Talk:Bell's spaceship paradox/Archive 2

New Version
I have now written an (almost completely new) version of Bell's spaceship paradox offering a simpler explanation of the mainstream viewpoint and a brief discussion of dissident viewpoints. Since the kinematic decomposition is a standard method in the differential geometry of curves in Lorentzian manifolds, the simple computation offered in the new version should clear up Rod's confusion, provided he is willing to learn to compute with frames. I plan to add some figures to the new version when I get a chance which should make the discussion more geometrically vivid. Time permitting, rewrite of Ehrenfest paradox and new Born coordinates articles will be forthcoming as well.---CH 02:40, 9 May 2006 (UTC)

Later today I plan to add the promised figures. I will also rewrite a bit to better bring out the reason why frame fields are the appropriate tool here: as discussed in Rindler coordinates, there are various distinct notions of distance which can be used by accelerating observers, but of course these all agree to first order, so it makes good sense to introduce a family of observers with similar motion. In addition, some analyses of spaceship and string paradoxes use Bell observers and some use Rindler observers; if the reader doesn't realize that these are very different congruences, serious misunderstanding is inevitable. I sense some confusion of this type in Rod's diagrams above. ---CH 17:12, 9 May 2006 (UTC)


 * If the space ship captains distrust both the theoreticans and the rope, they can program their autopilots to stop the drive after N seconds proper time. Then the distance measurement can be done in intertial frames. Pjacobi--17:18, 9 May 2006 (UTC)

Good idea, I'll use that. ---CH 17:28, 9 May 2006 (UTC)

I have just finished the promised new article on Born coordinates, which stands to Ehrenfest paradox and Rindler coordinates stands to this article. ---CH 01:32, 19 May 2006 (UTC)

User:Rod Ball's edits to new version
Rod, you removed the statement of the first version (with the Bell observers). Why? It is true that the entire subsequent discussion refers to the second version of the "paradox", but I see no reason to remove the brief mention of the first version.

It seems you also want to heavily modify the last section. Can you clearly explain here what is troubling you in that section? Maybe I will agree to modify it, although I'd prefer to do this myself because you seem to be ignoring the fact, already stressed by both Pjacobi and myself, that a simple computation shows your view about whether the string will break is not only not the mainstream view, it is manifestly incorrect since a short computation resolves the question unambiguously.

I have reverted to the previous version of the article until such time as you can explain/justify the changes you wish to make.---CH 16:36, 9 May 2006 (UTC)


 * You make the same error I've been complaining about since the very start. Your "Bell" observers are assumed to be at const.dist. w.r.t. launchsite. The counter argument (correct as regards SR) would require "Rindler" observers ! So my diagrams are not wrong but simply consistent with the widely held counter opinion of non-string breaking.


 * Why can't you acknowledge any but your own blinkered views. The non-breaking is not just "my" view but a widely held one by "relativity-wise" academics and others. "A short computation" certainly does not, repeat not, resolve the issue (in your favour). The computations you mean ALL assume the s'ship distance does not show Lorentz contraction for launch observer before even calculating anything. This simply assumes the conclusion since it trivially follows that the proper accelerations cannot then be the same ! [One could almost reverse the argument - since the proper accelerations are known to be equal, it follows that the s'ships must appear to get closer by LT from the launch observer pov.]


 * Why not address your criticism to the Hsu & Suzuki paper ? That should give you something to chew on. Rod Ball 08:37, 10 May 2006 (UTC)

Rod: I hope that that the latest version will convince you to change your mind. I have added new figures and I have discussed the same thought experiment as Matsuda and Kinoshita, as per User:Pjacobi's suggestion above. The new computations I added should be easily understood even by those unwilling to learn about frame fields. However, the frame field computation has additional value. ---CH 02:41, 11 May 2006 (UTC)
 * 1) "Your "Bell" observers are assumed to be at const.dist. w.r.t. launchsite."  I could not guess what you might mean by this.  (Spatial distance?  In some inertial frame?  If not, there are many distinct notions of spatial distance for accelerating observers.  Launch site?  Is this represented by a world line at rest in some inertial frame?) but this claim doesn't sound right.
 * 2) Hsu & Suzuki: why do you assume that I haven't examined the eprints you cited?  Although this particular paper is so poorly written that it is virtually incomprehensible.  (From their (a) it seems that they are discussing the thought experiment discussed by Matsuda and Kinoshita, which is illustrated in the first figure of the latest version, but from (b) it is not clear whether they are talking about A&prime;B&prime; or A&prime;B&Prime; in that figure.  It appears that the authors never clarify this ambiguity.)  The earlier paper by Matsuda and Kinoshita is also badly written but at least their position is clear (and do they come to the correct answer to the question of whether the string will break).


