Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[pervect]

I have no idea what "stress logic" is. I find it a bit disturbing that our poster seems to be rather demanding and attempting to have others do things the hard way. I'm not quite sure what the point of this all is - I don't get the sense that he's actually interested in learning relativity, but I don't get a clear sense of what he's actually after.

There isn't any such thing in relativity as rigid motion - any actual measuring rod will have a rigidity limited by the speed of sound in the material. The behavior of a non-ideal measuring rod being "squished" isn't particularly interesting, though.

The most possible rigid physical body would probably be the SI standard for length - which is defined in terms of a light beam. (This is an offhand remark, but I believe it to be a correct observation).

The ideal of rigidity, as SR understands it, is widely understood to be "Born rigidity",

It would be relatively easy to post some calculations about what happens to a Born rigid body being accelerated, but I'm not sure there's any purpose in doing so at this point. Unless we have someone who is actually interested in this and would be convinced by the results...

 

[Fredrik]

Well said Pervect. I was thinking the same thing. I especially liked this part:

I don't get the sense that he's actually interested in learning relativity, but I don't get a clear sense of what he's actually after.

 

[Al68]

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.

Nobody has had any problem using the launch frame that I can see.

If the string stays attached to both ships, its proper length as calculated in the launch frame is coordinate length/sqrt(1-v^2/c^2), where coordinate length is equal to the distance between the ships in the launch frame.

If the distance between the accelerating ships is constant in the launch frame, then the string's proper length will increase (stress) with time per the above calculation in the launch frame.

Just in case its not clear, all of the above is determined in the launch frame.

What else would you like to determine in the launch frame with SR?

 

[matheinste]

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

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

Matheinste

I hope the capital 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).

Thanks for your continued efforts to help. Perhaps I could make it a little clearer. Can I take one point at a time. Of course when I do it may still be incorrect but here goes.

My original words----------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 . ----------

What this was meant to convey is that the constancy of the length of the gap in the launch frame means that the length of the gap in the ship's frame(s) is increasing. If the length of the gap was constant in the ship's frame(s) it would be increasingly contracted in the launch frame because of the continued increase in relative velocity between the launch frame and the ship's frame(s).

I accept that your comments about "in a frame" and "relative to a frame" and "appears" are correct and helpful. That usage was sloppiness on my part.

Is that Ok for starters?

Thanks. Matheinste.

 

[A.T.]

Let us abandon the logic of breakage.

But why? Just because it has been shown to you that the launch frame will also conclude that the string breaks, if it applies SR consequently?

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

Wrong. SR applied consequently in the launch frame concludes a change of the string: All elements the string is made of are moving and are therefore contracted, so they cannot fill the constant distance between the ships anymore.

...implied by this post I was a crackpot.

You are not even a crackpot. Let me summarize this discussion:

"Show me the string breaks, in the launch frame"

It breaks in it's own frame due to elongation, so it breaks in every frame.

"But you are not allowed to use other frames, just launch frame"

In the launch frame all elements the string is made of are moving and are therefore contracted, so they cannot fill the constant distance between the ships anymore

"But you are not allowed to use 'material science'"

"Without some 'material science' you cannot show that a material will break"

"Then I don't care anymore if the string breaks. I just want to repeat the same nonsense, that SR concludes no change of the string in the launch frame, over and over again, and ignore all explanations. La la la, I don't see them ..."

 

[cfrogue]

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.

I am not really sure what you mean.

 

[cfrogue]

I have no idea what "stress logic" is. I find it a bit disturbing that our poster seems to be rather demanding and attempting to have others do things the hard way. I'm not quite sure what the point of this all is - I don't get the sense that he's actually interested in learning relativity, but I don't get a clear sense of what he's actually after.

There isn't any such thing in relativity as rigid motion - any actual measuring rod will have a rigidity limited by the speed of sound in the material. The behavior of a non-ideal measuring rod being "squished" isn't particularly interesting, though.

The most possible rigid physical body would probably be the SI standard for length - which is defined in terms of a light beam. (This is an offhand remark, but I believe it to be a correct observation).

The ideal of rigidity, as SR understands it, is widely understood to be "Born rigidity",

It would be relatively easy to post some calculations about what happens to a Born rigid body being accelerated, but I'm not sure there's any purpose in doing so at this point. Unless we have someone who is actually interested in this and would be convinced by the results...

If "Born rigidity" is the standard, then what are the calculations from the launch frame that would break the rod?

