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Natural Science Forum / Physics / Research / March 2008Bell's Spaceship paradox
This famous paradox is about the distance between two identicaly accelerating rockets starting from rest from an inertial lab frame. It is described i.e in: http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html http://en.wikipedia.org/wiki/Bell's_spaceship_paradox It illustrates the problem of defining a "physical"distance (something we would call"proper distance") in non inertial frames due to the breakdown of simultaneity. There is not only one definition and they do not give always the same result:(which one is correct?). In the Wiki article, one tries to avoid the difficulty in considering that the two rockets will stop their engine after the same ellapsed proper time continuing flight in inertial frames. So one can perform easily the distance "d" between rocket 1 and 2 in lab frame and this distance "D" in rocket 1 frame using plain Lorentz transform group. The result is that (D = d* gamma) which looks fine, but the conclusion looks quite odd to me, as it is said that a string linking the 2 rockets should break according to this formula. I thought that, in SR, the Lorentz "contraction" between two inertial systems was not physical and would not involve the string to break. Can someone help me to understand whether and in case where I am wrong? Notice also that this solution does not describe the situation when the 2 rockets are accelerating, but the result of such situation when freezed.. > This famous paradox is about the distance between two identicaly > accelerating rockets starting from rest from an inertial lab frame. It > is described i.e in: > > http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.... http://en.wikipedia.org/wiki/Bell's_spaceship_paradox > It illustrates the problem of defining a "physical"distance (something > we would call"proper distance") in non inertial frames due to the [quoted text clipped - 19 lines] > the 2 rockets are accelerating, but the result of such situation when > freezed.. Forget my first post, Meanwhile, I found my error . "D" is the "proper" distance measured between rocket in their boosted rest frame (at the end of acceleration), and "d" was the "proper"distance between rockets measured before to start motion in the lab frame (rest frame at that time). So the relation is between two measures in their respective rest frames. The string would be stretched (according to SR length measurement using simultaneity SR rules). So the conclusion of Wiki looks correct.. But I guess that if from this status, the two rocket now deccelerate at the same rate for the same time this stretch would be cancelled which shows some antisymetry in the process depending on relative directions of motion and acceleration, which looks quite odd in SR (motion is usually considered as not absolute). If we close the loop ( by proceeding the reverse operation) to get back in the lab frame at rest, the distance between rockets would be the original distance but obviously the proper time of the rocket observers would be different from the proper time of static observers remained in the lab frame (twin paradox). I find this disymmetry between distance and time intriguing and I wonder how physical is a distance (therefore this stretch) in Relativity. The only physical thing in relativity including SR looks to be "length" of worldline of observers, the "s^2" as measured by clocks carried by the observers. So I still wonder how physical is this stretch ? >> This famous paradox is about the distance between two identicaly >> accelerating rockets starting from rest from an inertial lab frame. It [quoted text clipped - 33 lines] > frame at that time). So the relation is between two measures in their > respective rest frames. Sure. It is customary in SR to do so. > The string would be stretched (according to SR > length measurement using simultaneity SR rules). > So the conclusion of Wiki looks correct.. Wikipedia is meant to only describe the opinions of the literature. The literature's conclusion looks correct indeed. > But I guess that if from this status, the two rocket now deccelerate > at the same rate for the same time this stretch would be cancelled > which shows some antisymetry in the process depending on relative > directions of motion and acceleration, which looks quite odd in SR > (motion is usually considered as not absolute). Interesting variant! And some adherents to SR considered motion as absolute. However, me thinks you overlooked the simultaneity issue. Which reference system do you use? That changes everything! There is no asymmetry. > If we close the loop ( by proceeding the reverse operation) to get > back in the lab frame at rest, the distance between rockets would be > the original distance OK, always simultaneous in the earth frame. > but obviously the proper time of the rocket > observers would be different from the proper time of static observers > remained in the lab frame (twin paradox). Yes, that's SRT. > I find this disymmetry > between distance and time intriguing and I wonder how physical is a > distance (therefore this stretch) in Relativity. The only physical > thing in relativity including SR looks to be "length" of worldline of > observers, the "s^2" as measured by clocks carried by the observers. > So I still wonder how physical is this stretch ? In SR, the Lorentz contraction is supposed to be physical, all originators of SR agreed on that. And the very purpose of Bell's Spaceship paradox was to illustrate the "physical reality" of length contraction. Of course, different people mean different things with "physical reality". :-) Cheers, Harald > "jacques" <jacques.f...