 * Really, your quibbles make me chuckle sometimes. You can't possibly have been totally flummoxed by "constant distance w.r.t. launchsite". "With respect to" is widely abbr. to w.r.t. and understood as "as measured from" in an SR context. Launchsite is obviously an inertial RF, it is the RF for which we are drawing Minkowski diagrams and from which the observations that we (mostly) agree about, are made. Distance is obviously spatial unless otherwise indicated. The meaning is perfectly clear - your "Bell observers" are on parallel hyperbolic trajectories such that their horizontal X1-X2 distance, at the same time t, is constant.


 * This you assume before writing anything down and are therefore on the wrong track from the very start. As I said, the s'ship trajectories should be "nested" hyperbolae, as you put it ( one author uses 'homothetic' but I'm not sure if this is correct ), so that the s'ship passengers should be your "Rindler observers". If you want to see why this is so you could do worse than examine my pair of diagrams immediately above "Guaranteed to fail"....


 * I agree the Matsuda & Kinoshita is badly written.


 * Lastly, I reject your unsupported claim that there are many variants of "Bell's problem". It did not exist before Dewan & Beran's 1959 note, and that version is exactly the same form the problem has taken ever since. Indeed, your introduction is entirely false in suggesting the existence of some primary "Ur-paradox" that considered impulsive motion and from which Dewan & Beran's, or Bell's, is a "variant". If you still imagine there are "impulsive" or other forms, let's have specific references, please. ( And bear in mind that I have to hand a large collection, gathered over the years, of literature on Ehrenfest's Paradox to check vague allusions in that direction.) Rod Ball 13:23, 11 May 2006 (UTC)

Rod, you wrote:
 * 1) "You can't possibly have been totally flummoxed": please try to bear in mind WP:AGF.
 * 2) "Quibbles": since I have stated that my view is that confusion in discussing this "paradox" arises in great part from confusing distinct notions which happen to agree in simpler situations, if you want to understand my point of view (so that you can argue in good faith), it seems to me that you cannot simply dismiss my attempts to write precisely as "quibbling". Please note that I have (clearly, yes?) been attempting to understand the nature of your own argument.
 * 3) In retrospect I should have specifically asked what you meant by "launchsite" (which inadvertently illustrates my point about the value of being precise). I had genuine trouble understanding what you wrote just above, but I think I understand now: by "distance wrt launchsite" you just meant differences in X coordinate values in my figures, or lengths of horizontal line segments in your figures on this talk page.
 * 4) It seems we agree about the geometric character of the Bell observers. I am not sure whether we agree about the geometric character of the Rindler observers, however. Did you see my comment up above on your first figure?  Do you understand my euclidean analogies?  Bell congruence is analogous to a congruence of semicircles of equal radius all orthogonal to some line, while Rindler congruence (presumably this is what you mean by "nested hyperbolae") is analogous to congruence of concentric circles.  Hyperbolae asymptotic to null lines or pseudocircles (constant Minkowski path curvature curves) are analogous to circles (constant euclidean path curvature curves).
 * 5) In constructing the Bell congruence, I didn't assume that neighboring curves in this congruence maintain constant X coordinate difference. This is a consequence of how they are defined: they all are comoving with $$\dot{X}_0 = 0$$ and they all have constant path curvature k (with the two inessential spatial dimensions suppressed); that is, constant acceleration in same direction and same magnitude).  This should be immediately clear from the euclidean analogy.
 * 6) You seem to feel that it is somehow misleading to discuss both Bell observers and Rindler observers. If so, can you explain why you feel that way?  If not, can you explain the nature of your objection to discussing Bell observers in the article?
 * 7) Unsupported claim that there are many variants of "Bell's problem": several versions of accelerated rod models are treated in Nikolić's paper, as User:Pjacobi already pointed out above, and while I haven't had a chance yet to rewrite Ehrenfest paradox, this has a larger literature, and papers on this topic often discuss accelerated-rods/spaceships-&-string in passing. At least one of these papers mentions the first version of the spaceship-&-string "paradox" and attributes it to Bell: see section 9.3 of  Olaf Wucknitz, Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times.
 * 8) Examine my pair of diagrams immediately above "Guaranteed to fail": I am willing to discuss those with you, but to move this discussion forward, I think I need to demand some (not unreasonable?) conditions. First, please make a good faith attempt to verify the computations in the current version of the article, to make sure you understand correctly what I wrote there.  Second, please archive (or let me archive) the very very long first two sections in this talk page.  Third, after having carefully studied the current version, please copy the figures to a new section at the bottom of this talk page and as carefully and precisely as possible explain what these figures are intended to show and how precisely they contradict or mitigate the arguments given in the article.  Fourth, it would be helpful if you clarified your math/physics educational background.

