Does Bell's Paradox Suggest String Shouldn't Break Due to Length Contraction?

[yuiop]

A question has occurred to me:
If we assume a realistic rod connecting the ships so there is some degree of flex without breaking. The ships are spaced prior to acceleration such that there is a small degree of arc,
Say a drop of 5 cm. in the middle
In the launch frame, after an initial extremely short period as the momentum propagates through the rod, there should be no decrease in the deviation of the middle of the rod until eventually enough velocity is achieved to cause measurable contraction.
But in a frame moving in the same direction as the acceleration, the lead ship begins accelerating/moving first.
This would seem to indicate that the slack in the rod must instantly diminish to some extent. In a frame with a high velocity and therefore a greater interval between initiation of the front and rear ships, this seems like it would be significant.
Of course any changes in measurement of the difference in deviation would be transverse to motion, so the relative velocities would not affect this measurement in any frame.

Besides an actual coordinate displacement of the front ship relative to the rear , there would also be the resulting velocity away from the rear ship which would immediately continue taking up the slack and reducing the arc deviation from straight.
Without setting numbers, it still seems safe to say that a very small reduction of distance in the launch frame from the separation that would draw the rod taut would result in the small degree of sag I am talking about.
So it seems reasonable to suppose that a very small increase in the coordinate separation would then remove the deviation and render the rod straight between the ships.
Any thoughts??

Hi Austin. I like the basic premise of your scenario and would like to present my slightly exaggerated version with a small twist that should hopefully make the physical nature of length contraction startling clear.I particularly like that in your variation the length contraction is clearly visible as a change in shape, rather than an invisible change in tension of a straight string.

Initially the two rockets on the ground are 1km apart and joined with a loose chain that is 2km long. Clearly there is very visible sag in the chain. When the rockets take off to the right and accelerate they maintain a separation of 1km as measured in the ground based reference frame.

Prediction: As the rockets accelerate the chain gradually tightens up until at 0.866c relative to the ground the chain lies in straight line between the two rockets. At some velocity greater than 0.866c relative to the ground, the chain will snap as long as the rockets can maintain the acceleration profile and as long as the chain is not infinitely strong.

Twist: To an observer going to the left the rear rocket appears to take off first and the two rockets appear to be getting closer together and yet this observer still sees the connecting chain getting straighter, just like all the other observers. The only explanation is that the chain is length contracting faster than the rockets are approaching each other in this frame. From this view point, length contraction is a physical phenomena. This is the part that I think Eli has difficulty accepting.

 

[harrylin]

A question has occurred to me:
If we assume a realistic rod connecting the ships so there is some degree of flex without breaking. The ships are spaced prior to acceleration such that there is a small degree of arc,
Say a drop of 5 cm. in the middle
In the launch frame, after an initial extremely short period as the momentum propagates through the rod, there should be no decrease in the deviation of the middle of the rod until eventually enough velocity is achieved to cause measurable contraction.
But in a frame moving in the same direction as the acceleration, the lead ship begins accelerating/moving first.
This would seem to indicate that the slack in the rod must instantly diminish to some extent. In a frame with a high velocity and therefore a greater interval between initiation of the front and rear ships, this seems like it would be significant.
Of course any changes in measurement of the difference in deviation would be transverse to motion, so the relative velocities would not affect this measurement in any frame [..]

While I like yuiop's post, he didn't really discuss your question. If I understand you correctly, in the launch pad frame we expect to observe that at first the rod is still sagging a bit, while in a frame that is moving fast along X we expect to observe that the front rocket will pull the rod straight as the rear rocket is not yet moving, which would be a contradiction.

I guess that this is where the dynamics seriously kick in and have to be examined (calculated). From both perspectives the front rocket pulls on the rod; and as long as the rod molecules are accelerating the rod will be under push&pull tension so that it is partly stretched and partly compressed. And necessarily the rear rocket takes off before the tension wave of the front rocket reaches it.

Thus, the idea that the slack in the rod must "instantly diminish" as viewed from the moving frame is certainly wrong, and I'm not so sure that at first nothing happens to the slack in the launch pad frame. So far my first 2cts.

 

[A.T.]

And no, I’m not, as A.T. suspects, considering an elastic string.

If you talk about proper length of the string changing over time, then you are talking about an elastic string.

The rigid body’s “proper” length can be altered by intermediate accelerations.

Then it is not a rigid body. A rigid body breaks, if you try to change it's proper length.

 

[Dale]

DaleSpan:
In answer to 140:
I find no serious fault with your A,B,C

OK, then since you admit that my proof was a valid conclusion from my assumptions, and since you now understand and accept my assumptions, then you must logically agree that it can be proven using SR that the string breaks.

What I find unsettling is using the string's proper length in the co-moving frame when that length applies only to the ground frame. Note that the ships' proper separation changes by the SR transformation equations in either ship's co-moving frame (which is a momentary rest frame). Why is not the string length subject to that transformation?

I guess its your rejection of my view on proper length that I expounded recently ;-)

The strings length is subject to the same relativistic effects as everything else. I will write some more details, but it will have to be later. Perhaps I can help you feel less unsettled.

 

[Eli Botkin]

yuiop:

I know I said goodbye so very recently but your reply 150 just drew me back for this reply. I don’t know how you draw your Minkowski diagram so I’ll try to explain mine to you.

Draw the orthogonal axes (vertical T and horizontal X} to represent inertial frame A. Draw the worldlines of a rod’s endpoints in the region T < 0. Say the rod has length L and is at rest in frame A, then the two worldliness are vertical, say the left endpoint at X= 0, the right at X = L.

At T = 0 (in frame A) the rod is impulsively accelerated to a velocity V (relative to frame A) toward the right. Therefore both endpoint worldlines in the region T > 0 will slant toward the right at an angle = arctan(V/c) with respect to the T-axis.. Now note that in frame A the spatial separation between endpoints at any time T > 0 is still L, EVEN THOUGH THE ROD IS IN MOTION RELATIVE TO FRAME A.