 

[cfrogue]

But why? Just because it has been shown to you that the launch frame will also conclude that the string breaks, if it applies SR consequently?

Wrong. SR applied consequently in the launch frame concludes a change of the string: All elements the string is made of are moving and are therefore contracted, so they cannot fill the constant distance between the ships anymore.

You are not even a crackpot. Let me summarize this discussion:

"Show me the string breaks, in the launch frame"

It breaks in it's own frame due to elongation, so it breaks in every frame.

"But you are not allowed to use other frames, just launch frame"

In the launch frame all elements the string is made of are moving and are therefore contracted, so they cannot fill the constant distance between the ships anymore

"But you are not allowed to use 'material science'"

"Without some 'material science' you cannot show that a material will break"

"Then I don't care anymore if the string breaks. I just want to repeat the same nonsense, that SR concludes no change of the string in the launch frame, over and over again, and ignore all explanations. La la la, I don't see them ..."

Then I don't care anymore if the string breaks. I just want to repeat the same nonsense, that SR concludes no change of the string in the launch frame, over and over again, and ignore all explanations. La la la, I don't see them ..."

I want to see the math from the launch frame.

Why is this wrong to ask?

 

[cfrogue]

Nobody has had any problem using the launch frame that I can see.

If the string stays attached to both ships, its proper length as calculated in the launch frame [/B]is coordinate length/sqrt(1-v^2/c^2), where coordinate length is equal to the distance between the ships in the launch frame.

If the distance between the accelerating ships is constant in the launch frame, then the string's proper length will increase (stress) with time per the above calculation in the launch frame.

Just in case its not clear, all of the above is determined in the launch frame.

What else would you like to determine in the launch frame with SR?

The latest paper posted shows the ships drift apart as the explanation.

How is the consistent?

Though, I would tend to agree with your analysis above.

 

[cfrogue]

Nobody has had any problem using the launch frame that I can see.

If the string stays attached to both ships, its proper length as calculated in the launch frame is coordinate length/sqrt(1-v^2/c^2), where coordinate length is equal to the distance between the ships in the launch frame.

If the distance between the accelerating ships is constant in the launch frame, then the string's proper length will increase (stress) with time per the above calculation in the launch frame.

Just in case its not clear, all of the above is determined in the launch frame.

What else would you like to determine in the launch frame with SR?

I have an additional problem thinking with this.

The space between the rockets is that of the launch frame space since the distance between the ships does not change and thus is Euclidian.

However, if any rod exists between the space of the two ships, that is Minkowsky space-time.
Thus, it appears that the space between the two ships is Euclidian while at the same time it is Minkowsky also.

Does this seem correct?

 

[JesseM]

I want to see the math from the launch frame.

Why is this wrong to ask?

And if the math was a detailed calculation of the electromagnetic forces between atoms in a rod moving at different velocities in the launch frame, do you think you'd be able to follow it? Or are you just challenging people to show that the calculations have been done?

As mentioned several times before, if you look at the book linked to in post #5 and check pages 68-74, you'll see that Bell does provide some calculations from classical electromagnetism to show that a simplified solid object like a rod or string will have a shorter equilibrium length at higher velocities, with the calculations done not from the object's rest frame but from the frame in which it is moving (the launch frame).

 

[yuiop]

Thus, it appears that the space between the two ships is Euclidian while at the same time it is Minkowsky also.

You have discovered that observers in different reference frames have different points of view. Welcome to relativity :)

 

[cfrogue]

And if the math was a detailed calculation of the electromagnetic forces between atoms in a rod moving at different velocities in the launch frame, do you think you'd be able to follow it? Or are you just challenging people to show that the calculations have been done?

As mentioned several times before, if you look at the book linked to in post #5 and check pages 68-74, you'll see that Bell does provide some calculations from classical electromagnetism to show that a simplified solid object like a rod or string will have a shorter equilibrium length at higher velocities, with the calculations done not from the object's rest frame but from the frame in which it is moving (the launch frame).

Oh, this is not correct about what I am doing.

What do you think about the overlapping space argument?

 

[cfrogue]

You have discovered that observers in different reference frames have different points of view. Welcome to relativity :)

Can you specify the conditions under which this statement is true?


Is this statement true when considering the light cone and events in the absolute past?

 

[JesseM]

Oh, this is not correct about what I am doing.

Then what are you doing? Do you disagree that the only way to show why it snaps in the rest frame would be to calculate the changing electromagnetic forces between atoms in this frame? If you agree that such a calculation would show it snaps, but you are somehow unsatisfied with this, can you explain why?