@neuf.fr> wrote in message > > > Interesting variant! And some adherents to SR considered motion as absolute. > However, me thinks you overlooked the simultaneity issue. Which reference > system do you use? That changes everything! There is no asymmetry. Roughly speaking, I use the method used in the FAQ originated by M. Weiss : In Minkowski diagram in the lab frame (t, x), I draw the 2 rockets worldlines and local comoving frames at some point P (t', x'), according to the rules of SR. In the FAQ, only accelerating world lines are anlyzed. This exhibits a "strech" as measured on the x' axis (line of simultaneity at P) of the co moving frame at P. In the Wiki variant the rockets stopped their engine and inertial flight is (correctly) analyzed exhibiting some "strech" in the "rest frame" of rockets, so we would tempted to say that this strech is physical. I added a deceleration step. After accelerating during the same proper time, the rocket reverse the thrust. I draw the corresponding world lines and analyze according to the same method. You find opposite conclusions, the distance between the 2 rockets decrease in the same ratio than the increase during acceleration. So if deceleration time is equal to acceleration time, the 2 rockets are at rest (but at some distance from the taking off point) in the lab frame (no contraction). Obviously you may continue to play, in the same way, in order to close the loop (round trip) by symmetrical operation. You will enjoy the twin paradox (about proper time) , but the distance would be equal to the initial distance....So the conclusion is interesting. In such round trip, distance may encounter temporary streching and contracting but as it is antisymmetrical regarding motion/acceleration sign, at the end, all of this cancels. But for proper time, the story is different, whether you compute it, which easy on such world line, you see that you have to add all the legs of the world line and no cancelation occurs. Ellapsed proper time of travelers are shorter than these of guys remained at rest in the lab frame. So the conclusion of the conclusion would be: Is the distance a physical entity or just something conventional measured by exchange of light signals, using simultaneity rules of SR? Isn't the proper time which is the world line affine parameter by correct parametisation the only physical entity in Relativity as only worldline have a physical content in Relativity.? Jacques > > Cheers, > Harald- Masquer le texte des messages précédents - > > - Afficher le texte des messages précédents - > > "jacques" <jacques.f...@neuf.fr> wrote in message > > [quoted text clipped - 13 lines] > frame" of rockets, so we would tempted to say that this strech is > physical. The stretch is physical for sure, and depending on one's perspective it's the result of a certain combination of length contraction and lack of simultaneity of time of departure. > I added a deceleration step. I meant: in what frame do the rockets decelerate simultaneously? Apparently, you chose the lab frame S. > After accelerating during the same proper time, The use of the word "proper" gives the impression that instead you chose to start simultaneously in S'. > the rocket reverse the thrust. I draw the corresponding world > lines and analyze according to the same method. You find opposite > conclusions, the distance between the 2 rockets decrease in the same > ratio than the increase during acceleration. I tend to analyse things from inertial frames only - no need to complicate matters. :-) - The distance between the rockets remains constant as measured in S if their deceleration was synchronous in S. The end result will be a return to the original state. - If they decelerated synchronously in the frame S', then their distance will remain constant in S'. That moving (stretched) distance in S' will then look (doubly) stretched at rest in S. > So if deceleration time > is equal to acceleration time, the 2 rockets are at rest (but at some [quoted text clipped - 11 lines] > travelers are shorter than these of guys remained at rest in the lab > frame. Yes, an interesting feature of elapsed time is that it accumulates history. What is similar to length is clock frequency. > So the conclusion of the conclusion would be: Is the distance a > physical entity or just something conventional measured by exchange > of light signals, using simultaneity rules of SR? That's a question and not a conclusion. IMHO, it's a mixture of physical entity and convention. > Isn't the proper time which is the world line affine parameter by > correct parametisation the only physical entity in Relativity as only > worldline have a physical content in Relativity? Both Lorentz and Einstein were of the opinion that clock time and