Now draw the axes for an inertial frame B which will be the rest frame for the rod when T > 0. Call those axes (t, y). Of course these axes will not be orthogonal on this Minkowski diagram. The t-axis, drawn at the same clockwise angle as the worldlines will be parallel to the slanted worldlines, The y-axis is drawn at that same angle but counter-clockwise from the X-axis.

The time scale on the t-axis and the distance scale on the y-axis are set by the intersection of the t-axis with the family of hyperbolas T^2 – X^2 = a^2, and the intersection of the y-axis with the family of hyperbolas X^2 – T^2 = a^2.

I’ve selected the two sets of axes so that the two origins coincide.
In frame B, which is the rod’s new rest frame the separation between endpoints at any time t > 0, is measured along the y-axis. And if you do the math for this diagram you will discover that the rod’s length, as measured in this new rest frame B, has a length > L, showing that THE ROD HAS A DIFFERENT "PROPER" LENGTH AFTER THIS ACCELERATION.

Hopefully this explanation will help you and others who believe that the “proper” length of a rigid rod is universally fixed forever regardless of its acceleration history.

I know I had said that I would review your posts but our fundamental difference in viewing SR makes that a mute issue. Thanks again for participating.

 

[Doc Al]

Hopefully this explanation will help you and others who believe that the “proper” length of a rigid rod is universally fixed forever regardless of its acceleration history.

You still seem to think that accelerating a 'rigid' rod so as to preserve its length in its original frame can be done without destroying it.

 

[Eli Botkin]

Doc Al:
Do we not "preserve" the separation between ships in its original frame though the ships have the same acceleration history in that frame? If you believe it is otherwise for the rod, even for mild accelerations, then it is not due to SR transformation between frames.
Rather you are positing other physical happenings to the rod. I have no problem with that since I don't know what you have in mind that is causing the destruction.

 

[Doc Al]

Doc Al:
Do we not "preserve" the separation between ships in its original frame though the ships have the same acceleration history in that frame?

Yes, of course!

If you believe it is otherwise for the rod, even for mild accelerations, then it is not due to SR transformation between frames.

You seem to think that accelerating the rod in your destructive manner is simply equivalent to doing a Lorentz transform between inertial frames. Far from it!

Rather you are positing other physical happenings to the rod.

Absolutely! This is not just viewing things from another frame--it is ripping the rod apart!

I have no problem with that since I don't know what you have in mind that is causing the destruction.

Whatever mechanism you used to accelerate the rod so as to preserve its length in the original frame is what is causing the destruction.

 

[A.T.]

THE ROD HAS A DIFFERENT "PROPER" LENGTH AFTER THIS ACCELERATION.

If it was a rigid rod then it is broken, and doesn't have a proper length any more.

Hopefully this explanation will help you and others who believe that the “proper” length of a rigid rod is universally fixed forever regardless of its acceleration history.

Isn't that the definition of "rigid"? How do you define "rigid", if not by constant proper length?

 

[Eli Botkin]

To All:
It may be too easy to lose track of SR’s message. That message is that reality exists only in the worldlines that are engraved on the spacetime manifold. Any observer’s measurement of multi-worldline relations (such as time intervals or spatial separations, or even what we’ve chosen to call proper length) requires the observer to lay down a coordinate frame on the manifold.

In a real sense that is an arbitrary choice by the observer, except that we’ve choosen to follow SR rules in laying down frames because we are convinced that the SR postulates encompass reality. By laying down a coordinate frame the observer is just assigning an address to each worldline event. How those addresses change between observers follows the SR rules. The addresses we assign are not universal truths, so the computed time-intervals, distances and even proper lengths, cannot be.

 

[Eli Botkin]

Doc Al:
You say "Whatever mechanism you used to accelerate the rod so as to preserve its length in the original frame is what is causing the destruction."

Is this so by your pronouncement? Or is there some physics proof you can share with us?

 

[harrylin]

[..] I don't know what you have in mind that is causing the destruction.

Well I think, just like Bell, that it's in principle quite simple, especially* if we limit ourselves to the perspective of the launch pad frame: "the [contraction] hypothesis of H.A. Lorentz and G.F. Fitzgerald appears [..] as a necessary consequence of the theory" -Einstein 1907.

According to that hypothesis, moving objects will have an equilibrium length that is reduced by the factor γ because the EM fields that hold the matter together contract along that direction.

* from the perspective of other frames, non-synchronous departure plays a role as well but while complicating the explanation, this doesn't alter the physical interpretation of Lorentz contraction

 

[A.T.]

Doc Al:
You say "Whatever mechanism you used to accelerate the rod so as to preserve its length in the original frame is what is causing the destruction."

Is this so by your pronouncement? Or is there some physics proof you can share with us?

You just proved it yourself, by showing that the proper length of the string would have to increase to preserve its length in the original frame. For a rigid string that means breakage.

 

[Doc Al]

You just proved it yourself, by showing that the proper length of the string would have to increase to preserve its length in the original frame. For a rigid string that means breakage.

Simple as that.

 

[A.T.]

It may be too easy to lose track of SR’s message: ... The addresses we assign are not universal truths, so the computed time-intervals, distances and even proper lengths, cannot be.

If by "universal truths" you mean frame invariant, then you are wrong. Proper lengths, just like proper time intervals and proper accelerations are frame invariant. What you call "SR’s message" is the common misinterpretation of SR, that everything is relative (frame dependent). It's not. And Einstein originally called SR the "Theory of Invariants", to put emphasis on those "proper" quantities, that are frame invariant.

 

[Eli Botkin]

A.T.:
The frame-invariant in SR is dS^2 = [T^2 - (X^2 + Y^2 + Z^2)]. All else in SR must follow from that.

 

[Eli Botkin]

A.T. & Doc Al:
"You just proved it yourself,..." & "Simple as that."

