV of the ship observers.

Yes, but what about the launch frame?

 

[Al68]

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?

Still not correct. The rod is contracted by the lorentz factor.

In the launch frame, the rod is contracted because it is in relative motion. That's just basic SR. It is length contracted in the launch frame regardless of whether it stretches and stays the same length, or breaks and gets shorter. Either way its length in the launch frame is less than its proper length.

In the launch frame, there is an increasing difference between the relaxed length of the string (or rod) and the distance between the ships.

 

[yuiop]

This is the latest mainstream paper.
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

The person that wrote that paper seems a bit confused. He seems to think that in the accelerating frame there is a series of instantaneous rest frames, but how can one ship consider itself to be in the same rest frame as the other ship fs it observes the other ship to be moving with respect to his own ship?

Also, the author seems to think there is only one possible physical explanation for an observation. In fact observers with different relative velocities will have different physical explanations for the same outcome.

Explanation 1)

From the POV of the accelerated observer the gap is increasing and the string is being stretched.

Explanation 2)

From the POV of the launch frame observers the gap remains constant and the string is length contracting.

Explantion 3)

From the POV of an observer moving with constant velocity relative to the launch frame in the same direction as the intended path of the rockets, the front rocket takes off before the rear rocket and it is the difference in synchronicity that causes the string to break.

All the explanations are equally valid and all reach the same conclusion that the string will snap. Every observer must have an explanation for why the string snaps from their point of view even if that explanation differs from the explanation of other observers in different reference frames. The author of the paper claims that only the accelerating observers can provide a physical explanation for why the string snaps and leaves observers in other frames without an explanation.

 

[matheinste]

These comments referred to the cited paper.

On page 3 the author seems to be using the fact that Lorentz transforms (coordinate transforms) do not induce stress in an object as proof that Lorentz contraction, in the original Lorentzian use of the term, do not either.-----"-One other point to be considered is whether strains and stresses can be induced by Lorentz contraction, as is contended in Refs. [1,2,4,5]. Our answer to this is clear from the previous discussion. Just as a 3D rotation of an object does not induce strain, a 4D rotation (Lorentz transformation) will not induce strain and consequent stress."--------

Also the fact that he describes the apparent relativistic contraction of length as illusory is a bit unusual.-------"And, just as the “shortening” of a stick that is rotated in three dimensions is an illusion, we now can see that the “shortening” of a stick that is rotated in four dimensions by a Lorentz transformation is also illusory."----------

I have not read the rest of the article closely yet but the above points disturb me a little.
Of course it may just be my reading of the text that is in error.

Matheinste.

So I am not alone in finding it a bit unusual.

 

[cfrogue]

The person that wrote that paper seems a bit confused. He seems to think that in the accelerating frame there is a series of instantaneous rest frames but how can one ship consider itself to be the same rest frame as the other ship is it observes the other ship to be moving with respect to his own ship?

Also, the author seems to think there is only one possible physical explanation for an observation. In fact observers with different relative velocities will have different physical explanations for the same outcome.

Explanation 1)

From the POV of the accelerated observer the gap is increasing and the string is being stretched.

Explanation 2)

From the POV of the launch frame observers the gap remains constant and the string is length contracting.

Explantion 3)

From the POV of an observer moving with constant velocity relative to the launch frame in the same direction as the intended path of the rockets, the front rocket takes off before the rear rocket and it is the difference in synchronicity that causes the string to break.

All the explanations are equally valid and all reach the same conclusion that the string will snap. Every observer must have an explanation for why the string snaps from their point of view even if that explanation differs from the explanation of other observers in different reference frames. The author of the paper claims that the only the accelerating observers can provide a physical explanation for why the string snaps and leaves observers in other frames without an explanation.

You did not specifify the launch frame.

How about that?

The person that wrote that paper seems a bit confused. He seems to think that in the accelerating frame there is a series of instantaneous rest frames but how can one ship consider itself to be the same rest frame as the other ship is it observes the other ship to be moving with respect to his own ship?

This is the way all of them look at the problem.

They all consider an artificial at rest frame for an infinitesimally small segment of the "string" vs the accelerating frame.

Here is another analysis using the same "at rest frame" with a different conclusion.

http://www.mathpages.com/home/kmath422/kmath422.htm


Please read it again.

 

[JesseM]

You did not specifify the launch frame.

How about that?