rod length had "physical meaning" (see for example Einstein's 1905 article, paragraph 3) http://www.fourmilab.ch/etexts/einstein/specrel/www/ And of course, the purpose of Bell's spaceship "paradox" was to argue that something physical happens when changing speed (it may look paradoxical for those who disagree). Regards, Harald > I find this disymmetry > between distance and time intriguing and I wonder how physical is a > distance (therefore this stretch) in Relativity. The only physical > thing in relativity including SR looks to be "length" of worldline of > observers, the "s^2" as measured by clocks carried by the observers. > So I still wonder how physical is this stretch ? Distances in SR (and hence stretching) are very physical. Give me two points in space-time and I'll give you the distance between them. Notice the careful wording. In SR, distances (or rather space-time intervals) are defined between events (space-time points) rather than "objects" (which occupy world-lines). Most of the confusion in puzzles like Bell's spaceship paradox comes from forgetting this fact. When two observers are stationary with respect to each other, it is easy to decide how to define the distance between their world lines: just take any space-like segment orthogonal to their world-lines (that is, make use of their common notion of simultaneity). However, to actually measure this distance, one has to come up with a local experiment for either or both observers. One common trick is to stretch a string between them. Assuming linear elasticity, the tension in the string will be proportional to this distance. One must still not forget that tension is a local quantity, which can potentially vary with time and along the length of the string. Since the situation is completely static, the tension is uniform both in time and along its length (over the entire space-time world-sheet swept out by the string). So, a measurement of the tension by either observer, that is at either end of the string, is enough to determine the distance between them. Unfortunately, once the observers are no longer mutually stationary, the situation is no longer static. The string's tension becomes non- uniform in time and along its length. So, the local tension measured by either observer is no longer a reliable measure of any sort of "instantaneous" distance between the observers. The moral of the story is that, to correctly predict the outcome of a scenario such as Bell's spaceship paradox, one must worry more about the tension suffered by different parts of the string at various times rather than than what the correct notion of simultaneity or distance should be (precisely because these cannot be defined in a unique way between the two observers). So, here's an analysis of Bell's paradox from the above perspective. The first step is to pick a model for the string, otherwise we cannot say anything about its tension. The model does not need to be terribly precise. For example, we can assume that the tension of a string stretched between two mutually stationary observers is uniquely and monotonically determined by the simultaneous distance between them (which is perfectly well defined). Further, assume that there is a critical tension, such that the string breaks as soon as this critical tension is reached anywhere along its length (the breaking point will be some space-time event). Finally, assume that disturbances propagate along the string at some speed of sound, smaller than that of light, and that their propagation is damped (so that we can recover the previously described stationary behavior in case the ships eventually become mutually stationary). Consider the spaceship world-lines as shown in the Analysis section of the Wikipedia article [1]. They consist of three segments: (a) stationary in the lab frame, (b) accelerating, and (c) mutually stationary but moving wrt the lab frame. In part (a), the string tension is uniform and smaller than critical. In part (c), long after the ships have stopped accelerating and the string vibrations have died down, the string tension should again be uniquely determined by the simultaneous distance between the ships' world-lines. So, if the tension necessarily suffered by the string after a long enough time exceeds critical, then the string must have broken at some prior point, which would have been somewhere in the (b) or early stages of the (c) sections. To be any more precise about when and where the string breaks, one would have to assume a specific dynamical model for the string and solve its equations of motion with boundary conditions given by the motion of the spaceships. For an extensive discussion of how this is done (albeit in a more complicated context), see Greg Egan's thread "Why is this model of relativistic elasticity flawed?" in the group archives from last summer. [1] http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox Hope this helps. Igor > > Notice the careful wording. In SR, distances (or rather space-time > intervals) are defined between events (space-time points) rather than > "objects" (which occupy world-lines). That's the point, SR (GR) defines spacetime interval between two events. In my opinion only spacetime intervals have a physical