The physics of the rigid body hasn't changed because the observer has changed. They just have different views of what they observe ;-)

 

[A.T.]

The frame-invariant in SR is dS^2 = [T^2 - (X^2 + Y^2 + Z^2)]. All else in SR must follow from that.

Wrong. The number of my legs is frame-invariant too, but that doesn't follow from the above.

The physics of the rigid body hasn't changed because the observer has changed.

Exactly. That's why proper length is frame-invariant. It doesn't change because the observer has changed.

But if that proper length would have to increase over time (as you have shown), then a rigid string would break for every observer.

Try to keep "frame invariant" and "time invariant" apart.

 

[Nugatory]

Doc Al:
Do we not "preserve" the separation between ships in its original frame though the ships have the same acceleration history in that frame? If you believe it is otherwise for the rod, even for mild accelerations, then it is not due to SR transformation between frames.
Rather you are positing other physical happenings to the rod. I have no problem with that since I don't know what you have in mind that is causing the destruction.

The two "separations" are different.

The separation between the spaceships is being preserved in the ground observer's frame, which is not the rest frame of either ship. In the ship frame (frames while they're accelerating, frame after the acceleration has ended and they've both stabilized at the same speed) that separation increases.

The rod's length is preserved in the rest frame of the rod, which is moving with the spaceships, at least until it breaks. So the length of the rod is constant in the frame in which the spaceships are separating.

I'll get the space-time diagram I promised you, the one showing the whole thing from the point of view of the left-moving observer, cleaned up and posted in the next day or so. It is seriously illuminating not just because it's the left-moving observer you asked for, but because both ground observer an ship is moving.

 

[Eli Botkin]

A.T.:
I forgot to tell you that your legs are frame-invariant because all events in one frame are also there in other frames, and I would guess that your legs are events ;-)

 

[A.T.]

I forgot to tell you that your legs are frame-invariant because all events in one frame are also there in other frames, and I would guess that your legs are events ;-)

And the proper-length of my legs is frame-invariant, because it's per definition the length that I measure in my rest frame. Just like my proper-time is frame-invariant, because it's per definition the time that I measure in my frame.

 

[Nugatory]

A.T.:
I forgot to tell you that your legs are frame-invariant because all events in one frame are also there in other frames, and I would guess that your legs are events ;-)

Your guess would be wrong. An event is a single point, defined by four coordinate (x, y, z, and t) in the most obvious Minkowski coordinates. A.T.s legs are a collection of multiple events, and statements about their size, shape, and length are in fact statements about relationships between these events. For example, the measured length is the spatial distance between two events, one at the heel and one the hip - choosing two events that have the same value of t in the reference frame in which the measurement is made, of course.

Some of these relationships are frame-invariant, meaning that they hold in all frames. For example, the quantity \int_P \sqrt{g_{\mu \nu} dx^{\mu} dx^{\nu}} integrated along a straight line between the heel and hip events will be the same in all frames, even though the coordinates may be wildly different. (It will also be equal to the measured length of the leg in a frame in which AT and his legs are at rest).

Other relationships, such as the measured length of AT's leg, are not frame-invariant. You'll get different answers in frames moving at different velocities relative to AT. However, these lengths can be calculated from the known velocity and the frame-invariant quantity above (which is, BTW, the infamous "proper length" of another fork in this thread), using the well-known formula for length contraction.

 

[Dale]

Hi Eli,

Let me explain some of the core concepts of relativity and how they relate to length.

First, even more basic than relativity is the form of the laws of physics. The laws of physics are expressed in terms of differential equations. A differential equation explains how something changes over space and time. In order to use them you also need to provide a set of initial conditions, or boundary conditions. Once you have that set of boundary conditions you can use the differential equations of the laws of physics to predict how the situation changes over time and space.

The principle of relativity means that the laws of physics are the same in all frames. That means, if there is some specific experimental measurement we perform and we use two different frames to predict the measurement then both frames must use the same laws of physics and get the same number. Furthermore, since both frames are describing the same experiment there must be some well defined equation relating the boundary conditions in one frame to those in another frame. That mathematical relationship is called a coordinate transform.

The principle of relativity can then be taken to mean that the form of the laws of physics is not changed under the transformations that relate the boundary conditions in one frame to those in another frame.

So far so good? Any questions so far?

 

[Dale]

So, now for a few definitions:

Clocks are experimental measuring devices which measure a quantity called proper time. Because proper time is the measured outcome of a physical device it must be frame invariant.

This is contrasted with coordinate time (often just called time). Coordinate time cannot be directly measured, but instead requires a frame-dependent convention for which events are simultaneous. However, in a given frame changes in coordinate time are equal to changes in proper time for clocks which are at rest.

This nomenclature is pretty common, a directly measurable frame invariant quantity designated as proper and a related frame variant quantity designated as coordinate or undesignated. Usually the two are equal in the rest frame of the measuring device.

Another example is acceleration. An accelerometer is an experimental device which measures a quantity called proper acceleration. It is equal to the coordinate acceleration in a reference frame where it is momentarily at rest.

Finally, proper length is the frame invariant quantity measured by a rod. It is equal to the coordinate length in the frame where the rod is at rest.

Is that clear?

 

[yuiop]

... Now note that in frame A the spatial separation between endpoints at any time T > 0 is still L, EVEN THOUGH THE ROD IS IN MOTION RELATIVE TO FRAME A.

You seem to think that the only way to increase the stress on a rod is to increase the spatial separation of its endpoints. Here is a counterexample. Imagine you have metal rod that is 1m long at room temperature. It is heated to 1000 degrees C so that it expands by about 1cm and then the ends are clamped so that they cannot move. It is also clamped in such a way that while it is still hot, the metal rod is unstressed. When the metal is cooled back to room temperature, it will try to regain its unstressed length (1m) but since it cannot, it becomes stressed and may even break. This destructive stress comes about with no change in the spatial separation of the rod endpoints.