Yes he did:

Explanation 2)

From the POV of the launch frame observers the gap remains constant and the string is length contracting.

This is the way all of them look at the problem.

They all consider an artificial at rest frame for an infinitesimally small segment of the "string" vs the accelerating frame.

Yes, but I think kev's point was that in the instantaneous rest frame of one of the ships, the other ship is moving--there is no common rest frame for the entire ship/string combo. The paper seems to say otherwise on p. 4 when the author writes "If, at a time when each spaceship has a velocity v, we make a Lorentz transformation with velocity v, each spaceship will be at rest".

Here is another analysis using the same "at rest frame" with a different conclusion.

http://www.mathpages.com/home/kmath422/kmath422.htm

This page isn't talking about the type of acceleration seen in Bell's spaceship paradox, it's talking about Born rigid acceleration, where the two ships would not have the same acceleration in the launch frame (nor would they have the same proper acceleration), instead the acceleration of the ship in the rear would be greater than that of the ship in front. Born rigid acceleration is specifically designed so that if you pick the instantaneous inertial rest frame of one point on the accelerating object (like the front ship), at that moment in that frame every other part of the object (like the back ship) is at rest too.

 

[cfrogue]

Yes he did:

How?

This page isn't talking about the type of acceleration seen in Bell's spaceship paradox, it's talking about Born rigid acceleration, where the two ships would not have the same acceleration in the launch frame (nor would they have the same proper acceleration), instead the acceleration of the ship in the rear would be less than that of the ship in front. Born rigid acceleration is specifically designed so that if you pick the instantaneous inertial rest frame of one point on the accelerating object (like the front ship), at that moment in that frame every other part of the object (like the back ship) is at rest too.

It is talking about how to reduce the acceleration of the front ship to equate the effect of the string's length contraction.

On the other hand, with respect to the original inertial coordinates x,t, the two branch families represent two widely separate clusters of particles, initially both approaching the pivot event at near light speed and highly contracted spatially. As they approach, each cluster slows down and expands, until finally the two clusters both come to rest at time t = 0, just as they touch each other and achieve their maximum lengths. Then they separate again, each accelerating away and contracting.
http://www.mathpages.com/home/kmath422/kmath422.htm

Am I wrong?

 

[yuiop]

Here is another analysis using the same "at rest frame" with a different conclusion.

http://www.mathpages.com/home/kmath422/kmath422.htm

As Jesse mentioned, the author of that webpage (Kevin Brown) is talking about a different kind of acceleration that keeps the gap between the rockets constant from the point of view of the rocket observers, but now observers in the launch frame see the gap as length contracting at the same rate as anything connecting the two rockets. In that context an instantaneous rest frame for the accelerating rockets makes sense.

As an aside, I like this interesting quote from Kevin Brown in the linked webpage:

More fundamentally, it's worth recognizing that, even in circumstances when Born rigid motion of a configuration of particles is feasible, it does not actually represent perfectly "stressless" motion, because although the proper distances with respect to the instantaneously co-moving reference frames remain constant, the proper times of the different parts of the object do not remain coherent. In other words, if we contrive to hold the spatial relations fixed during an acceleration, a phase shift is introduced between different parts of the object, just as, if the phase is held constant, there is spatial stretching. (This is even more obvious in the case of angular acceleration, because in that case both spatial and temporal distortions are unavoidable.) This raises the question of whether material particles and their associated fields resist changes in their temporal as well as their spatial relationships. Typically we regard the equilibrium conditions as dependent only on the latter, and ignore differences in elapsed proper time, probably because such differences are extremely slight for the motions of ordinary macroscopic objects. Also, once a phase shift has been introduced, the assumed memorylessness of elementary entities ensures that the new equilibrium configuration will have the same spatial relations as the old. Nevertheless, it may still be the case that entities resist changes in their proper phase relations.

This leads to the intriguing idea that inertia, i.e., the resistance of objects to acceleration, may be partly or totally due to self-stresses of extended configurations. When we push on an object, it seeks to maintain not only the pre-existing spatial relations between its parts, but also the temporal phase relations. As we've seen, a direct consequence of the Minkowskian structure of spacetime is that if all these relations are held constant, the object cannot be accelerated. In order for the object to be accelerated, it is necessary to overcome the object's intrinsic resistance to changes in these relations (spatial, temporal, or both), and this resistance might be identified with the resistance of inertial bodies to acceleration. The only truly stressless "acceleration" would be of objects in a perfectly uniform gravitational field, in which case the intrinsic curvature of spacetime conforms identically to the skewed spatio-temporal relations usually associated with acceleration, so that in a local sense the object is actually moving inertially.