Speaking of which, here are my own qualifications: I am entirely self-taught in physics, but have extensively studied the literature related to classical gravitation. I earned a Ph.D. in Mathematics (1998 University of Washington); my diss concerns a generalization of a dynamical system constructed using Penrose tilings, a topic in symbolic dynamics. (I once pointed out to RP that these have a somewhat silly spacetime interpretation allied to Regge calculus, which made him laugh.)  My mathematical interests and knowledge are fairly broad--- see my user page. When I was a graduate student, I created a website which is now archived at http://math.ucr.edu/home/baez/relativity.html, and I am one of the coauthors of the Usenet Relativity FAQ and in fact briefly coedited with Nathan Urban the relativity subsection of the FAQ. HTH ---CH 18:54, 11 May 2006 (UTC)

This may now be moot: see the next section. ---CH 20:09, 11 May 2006 (UTC)


 * Just a couple of points before I get round to digesting your latest. You said you couldn't guess - and then guessed correctly. My "totally flummoxed" was a different kind of "hyperbolae"! I'm really very pleased that your attention is at last focussed on the topic at hand.


 * I had meant to add to previous that the a) and b) you refer to in Hsu & Suzuki are direct quotes from Matsuda & Kinoshita. (They use single quote marks - not so obvious). The a) is the last para at bottom right of M&K's front page and b) is the very next para at top left of next page. I don't think there is any ambiguity - their length L or is A'-B' on your diagram. H&S are asserting the same length should be L' contracted like B1-B2, C1-C2 etc. on my rigidrod diagram.


 * However, I would complain about H&S that they absolutely should have used a diagram or two of their own to clarify what they're talking about but most of all they should have provided a good deal of detailed explanation to justify the "Moller-Wu" transformation. It is a very little used technique with hardly any exp. elsewhere, but here's one of the best I (eventually) found....http://psroc.phys.ntu.edu.tw/cjp/v40/265.pdf Rod Ball 10:18, 12 May 2006 (UTC)

User:Rod Ball's mistakes?
I think I have found the precise location of Rod's mistake. It seems to involve a point in hyperbolic trigonometry. The analogous point in circular trigonometry would amount to confusing arc length with angle.

