Comments about "Bell's spaceship paradox" in Wikipedia

This document contains comments about the document "Bell's spaceship paradox" in Wikipedia
In the last paragraph I explain my own opinion.

Contents


Introduction

The article starts with the following sentence.
Bell's spaceship paradox is a thought experiment in special relativity.
In order to understand physics the use of thought experiments can be very misleading. They should be avoided.
In this particular case the thought experiment can not be used to either validate or invalidate special relativity.
Both spaceships now start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times in S.
If that is the case the distance between them stays constant and clocks on board the space ships stay synchronised in S.
Therefore they are all subject to the same Lorentz contraction, so the entire assembly seems to be equally contracted in the S frame with respect to the length at the start.
You have to perform a real experiment to validate this claim.
You have to place synchronised clocks on the surface at equal distances on the surface above which these spaceships (airplanes) are supposed to fly.
It is also by using these clocks that the pilots are controlling the speed of their planes.
Therefore at first sight it might appear that the thread will not break during acceleration.
Again you have to perform a real experiment to validate this claim.
The real issue is that the thread will not bend either but always be straight.
The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start, because in S it is effectively defined to remain the same, due to the equal and simultaneous acceleration of both spaceships in S.
i.e. there is no length contraction in the rest frame S.
It also turns out that the rest length between the two has increased in the frames in which they are momentarily at rest (S'), because the accelerations of the spaceships are not simultaneous here due to relativity of simultaneity.
What is happening from the point of view of a moving frame S' is not important for the outcome of the experiment.
The thread, on the other hand, being a physical object held together by electrostatic forces, maintains the same rest length.
That means it's length in frame S does not change?
So the thread must break in both frames: In S' due to the non-simultaneous acceleration and the increasing distance between the spaceships, in S due to length contraction of the thread.
The outcome in both frame's must always be the same. The issue is what happens in frame S.
See also Reflection

1. Dewan and Beran

"According to the special theory the thread must contract with respect to S because it has a velocity with respect to S. However, since the rockets maintain a constant distance apart with respect to S, the thread (which we have assumed to be taut at the start) cannot contract: therefore a stress must form until for high enough velocities the thread finally reaches its elastic limit and breaks."
The outcome of the experiment in this case is described that the thread should break. The question is what caused this? Is this caused by length contraction or something else.
The most obvious physical effect will be that the thread will stretch and become longer not shorter.
The problem is that such an experiment can not be performed in reality so what do we learn.
Dewan and Beran also discussed the result from the viewpoint of inertial frames momentarily comoving with the first rocket, by applying a Lorentz transformation:
You can not uncover the outcome of an experiment by changing to a different frame.

2. Bell

In Bell's version of the thought experiment, three spaceships A, B and C are initially at rest in a common inertial reference frame, B and C being equidistant to A.
In the picture it seems more logical that spaceship A should be drawn vertically.
Then a signal is sent from A to reach B and C simultaneously, causing B and C starting to accelerate in the vertical direction (having been pre-programmed with identical acceleration profiles), while A stays at rest in its original reference frame.
All in this common frame.
According to Bell, this implies that B and C (as seen in A's rest frame) "will have at every moment the same velocity, and so remain displaced one from the other by a fixed distance."
Correct. The same reasoning as above,
Now, if a fragile thread is tied between B and C, it's not long enough anymore due to length contractions, thus it will break. He concluded that "the artificial prevention of the natural contraction imposes intolerable stress".
The issue is here that the thread will break because of physical limitations.
At the same time it seems more obvious that the thread will strech and become longer.
Bell reported that he encountered much skepticism from "a distinguished experimentalist" when he presented the paradox.
A paradox is something when there are two different descriptions of the outcome of the same thought experiment
According to Bell, there was "clear consensus" which asserted, incorrectly, that the string would not break.
The only way to solve this is by performing a real experiment.