 

[matheinste]

cfrogue,

Try thinking of what happens in the launch frame as a sort of inverse of what happens in the ship frame.

In the ships frame(s) the gap is increasing but the length of the thread is not. In the launch frame the gap is constant but the length of the thread is not. Both cases lead to stress in the thread and eventually breaksge.

Matheinste.

 

[cfrogue]

cfrogue,

Try thinking of what happens in the launch frame as a sort of inverse of what happens in the ship frame.

In the ships frame(s) the gap is increasing but the length of the thread is not. In the launch frame the gap is constant but the length of the thread is not. Both cases lead to stress in the thread and eventually breaksge.

Matheinste.

Can't.

There exists the SR acceleration equations.

These equations show the ships do not change in distance.

Further, there does not exist a mainstream paper that proves the string breaks in the calculations of the launch frame.

If you can prove this within the launch frame, please show me.

 

[cfrogue]

As Jesse mentioned, the author of that webpage (Kevin Brown) is talking about a different kind of acceleration that keeps the gap between the rockets constant from the point of view of the rocket observers, but now observers in the launch frame see the gap as length contracting at the same rate as anything connecting the two rockets. In that context an instantaneous rest frame for the accelerating rockets makes sense.

As an aside, I like this interesting quote from Kevin Brown in the linked webpage:

Interesting, this author draws the following conclusion.

4 Conclusion
We have seen that the physical length of an object is the rest frame length as
measured in the instantaneous rest frame of the object. For two spaceships
having equal accelerations, as in Bell’s spaceship example, the distance between
the moving ships appears to be constant, but the rest frame distance between
them continually increases.

http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v2.pdf

 

[JesseM]

How?

You said he did not specify the launch frame, but he did specify what he thought was going on in the launch frame. I didn't say I agreed that his explanation was complete.

It is talking about how to reduce the acceleration of the front ship to equate the effect of the string's length contraction.

What do you mean by "equate the effect"? In Born rigid acceleration the distance between the ships does not change in their own instantaneous rest frame from one moment to the next, if that's what you mean, so a string between them which was also accelerating in a Born rigid way would not snap. But if you meant something else, please explain.

 

[matheinste]

Can't.

There exists the SR acceleration equations.

These equations show the ships do not change in distance.

Further, there does not exist a mainstream paper that proves the string breaks in the calculations of the launch frame.

If you can prove this within the launch frame, please show me.

But the specifications of the problem state that the distance is constant in the launch frame and it follows logically in SR that if the gap is constant in the launch frame then it cannot be constant in the ship frame(s).
The usual acceleration laws do not apply because the constancy of the gap is is not a natural consequence of normal acceleration but imposed upon the system by the problem designer.

Matheinste.

 

[cfrogue]

You said he did not specify the launch frame, but he did specify what he thought was going on in the launch frame. I didn't say I agreed that his explanation was complete.

What do you mean by "equate the effect"? In Born rigid acceleration the distance between the ships does not change in their own instantaneous rest frame from one moment to the next, if that's what you mean, so a string between them which was also accelerating in a Born rigid way would not snap. But if you meant something else, please explain.

Did you operate on the launch frame yet, that is what I am after.

If you look early on, you said I was "a crackpot" for considering the launch frame.

Yet, we are all finding a problem using this frame.

 

[matheinste]

Did you operate on the launch frame yet, that is what I am after.

If you look early on, you said I was "a crackpot" for considering the launch frame.

Yet, we are all finding a problem using this frame.

Not all.

 

[cfrogue]

But the specifications of the problem state that the distance is constant in the launch frame and it follows logically in SR that if the gap is constant in the launch frame then it cannot be constant in the ship frame(s).
The usual acceleration laws do not apply because the constancy of the gap is is not a natural consequence of normal acceleration but imposed upon the system by the problem designer.

Matheinste.

Sorry, acceleration is absolute motion under SR.

You are trying to apply relativity for acceleration which is not applicable.
Thus, most solutions apply a "theoretical rest frame logic" to prove consequences of the accelerating frame.

Yet, none operate from the launch frame which does not calculate a distance differential.