 

[yuiop]

Can you specify the conditions under which this statement is true?

Is this statement true when considering the light cone and events in the absolute past?

Observers in different reference frames will disagree on some measurements such as the length of an object or the time interval between two spatially separated events. Other measurement are invariant such as the speed of light or the spacetime interval between two events. I am not sure, but I think invariant quantities are usually four vectors. As for the light cone, all observers will agree which event occurred first if the events are causally connected i.e. one event is within the light cone of the other event, but they might disagree on which event happened first if the events are outside each others light cones.

Also, as I explained in an earlier post, observers in different reference frames will have different explanations for an observation such as in the famous pole and barn paradox. The observer at rest with the barn says the the pole fits within the barn because it is length contracted. The observer at rest with the pole says the pole is longer than the barn, but gets through the barn unscathed, because the doors at the back and front do not open and close simultaneously from his point of view.

 

[cfrogue]

Then what are you doing? Do you disagree that the only way to show why it snaps in the rest frame would be to calculate the changing electromagnetic forces between atoms in this frame? If you agree that such a calculation would show it snaps, but you are somehow unsatisfied with this, can you explain why?

Oh, this is a valid question.

I am not satisified that it appears that the space between the two ships is Euclidian while at the same time it is Minkowsky also.

 

[cfrogue]

Observers in different reference frames will disagree on some measurements such as the length of an object or the time interval between two spatially separated events. Other measurement are invariant such as the speed of light or the spacetime interval between two events. I am not sure, but I think invariant quantities are usually four vectors. As for the light cone, all observers will agree which event occurred first if the events are causally connected i.e. one event is within the light cone of the other event, but they might disagree on which event happened first if the events are outside each others light cones.

Also, as I explained in an earlier post, observers in different reference frames will have different explanations for an observation such as in the famous pole and barn paradox. The observer at rest with the barn says the the pole fits within the barn because it is length contracted. The observer at rest with the pole says the pole is longer than the barn, but gets through the barn unscathed, because the doors at the back and front do not open and close simultaneously from his point of view.

I agree with all you said above.

But, one cannot just blindly conclude because there exists a difference under SR that it is normal.

In particular, one solution has the ships drifting apart.

Yet the launch frame does not see this.

This is not a normal disagreement.

Now, if R of S is involved and two light cones are involved, then many observers can disagree on the ordinality of when they are struck by the two different light cones and that is normal.

But, the above is not normal.

 

[JesseM]

I am not satisified that it appears that the space between the two ships is Euclidian while at the same time it is Minkowsky also.

Space cannot be Minkowski, Minkowski describes a geometry of spacetime. The spatial part of Minkowski spacetime is Euclidean in the sense that all of Euclid's laws work (parallel lines never meet, the sum of angles in a triangle is 180 degrees, etc.)

 

[atyy]

I have an additional problem thinking with this.

The space between the rockets is that of the launch frame space since the distance between the ships does not change and thus is Euclidian.

However, if any rod exists between the space of the two ships, that is Minkowsky space-time.
Thus, it appears that the space between the two ships is Euclidian while at the same time it is Minkowsky also.

Does this seem correct?

Space is Euclidean in every inertial frame. The crucial thing is that the relation between inertial frames is given by Lorentz transformations, not Galilean transformations, so eg. what is deemed simultaneous in one inertial frame isn't in another.

 

[cfrogue]

Space cannot be Minkowski, Minkowski describes a geometry of spacetime. The spatial part of Minkowski spacetime is Euclidean in the sense that all of Euclid's laws work (parallel lines never meet, the sum of angles in a triangle is 180 degrees, etc.)

OK, you are right, but they must overlap and cannot which is my point based on the launch frame and the accelerating frame.

You must know, when operating from A to relative frame B, Minkowski must be used.

That presents a problem under this analysis.

 

[JesseM]

OK, you are right, but they must overlap and cannot

What must overlap and cannot?

 

[A.T.]

I want to see the math from the launch frame.

That is the math:

In the launch frame all elements the string is made of are moving and are therefore contracted, so they cannot fill the constant distance between the ships anymore

Just because something is not a symbolic formula, doesn't mean it is not math. But here the symbolic version anyway:

N : number of string elements placed in line to span the length of the string
d : distance a single resting string element can span
D = N*d: distance the string can span at rest = distance between the rockets in the launch frame

When the string moves at velocity v all elements are contracted by 1/gamma(v) (Lorentz factor) according to SR:

D' = N*d/gamma(v): distance the string can span while moving at v

since

gamma(v) > 1

it follows that

D' < D

So the string cannot span the distance between the rockets D when it is moving. It brakes.