Referring point by point to Rod's diagram depicting a pair of Rindler observers

and his subsequent discussion from his 25 April 2006 edit: In any case, it seems clear that Rod's key error is the hyperbolic trigonometric analog of confusing (in circular trigonometry) the (unequal) arc lengths of inner and outer circular arcs between two given radii with the angle subtended by the pair of radii. HTH---CH 20:09, 11 May 2006 (UTC)
 * 1) The two hyperbolae with the same vertex at O and common asymptote c=1 (dashed line) show the locus of points representing each end of the rod starting at x=3 and 5 when t=0 and v=0. Agreed. In the article, referring to the Rindler congruence, not the Bell congruence, this is written $$T=0, \; \dot{X}_0 = 0$$.
 * 2) The dotted lines parallel to the x axis B1-B2, C1-C2 and D1-D2 show successive position and length of the rod as observed by measurement from the launchsite frame. The word "observed" is a possible source of confusion, but if we take this to mean that the dotted horizontal line segments are spacelike and have length given by X coordinate differences (i.e. in the Cartesian chart), agreed.
 * 3) The sloping lines are lines of simultaneity along the rod itself and show the orientation of the x' axis of the rod's comoving frame as v increases with successive time intervals. This is a bit tricky, since each event on the world line of a bit of matter in the rod is associated with a distinct infinitesimal Lorentz frame; this is why we need to use the Rindler frame field to discuss this accelerated rod model; we cannot get away with the elementary language of special relativity.  But is true that in the article, the vectors $$\vec{f}_1$$ are aligned with the sloping lines mentioned by Rod.
 * 4) All points along each hyperbola are a constant "proper" distance from the origin, so A1-B2, B1-C2, C1-D2 and D1-E2 are all the same "proper" length of the rod, which is constant. After replacing constant proper distance by constant spacetime interval, agreed.
 * 5) Furthermore, each end of the rod has the same velocity at any given "proper" time. That is to say Va1=Vb2, Vb1=Vc2, Vc1=Vd2 and Vd1=Ve2 as the tangents are equal at each pair of points. The second half is analogous to saying that in a congruence of concentric circles in E2, if we consider an inner and outer circular arc lying between two given radii, corresponding points on each radius have the same slope in a Cartesian chart.  I agree with this bit.  But the first half appears to be analogous to claiming that in our congruence of concentric circles, if we consider an inner and outer circular arcs lying between two given radii, these arcs not only make the same angle but have the same arc length!  That would be correct about angle but of course would be incorrect about arc length.
 * 6) The last two alone (constant length plus same velocity at each end) are sufficient to show that the "proper" acceleration at each end of the rod is always equal, in its own "proper" comoving reference frame. Agree about the conclusion, with proper acceleration replaced with path curvature of the world line, i.e. the acceleration as measured by the accelerating observer himself, but suspect an error in the reasoning which led Rod to this conclusion.
 * 7) Thus the accelerations along the diagram hyperbolae at t=0 are 1/3 and 1/5. Better yet, as the article says: the left endpoint (trailing Rindler observer) has constant acceleration with magnitude (path curvature) 1/3 and the right endpoint (leading Rindler observer) has constant acceleration with magnitude (path curvature) 1/5, so the trailing observer is accelerating harder to keep up.
 * 8) Each end of the rod has the same velocity at any given "proper" time. That is to say Va1=Vb2, Vb1=Vc2, Vc1=Vd2 and Vd1=Ve2 as the tangents are equal at each pair of points. Agreed, as the euclidean analogy makes clear, analyzing the Rindler congruence is an exercise in the hyperbolic trigonometry of the Minkowski plane E1,1.
 * 9) Each sloping line of simultaneity is a line of constant "proper" time ( tau ). Aha!  This is incorrect!  This is analogous to saying that in a family of concentric circles, if we consider the inner and outer circular arcs lying between a pair of radii, the arc length of the inner and outer arcs agree.  In fact of course the arc length of the outer arc is larger by the ratio of the large radius to small radius.  That is the whole point of trigonometry!  In the Rindler coordinates, note that the line element reads $$ds^2 = -x^2 \, dt^2 + dx^2 + dy^2 + dz^2$$, so the vectors $$\vec{e}_0$$ appear to have different heights in this chart, if one incorrectly interprets the Rindler chart description in terms of euclidean geometry.  In the figures I created for Rindler coordinates, I scaled the light cones properly to suggest this property.
 * 10) To propel a rigid rod to relativistic speeds without distorting or stressing it, only equal forces uniformly distributed along its length are required. This is incorrect. As the computations in the article demonstrate, and as Rod himself said above, the accelerations are constant for each point on this "rigidly accelerated rod". This phrase is problematic, as I have been trying to explain. It seems that we do all agree that only the Rindler congruence is a reasonable model for the closest we can come in relativistic physics to a rigidly accelerated rod, but the accelerations in fact vary along the rod.  Specifically, as the article says, trailing points must accelerate harder to keep up with leading points.  Also, the rod is in fact stressed because every bit of matter is being accelerated, with accelerations varying along the rod.  However, analyzing the details would appear to require a material model, and it is clear that depending upon where the body force is applied to the rod, any reasonable model of an accelerated rod will not result in all parts of the rod simultaneously beginning to accelerate at the start of the acceleration phase.