This is the issue at hand.

SR must solve this problem logically from both the launch frame and the accelerating frame and arrive at the same exact conclusion.

 

[matheinste]

Sorry, acceleration is absolute motion under SR.

You are trying to apply relativity for acceleration which is not applicable.
Thus, most solutions apply a "theoretical rest frame logic" to prove consequences of the accelerating frame.

Yet, none operate from the launch frame which does not calculate a distance differential.

This is the issue at hand.

SR must solve this problem logically from both the launch frame and the accelerating frame and arrive at the same exact conclusion.

Point 1. Sorry, acceleration is absolute motion under SR.

Aceleration is absolute.

Point 2.You are trying to apply relativity for acceleration which is not applicable.
Thus, most solutions apply a "theoretical rest frame logic" to prove consequences of the accelerating frame.

No idea what you are talking about. Too technical for me.

Point 3. Yet, none operate from the launch frame which does not calculate a distance differential.

No idea what operate from the launch frame means.

Point 4. SR must solve this problem logically from both the launch frame and the accelerating frame and arrive at the same exact conclusion.

That is obvious, and several people have already done it for you.

Matheinste

 

[JesseM]

Did you operate on the launch frame yet,

Why do you keep repetitively asking me this question when I've already told you several times that proving it in the launch frame would require calculating the electromagnetic forces between atoms, and I don't have the specific math for this but I am sure it would work? Did you really forget that I already gave you this answer more than once, or do you have problems comprehending it or something?

If you look early on, you said I was "a crackpot" for considering the launch frame.

Um, no I did not, your memory is playing tricks on you. I just did a search for posts by me using the word "crackpot", there were none on this thread.

 

[cfrogue]

Why do you keep repetitively asking me this question when I've already told you several times that proving it in the launch frame would require calculating the electromagnetic forces between atoms, and I don't have the specific math for this but I am sure it would work? Did you really forget that I already gave you this answer more than once, or do you have problems comprehending it or something?

Are you claiming SR does not answer this?

Um, no I did not, your memory is playing tricks on you. I just did a search for posts by me using the word "crackpot", there were none on this thread.

LOL

 

[JesseM]

Are you claiming SR does not answer this?

SR alone does not answer questions about strings breaking in any frame, you always need additional assumption about materials science. This is what I already told you in post 114 and post 122, and A.T. said the same thing in post 125.

 

[cfrogue]

Um, no I did not, your memory is playing tricks on you. I just did a search for posts by me using the word "crackpot", there were none on this thread.

Post $21

If you think there is any dispute among modern physicists about what would happen in this thought-experiment, you need to post some actual peer-reviewed literature, not a reference to an informal poll taken back when the idea was totally new. I am quite confident that there is no actual dispute about the fact that the stress increases, although as I said there could be types of accelerations where the stress increases but the string doesn't break (maybe because in certain types of accelerations the stress would approach a fixed limit rather than increasing without bound).


The implication you presented is that the consensus all agree the string will break. You implied by this post I was a crackpot. Thank goodness science is not conducted by consensus.

Yet no one thought to ask the launch frame.

I am doing that and with validity I might add.

 

[cfrogue]

SR alone does not answer questions about strings breaking in any frame, you always need additional assumption about materials science. This is what I already told you in post 114 and post 122, and A.T. said the same thing in post 125.

I posted mainstream papers that decide the string can only stretch or contract from the context of the accelerating frame.

Yet, we are trying to evaluate the launch frame.

 

[JesseM]

The implication you presented is that the consensus all agree the string will break.

Yes, and that is entirely true.

You implied by this post I was a crackpot.

How did I "imply" that? You're being oversensitive, I just thought you weren't aware that this was a settled issue among scientists so I was pointing that out to you.

Yet no one thought to ask the launch frame.

Sure they did. As pointed out by atyy (see posts 89-93), Bell did a calculation of the expected equilibrium length in the launch frame using classical electromagnetism, and showed the equilibrium length would shrink with increasing velocity, showing that if the string's length in the launch frame is constant it must be getting further and further past its equilibrium length.

Also, since we know that the electromagnetic laws governing atomic bonds are Lorentz-symmetric, that shows a priori that electromagnetic calculations done in different frames must always arrive at the same conclusions about local events, like whether the atomic bonds in the string are broken.

 

[JesseM]

I posted mainstream papers that decide the string can only stretch or contract from the context of the accelerating frame.

