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Bell's Spaceship Paradox and Length Contraction

  1. Jun 9, 2013 #1
    Can someone please clarify for me whether length contraction in special relativity is considered a physical effect (a contraction of a cohesive material) or a kinematic effect (applied to the space the material occupies)? I've been thinking about Bell's Spaceship Paradox this week and realised that it stems from a discrepancy between these two different viewpoints.

    The spaceships are identically accelerated from rest to some speed. Therefore they will keep their separation, L, before and after acceleration (as observed in their original rest frame); although, each spaceship will be length contracted due to its speed relative to the rest frame.

    The paradox arises from the following. If the experiment is repeated with an inelastic string attached to the same point on each spaceship (say the back, near the rocket), then the entire connected setup can be considered as one large spaceship and so should under go length contraction as a whole, causing the string and hence the distance between the string attachment points to decrease. However, Bell poses the paradox in such a way that the string is too weak to draw the spaceships closer, and hence breaks.

    If length contraction is purely kinematic, then the string should feel no stress as the entire setup contracts; but then why are the spaceships not drawn closer when accelerated without a string present? A notion that resolves the issue is that the interatomic forces of the contracting string draw the spaceships closer as the string contracts, but I think this is at odds with standard interpretations of what length contraction means in special relativity (or is it?).

    I've seen some proposed solutions to this which move from the rest frame to the frame of the spaceships, but this does not seem necessary, as the paradox occurs in the original rest frame, so it should be possible to resolve it without changing frames.

    Any ideas? Thanks.
  2. jcsd
  3. Jun 9, 2013 #2


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    "Identically accelerated" means that both of them change their speed by the same amount at the same time, does it not?

    And the words "at the same time" are a sure sign that we're talking in frame-dependent terms and haven't adequately considered relativity of simultaneity. Choose a frame in which either spaceship is momentarily at rest, and you'll see that the other spaceship is not at rest; it's moving away from the first.

    (Search this forum and you'll find some other threads, as well as references to some pretty decent papers on this problem)
  4. Jun 9, 2013 #3


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    It's so easy to resolve Bell's Spaceship Paradox by setting up the scenario in one frame and then transforming it to another frame, so that's what I'm going to do. First, I show the situation of two spaceships at rest with respect to each other and separated by 5000 feet. Then, at the Coordinate Time of 4 microseconds, they both instantly accelerate to a speed of 0.6c while maintaining their separation of 5000 feet in the original frame:


    Now we transform the coordinates of all the events in the original frame to the frame in which the spaceships are at rest after they accelerate:


    As you can see, the spaceships end up farther apart, separated by 6250 feet, so if there had been an inelastic string connecting them prior to acceleration, it would be broken after the acceleration. The Length Contraction factor at 0.6c is the inverse of gamma which is 0.8 and it applies by comparing the separation of the spaceships in the frame in which they are in motion to their separation in the frame in which they are at rest. If we multiply 6250 feet by 0.8 we get 5000 feet so the separation after they accelerate is Length Contracted according to theory. If we multiply 5000 feet by 0.8 we get 4000 feet so the separation before the accelerate is also Length Contracted according to theory.

    Does this all make sense to you?

    Attached Files:

  5. Jun 9, 2013 #4
    During the acceleration of a spaceship the nose has less kinetic energy than the rear.

    After the acceleration the nose and the rear have the same kinetic energy.

    (We are assuming a homogeneous spaceship)

    During the acceleration some energy was on its way from the rear to the nose.

    (we are assuming the rocket motors push the rear)

    If we assume the spaceship is as rigid as possible, it still takes time for energy to reach the nose. So it follows that the spaceship must contract.

    If that does not follow, then I'll add that the rigidity of the spaceship kind of decreases as the velocity increases, I mean it takes more time for a signal to travel from the rear to the nose.
    Last edited: Jun 9, 2013
  6. Jun 9, 2013 #5


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    Look at http://arxiv.org/pdf/0906.1919.pdf
    "Lorentz contraction, Bell’s spaceships, and rigid body motion in special relativity".
  7. Jun 9, 2013 #6
    Thanks, Nugatory and ghwellsjr. I agree with both of your interpretations, but it doesn't resolve the problem. Saying the ships stay the same distance and the string contracts in the orginal frame, is the same as saying the distances increase and the string stays the same length in the frame of either ship. That's not the issue.