meaning. They can be timelike intervals where spatial contribution to the ds^2 is zero (affine parameter of the world line of an observer measured by a co-moving clock). They can be spacelike intervals (what we try to define in the Bell paradox) where time contribution to the ds^2 is zero. > When two observers are stationary with respect to each other, it is > easy to decide how to define the distance between their world lines: I agree. In the same inertial frame, simultaneity is perfectly defined in an unique way. > Unfortunately, once the observers are no longer mutually stationary, > the situation is no longer static. That's the point. For simultaneity in SR, the basic concept is to use light signals (3 events in space time + middle of worldline ) usually called radar method. When nothing is static we may wonder about the physical meaning of this result. > [1]http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox > > Hope this helps. Yes, this helps, this discussion has been very fruitful, thanks a lot for your explainations and those of the other contributors. In fact I think I understand better the trick. In "pure" SR, we suppose the existence of an infinity of inertial frames having relative velocity. These frames are supposed to have always existed and will exist forever (static). We do not know how this is physically possible but we assume it. So for observers having these inertial frames for worldlines there are all equivalent for the laws of physic. Their "Lorentz"contraction,(this contraction is the result of a measurement method ) is symmetrical as in such SR there is no prefered frame but we do not know whether it is physical or not (this question may have no sense as we should have to break it for knowing it). But in a problem such as the Bell paradox the symmetry is broken, the spaceships observer worldlines originate from the lab frame and are not (fully) inertial so a physical contraction is not in contradiction with SR and the lab frame may play the role of an"absolute space time frame" for the problem. In an ultimate version whether we assume an infinity of spaceships in inertial flight but all originated from the same lab frame (by different acceleration steps), would'nt this be equivalent to define an absolute spacetime for all inertial frames?. Just a naive question: Would it be then sensible to use these Lorentz transforms on a non fully inertial worldline as these transforms have been demonstrated in assuming that these frames were (fully) inertial (especially homogeneity is supposed)? . Jacques > Igor > This famous paradox is about the distance between two identicaly > accelerating rockets starting from rest from an inertial lab frame. It [quoted text clipped - 7 lines] > There is not only one definition and they do not give always the same > result:(which one is correct?). Which is correct depends on what you mean by "correct". That is, what are you trying to do? Or more directly: what are you MEASURING? There is no "correct" in the abstract here, there is only a set of different possible measurements which obtain various different results for "proper distance" in non-inertial coordinates. Because, as you mentioned above, there is no definitive simultaneity in such coordinates. > I thought that, in SR, the Lorentz "contraction" between two inertial > systems was not physical and would not involve the string to break. Yes. Length contraction is purely observational, and the fact that some other observer moving past your rocket sees it as shorter than you do does not affect the rocket at all. Just like looking at a building from different points of view changes how you see it but does not affect the building itself. The difference between that and the Bell paradox is that in the latter a PHYSICAL SITUATION was constructed (well, imagined) that breaks the string. It is not some other observer measuring the string, it is two rockets PULLING on it. > Notice also that this solution does not describe the situation when > the 2 rockets are accelerating, but the result of such situation when > freezed.. One can imagine the two rockets stopping (briefly) in successive inertial frames. Thus one sees that the string breaks as they are accelerating, and there is no need to stop in any inertial frame for it to break. Tom Roberts Jacques, > > Notice also that this solution does not describe the situation when > > the 2 rockets are accelerating, but the result of such situation when > > freezed.. Freezing the situation is not really necessary. > But I guess that if from this status, the two rocket now deccelerate > at the same rate for the same time this stretch would be cancelled > which shows some antisymetry in the process depending on relative > directions of motion and acceleration, which looks quite odd in SR > (motion is usually considered as not absolute). Look at http://grenouille-bouillie.blogspot.com/2007/10/how-to-teach-special-relativity.html, maybe it will make the whole thing a little more intuitive. If you consider the diagram at the bottom of the blog post (direct link for the diagram: http://bp1.blogger.com/_UnfX9_V6JCc/RwxwoODXZJI/AAAAAAAAAAk/SkOsruyghSs/s1600-h/ Bells+Paradox.jpg), it shows the equivalent "paradox" in the Euclidean case. Hopefully