Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[cfrogue]

That would be wrong, you can only use the length contraction equation for an object with a constant length in its rest frame, but the string's length in its rest frame is changing because its ends are attached to the ships..

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

So, how would the launch frame conclude the string contracts when the launch frame concludes the distance between the ships does not change?

Well, only in the launch frame, not in other frames.

Understood

 

[JesseM]

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

In the launch frame? No, of course the string does not contract in this frame (at least not until it breaks), how could it when it's attached to the ships? Why would you think it should? Did you read what I just said about the length contraction not applying when the rest-frame length of an object is not constant? There is no question that the rest-frame length of the string in this scenario is not constant.

 

[cfrogue]

In the launch frame? No, of course the string does not contract in this frame (at least not until it breaks), how could it when it's attached to the ships? Why would you think it should? Did you read what I just said about the length contraction not applying when the rest-frame length of an object is not constant? There is no question that the rest-frame length of the string in this scenario is not constant.

OK, then how does the string break from only the solution of the launch frame?

 

[Al68]

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

So, how would the launch frame conclude the string contracts when the launch frame concludes the distance between the ships does not change?

In a co-moving reference frame, the ships are getting farther apart with time. In the launch frame the distance between the ships stays the same. That's length contraction, by a factor that increases with time.

In a co-moving reference frame, the ships get farther apart while the length of the string stays the same. The string breaks.

In the launch frame, the ever increasing length contraction factor results in a constant distance between the ships, while the string gets "shorter". The string breaks.

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite being stretched in its own frame.

Length contraction doesn't mean contracting with time, it means contracted relative to the proper length.

 

[JesseM]

OK, then how does the string break from only the solution of the launch frame?

You asked this question before, I gave my answer in post #42 when I said:

As I've said before, if you wanted to do the calculation solely from the perspective of the launch frame I think you would need to actually do some detailed calculation of the inter-atomic forces in this frame. Even though the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame), as A.T. said the way the electromagnetic field between atoms varies as a function of distance would change, and from this you could presumably show that the stress in the string was increasing. The details of such a calculation are beyond me though.

 

[JesseM]

In the launch frame, the ever increasing length contraction factor results in a constant distance between the ships, while the string gets "shorter". The string breaks.

This explanation seems confused to me...why do you say it's the "length contraction factor" that results in a constant distance? And why do you say the string gets shorter? Both the distance between ships and the length of the string are constant in the launch frame, because both ships have identical velocity as a function of time in this frame.

 

[cfrogue]

You asked this question before, I gave my answer in post #42 when I said:

Can you explain how this is consistent with the SR acceleration equations?

 

[Al68]

This explanation seems confused to me...why do you say it's the "length contraction factor" that results in a constant distance?

Because while the distance is constant in the launch frame, it is contracted by the ever increasing lorentz factor. I didn't intend to imply a causal relationship by the words "results in".

And why do you say the string gets shorter? Both the distance between ships and the length of the string are constant in the launch frame, because both ships have identical velocity as a function of time in this frame.

That part of my post was assuming the string wouldn't stretch, and therefore break. I addressed a stretchy string in the next part:

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite its proper length increasing with time.

 

[JesseM]

Can you explain how this is consistent with the SR acceleration equations?

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

 

[JesseM]

Because while the distance is constant in the launch frame, it is contracted by the ever increasing lorentz factor.

Presumably you mean it's "contracted" relative to the distance in some other frame, like the instantaneous rest frame of one of the ships? But this is still a bit murky, because in every other frame the distance between the ships is changing, and you can't really compare the distance between ships at a given moment in one of these frames with the distance between them in the launch frame "at the same moment" without running into simultaneity issues. I suppose we can say that if we look at the length L' in the launch frame at any given time t, and then imagined the ships shutting off their engines simultaneously at time t in the launch frame and coasting inertially thereafter, and then we looked at the length L in the inertial frame where the ships were at rest once they had both shut off their engines, then it would make sense to say that L' is related to L by the length contraction equation L' = L/gamma.