 * Gosh, there's a lot to deal with. I'll very briefly touch on a few of your 8 points between "Rod you wrote" and "Rod's mistake". Point 4) What I mean by "nested hyperbolae" is I hope the same as you meant. I thought it would help clarity if I used the same term you did in the article page alongside the diagram immediately above "dissident views". Point 5) Alright but then I would say that assuming constant path curvature in Minkowski x-t diagram of launchsite POV (which I would disagree with) amounts to the same thing. Point 6) I didn't mean misleading, just that you consider what you call "Bell observers" as appropriate to the two s'ship problem and contrast that with Rindler observers, whereas I consider Rindler observers appropriate to the problem and "Bell observers" not. Point 7) versions of accelerated rods are neither here nor there but I should have remembered Olaf Wucknitz's article as I had a lively & friendly email discussion (unresolved) with him on Bell's problem before I came to Wikipedia. Note, however that his is not an "earlier version", but what he calls an "idealized" version, of Bell's thought experiment. Aside from the fact that it was Dewan & Beran's thought experiment, Olaf is obviously deriving his from Bell's and it is rather oversimplified like Matsuda & Kinoshita's. Olaf, incidentally, also considers the comoving POV and expects breakage due to different starting times which, as I mentioned before in connection with my diagram pair before "Guaranteed to fail", doesn't really work.


 * Now to the 10 most recent points.
 * OK
 * OK just "measured from" then.
 * I think you're agreeing here. The customary practice of invoking an "instantaneously comoving" inertial observer does, I feel, justify simple SR arguments in this case.
 * Because of point 9) which I will come to, the two are synonymous due to constant tau along the rod.
 * Again related to point 9) I disagree. The path length is relevant to "t" but not to tau. Otherwise starting with tau=t=0, tau would equal t from then on which it obviously doesn't.
 * This one startled me ! I thought you were agreeing with the very point my whole demonstration was aimed at. Then I realised you were thinking again of "Bell observers" for whom the equal accelerations would not be at the same proper time, whereas I contend that they are.
 * Er, no. The trailing observer only "appears" to be accelerating harder from launchsite POV. Exactly as the relativistic train "appears" to be shorter from the platform while its true "proper" length remains unaltered, the proper acceleration is the same front and back. I use "appears" here to mean "as measured from" (a relatively moving inertial reference frame).
 * OK
 * As I said in point 5) the diagram path lengths apply to "t" not tau. Constant tau along rod can be shown straightforwardly. Take the two points at each end of the proper length ie. A1-B2 or B1-C2 etc. in my diagram, say (x1,t1) and (x2,t2) and then put (x2-x1)=(delta)x and (t2-t1)=(delta)t. Then:


 * &delta;&tau; = &gamma; (&delta;t + v &delta;x)  and   L = &gamma; (&delta;x + v &delta;t)


 * squaring both equations and subtracting the first result from the second result gives:


 * L² - &delta;&tau;² = &gamma;² &delta;x² (1 - v²) - &gamma;² &delta;t² (1 - v²)


 * L² - &delta;&tau;² = &delta;x² - &delta;t² = L²


 * &delta;&tau;² = 0 and &delta;&tau; = 0


 * Thus the proper time is the same at each end of the proper length, and by extension along the entire sloping line of simultaneity.