    They question is not one of changing frames. The question is, why would a material's length contract when its speed is increase, but two ships with only space in between accelerating identically (in the frame of the stationary observer) would not see the space between the ships decrease? Changing frame does not help our understanding of the situation, it only changes our perception of it.

    jartsa: So you are saying it's related to the interatomic distances being compressed because of the acceleration?

    clem: The article you refer to also seem to use the transformed frame to prove the distance between the ships increases in that frame. That's a given. The question is as above: in the original frame why does the distance between the ships remain unchanged when unconnected, but contracted when the string is attached. Do the atomic forces holding the contracting string together pull the ships closer?
  8. Jun 9, 2013 #7


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    Length Contraction is a frame dependent effect. It is not directly observable by any observer in any scenario. They can make an assumption (like we do when we establish frames) which is that the speed of light is a constant for them, and then they can make some measurements (using radar methods), and then do some calculations and they will arrive at the same conclusion, that the distance between the two ships has increased (not decreased or stayed the same) after they accelerated. Maybe you should do this little exercise. Do you know how to do it?
  9. Jun 9, 2013 #8
    ghwellsjr, I'm not sure what you're getting at. In the single frame I'm choosing I watch the ships accelerate while maintaining a fixed separation. I also watch the string contract and break. In this frame and no other, why does the material of the string contract but the distance between the ships not? Do the laws of physics not apply in this frame?
  10. Jun 9, 2013 #9
    In the Bell set up, a system is accelerated. This means that forces must be applied to the system. If the string will be stressed, it will try to pull the spaceships together, and it will break. In this sense, there is a contraction of a cohesive material.

    If, in Relativity, we talk about what happens to genuine, actual, physical, metre rulers when they are accelerated, there will be no reason to think that they experience any kind of contraction without taking into account the forces that serve to restore the body's original dimensions after it has been accelerated. But the situation is quite complex, because there are many different ways to accelerate an extended body, applying forces at different parts of the body, and using the forces that maintain its cohesion to bring the body back to equilibrium. Normally, one ignores the details, and just assumes one's dealing with a `rigid' body: one that will go back to its original length after the forces have ceased to act on it.

    But to see that the forces must be taken into consideration, consider what happens when we accelerate a `system' consisting of two tennis balls that exert no force on each other. We could call the distance between the two balls the length of the system. Let, at the same moment (in my frame) the same impulse (in my frame) be given to both balls. The system now has velocity v (wrt to my frame.) But the resulting distance between the two tennis balls will not be different after the acceleration. (From another frame, it will simply look as though one ball was accelerated before the other). Only when restorative forces are involved can say that a rod will experience length contraction. For Bell-like examples, the dynamics has to be part of the story. However, I think one doesn't need to analyse the laws in detail -- as long as they obey the Lorentz transformations, rigid rods should contract.

    This is not to claim that the general concept of length contraction in general is always dynamical.

    I believe that there have been some -- Lorentz? Fitzgerald? Early Einstein?? (I'm thinking of his early scepticism to Minkowski's introduction of space-time, and his scepticism resurfaces in later work too) -- who argue that it is the laws of the theory -- kinematical and dynamical -- which really underpins the structure of Minkowski space-time.
  11. Jun 9, 2013 #10


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    What do you mean by "I watch"? Do you mean that you are an observer in the scenario? If so, where are you? Are you inertial? Or do you mean that you are not in the scenario but are watching the diagram and imagining that you are seeing the ships accelerate and the string break?
  12. Jun 9, 2013 #11
    Thanks, yossell. Your understanding seems to agree with mine. What's getting me confused is why the increase in speed would cause the atomic forces to contract the rod in your example? If it was your rod connecting the ships of my example together, they would be drawn closer. So some law of physics is being affected by the fact that the lot is moving, which causes a contraction.
  13. Jun 9, 2013 #12
    ghwellsjr, I would be imagining I'm the observer in the frame that the ships occupied before they accelerated. My position in this frame is not relevant. In fact, perhaps I need not observe all events, but just setup the experiment, close my eyes, collect the two halves of a broken string later and wonder why it contracted.
  14. Jun 9, 2013 #13
    I wouldn't say that the laws of physics are being affected. Rather, there will be laws that govern the cohesion of the body -- which laws (electromagnetic, quantum, Gravitational) depending on the way the body is constructed.