 

[cfrogue]

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

 

[JesseM]

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

I didn't prove it from the perspective of the launch frame, I just outlined how I think you would go about proving it in terms of changing electromagnetic forces between atoms, and then said "The details of such a calculation are beyond me though."

However, if we're just interested in the question of whether it snaps or not, we don't actually have to prove it from the perspective of multiple frames, proving it snaps using the perspective of anyone frame is good enough, since in relativity all frames always agree on localized physical events.

 

[Al68]

Presumably you mean it's "contracted" relative to the distance in some other frame, like the instantaneous rest frame of one of the ships? But this is still a bit murky, because in every other frame the distance between the ships is changing, and you can't really compare the distance between ships at a given moment in one of these frames with the distance between them in the launch frame "at the same moment" without running into simultaneity issues. I suppose we can say that if we look at the length L' in the launch frame at any given time t, and then imagined the ships shutting off their engines simultaneously at time t in the launch frame and coasting inertially thereafter, and then we looked at the length L in the inertial frame where the ships were at rest once they had both shut off their engines, then it would make sense to say that L' is related to L by the length contraction equation L' = L/gamma.

Yes, and gamma would depend on t, which is what I meant by the lorentz factor increasing with time.

My only point was that the constant distance between the ships in the launch frame is "contracted" distance, and that "constant distance" and "increasingly lorentz contracted distance" aren't contradictory since the proper distance between the ships is increasing with time.

 

[Al68]

Where did you prove the string would snap from the POV of the launch frame?

Are you suggesting that the string's proper length can stretch indefinitely without snapping?

 

[cfrogue]

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

Oh, this is where I am confused.

Where is the string snap logic in the context of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

 

[cfrogue]

Are you suggesting that the string's proper length can stretch indefinitely without snapping?

I am trying to look at the problem from the launch frame.

This is legal.

There is no distance differential in the launch frame for the ships.

 

[Al68]

I am trying to look at the problem from the launch frame.

This is legal.

There is no distance differential in the launch frame for the ships.

Right, but the length of the string is lorentz contracted. So either the string's proper length stretches or it is shorter than the (increasing) proper distance between the ships and breaks.

Are you asking for proof that a string's proper length can only increase so much before it breaks?

 

[JesseM]

Oh, this is where I am confused.

Where is the string snap logic in the context of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

I already told you, you would need to calculate the changing electromagnetic force between atoms and I don't have the specific math for that, but I am totally confident this approach would show that the stress increases in the string until it snaps, because we already know that's what must happen from the analysis in other frames and it's impossible in SR for different frames to disagree in their predictions about localized events like whether a string snaps or not.

 

[atyy]

I mean, where is the math?

See the link in post #5.

 

[JesseM]

See the link in post #5.

In that chapter he does discuss how the electromagnetic field of a moving atom is altered by its velocity, but from skimming it, it doesn't look like he actually goes so far as to apply that to the case of the accelerating string to show how the stress increases until the electromagnetic forces between atoms are no longer strong enough to hold the string together and thereby show at what point it will snap.

 

[atyy]

In that chapter he does discuss how the electromagnetic field of a moving atom is altered by its velocity, but from skimming it, it doesn't look like he actually goes so far as to apply that to the case of the accelerating string to show how the stress increases until the electromagnetic forces between atoms are no longer strong enough to hold the string together and thereby show at what point it will snap.

Some crucial pages are missing to me on Google, but I think he goes far enough to show that the equilibrium state of a moving rod will be shorter - wouldn't that be enough?

 

[JesseM]

Some crucial pages are missing to me on Google, but I think he goes far enough to show that the equilibrium state of a moving rod will be shorter - wouldn't that be enough?

Ah, I didn't think of calculating the equilibrium length in the launch frame (which will get increasingly short relative to the actual length of the rod in this frame) rather than calculating the stress in the launch frame...that seems like a good approach.

