spaceship

What Is the Bell Spaceship Paradox, and How Is It Resolved?

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Bell describes two spaceships that start out at rest relative to each other, with an elastic string between them, one end attached to each ship, which is initially just long enough to span the distance between the ships without being stretched.

bell-spaceship-description

The ships’ clocks are synchronized in their common initial rest frame K. At some time t by their clocks, both of them turn on their rockets and start accelerating in the positive x direction, with exactly the same constant proper acceleration (meaning that the crew of each ship feels the same constant acceleration). What happens to the string?

The correct answer is that, as the ships accelerate, the string is stretched more and more until it breaks. The “paradox” arises from the following (erroneous) line of argument: since both ships start accelerating at the same time, and they both have exactly the same acceleration, the distance between them should remain constant, so how can the string be stretched and break?

Of course, in relativity, whenever you see an argument using the terms “time” and “distance,” you know you need to be careful, since time and distance are frame-dependent. To resolve the “paradox,” we look at the distance between the ships, as seen from the instantaneous rest frame K’ of either ship. If you do this at different points along either ship’s worldline, you will see that the distance gradually increases, e.g., L2>L1 for the two lengths shown in the figure, as measured on the respective graph-paper grids. These lengths are measured parallel to the spacelike axes of frames K and K’, so that they are distances between events that are simultaneous as measured in these frames. Since the string remains attached to both ships, increasing distance between the ships in either ship’s instantaneous rest frame means the string has to stretch (and eventually it will stretch enough to break).

bell-spaceship-spacetime-diagram-4.

Some treatments of the paradox attribute the breaking of the string to “length contraction,” and you may be wondering how that comes into play, since the discussion so far has not mentioned it. In the article by John Bell in which he originally introduced the paradox, he gave an alternate line of reasoning to get the correct answer. The ships and the string are originally set up at rest relative to one another, with the string unstressed. As the ships accelerate, the string moves faster and faster relative to frame K, so its “natural” (i.e., unstressed) length, as measured in K, gets shorter and shorter due to length contraction. But its actual length as measured in K does stay the same, so the string will be increasingly stretched until it breaks. (It may be helpful to consider the case where an additional, identical string is attached only to the front spaceship. As the ships accelerate, it retains its original length, as measured in its own rest frame, and is therefore too short to span the increased distance between the ships. The shorter line marked L/γ in the diagram below can be thought of as representing this second string.)

bell-spaceship-natural-length

Bell’s argument is valid. However, note that what is “contracted” is not the actual length of the string in the original rest frame, but its “unstressed” length in that frame. The actual length of the string in the original rest frame stays the same, so the use of the term “length contraction” in this connection is a little unusual: normally that term is used to refer to a length that is actually measured in some frame. That’s why, if you ask whether the string in the Bell spaceship paradox breaks due to “length contraction,” the answer is not a straightforward “yes.”

References

J.S. Bell, “How to teach special relativity,” in Speakable and unspeakable in quantum mechanics, 1988, Cambridge University Press

http://en.wikipedia.org/wiki/Bell_spaceship_paradox

http://www.lightandmatter.com/sr/ (original source for diagrams, somewhat modified for this FAQ)

The following forum members contributed to this FAQ:

PeterDonis
WannabeNewton
PAllen
ZapperZ
jtbell
bcrowell
tiny-tim

PhD in physics. I teach physics at Fullerton College, a community college in Southern California. I enjoy writing, playing viola, brewing beer, climbing and mountaineering.
8 replies
  1. JD96
    JD96 says:

    Great article and since I am trying to get my head around this seeming paradox I found this insight to be very useful to get an overview. Nonetheless I would like this opportunity to ask a few additional questions:1. Since forces are involved, why does the rest length of the string stay the same in the K'? 2. Replacing the second spaceship with one whose proper acceleration has a different value, namely one corresponding to a hyperbolic worldline (where the center of the hyperbola is located in the origin), will the distance between the spaceships in K' now be constant? 3. In a relativity book I am reading a connection is drawn to the equivalence principle by showing that in K' the proper time of the second spaceship is ahead of the first one's proper time by the amount expected using the weak field formula for gravitational time dilation (when considering the first order approximation of the ratio of each spaceships proper time). This makes me curious whether using the equivalence principle the increasing distance in K' would be attributed to tidal force? I would be grateful for any help regarding one of the questions.Also thanks to all participants for the well written article.