... And if you do the math for this diagram you will discover that the rod’s length, as measured in this new rest frame B, has a length > L, showing that THE ROD HAS A DIFFERENT "PROPER" LENGTH AFTER THIS ACCELERATION.

This is your fundamental misunderstanding. The only way you can increase the proper length of a rod is by applying stress (forces) to the rod. For example I can take an elastic band in my fingers and change its proper length, at will, by simply stretching the elastic band. The important point is that the change in the proper length is accompanied by a change in the tension of the elastic band. If an object remains unstressed then its proper length cannot change. I notice you almost never mention words like "unstressed" or "tension" in your posts, so perhaps you do not realize those physical aspects involving forces are important?

... Hopefully this explanation will help you and others who believe that the “proper” length of a rigid rod is universally fixed forever regardless of its acceleration history..

I will state my belief as "The proper length of a rod that remains unstressed (or under constant tension) does not change, regardless of its acceleration history and regardless of its velocity relative to the observer". On the other hand, the coordinate length (the length measured by an observer moving relative to the rod) of the rod under constant tension, does depend on the velocity of the rod relative to the observer and under those conditions the coordinate separation between the endpoints becomes smaller with increasing relative velocity. If the endpoints are not getting closer together (as measured in the initial frame that sees the rod as moving), then the tension on the rod must be increasing and eventually break, as does the string in Bells rocket paradox.

 

[Eli Botkin]

harrylin (162):
This hypothesis (which turned out to be a good guess) of a Lorentz contraction to explain the Michelson-Morley experimental result was overshadowed by SR’s explanation that this same contraction could be derived from the hypothesis of an invariant speed of light. SR is what leads us more firmly to understanding the reason for length contraction, and SR does not say that it is “because the EM fields that hold the matter together contract…”

 

[Eli Botkin]

Nugatory (172):
Yes, I know how en “event” is defined in SR. Forgive me for my awkward attempt at levity.

 

[Eli Botkin]

DaleSpam (173 & 174):
You really needn’t have spent so much of your time enlightening me about those “core concepts.” But thanks for your concern. PS, my degree in physics is 61 years old, undilated ;-)

 

[Eli Botkin]

Yuiop (175):
Be assured that I was careful in assuring that the string (or rod, if you like) was not heated or cooled ;-)

You add: I will state my belief as "The proper length of a rod that remains unstressed… does not change, regardless of its acceleration history...”

This is a worthy belief if you can show it on a Minkowski diagram (or through SR transformation equations).

 

[yuiop]

Hi Eli,
I would like to try a different approach and see what you think. Let us say we have two rockets, A and B, with the same proper length L that are at rest alongside each other. Rocket B accelerates off in the x direction, until it reaches a velocity of 0.8c relative to rocket A and then switches off the drive. The observers onboard rocket B measure the coordinate length of rocket A to be 0.6L. The observers on board rocket A report the proper length of rocket A to still be L. There is no reason for the proper length of rocket A be anything other than L because we have not done anything to rocket A. If we had stress gauges on the rockets, then they would indicate that rocket A is unstressed. If you agree with all the above then you should agree that in order for rocket A to be unstressed when it has a velocity relative to rocket B, then the rocket B observers must measure the coordinate length of rocket A to less than L. For the length of rocket A to still be L when it has motion relative to B then rocket A would have to be physically stretched and probably break. Agree?

You might argue that it would be different if we accelerated rocket A instead of rocket B, but once the rockets engines are switched off and the stresses are allowed to settle down, then SR tells us that the proper length of both rockets is still L and they each measure the coordinate length of the other ship to less than L, so it makes no difference which rocket actually accelerates.

If you agree with the above, then you should conclude that if you accelerate an object while maintaining its coordinate length in the initial reference frame, then its proper length must be increasing (which I think you have already figured out) AND it must be under increasing stress and eventually break.

 

[yuiop]

You add: I will state my belief as "The proper length of a rod that remains unstressed… does not change, regardless of its acceleration history...”

This is a worthy belief if you can show it on a Minkowski diagram (or through SR transformation equations).

I mentioned it before and I will mention it again. Google "Born rigid" motion or acceleration. It will give you the equations and Minkowski diagrams for how to accelerate an object without introducing stresses and maintaining the proper length of the object. Any other acceleration scheme (e.g. the method used in Bell's rocket paradox) will subject the object to stress.

 

[Eli Botkin]

To All:
A bit more about our connection to SR’s message.

The reason we call these SR results paradoxes is because they seem so strange to us as beings who never got even close to leaving the frame we live in and entering a frame that has a great velocity relative to the one we left. All our experiences drive us to believe that the length of a rod will always retain that length no matter how we toss it, move it , or whatever, so long as we don’t heat or cool it, or squeeze or pull on it. That’s what the experience in our cocoon tells us. Remember how difficult it was for people to accept, when you tried to tell them, that if they had left their current place and then returned, their age would then be less than it would be if they had stayed and not traveled. We still don’t have an internalized feeling that this occurs, but we accept it because of our confidence in SR. Maybe after much time we will begin to accept (though un-internalized) that rest-frame lengths also depend on previous travel history.

As to the Bell paradox, I tend to be fixated on the need to show that if the string does break, it must do that for all observer. For some observers the ship separation increases and we can happily say that the string breaks, providing we also assume that the string doesn’t follow the same SR transformation which would avoid breakage. Some claim it doesn’t follow those transformations because it’s “rigid”. But there is nothing in SR that says thou shalt not transform “rigid” items.

For those observers who see the ships approaching each other, the string, if again deemed to be “rigid” would break under compression (or otherwise sag). Here again SR transformation would avoid compression.

Also the oft stated EM field contraction, drawing the object’s atoms closer, as reason for breakage, would have a frame relative-velocity dependence and would still leave the question: Does every observer see the same event (breakage, if it takes place)?

It may turn out that if breakage does occur in every frame’s view, it is because the string is undergoing an acceleration as viewed from all frames.