 

[atyy]

Ah, I didn't think of calculating the equilibrium length in the launch frame (which will get increasingly short relative to the actual length of the rod in this frame) rather than calculating the stress in the launch frame...that seems like a good approach.

Yes, exactly the same approach you suggested earlier in the thread - but just without switching frames.

 

[yuiop]

I am surprised this thread is still going. Is there anyone here who still has any shadow of a doubt that the string connecting the rockets WILL break??

Where did you prove the string would snap from the POV of the launch frame?

Dr Greg demonstrated that the string would snap from the POV of the launch frame way back in post #33 of this thread here: https://www.physicsforums.com/showpost.php?p=2443127&postcount=33

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

It is not in the SR equations because SR is specifically about reference frame that are NOT accelerating. However, SR does tell us that if an object is moving relative to an observer (accelerating or not) then that object should be length contracted. If the moving object is not length contracted then it MUST be under stress.

 

[yuiop]

There is a way to demonstrate that the string will break without invoking electromagnetic forces or even length contraction. Imagine rockets A and B are moving to the right with constant velocity v relative to observer C. The rockets are separated by a distance (d) and joined by an unstressed string. Rockets A and B are instructed to launch simultaneously according to the clocks in their inertial frame and thereafter accelerate to the left with constant acceleration until they come to rest in C's frame. In this scenario they are slowing down according to observer C and so length contraction of the string is not a factor according to observer C. (if anything he expects the string to expand.) When the rockets take off, C notices that the rear rocket takes of first because the clocks of A and B are not synchronised from C's point of view and the string snaps because the rear rocket slows down to a stop while the front rocket is still going to the right. In this case it is the relativity of simultaneity that snaps the string rather than length contraction, but the end result is still the same - the string snaps.

 

[matheinste]

It is not in the SR equations because SR is specifically about reference frame that are NOT accelerating. However, SR does tell us that if an object is moving relative to an observer (accelerating or not) then that object should be length contracted. If the moving object is not length contracted then it MUST be under stress.

Would it be OK to say that if an object whose unstressed rest length is known, appears to have, at a later time, the same length viewed from a frame moving relative to the one at which it is at rest, then it must at this later time be stressed.

Whatever its rest length, stressed or unstressed, it must in, real time, appear contracted when viewed from a relatively moving frame.

Matheinste.

 

[A.T.]

There is no distance differential in the launch frame for the ships.

So? Where is your problem? The condition for the string to snap is that the distance between the rockets is greater than the maximal length of the string. In the launch frame the distance between the rockets is constant, but the maximum length of the string reduces, because all elements the string is made of (fibers, chain links, down to individual atoms) are contracting as they accelerate.

 

[JesseM]

So? Where is your problem? The condition for the string to snap is that the distance between the rockets is greater than the maximal length of the string. In the launch frame the distance between the rockets is constant, but the maximum length of the string reduces, because all elements the string is made of (fibers, chain links, down to individual atoms) are contracting as they accelerate.

Well, the individual atoms contract, as do the electromagnetic fields surrounding them, but since the average distance between atoms remains constant until the string snaps in the launch frame, the fibers or chain links can't actually contract. The equilibrium length of the string (that is, the length it would be if its ends were free) does contract though, and the maximum length the string can reach without snapping is just some multiple of the equilibrium length.

 

[yuiop]

Would it be OK to say that if an object whose unstressed rest length is known, appears to have, at a later time, the same length viewed from a frame moving relative to the one at which it is at rest, then it must at this later time be stressed.

Whatever its rest length, stressed or unstressed, it must in, real time, appear contracted when viewed from a relatively moving frame.

Matheinste.

I agree. In the launch frame the (stressed) string between the accelerating rockets has length (d') and in an frame instantaneously co-moving with the average velocity of the accelerating rockets, the length of the string by their measurements (d) is greater than d' by aproximately the average gamma factor. (In the frame of one rocket the other rocket is moving so a sort of average velocity has to be used to estimate what is going on.)