that makes things clearer. In the diagram, the equivalent of deceleration is when the green and red curve both become horizontal again, meaning that the distance between them returns to its original value. > I find this disymmetry > between distance and time intriguing and I wonder how physical is a > distance (therefore this stretch) in Relativity. The stretch is as physical as the loss of tension you would get if you put a piece of string with one end on the red curve and one end on the green curve along one of the arrows in the Euclidean case. In reality, the string will follow a slightly more complicated curve than the straight lines I drew, since it will be perpendicular to the local time line at every point, but the difference in length is really minor and can be ignored in first approximation. > The result is that (D = d* gamma) which looks fine, but the conclusion > looks quite odd to me, as it is said that a string linking the 2 [quoted text clipped - 3 lines] > Can someone help me to understand whether and in case where I am > wrong? The position that Lorentz contraction is not physical is simply wrong. Time dilation is also physical, it gives a real time difference in the case of the twin "paradox". This is the point Bell has made in this thought experiment. > Notice also that this solution does not describe the situation when > the 2 rockets are accelerating, but the result of such situation when > freezed.. The same result holds for accelerating rockets as well. Ilja On 22 f=E9v, 02:09, Ilja Schmelzer <ilja.schmel...@googlemail.com> wrote: > On 15 Feb., 21:47, jacques <jacques.f...@neuf.fr> wrote: > The position that Lorentz contraction is not physical is simply wrong. > Time dilation is also physical, it gives a real time difference in the > case of the twin "paradox". > > > Ilja Length and time (x,t) or (x',t') are coordinates (respectively in lab frame, and rocket co-moving frame) in the paradox. They are measured according to the rules of SR, based on the SR simultaneity concept which is not absolute, making he measure quite "conventionnal". In the round trip, at the end of the loop, they recover their initial values. But elapsed proper time is not a coordinate it measures the "length" (affine parameter, with right parametrization) of the world line by a clock associated to this line. In my understanding, that makes a conceptual difference which may explain why at the end of the round trip the elapsed proper time of traveler and lab observer (integrated on their respective worldline) are different but not their size. Jacques > On 22 f=E9v, 02:09, Ilja Schmelzer <ilja.schmel...@googlemail.com> > wrote: > > > On 15 Feb., 21:47, jacques <jacques.f...@neuf.fr> wrote: > In my understanding, that makes a conceptual difference which may > explain why at the end of the round trip the elapsed proper time of > traveler and lab observer (integrated on their respective worldline) > are different but not their size. The analogon of the size is the current speed of the clock. It is unchanged as well. What would be an analogon of proper time? Imagine two Bell spaceships which, during their round-trip, measure the tension of the string and integrate. Then the result will differ from similar spaceships in rest. Even better. Assume that the spaceships are not exactly ideal Bell spaceships. They use the same power program, but the string influences in an infinitesimal way their acceleration, which, therefore, is no longer identical. The "size" of the spaceship pair decreases, and for greater string tension decreases faster. Then, after the round trip, the "size" will be smaller for the travelling device. Thus, no asymmetry between space and time. And, in above cases, length contraction and time dilation are physical effects. > > On 22 f=E9v, 02:09, Ilja Schmelzer <ilja.schmel...@googlemail.com> > > The analogon of the size is the current speed of the clock. It is > unchanged as well. I agree > What would be an analogon of proper time? Imagine two Bell spaceships > which, during their round-trip, measure the tension of the string and > integrate. Then the result will differ from similar spaceships in > rest. I agree. > Even better. Assume that the spaceships are not exactly ideal Bell > spaceships. They use the same power program, but the string influences > in an infinitesimal way their acceleration, which, therefore, is no > longer identical. The "size" of the spaceship pair decreases, and for > greater string tension decreases faster. Then, after the round trip, > the "size" will be smaller for the travelling device. Even better. Assume perfect elasticity of the string, and that after an accelerating step within the elasticity limit, we stop the engines. In inertial flight we experience no stress so the string would recover its original length. Repeat this as many time you like, you may reach any celerity within speed of light without breaking the string. > Thus, no asymmetry between space and time. For "instant" measurement, I agree at the end of the round trip, both clock and distance would recover the current value in the frame lab. My comment was about the integrated entity along the worldline taking into account the history of the trip. For time it's clear you have an integrated value of total elapsed proper time. A nice feature of the clocks is that they are integrators of time. According to your argument, I agree you may record the stress on the string on the world lines , but this looks to be of a smaller physical significance than the elapsed time. In addition we are used to deal with some stress (i.e Earth gravity) elasticity looks to neutralyze stress but nothing neutralyzes time. For the twin paradox, whether this acceleration is moderate (i.e 1g), the twin returns younger but not smaller or taller. And, in above cases, length > contraction and time dilation are physical effects. Thanks for your comments. Jacques On 25 fév, 20:03, Ilja Schmelzer <ilja.schmel...