    What's causing the atomic forces to contract? Whatever it is about the rod that makes it rigid. Rods, unlike pairs of tennis balls, are designed to keep their original shape. Nothing is truly rigid in SR, a rod necessarily deforms while accelerated -- but to be a good measuring device, it should be constructed so that, provide it's not pressurised past breaking point, it tries to regain its original dimension. Whatever these forces are, these are the forces responsible for the contraction in these examples.

    So -- by construction of a rod, the dynamical laws responsible for its `rigid' properties, will act so that a system, if acted on by a force (which doesn't break or damage the rod) will, after the force has stopped acting, recover its original length - say L. This is what we want from a ruler, as opposed to a couple of tennis balls.

    Suppose such a rod, at rest in Frame F, has length L in frame F. Exert a force on the rod so that, after the restorative forces in the rod have acted, it is moving with velocity v. Call this new frame F'. In this new frame, the rod is at rest. Accordingly, by the construction of the rod, it is trying to recover its original length L -- but now, L is referred to the new frame F'. From the point of view of F, there is length contraction. From the point of view of F, the forces on the rod are acting to restore it to a length less than L (in frame F).

    If, in frame F, I try to the rod the same length apart -- L in my frame -- it will eventually snap. As Bell points out.

    There's no need for a detailed analysis of the exact laws here -- we just needed the familiar facts about (kinematical) length contraction, plus the fact that the dynamical laws that keep the rod rigid obey relativity.
  15. Jun 9, 2013 #14
    But that's what I'm asking, by what mechanism do dynamical laws obey relativity? That question strikes at heart of it. To simply trust that they obey relativity seems unsatisfactory (to me, I guess!).
  16. Jun 9, 2013 #15
    Oh -- there's no *mechanism* in relativity by which laws obey relativity. And I'm not sure what you mean when you say `to simply trust that they obey relativity seem unsatisfactory.' No more than trusting a law of any well confirmed theory. That just sounds like you don't trust the postulates of SR.

    At some level, the world obeys fundamental laws. Relativity posits that the fundamental laws have a certain symmetry -- they're Lorentz invariant. Other things are explained in terms of this, but relativity gives no deeper explanation. And, empirically, it has so far turned out that the laws are of this form.

    I don't see anything problematic or unsatisfactory about this though. You can keep asking `why' questions of this sort for any theory. That's not to say there can't be a deeper answer -- perhaps the symmetry emerges elsewhere. But the explaining theory itself will take other things as unexplained.

    Hmm -- perhaps your worry is why different theories should all have the same symmetry in their laws? That the laws governing light, electrons, etc. etc. are ALL Lorentz invariant. Is that your worry?
  17. Jun 9, 2013 #16
    Such postulates may well generate an excellent theory, but such a curious question of why matter contracts when moving is too tantalizing to submit to a postulate. I think we should keep asking why - we might learn something.

    Besides, it goes to the core of why the experiments like Michelson-Morely might fail, even if there was an absolute reference frame. So it is an important "why". [Count to 10 before replying. ;) ]
  18. Jun 9, 2013 #17
    Sure, I think it's always good to ask why. And to be open-minded about what may eventually get a deeper explanation.

    Also, I thought I had given an explanation, in terms of relativity, of `why matter contracts' -- at least in the case of Bell's spaceship. Your why question now seems to be of the form -- `why are these the laws and not those', or `why do the laws have *these* symmetries and not those.' It seems to me that such a question can be asked about any bedrock law of any theory. It no longer seems to have much to do with Bell's spaceship, or the details of relativity.