It’s been fun.

 

[harrylin]

harrylin (162):
This hypothesis (which turned out to be a good guess) of a Lorentz contraction to explain the Michelson-Morley experimental result was overshadowed by SR’s explanation that this same contraction could be derived from the hypothesis of an invariant speed of light. SR is what leads us more firmly to understanding the reason for length contraction, and SR does not say that it is “because the EM fields that hold the matter together contract…”

Hi Eli, surely you realize that many different "because" answers on a single question can be correct. In particular, SR is based on the assumption that Maxwell's laws are valid. According to those laws the EM fields that hold the matter together contract (and that was the basis for Fitzgerald's assumption of length contraction. I don't have the reference, but I think that the correctness of that assumption has been verified in more recent times with computer simulation aids.
SR is not magic, every physical principle must relate to physical means by which it works.

 

[harrylin]

[...] As to the Bell paradox, I tend to be fixated on the need to show that if the string does break, it must do that for all observer.

We all agree that the string must break according to every inertial reference system - and this is indeed the case for any perspective that I analyzed (happily so, for else SR would be defect!).

For some observers the ship separation increases and we can happily say that the string breaks, providing we also assume that the string doesn’t follow the same SR transformation which would avoid breakage. Some claim it doesn’t follow those transformations because it’s “rigid”. [..]

For sure all people in this discussion claim that the string does obey the Lorentz transformation between inertial frames - that's the "SR transformation" that must be obeyed. If the distance between the space ships is 100 m before departure, it will remain 100 m after departure until cruising speed according to measurements in the launch pad frame S. Consequently, according to SR this distance will be measured as γ*100 m in the new co-moving frame S' in which also the string is in rest. This distance defines the length of the string, which now is greater than that of the same string without constraints.

Thus it's just the contrary of what you think. For those co-moving observers the ship separation has increased and they can happily confirm that the string may have been broken, providing we also assume that the string does follow the same SR transformation which in this case implies proper stretching (and ultimately breakage).

Did that help?

 

[A.T.]

For some observers the ship separation increases and we can happily say that the string breaks, providing we also assume that the string doesn’t follow the same SR transformation which would avoid breakage.

There are no SR transformations which would avoid breakage.

Some claim it doesn’t follow those transformations because it’s “rigid”

Nobody claimed this here.

Also the oft stated EM field contraction, drawing the object’s atoms closer, as reason for breakage, would have a frame relative-velocity dependence

Yes, "reasons" for something can be frame-dependent. In my frame I die because the bullet hits me. In the bullets frame I die because I hit the bullet.

and would still leave the question: Does every observer see the same event (breakage, if it takes place)?

No, it doesn't leave leave that question. Physics predicts that all observes observe it to break. But physics doesn't care what informal "reasons" human observes come up with, to rationalize the result in terms of their intuitive cause-effect thinking.

 

[Dale]

DaleSpam (173 & 174):
You really needn’t have spent so much of your time enlightening me about those “core concepts.” But thanks for your concern. PS, my degree in physics is 61 years old, undilated ;-)

OK, so you understand that proper length is frame invariant now? In particular you understand the difference between constant (over time) and invariant (across frames)? You pretty clearly did not understand it a couple of days ago, so I wasn't sure how far to go.

 

[A.T.]

In particular you understand the difference between constant (over time) and invariant (across frames)?

He obviously doesn't, because in post #182 he still confuses rigidity (constancy of proper length over time) with coordinate transformations (which relate lengths across frames).

 

[ghwellsjr]

To All:
A bit more about our connection to SR’s message.

The reason we call these SR results paradoxes is because they seem so strange to us as beings who never got even close to leaving the frame we live in and entering a frame that has a great velocity relative to the one we left.

No, the reason we have so-called paradoxes in SR is because of an incorrect assumption about what happens when the same scenario is viewed from different frames. If you correctly apply the Lorentz Transformation process, not just the idea of Length Contraction or Time Dilation while ignoring the Relativity of Simultaneity, then there can never be a paradox.

All our experiences drive us to believe that the length of a rod will always retain that length no matter how we toss it, move it , or whatever, so long as we don’t heat or cool it, or squeeze or pull on it.

Yes, if we move a rod by accelerating it at one point then we won't squeeze or pull on it but if we accelerate one end of rod separately from accelerating the other end of the rod, we can end up squeezing it or pulling it apart. Isn't that obvious?

That’s what the experience in our cocoon tells us. Remember how difficult it was for people to accept, when you tried to tell them, that if they had left their current place and then returned, their age would then be less than it would be if they had stayed and not traveled. We still don’t have an internalized feeling that this occurs, but we accept it because of our confidence in SR. Maybe after much time we will begin to accept (though un-internalized) that rest-frame lengths also depend on previous travel history.

Rest-frame lengths don't depend on previous travel history but if you pull separately on both ends of an object, you can change its rest-frame length or break it if it won't stretch which is the meaning of rigid.

As to the Bell paradox, I tend to be fixated on the need to show that if the string does break, it must do that for all observer. For some observers the ship separation increases and we can happily say that the string breaks, providing we also assume that the string doesn’t follow the same SR transformation which would avoid breakage. Some claim it doesn’t follow those transformations because it’s “rigid”. But there is nothing in SR that says thou shalt not transform “rigid” items.

You are overlooking the fact that although the ship separation can be different in different frames, these events have different times associated with them. If you instead calculate the separation in these different frames with events that have the same time associated with them, then you will understand why they all show that the string breaks. You can't ignore the Relativity of Simultaneity.

For those observers who see the ships approaching each other, the string, if again deemed to be “rigid” would break under compression (or otherwise sag). Here again SR transformation would avoid compression.

Also the oft stated EM field contraction, drawing the object’s atoms closer, as reason for breakage, would have a frame relative-velocity dependence and would still leave the question: Does every observer see the same event (breakage, if it takes place)?

It may turn out that if breakage does occur in every frame’s view, it is because the string is undergoing an acceleration as viewed from all frames.