 * 10. I think you are wrong. The statement is correct because "proper" accelerations, as would be provided by identical propulsion in the frame of the rod, are identical at each end of and along the proper length. The increasing shift of simultaneity between the rod and the launch frame means that launch frame measurements at same "t", exhibit shortening apparent length and correspondingly different apparent accelerations, just as they do for Bell's spaceships. It goes without saying by now that I consider "relativistic stresses" don't exist. There are inertial stresses and mechanical stresses etc. but simply no such thing as "relativistic stresses" because no real contraction actually takes place in SR. [BTW I've taken the liberty of adding a ? to your title...]
 * Rod Ball 11:07, 13 May 2006 (UTC)

Rod: Rod, we need to wrap up this discussion. If you are unwilling to you will not be able to properly understand your own diagram. In that case, we will have to leave it at this: you disagree with the view expressed in the article and in the sci.physics FAQ. (Note that in the FAQ essay, Michael Weiss offers three different ways to see that the string will break.) I don't care whether you continue to maintain (incorrectly) that this does not represent the mainstream viewpoint, but you will need to concede, I think, that the most (possibly all) of WikiProject Physics members are with me on this point, so that the article should stand as I have rewritten it. ---CH 19:31, 13 May 2006 (UTC)
 * 1) See the wikicode to see how to indent your comments by one tab in this section, to distinguish from mine. The idea is that if a third person comments, he/she will indent by two tabs, and so on.  This makes it much easier to read talk spaces.  (IMO tabbing in talk pages should be automatic, but that is something for the developers of WikiMedia to consider.)
 * 2) A lot to deal with: you don't really need to respond point by point. The key point was my point 9: you are confusing hyperbolic angle with hyperbolic arc length. I just found it easier to comment point by point to assure myself that the basic problem concerns confusing angle and arc length.  (As we see below, there is also a problem with your notion of what you call "proper acceleration", by which everyone else would mean path curvature vector, with components given in a suitable local Lorentz frame for the accelerating observer himself, which is precisely what I have been computing all along; note again that I am still trying to get you to learn about frame fields).
 * 3) Re your 3, I am not "agreeing"; I am still trying to get you to appreciate a key point. Read what I said above again, please (or else just take my word for it, since this is not the main point right now).
 * 4) Re your 6, no, I am commenting on your discussion of Rindler observers. An easy computation shows that a pair of Bell observers will measure the same magnitude acceleration at a pre-agreed lapse of proper time after the start of the acceleration phase (which happens at T=0 in terms of the coordinate time of the Minkowski chart, as discussed in the article).  Likewise, an easy computation shows that leading Rindler observers measure a constant magnitude acceleration which is smaller than the constant magnitude acceleration measured by trailing Rindler observers.  Here, acceleration is the path curvature vector, obtained by covariant differentiation, as discussed in most standard textbooks.  See for example the classic monograph by Hawking & Ellis, the classic textbook MTW, the new textbook by Eric Poisson, etc. (you can find full citations here).  I stress again that an essential point here is that the kinematic decomposition belongs to the differential geometry of congruences of curves in Lorentzian manifolds whether curved or not.  In Minkowski spacetime, this is simply an analog of the well-known Frénet-Serret frame description in classical differential geometry of curves in E3.  And of course frame fields are simply "moving frames" in the terminology of Élie Cartan.  You need to learn to use frame fields.   You simply cannot discuss this topic without knowing this essential and standard technique.
 * 5) Re your 9, you write "Thus the proper time is the same at each end of the proper length, and by extension along the entire sloping line of simultaneity."  Is it possible that you are confusing Rindler coordinate time with elapsed proper time from the start of the acceleration phase, as measured by individual Rindler observers?  Naturally I have been interpreting your statement to refer to the latter, not the former.  If you mean the former, you have simply misunderstood Rindler coordinate time, which would be fatal to your chance of understanding Rindler observers.  If the latter, you have simply misunderstood the whole idea of hyperbolic trigonometry, which would also be fatal.
 * 6) Regarding both the Rindler and Bell observers, I have proven all the claims made in the article with simple computations. Your only remaining "wiggle room" regarding Bell's question (does the string break?) would require you to argue that the standard notion of acceleration in relativistic physics is somehow inappropriate. But you have no wiggle room at all regarding who is presenting the mainstream view in this discussion: I am presenting the mainstream view; you are arguing for a dissident conclusion.
 * 7) To see why your claim that "the proper time is the same at each end of the proper length, and by extension along the entire sloping line of simultaneity" is obviously incorrect, just draw a diagram of the euclidean or circular trigonometry analog: a family of concentric circles with some equally spaced radii. Two radii make some angle, but the arc length along each circular arc is not constant, it depends upon the radius.  This is the whole point of angular measure: angle is the ratio of arc length along a constant distance curve are to that constant distance.  The analogous statements for hyperbolic trigonometry are precisely what I've been trying to explain to you.
 * BTW, the simple notion of angular measure which I just mentioned also generalizes to parabolic trigonometry. From the three possible notions of angular measure and constant positive, zero, or negative curvature one obtains the nine plane homogeneous geometries of Felix Klein.  Parabolic trigonmetry is the kind you need in Newtonian spacetime, which has a degenerate metric.   Hyperbolic trig is the kind you need for Lorentzian manifolds (special relativity and the curved spacetimes used in metric theories of gravitation).  You don't need to understand the analogy between parabolic and euclidean trig for this discussion, but you do need to have mastered the analogies between hyperbolic and circular trig.
 * 1) master hyperbolic trigonometry, or to
 * 2) read Rindler coordinates and verify the existence of numerous competing notions of distance for even linearly accelerating observers even in flat spacetime, or to
 * 3) master standard notions including congruences, frame fields, and the kinematic decomposition,