  2. Ken G
    Ken G says:

    What I think this "paradox" is best at bringing out is that there really does not exist a "correct" language to say "why" the string breaks.  The physics prediction is only that the string breaks– the why question simply does not have a unique answer.  One can say it is "because of length contraction", one can say it is "because of the equivalence principle", and one can even say it is "because the forward ship blasts off first," etc., it all depends on the reference frame chosen.  This is natural– we don't answer physics questions about "why" until after we choose a reference frame, indeed I would argue that the main reason we choose reference frames in the first place is to be able to create a language capable of answering "why" questions at a more experiential level.  Since relativity is the one place where we advance a certain agnosticism about reference frames, this gives "why" questions an especially interesting status there.

  3. Semyuel
    Semyuel says:

    Who called this phenomena "paradox"?? Why we make things even more complex than they are? This topic is a great example of pseudoscientific confusion, means opposite to the science. Let me sort it out. If two ships start moving at exactly same point in time, with exactly same acceleration, than whats the difference if it were a single unit? The conditions are the same if it were a single ship. According to the predictions and experiments which were done since the middle of the past century, length of the body, which is moving at high speed (near the speed of light) is NOT CONSTANT. This is well known phenomena, called "contraction of length". From the ships point of view or from the string frame of reference – they are standing still, it's the entire universe who is accelerating and shrinking, causing elastic string to break, This is the point ))))) so funny, nothing mysterious or paradoxical.

  4. Semyuel
    Semyuel says:

    Hi @JD96,1) there is no issue with the length of the string. The entire universe is accelerating and shrinking, breaking the string.The paradox description causes some confusions in understanding. Since two spaceships are starting at exactly the same point of time and always are moving with the same acceleration, they can be considered as a single unit. BUT, there is a space between them (this is the condition of string breaking). Now, from the string frame of reference, two ships and string are standing still, but it's the entire universe who is accelerating and shrinking. So, from the elastic string point of view, the space between ships is shrinking, causing it to break. Take it easy, don't make things more difficult than they are. Read about "length contraction" http://newt.phys.unsw.edu.au/einsteinlight/jw/module4_time_dilation.htm

  5. Gary Feierbach
    Gary Feierbach says:

    There is not much of a paradox here. As the pair of ships approach the speed of light the string won’t break unless you mechanically try to make it the same length as it would have been at rest. Everything contracts uniformly including the string and the space between the ships and the ruler you measure everything with. If there really was a problem with the string breaking then spaceships themselves would have to be infinitely short or they would break apart at near light speed.

    Imagine a ship with a nuclear thruster that would have to be far from the human pilots. It might be very long with a giant nuclear engine in the rear connected by a tubular frame to the human occupied module in front. The entire assembly including the connecting tube would be subject to the Lorentz contraction as it approached the speed of light. Nothing unusual would happen from the perspective of the crew. The ship and everything in it would look perfectly normal to the crew and the ship would not be subject to any unusual forces other than the engine thrust. Of course running into micro meteorites or something larger would not be fun but that has little to do with the so called paradox.

  6. Vergil_gerbil
    Vergil_gerbil says:

    Excuse the noob observation here – but isn’t this to do with the problem of observation and measuring everything in terms of ‘light’ and observed effects?

    What happens in actuality to the string? Surely it stays at the same length in reality. If the ships accelerate to light spped, then decelerate to rest, would the string physically be broken or would it still be intact?

  7. Olegas
    Olegas says:

    Just for fun: you can compare how the Bell spaceship paradox is resolved in the both SR and Lorentz theory. Just consider that after initial acceleration the spaceships turn their engines a 180 degrees and start their engines running again. Will that bring them back into the original position? http://Www.theoryrelativity.com

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