Bell Paradox Questions - Diagram help!

  • #1
Consider the following thought experiment. Two spaceships are initially floating in
a region of space far removed from other matter. They are at rest with respect to each
other, and with respect to some inertial reference frame F. There is a distance L between
them. At some time, t=0, as measured by this reference frame F, they both turn on their
engines and start accelerating very gently in the same direction (see the figure below).
The two spaceships have identical engines and are both programmed by identical
software to maintain this gentle constant acceleration for a long time T, until they
reach half the velocity of light, after which the engines are turned off simultaneously.
Furthermore, the rockets are initially connected by a fragile thread, just long enough
to cover the distance L between the two ships. (The material of which this thread is
made is heat resistant and will not suffer from the exhaust of the engines.) The
question is what will happen to this thread due to the relativistic length contraction.

Sketch a Minkowski diagram, clearly and completely labeled, of the world-lines
of both spaceships. (You are allowed to suppose their size is negligible.)



Here is what happens to the thread from the point of view of reference frame F: the
ships started accelerating from rest with the same acceleration at time t =0, and thus, at
all later times, gained the same speed. Hence the distance between them has not
changed: at time T, they are still a distance L apart. However, since they, and the
thread that connects them, are now moving at a very large velocity, relativity predicts
that thread will suffer length contraction. It will become shorter than the distance L it
needs to span, and build up tension and eventually break.

Sketch how the situation would look like from the perspective of a pilot in one of
the spaceships, say from the inertial frame in which he/she is at rest after time T. In this
frame, the thread is at rest too and there is no length contraction. Consider in particular
the questions whether in this frame of reference the moments their engines are turned
of is simultaneous, and whether the distance between the ships remain constant.
 
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Answers and Replies

  • #2
haruspex
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Here is what happens to the thread from the point of view of reference frame F: the
ships started accelerating from rest with the same acceleration at time t =0, and thus, at
all later times, gained the same speed. Hence the distance between them has not
changed: at time T, they are still a distance L apart.
I'm no relativity expert, but on the limited understanding I have the two spaceships will, in frame F, appear closer than L. I don't believe the string will snap.
Consider a flat plate moving at relativistic speeds. It appears shorter. Now the same plate with a long hole cut out that nearly reaches the ends. Will the hole punch through the ends when the plate goes fast enough? That seems to me to be the same situation as for the gap between the spacecraft.
 
  • #3
ghwellsjr
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Consider the following thought experiment. Two spaceships are initially floating in
a region of space far removed from other matter. They are at rest with respect to each
other, and with respect to some inertial reference frame F. There is a distance L between
them. At some time, t=0, as measured by this reference frame F, they both turn on their
engines and start accelerating very gently in the same direction (see the figure below).
The two spaceships have identical engines and are both programmed by identical
software to maintain this gentle constant acceleration for a long time T, until they
reach half the velocity of light, after which the engines are turned off simultaneously.
Furthermore, the rockets are initially connected by a fragile thread, just long enough
to cover the distance L between the two ships. (The material of which this thread is
made is heat resistant and will not suffer from the exhaust of the engines.) The
question is what will happen to this thread due to the relativistic length contraction.

Sketch a Minkowski diagram, clearly and completely labeled, of the world-lines
of both spaceships. (You are allowed to suppose their size is negligible.)



Here is what happens to the thread from the point of view of reference frame F: the
ships started accelerating from rest with the same acceleration at time t =0, and thus, at
all later times, gained the same speed. Hence the distance between them has not
changed: at time T, they are still a distance L apart. However, since they, and the
thread that connects them, are now moving at a very large velocity, relativity predicts
that thread will suffer length contraction. It will become shorter than the distance L it
needs to span, and build up tension and eventually break.

Sketch how the situation would look like from the perspective of a pilot in one of
the spaceships, say from the inertial frame in which he/she is at rest after time T. In this
frame, the thread is at rest too and there is no length contraction. Consider in particular
the questions whether in this frame of reference the moments their engines are turned
of is simultaneous, and whether the distance between the ships remain constant.
This post is plagiarized from this website from a year ago:

http://forums.xkcd.com/viewtopic.php?f=18&t=75960&view=next

Jamie also reposted this question with a slight modification here:

https://www.physicsforums.com/showthread.php?t=649973

However, she has since been trying to erase the evidence of her activity.
 
  • #4
vela
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I'm no relativity expert, but on the limited understanding I have the two spaceships will, in frame F, appear closer than L. I don't believe the string will snap.
Consider a flat plate moving at relativistic speeds. It appears shorter. Now the same plate with a long hole cut out that nearly reaches the ends. Will the hole punch through the ends when the plate goes fast enough? That seems to me to be the same situation as for the gap between the spacecraft.
That's a different situation. The edges of the plate won't accelerate at the same rate as seen by an observer at rest in F whereas the ships do have the same acceleration.
 
  • #5
haruspex
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That's a different situation. The edges of the plate won't accelerate at the same rate as seen by an observer at rest in F whereas the ships do have the same acceleration.
Aha - so the answer to the paradox is that in frame F the spaceships will appear to become further apart. Their fronts stay a constant distance apart, as do their rears, but since each shrinks lengthwise the gap between them appears to grow. So the thread snaps explicably for everyone.
 
  • #6
vela
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It doesn't really have anything to do with the finite size of the ships. Even with pointlike ships, the thread would snap. Because they remain the same distance apart in F, in their own frames, they must be moving apart.
 

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