    *Full disclosure* I didn't count to 10 before replying. Sorry -- I didn't feel the need.
  19. Jun 9, 2013 #18
    From a relativity viewpoint your answer was fine. You're correct of course, if theories are consistent, then it's down to whatever sits right with you. I've always felt dismissing a medium which light (a wave) propagates in, to explain a phenomenon, when the medium can be retained and still explain the same and yet pose more tasty questions is the path I'd choose.

    .. The medium is a latter day Voldemort, whose name shall not be spoken, if you hadn't guessed! :)

    Thanks for your answers, yossell. Much appreciated.
  20. Jun 9, 2013 #19


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    The distance between the ships in this frame remains the same because you stipulated that when you constructed the scenario. More precisely, you stipulated that the two ships' rocket engines fire in such a way that the distance between them, in the frame you specified, stays the same. That is sufficient to completely determine the worldlines of both rockets, regardless of any other considerations. So asking why the distance between the ships doesn't change is getting things backwards: you don't ask why a scenario is the way you stipulated it to be.

    The real question, as others have indicated, is your first one: why does the string contract in this scenario? But the way you've asked the question invites confusion: what does "contract" mean, exactly? Does it mean the length of the string changes, in the same frame in which the distance between the ships is unchanged? If so, your question is simply based on a false premise: the string's length does *not* change in this frame! How can it? Its ends are attached to the ships, and the distance between the ships is unchanged, therefore the length of the string is unchanged as well.

    A better way to put the question is, why does the tension in the string increase as the ships continue to accelerate? *That* is the question that Nugatory and ghwellsjr were really answering, because that is the question that relativistic kinematics can answer: the tension in the string increases because, in the instantaneous rest frame of any small piece of the string, at any instant of time after the ships start accelerating, the distance between the ships is increasing. The fact that the distance between the ships in your chosen frame is unchanged is irrelevant, because that's not the appropriate frame for evaluating the forces to which the string is being subjected.

    More precisely, it's not the appropriate frame for evaluating the forces in the simple way you're trying to (by just looking at the rate of change of distance between the ships). You could, I suppose, write down the equations governing the forces involved in relativistically covariant form, and then transform them to your chosen frame and see what they look like; but they won't look as simple in that frame as you seem to be assuming.
  21. Jun 9, 2013 #20


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    Your position will make a difference as to what you are watching and what you actually see but, you're right, if you mean that you're going to use the radar method to construct a diagram of what happened (long after it happened).
    I ask you what you mean by, "I watch" and your answer is, "close my eyes"??? No wonder you wonder. Just kidding.

    But I'm sure you noticed that I didn't include the string in my diagrams and that was on purpose. That's because you didn't specify how the different parts of the string were going to accelerate. All you said is that it was inelastic. Presumably, the ships are also inelastic but I treated them as points so as not to have to deal with details that won't matter for explaining why the string must break and that's because in the final common rest frame of both ships they are farther apart than they were in their initial rest frame.
  22. Jun 9, 2013 #21
    PeterDonis: There is nothing wrong with assessing the situation from any frame. All are equally valid. I'm preferring the frame where I don't move with the ships. If I do as you say and transform the "equations governing the forces" to my chosen frame, they will no doubt look strange. My question is why? Relativity seems to say "just because, that's why". That's fine, but personally feels like throwing the towel in.

    ghwellsjr: You've returned to the frame of the ships again. It's not in dispute that the string breaks in both frames, but for apparently different reasons. In my preferred frame what seems to cause the string to contract and break? Was the whole point of relativity not to ensure the law of physics seem the same in each reference frame? So this strange contraction would now appear to defy our understanding of the laws governing how materials remain cohesive.
  23. Jun 9, 2013 #22


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    This is true, but it does not mean that all frames give an equally simple picture of the physics. Nor does it mean that the physics can only be described by using frames. See below.

    No; relativity says that looking at things from a specific frame is not the fundamental way to look at them. The fundamental way to look at them is the frame-independent formulation of the physics in terms of invariants. That frame-independent formulation has to be translated regardless of which frame you want to pick; it has to be translated to the instantaneous rest frame of a particular small piece of the string, just as much as it has to be translated to your chosen frame. It just so happens that the translation into the instantaneous rest frame of a piece of the string makes the physics look simpler to us, but that really tells us more about our cognitive abilities than about the physics.