 

[A.T.]

but since the average distance between atoms remains constant until the string snaps in the launch frame, the fibers or chain links can't actually contract.

If a stationary chain is already under maximal stress, then the chain links don't contract on acceleration, but simply break. But since atoms and bonding forces a difficult to grasp, it might be helpful consider a chain that still has some play when stationary, yet brakes at a certain speed, despite keeping a constant length:

 

[cfrogue]

Well, the individual atoms contract, as do the electromagnetic fields surrounding them, but since the average distance between atoms remains constant until the string snaps in the launch frame, the fibers or chain links can't actually contract. The equilibrium length of the string (that is, the length it would be if its ends were free) does contract though, and the maximum length the string can reach without snapping is just some multiple of the equilibrium length.

Yes, but the math from the launch frame should somehow make the string snap while the distance between the two ship remains constant.


Do you know how to do this from the launch frame only?

 

[yuiop]

Yes, but the math from the launch frame should somehow make the string snap while the distance between the two ship remains constant.


Do you know how to do this from the launch frame only?

Say we have string of length d and we know that if we stretch the string to twice its relaxed length that it will snap. If the string is connected to two rockets and its average velocity is such that we calculate its relaxed length to be d/2 and it is spanning a distance of d then it is about to snap, right?

 

[JesseM]

Yes, but the math from the launch frame should somehow make the string snap while the distance between the two ship remains constant.


Do you know how to do this from the launch frame only?

Why do you keep asking the same questions and not paying any attention to my answers? I already told you I didn't have the specific math in post #88, but that I'm confident the approach of calculating the changing electromagnetic force between atoms in the launch frame would show that it snaps.

Also, if you read posts 89-93 you'll see that atyy gave the alternate approach of using the electromagnetic force between atoms in the launch frame to calculate the relaxed length at different velocities in the launch frame, which shows that the relaxed length is shorter and shorter at higher velocities, implying that if the string is at constant length as its velocity increases in the launch frame, it is getting farther and farther past its relaxed length, so without actually calculating the stresses we can conclude it should eventually snap for this reason. The math for calculating the relaxed length as a function of velocity seems to be given in the book that atyy linked to.

 

[cfrogue]

I am surprised this thread is still going. Is there anyone here who still has any shadow of a doubt that the string connecting the rockets WILL break??



Dr Greg demonstrated that the string would snap from the POV of the launch frame way back in post #33 of this thread here: https://www.physicsforums.com/showpost.php?p=2443127&postcount=33





It is not in the SR equations because SR is specifically about reference frame that are NOT accelerating. However, SR does tell us that if an object is moving relative to an observer (accelerating or not) then that object should be length contracted. If the moving object is not length contracted then it MUST be under stress.

It appears the launch frame is having a problem WITHIN SR at proving the string will break.

The launch frame believes the distance between the two ships does not change and this is not disputed.

Thus, the accelerating frames believe the string will break within SR and the launch frame believes they will not under SR only.

It is this correct or do you have a proof or paper that shows the launch frame decides the string will break completely from the theory of SR.

 

[cfrogue]

Why do you keep asking the same questions and not paying any attention to my answers? I already told you I didn't have the specific math in post #88, but that I'm confident the approach of calculating the changing electromagnetic force between atoms in the launch frame would show that it snaps.

Also, if you read posts 89-93 you'll see that atyy gave the alternate approach of using the electromagnetic force between atoms in the launch frame to calculate the relaxed length at different velocities in the launch frame, which shows that the relaxed length is shorter and shorter at higher velocities, implying that if the string is at constant length as its velocity increases in the launch frame, it is getting farther and farther past its relaxed length, so without actually calculating the stresses we can conclude it should eventually snap for this reason. The math for calculating the relaxed length as a function of velocity seems to be given in the book that atyy linked to.

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?