 

[Al68]

I want to see the math from the launch frame.

Why is this wrong to ask?

The math for what? What specifically do you want to see the calculation for?

The latest paper posted shows the ships drift apart as the explanation.

How is the consistent?

Consistent with what? The distance between the ships increasing in the co-moving frame of the string is consistent with the distance between them being constant in the launch frame. In fact, the proper distance between the ships must increase with velocity to maintain their constant distance in the launch frame.

 

[Dale]

BTW everyone, there is a FAQ on Bell's spaceship: http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html

 

[atyy]

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

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

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

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

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

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

So perhaps there is something unsatisfactory with Bell's classical calculation after all, or at least of my understanding of it.

I didn't like his calculation because he omits the radiation that an orbiting charge should emit. I tried setting up a system of two unequal positive charges a fixed distance apart (in an inertial frame) with a negative charge in equilibrium between them. When the two positive charges moving at any constant velocity, the equilibrium position of the negative charge relative to the positive charges seems to be independent of the velocity of the positive charges. So it's not obvious to me that the equilibrium position will change depending on the velocity. If I haven't muddled my calculations (which I may well have), the system will only go out of equilibrium when it is accelerated, because the equilibrium position of the negative charge is not halfway between the unequal positive charges, so given the constant speed of light, the changes in the electromagnetic field propagating from each charge will reach the negative charge at different times, and thus the negative charge will not remain in equilibrium at all times if the whole system is "Born rigidly" accelerated. Comments and corrections please

 

[matheinste]

Hello all,

Perhaps I'm a bit naive but surely if you pull a thread hard enough it will break. Isn't it sufficient to show that the gap becomes physically longer than the thread by some relativistic effect or other.

Or is the debate about the breaking mechanism to do with the contraction in LET.

Matheinste.

 

[A.T.]

Perhaps I'm a bit naive but surely if you pull a thread hard enough it will break.

I agree. This thread is already to long and will break soon.

 

[atyy]

Hello all,

Perhaps I'm a bit naive but surely if you pull a thread hard enough it will break. Isn't it sufficient to show that the gap becomes physically longer than the thread by some relativistic effect or other.

Or is the debate about the breaking mechanism to do with the contraction in LET.

Matheinste.

Yes, of course. But it's interesting (or masochistic?) to try to see how to do it in the launch frame.

I agree. This thread is already to long and will break soon.

But not in the launch frame!

 

[matheinste]

I agree. This thread is already to long and will break soon.

Nice one.

Matheinste.

 

[matheinste]

But not in the launch frame!

Another nice one.

Matheinste.

 

[matheinste]

Yes, of course. But it's interesting (or masochistic?) to try to see how to do it in the launch frame.

But the breaking is a physical reality whichever frame you view it from. I can perhaps understand why the logic is a bit more involved in the launch frame but why is it so complicated physically in that frame. Surely the physics of the situation does not alter.

Matheinste

 

[JesseM]

But the breaking is a physical reality whichever frame you view it from. I can perhaps understand why the logic is a bit more involved in the launch frame but why is it so complicated physically in that frame. Surely the physics of the situation does not alter.

It would be complicated in other frames too if we did a full analysis of the electromagnetic forces between atoms in those frames...instead we rely on some implicit intuitive knowledge about how normal materials behave when stretched in their own rest frame (I suppose if we had a lot of daily experience with objects moving at relativistic velocities we might have implicit knowledge about them too)

 

[matheinste]

It would be complicated in other frames too if we did a full analysis of the electromagnetic forces between atoms in those frames...instead we rely on some implicit intuitive knowledge about how normal materials behave when stretched in their own rest frame (I suppose if we had a lot of daily experience with objects moving at relativistic velocities we might have implicit knowledge about them too)

I see your point and I am all for such analyses even though they are beyond me. But did the original problem/paradox designer have such detailed explanations in mind when he posed it.

Matheinste

 

[JesseM]

I see your point and I am all for such analyses even though they are beyond me. But did the original problem/paradox designer have such detailed explanations in mind when he posed it.

Again, the original paradox relied on implicit knowledge about how materials behave in their own rest frame, but I think anyone would acknowledge that to make the proof 100% rigorous you need to actually calculate how the material would behave in a given frame, based on the internal forces holding the material together.