No, none of the papers said it could only be decided in an accelerating frame and couldn't be decided in the launch frame, they just didn't bother to do a calculation in the launch frame since the problem is easier to evaluate in other frames. As with the GPS thread, you seem to have trouble distinguishing between denying that something is true vs. just not addressing it one way or another.

 

[cfrogue]

No, none of the papers said it could only be decided in an accelerating frame and couldn't be decided in the launch frame, they just didn't bother to do a calculation in the launch frame since the problem is easier to evaluate in other frames. As with the GPS thread, you seem to have trouble distinguishing between denying that something is true vs. just not addressing it one way or another.

OK, then do the calcs and prove your case from the launch frame.

 

[atyy]

OK, then do the calcs and prove your case from the launch frame.

As follows:

In the launch frame the laws of physics have Lorentz symmetry. Thus from the launch frame we can switch frames, and show it breaks.

 

[cfrogue]

Also, since we know that the electromagnetic laws governing atomic bonds are Lorentz-symmetric, that shows a priori that electromagnetic calculations done in different frames must always arrive at the same conclusions about local events, like whether the atomic bonds in the string are broken.

Now, I like this

But, the launch frame does not have stress logic.

Give yourself any point between the ships and the acceleration equations do not show contraction.

Maybe I am wrong.


Can you show me?

 

[cfrogue]

As follows:

In the launch frame the laws of physics have Lorentz symmetry. Thus from the launch frame we can switch frames, and show it breaks.

Show me the math please.

Thanks.

 

[atyy]

Now, I like this

But, the launch frame does not have stress logic.

Give yourself any point between the ships and the acceleration equations do not show contraction.

Maybe I am wrong.


Can you show me?

BTW, the acceleration equations are not SR equations.

I hope you are aware of phenomena such as:
http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html

Historically, electrodynamics gave rise to SR, not the other way round.

 

[atyy]

Show me the math please.

Thanks.

That was the math!

 

[JesseM]

OK, then do the calcs and prove your case from the launch frame.

I've told you a bunch of times that I don't know how to do the specific electromagnetic calculations, but I know this approach would work just based on the Lorentz-symmetry of electromagnetic laws. And if you read the link to Bell's book in post #5 as well as the comments in post 89-93, you'll see that Bell did apparently show that the equilibrium length of an unattached string would get shorter as its velocity increased in the launch frame, which shows that the string attached to the ships was steadily going farther and farther past its equilibrium length, a reasonable basis for concluding it will snap.

 

[atyy]

Now, I like this

But, the launch frame does not have stress logic.

Give yourself any point between the ships and the acceleration equations do not show contraction.

Maybe I am wrong.


Can you show me?

Compare the field lines for the linear (ie. constant velocity) case with v=0 and v=0.9 at http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html

 

[Fredrik]

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?

I hope the red "it" means "the string" and not the distance, which you have already said is constant, so it would be inconsistent to say that it should be contracting.

The blue "relative to", should be "in". An object has a velocity in a frame, and relative to another object. So you could also have said "relative to the launch platform" or something like that.

The last two sentences are a bit weird too. You shouldn't be saying that a length "appears" to have a certain value. It has that value in the frame you're talking about. It only makes sense to talk about how it "appears" if you're describing what it would look like in a photograph or something like that. What you should be saying instead is that since the problem specifies that the string won't influence the motion of the rockets, and that the endpoints of the string will remain attached to the rockets, the endpoints will be the same distance apart at all times in the launch frame.

And to finish it off you have to add that the string would be getting shorter as its speed increases if it hadn't been for the constraint that the endpoints remain a distance d apart, and explain why that is. (As I said in my previous post, it's an axiom in SR with matter added "manually", and a derived result in SR with matter added by specifying a Lagrangian or something equivalent to that).

 

[Fredrik]

OK, then do the calcs and prove your case from the launch frame.

Are you saying that you want someone to show you how to calculate how the length of an object in a particular inertial frame changes when the object's velocity in that frame changes? The easiest way to do this by far is to just take its length in the co-moving inertial frame and just Lorentz transform to the inertial frame you're interested in. What you're asking for is just a more complicated way to do the calculation. Why would you want to see a complicated way to find the same result when you already have an easy way to get it? Anyone who understands SR knows that the result can't be any different.

Are you also seriously suggesting that the string wouldn't really break?