It’s been fun.

If a string or any object undergoes an acceleration in one (inertial) frame, then it undergoes an acceleration in all (inertial) frames. If you accelerate the object at just one point (which means you apply a force at just one point), then you can use SR to determine how all the other points on the object accelerate so that the object maintains the same shape as it had before, as long as it is rigid. That's what we mean by rigid. If you separately accelerate the object at two different points (which means applying two forces at two different points), and that second point accelerates the object differently than what SR would have determined it to be if you had only applied one force, then the object is either rigid and will break, or it is not rigid and will be stretched or compressed.

 

[Eli Botkin]

yuiop (181):
Thanks loads for this suggestion. Though I am totally familiar with the math in Born rigidity, I confess that I hadn’t thought to apply it to adjacent particles in the rod. I clearly need to rethink my position which, at first glance, seems incorrect. If so, then I owe all breakage enthusiasts an apology.

 

[Dale]

I tend to be fixated on the need to show that if the string does break, it must do that for all observer.

And I have encouraged you multiple times to do so. All you have to do is to take the valid proof I posted and Lorentz transform into any other frame in order to get an equivalent proof in a different frame.

providing we also assume that the string doesn’t follow the same SR transformation which would avoid breakage. Some claim it doesn’t follow those transformations because it’s “rigid”. But there is nothing in SR that says thou shalt not transform “rigid” items

This statement is badly wrong. Again you seem to be not understand proper length and its relationship to strain and reference frames.

A strain is a change in the proper length of an object over time, so strain is frame invariant. Length contraction is a disagreement between two different frames about the coordinate length of an object at a single point in time.

The concept of "Born rigidity" defines a strain-free motion. A stiff object in SR is one which will remain Born rigid regardless of external forces, at least until it breaks.

Again, rigidity relates the length at one time to the length at another time, and length contraction relates the length in one frame to the length in another frame. They are independent concepts and it is incorrect that rigid objects are exempt from length contraction.

 

[Austin0]

Hi Eli,
I would like to try a different approach and see what you think. Let us say we have two rockets, A and B, with the same proper length L that are at rest alongside each other. Rocket B accelerates off in the x direction, until it reaches a velocity of 0.8c relative to rocket A and then switches off the drive. The observers onboard rocket B measure the coordinate length of rocket A to be 0.6L. The observers on board rocket A report the proper length of rocket A to still be L. There is no reason for the proper length of rocket A be anything other than L because we have not done anything to rocket A. If we had stress gauges on the rockets, then they would indicate that rocket A is unstressed. If you agree with all the above then you should agree that in order for rocket A to be unstressed when it has a velocity relative to rocket B, then the rocket B observers must measure the coordinate length of rocket A to less than L. For the length of rocket A to still be L when it has motion relative to B then rocket A would have to be physically stretched and probably break. Agree?

You might argue that it would be different if we accelerated rocket A instead of rocket B, but once the rockets engines are switched off and the stresses are allowed to settle down, then SR tells us that the proper length of both rockets is still L and they each measure the coordinate length of the other ship to less than L, so it makes no difference which rocket actually accelerates.

If you agree with the above, then you should conclude that if you accelerate an object while maintaining its coordinate length in the initial reference frame, then its proper length must be increasing (which I think you have already figured out) AND it must be under increasing stress and eventually break.

Yes, here you clearly exemplified the difference between the kinematic and physical EM field contraction interpretations of contraction.
If we assume the EM contraction of ship B it is clear that we cannot apply that explanation to the contraction of ship A as measured in B. We must assume a purely kinematic source in this case.
If we consider a third ship C with an inertial velocity equal to the final velocity of B then we see B expanding relative to that frame as it accelerates.


I myself find the, physical contraction as a consequence of EM and atomic light speed interactions hypothesis very convincing. But as you have shown here it is somewhat problematic in application to specific scenarios.

If we assume the EM interpretation in frame C then the expansion is a result of decreasing contraction from the initial velocity as the velocity decreases with deceleration.
This is not a problem with the kinematic interpretation but is an obvious contradiction if we assume actual physical contraction. If the contraction is the result of actual tensile forces due to light speed interactions within the structure, then it logically is directly dependent on the velocity, relative not to any frame, but to the absolutely invariant speed of light.
it then follows that if a system is changing velocity it must be either increasing or decreasing it's speed relative to light. Contracting or expanding but not both.
So assuming that the EM contraction is correct there is still no way to determine how it would apply. It could only be a partial cause for the observed phenomena with the necessary assumption of purely kinematic effects also.
With no way to tell which is which.

In the case here. Ship B is both contracting and expanding. EM contraction works fine if we assume frame A is at rest and B is actually increasing in velocity. EM expansion works fine in C if we assume C is at rest and B is actually decreasing in velocity.
But both depictions of the physics occurring in the ship during acceleration cannot be accurate.
Make sense??

 

[Eli Botkin]

To all who set me straight: Mea culpa!

I am now convinced that I erred in thinking that SR was not adequate to show that the string in Bell's Paradox will break. The clincher was the Born rigidity solution that yuiop had suggest that I review. My new understanding also removed my concern about inertial frames wherein the ships approach each other.

Thanks again,
Eli

 

[harrylin]

To all who set me straight: Mea culpa!

I am now convinced that I erred in thinking that SR was not adequate to show that the string in Bell's Paradox will break. The clincher was the Born rigidity solution that yuiop had suggest that I review. My new understanding also removed my concern about inertial frames wherein the ships approach each other.

Thanks again,
Eli

Nice to hear that this discussion was useful.

 

[harrylin]

[..]
I myself find the, physical contraction as a consequence of EM and atomic light speed interactions hypothesis very convincing. But as you have shown here it is somewhat problematic in application to specific scenarios. [..]

I did not see yuiop show such a thing... I see no such problem.