3. Importance of length contraction

In general, it was concluded by Dewan & Beran and Bell, that relativistic stresses arise when all parts of an object are accelerated the same way with respect to an inertial frame, and that length contraction has real physical consequences
The problem is that you cannot accelerate all parts of a large object the same way. As a result stress can happen which can result that the object physical becomes shorter if pushed.
The reverse can happen that the object can become longer when pulled.
The real question is if the object physical has the same length when the speed is constant.
Length contraction i.e. the Lorenz transformations don't supply the answer. Only real experiments do.
However, Petkov (2009) and Franklin (2009) interpret this paradox differently. They agreed with the result that the string will break due to unequal accelerations in the rocket frames, which causes the rest length between them to increase (see the Minkowski diagram in the analysis section).
All things can happen when the speed of the rockets are not the same in the rocket frame. Their distance can decrease or increase. The last will cause the thread to break.
However, they denied the idea that those stresses are caused by length contraction in S. This is because, in their opinion, length contraction has no "physical reality", but is merely the result of a Lorentz transformation, i.e. a rotation in four-dimensional space which by itself can never cause any stress at all
My interpretation of this sentence is that length contraction is only a mathematical concept.
If that is true what is then its purpose? Laws are there to describe physical phenomena.

4 Discussions and publications

However, in most publications it is agreed that stresses arise in the string.
The problem is what is meant with stress. At "rest" an object is considered in equilibrium. When you pull an object stress will be induced which will try to bring the object back to its original length. The reverse will happen when you push an object.
Generally speaking IMO when one airplane pulls a thread the thread will become longer and not break.

5 Analysis

5.1 Rotating disc

Historically, already Albert Einstein recognized in the course of his development of general relativity, that the circumference of a rotating disc is measured to be larger in the corotating frame than the one measured in an inertial frame
The important question is: is the circumference of a rotating disc physical larger or shorter as when at rest, observed in the rest frame.
A slightly different question is: is there a difference in radius of a disc at rest or when rotating?
To verify this you have to place the rotating disc about a fixed frame or grid of reference rods centered about the center of rotation. By preference the rods should have the same length as the radius of the disc.
That being the case it becomes easy to observe if the radius of the disc decreases. The same with the circumference.
Therefore it's impossible to bring a disc from the state of rest into rotation in a Born rigid manner. Instead, stresses arise during the phase of accelerated rotation, until the disc enters the state of uniform rotation.
The same problem arises with length contraction in a moving rod.

5.2 Accelerating ships

At the end of this paragrph we read:
As explained above, the same (it breaks) is also obtained by only considering the start frame S using length contraction of the string (or the contraction of its moving molecular fields) while the distance between the ships stays the same due to equal acceleration.
You can not use mathematics only to explain something. Nor a thought experiment.

5.3 Born rigidity

The mathematical treatment of this paradox is similar to the treatment of Born rigid motion.
The most important is the physical treatment.

6. See also

Following is a list with "Comments in Wikipedia" about related subjects


Reflection

In order to demonstrate the Bell's Spaceship paradox you should think about two airplanes, which fly one in front and the other behind. They should carry a long stick between each other.
The following sketch shows what happen. The line marked with the letter C is the ground and the C's are synchronised clocks which take care that the airplanes always perform the same operations.
 
 <--->|                                 |<--->
      |                                 |
      |                                 |
      |                                 |
      |                                 |
       ---------------------------------
C----C----C----C----C----C----C----C----C----C
            Figure 1    v = 0
 
  <-->\                                 /<-->
       \                               /
        \                             /  
         \                           /   
          ---------------------------

C----C----C----C----C----C----C----C----C----C
            Figure 1B    v > 0
Figure 1A shows the situation where the two airplanes are flying at very low speed.
Figure 1B shows the situation where the two airplanes are flying at very high speed.
The difference between the two is that there is length contraction in both airplanes and in the stick. The result is that the stick hangs higher above the ground.
The distance is the same because the two airplanes perform all tasks simulataneous using the Clock's (Beacons) on the ground.

The question is if this description is correct. The only way to test that is by performing a real experiment
If the description is correct and a moving observer thinks different than his reasoning is wrong.


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Created: 6 Februari 2015

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