    I think that's because you're not looking at the whole picture. You're focusing on the string, but you haven't even asked the question: what about the ships themselves? Do *their* lengths get shorter in your chosen frame? The answer is yes, at least in the simplest case where we assume that the rest length of each ship--its length at any given instant of ship time, as measured in the instantaneous rest frame of the ship at that instant--is constant. (Yes, I realize this sounds like a frame-dependent description when I just got finished saying that the fundamental way to look at it is a frame-independent formulation. But actually the description I just gave can be formulated in a frame-independent way, in terms of invariant properties of the congruence of worldlines that describe the ship. More on that in a moment.)

    So it's not as simple as just saying that the ship to ship distance remains constant yet the string is subjected to increasing tension. There's a corresponding puzzle about the ships: their lengths continually *decrease* in your chosen frame, yet the stresses experienced by each ship are *constant*. You would expect the ships to experience increasing *compression*, based on the "naive" view from your chosen frame, but they don't. Why not?

    The answer to both of these "puzzles" is to stop viewing things from any frame, and instead to look at the invariant properties of the congruences of worldlines that describe the ships and the string. The particular invariant property that is useful in this case is called the expansion scalar, which is described (along with the general concept of congruences of worldlines) here:

    https://en.wikipedia.org/wiki/Congruence_(general_relativity [Broken])

    The key point for this discussion is that the expansion scalar of the congruence describing each ship is zero; that's why the internal stresses of each ship are constant. But the expansion scalar of the congruence describing the string is positive; that's why the string is subjected to increasing tension.

    Now, it's true that in my previous post I said the instantaneous rest frame of a particular small piece of the string (or of one of the ships, for that matter) is the appropriate frame for analyzing the physics in this scenario. Why is that? Because that's the easiest frame in which to correlate the expansion scalar, an invariant, with quantities defined in that frame (in this case, the fact that the ships are moving apart with reference to any such frame). There's no corresponding easy way to correlate the expansion scalar invariant to quantities in your chosen rest frame.
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  24. Jun 9, 2013 #23


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    Your link doesn't quite work because the trailing parenthesis got left out of the link.

    Just fix it and don't respond and then I'll delete this post.
    Last edited by a moderator: May 6, 2017
  25. Jun 10, 2013 #24


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    It's also not in dispute that the string would contract in your preferred frame, assuming that what applies to the spaceships also applies to the string. Here's what you said in your OP:

    So why do you keep asking about the string? Don't you agree that if you had two rockets on each end of one spaceship and you accelerated both ends identically, it would not contract but would either stretch or break?

    It's not just that all the laws of physics are the same in each inertial reference frame, it's also that all the laws must remain the same when they are subjected to the Lorentz Transformation process, the same process that we derive the coordinates for one frame from another. This guarantees that the details of what happens to objects that are subjected to acceleration will be in accord with the conclusions of Special Relativity.
  26. Jun 10, 2013 #25
    I didn't know such a statement was a part of relativity's postulates - is that your opinion? By the way, if you want to use an invariant view (spacetime diagram, or whatever), consider a stronger string will contract as the rockets gather speed, so their world lines will be drawn closer as they gather speed; what would cause this attraction between world lines in this frame? It's the same question as what I'm asking, but is somewhat less accessible.

    I stated as much in my original post that the ships contract. This is not of concern that that the ships contract. If you hadn't realised, the two ships are there to demonstrate that it is the not the space that contracts but the material; hence, the string breaks. Consider ghwellsjr's idea of a spaceship with identical rockets at the front and back. When it accelerates it will physically contract. If you want to see why, cut the ship in the centre and watch as the two parts of the ship drift apart as it gathers speed. (This is just a reformulation of Bell's Spaceship Paradox.)

    The congruence stuff totally lost me. Can you explain in lay terms what it's trying to say?

    So my question again, when I obeserver an object (string, spaceship, etc) what is causing the atoms to appear to be pulled closer together, given that all I have to explain the phenomenon are my laws of physics (Maxwell's equations, gravity, quantum mechanics, etc)?
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