If we consider a third ship C with an inertial velocity equal to the final velocity of [ship] B then we see B expanding relative to that frame as it accelerates. If we assume the EM interpretation in frame C then the expansion is a result of decreasing contraction from the initial velocity as the velocity decreases with deceleration.
This is not a problem with the kinematic interpretation but is an obvious contradiction if we assume actual physical contraction. If the contraction is the result of actual tensile forces due to light speed interactions within the structure, then it logically is directly dependent on the velocity, relative not to any frame, but to the absolutely invariant speed of light.

Perhaps you forgot that the speed of light relative to an atom depends on the chosen frame? It's a direct result of our definition of simultaneity. The speed of light relative to an object (also called "closing speed") is not invariant but frame dependent, just as the EM effects.

it then follows that if a system is changing velocity it must be either increasing or decreasing it's speed relative to light.

Well, obviously this is necessarily so (as described from any inertial frame). That follows from the second postulate - no need for such a complex consideration! As interpreted from frame C, the rockets fly along with light that is going in one direction and counter the light going in the other direction.

Contracting or expanding but not both.
So assuming that the EM contraction is correct there is still no way to determine how it would apply. It could only be a partial cause for the observed phenomena with the necessary assumption of purely kinematic effects also.
With no way to tell which is which. In the case here. Ship B is both contracting and expanding.

It applies just the same as most other things in physics, such as electric and magnetic fields as well as energy. Kinetic energy is perhaps the clearest:

1. a rocket takes off, so that -according to the launch pad frame calculations- its length contracts and the rocket's kinetic energy increases.
2. you choose another reference frame, and in the new reference frame the rocket's length "is" contracted and the rocket "has" more kinetic energy.

Hopefully it is clear that 1. and 2. are physically completely different cases, and also that these effects are "relative" to the frame of observation; "is" and "has" are not absolutes.
Else you would have that the energy increases AND decreases, which is a contradiction.

It is similarly wrong to say that ship B is both contracting and expanding; that's an error due to flip-flopping reference systems (a major cause of errors, like mixing dollars and euros!). We should say that ship B is contracting and gaining energy according to system A, and expanding and loosing energy according to system B.

[..] EM expansion works fine in C if we assume C is at rest [..]

The laws of physics are defined relative to a reference system that is presumably "in rest".

But both depictions of the physics occurring in the ship during acceleration cannot be accurate.
Make sense??

They cannot both be "absolutely true". That makes perfect sense, and it's the starting point of SR and already of classical relativity (such as in Newton's mechanics) that we cannot determine "who is right". See the introduction here:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[Dale]

To all who set me straight: Mea culpa!

I am now convinced that I erred in thinking that SR was not adequate to show that the string in Bell's Paradox will break. The clincher was the Born rigidity solution that yuiop had suggest that I review. My new understanding also removed my concern about inertial frames wherein the ships approach each other.

Thanks again,
Eli

Awesome! That is the core educational purpose of PF at work then!

 

[Doc Al]

Yay!

 

[Austin0]

I myself find the, physical contraction as a consequence of EM and atomic light speed interactions hypothesis very convincing. But as you have shown here it is somewhat problematic in application to specific scenarios. [..]

I did not see yuiop show such a thing... I see no such problem.

You did not comment on the previous sentences.

If we assume the EM contraction of ship B it is clear that we cannot apply that explanation to the contraction of ship A as measured in B. We must assume a purely kinematic source in this case.

So would you say the relative contraction of ship A was the result of physical EM forces?
Or would you agree that purely kinematic changes due to the increasing relative velocity effected an equivalent contraction indistinguishable from the contraction of B as observed in A?

If we consider a third ship C with an inertial velocity equal to the final velocity of [ship] B then we see B expanding relative to that frame as it accelerates. If we assume the EM interpretation in frame C then the expansion is a result of decreasing contraction from the initial velocity as the velocity decreases with deceleration.
This is not a problem with the kinematic interpretation but is an obvious contradiction if we assume actual physical contraction. If the contraction is the result of actual tensile forces due to light speed interactions within the structure, then it logically is directly dependent on the velocity, relative not to any frame, but to the absolutely invariant speed of light.

Perhaps you forgot that the speed of light relative to an atom depends on the chosen frame? It's a direct result of our definition of simultaneity. The speed of light relative to an object (also called "closing speed") is not invariant but frame dependent, just as the EM effects.

I would say that any quantitative evaluation of the speed of light relative to an atom depends on the chosen frame.
But I made no such evaluation . I simply talked about a change in velocity with no implication that it was even determinable whether it was an increase or decrease.
When I said invariant wrt light I was not not talking about the frame invariance of measured speed but the independent isotropic constancy that we assume. That all photons absent the influence of gravity are moving at the same "absolute speed"
In your opinion does this or does this not imply some indeterminate, but actual, change in velocity relative to light resulting from a change of velocity through acceleration
??

it then follows that if a system is changing velocity it must be either increasing or decreasing it's speed relative to light.

Obviously the change can be either way according to relative frames but do you think it could be both increasing and decreasing relative to light /
Do you think that the fact that we can not determine the reality means that there is no definite condition?

Well, obviously this is necessarily so (as described from any inertial frame). That follows from the second postulate - no need for such a complex consideration! As interpreted from frame C, the rockets fly along with light that is going in one direction and counter the light going in the other direction.

Contracting or expanding but not both.
So assuming that the EM contraction is correct there is still no way to determine how it would apply. It could only be a partial cause for the observed phenomena with the necessary assumption of purely kinematic effects also.
With no way to tell which is which. In the case here. Ship B is both contracting and expanding.

It applies just the same as most other things in physics, such as electric and magnetic fields as well as energy. Kinetic energy is perhaps the clearest:

1. a rocket takes off, so that -according to the launch pad frame calculations- its length contracts and the rocket's kinetic energy increases.
2. you choose another reference frame, and in the new reference frame the rocket's length "is" contracted and the rocket "has" more kinetic energy.

