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.