@googlemail.com> wrote: [...] > Even better. Assume perfect elasticity of the string, and that after an accelerating step within the elasticity limit, we stop the engines. In inertial flight we experience no stress so the string would recover its original length. Repeat this as many time you like, you may reach any celerity within speed of light without breaking the string. Not sure if I understood what you meant, but the essential point of the "Bell's spaceship paradox" is that after we stop the engines, the string is under tension: it does not have its equilibrium length anymore. This stretch can be brought to zero by shrinking the distance between the spaceships in order to obtain the shorter equilibrium distance that belongs with its new state. And just as with the twin paradox, all inertial observers agree on these points Harald The links cited don't describe a paradox at all, just an incompletely stated question. If the two ships accelerate but remain at rest in the same inertial frame, then the distance between them in that rest frame will not change. If the frame (= both ships) is accelerated, then the distance between them in the frame with respect to which they are being accelerated will decrease, by the Lorentz formula. In their rest frame, the distance will not change. The "paradox" is because Newtonian absolute space is being assumed without thought of its relativistic implications. There is no such thing as "distance", unless one stipulates an inertial frame in which it is to be measured. Adding energy to an object in an inertial frame causes that object to be contracted in space and time in that frame, making general relativity an implication (integration) of special relativity. > This famous paradox is about the distance between two identicaly > accelerating rockets starting from rest from an inertial lab frame. It [quoted text clipped - 26 lines] > the 2 rockets are accelerating, but the result of such situation when > freezed.. > This famous paradox is about the distance between two identicaly > accelerating rockets starting from rest from an inertial lab frame. It [quoted text clipped - 6 lines] > we would call"proper distance") in non inertial frames due to the > breakdown of simultaneity. There seems to me to be an error in the Wikipedia article. The two spaceships are described as having the same proper acceleration. Later on it twice claims that the distance between the ships remains constant (by definition), as measured in the launch frame. For this to be the case, the two ships would need to have constant coordinate acceleration in the launch frame. -- Martin Hogbin >> This famous paradox is about the distance between two identicaly >> accelerating rockets starting from rest from an inertial lab frame. It [quoted text clipped - 18 lines] > -- > Martin Hogbin I don't see the problem. Please explain with an equation example. There is of course no need for the coordinate accelerations to remain constant in order to remain equal to each other. Harald > There seems to me to be an error in the Wikipedia > article. The two spaceships are described as having [quoted text clipped - 4 lines] > need to have constant coordinate acceleration in the > launch frame. If the two spaceships have identical proper acceleration profiles as a function of their proper time, then they also have identical coordinate acceleration profiles relative to the launch frame as a function of time in the launch frame (they start simultaneously in the launch frame). For this case the spaceships remain a constant distance apart in the launch frame, when looked at simultaneously in the launch frame. All of this is independent of the actual acceleration profile used, and it need not be constant -- all that matters is that the same profile is used by both spaceships. Tom Roberts > > There seems to me to be an error in the Wikipedia > > article. The two spaceships are described as having [quoted text clipped - 14 lines] > constant -- all that matters is that the same profile is used by both > spaceships. Thanks Tom, that answers my question. Is there a simple way of showing that? It does not seem obvious to me. -- Martin Hogbin > > > There seems to me to be an error in the Wikipedia > > > article. The two spaceships are described as having [quoted text clipped - 17 lines] > Thanks Tom, that answers my question. Is there a simple way of > showing that? It does not seem obvious to me. On reflection, maybe it does. -- Martin Hogbin |
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