Hopefully it is clear that 1. and 2. are physically completely different cases, and also that these effects are "relative" to the frame of observation; "is" and "has" are not absolutes.
Else you would have that the energy increases AND decreases, which is a contradiction.

I think there may be a bit of a typo in case 2. I assume you meant to write "is" expanded and "has" less kinetic energy.

In any case i don't think this is quite analogous. Momentum and KE are both inherently kinematic evaluations. Applying to interactions with external entities. Completely relative values that say nothing about the internal conditions of the particle in question.
As far as the contraction , as I stated previously; viewed kinematically there is no problem with the ship contracting relative to one frame and expanding relative to another.

It is similarly wrong to say that ship B is both contracting and expanding; that's an error due to flip-flopping reference systems (a major cause of errors, like mixing dollars and euros!). We should say that ship B is contracting and gaining energy according to system A, and expanding and loosing energy according to system C (I assume not B as you wrote).

Well I think if you look that is exactly what I did say (the bolded text without the reference to KE)

I was not flipping between reference frame but rather looking at the implications of the purely physical interpretation of contraction as applied to both frames at once.

Consider the fictitious paradox of contraction.
Length A is smaller than length B AND length B is smaller than length A
Obviously the correct application of the L transformation resolves this in a completely logically consistent way. But that resolution is a kinematic one. It includes the relativity of simultaneity.
Now we can say that some physical EM contraction is happening in addition to the kinematic factors and still be logically consistent.
But to propose that both A and B are physically contracted as a result of EM forces ,to me at least, brings it right back to a logical contradiction.

[..] EM expansion works fine in C if we assume C is at rest [..]

The laws of physics are defined relative to a reference system that is presumably "in rest".

Yes of course.But to my understanding the relevant physics in this case is the maths of the Lorentz transformation.This is a kinematic description that predicts the expected measurements of relative frames .
As I said I assume this to be a totally accurate description of reality. But the maths do not per se, directly describe or entail any physics interpretation. Does not make any statement regarding the physical cause of contraction or provide a definition to determine what is due to actual EM forces and what is a consequence of relative simultaneity or pure relative motion.
This is a matter of interpretation.

But both depictions of the physics occurring in the ship during acceleration cannot be accurate.
Make sense??

They cannot both be "absolutely true". That makes perfect sense, and it's the starting point of SR and already of classical relativity (such as in Newton's mechanics) that we cannot determine "who is right".

I never implied that it was a question of determining who was right.
What I was suggesting was that it was also impossible to determine what was due to actual EM forces and what was due to kinematic effects.

 

[Austin0]

Yes, if we move a rod by accelerating it at one point then we won't squeeze or pull on it but if we accelerate one end of rod separately from accelerating the other end of the rod, we can end up squeezing it or pulling it apart. Isn't that obvious?




If you accelerate the object at just one point (which means you apply a force at just one point), then you can use SR to determine how all the other points on the object accelerate so that the object maintains the same shape as it had before, as long as it is rigid. That's what we mean by rigid. If you separately accelerate the object at two different points (which means applying two forces at two different points), and that second point accelerates the object differently than what SR would have determined it to be if you had only applied one force, then the object is either rigid and will break, or it is not rigid and will be stretched or compressed.

this question is not related to the string scenario. i already stated long before that I assume the string will break.That was a case of two effectively independent systems
But you are apparently relating this to a single strong physical structure. SO I would like to clarify.
If we assume the two ships are connected with a massive cable or some structurally strong lattice and the ships have identical mechanical drives with equal thrust (remove the complication of equal proper acceleration), obviously there is going to be stress.
After initial application of thrust we can assume a stress gradient , compression at the rear transitioning to extension at the front. But after a stable equilibrium is achieved can you explain why there would be an overall net expansive force or why there would be an increasing expansive force over time?
Thanks

 

[yuiop]

Hi AustinO,

Have you tried analysing your question with Lorentz Ether Theory (LET)? LET and SR are mathematically identical and predict the same things but differ philosophically. In my Humble opinion, LET gives a more physical intuition of what is going on. Perhaps it might be worth starting a new thread as I think we are danger of going off topic here and I will contribute as and when I have time, although I am bit busy at the moment.

 

[ghwellsjr]

Yes, if we move a rod by accelerating it at one point then we won't squeeze or pull on it but if we accelerate one end of rod separately from accelerating the other end of the rod, we can end up squeezing it or pulling it apart. Isn't that obvious?


If you accelerate the object at just one point (which means you apply a force at just one point), then you can use SR to determine how all the other points on the object accelerate so that the object maintains the same shape as it had before, as long as it is rigid. That's what we mean by rigid. If you separately accelerate the object at two different points (which means applying two forces at two different points), and that second point accelerates the object differently than what SR would have determined it to be if you had only applied one force, then the object is either rigid and will break, or it is not rigid and will be stretched or compressed.

this question is not related to the string scenario. i already stated long before that I assume the string will break.That was a case of two effectively independent systems
But you are apparently relating this to a single strong physical structure. SO I would like to clarify.
If we assume the two ships are connected with a massive cable or some structurally strong lattice and the ships have identical mechanical drives with equal thrust (remove the complication of equal proper acceleration), obviously there is going to be stress.
After initial application of thrust we can assume a stress gradient , compression at the rear transitioning to extension at the front. But after a stable equilibrium is achieved can you explain why there would be an overall net expansive force or why there would be an increasing expansive force over time?
Thanks

First you say your question is not related to the string scenario and then you proceed to exactly describe the string scenario, except that it is replaced by a rigid rod, and then you agree that obviously there is going to be stress. So I'm not sure what you are looking for.

Nevertheless, even though this issue has been dealt with countless times in this and other threads, I will say succinctly that if you accelerate the two ships identically then they will maintain the same distance apart in their initial rest frame. But the connecting rod between them will be subject to length contraction in the same initial rest frame. Therefore, if it is rigid, it will break, if it is not rigid, it will stretch.