How weak must the string be?
Since encountering this problem several years ago, one assumption has worried me. It is assumed that the string can be weak enough to break instead of drawing the ships together. While this can be stated as a given assumption, it may be unphysical; there may be no material so weak as to fail under the tensions that arise. There is an article in Europhysics Letters that addresses this, but I do not have access to a copy yet. Gregory Merchan 14:34, 14 May 2006 (UTC)


 * Hi, Gregory, welcome to the wild and wooly world of Wikipedia (I see the above is the first and so far only edit by this user).


 * First things first: Europhysics Letters seems to be the merger of  Journal de Physique Lettres and Lettere al Nuovo Cimento. The UW physics library appears not to carry it, which if true would suggest that it must be very obscure or very new. Did you know that Nuovo Cimento is a favorite target of authors of papers in relativistic physics which rest upon elementary errors?  (As in, "if a graduate student did this, he'd be docked points on his homework problem!")  For example, papers by this author.  I see that the website you linked to is registered to "EDP Sciences" in France, but what is this company/organization?  Be aware that we've had quite a problem in Wikipedia with people citing cranky or vanity journals such as MetaResearch Bulletin, Hadronic Journal, Smarandache Notions Journal, or Journal of Scientific Exploration, possibly without realizing that these are not considered respectable journals.  Hence my caution.


 * Second, you may have noticed that earlier in this discussion, I tried to remind Rod Ball that paradoxes often arise because authors fail to notice that they are using an unsuitable idealization. In fact, I specifically challenged his notion of a "weak string".  Before I realized that I was apparently going over his head, I tried to draw his attention to the subtleties of the analogous Newtonian problem, including the fact that according to elementary textbooks on Lagrangian mechanics, it is natural to model a pulled rod as a pulled chain of n identical masses attached to n-1 identical springs in a linear array.  (Similarly for other locations for application of a force to a rod, whether "kicked" or steadily accelerated.)  Such finite chains become the harmonic lattice as n -> infinity.  Standard arguments suggest that the behavior of this lattice can be understood in terms of a certain dispersive wave equation.  But the symmetries of this equation include neither "Galilei boosts" (shears) nor Lorentz boosts--- at least, not in the space and time coordinates!  (In contrast, the symmetry group of the standard wave equation does of course include a transformation subgroup equivalent to the Poincaré group acting on spacetime in the standard way.) Indeed, careful attention shows that disturbances modeled with dispersive wave equations can behave paradoxically.  Since one traditional approach to treating accelerated rods even in Newtonian mechanics is connected to the Fermi-Pasta-Ulam paradox, which has a large literature of its own, we can hardly expect matters to be vastly simpler in relativistic physics if we want to know how reasonable models of "real rods" behave.  By the way, it may be more natural to consider damped harmonic motion in our chain of springs, and then more unexpected subtleties arise!


 * A general comment: some time ago I reviewed the existing literature on spaceships-and-string/accelerated-rods and on the Ehrenfest paradox. I found that investigation of this bloated literature is about as pleasant as the   autopsy of a drowned dog--- which tends to explain the sardonic tone of the following summary.


 * One finds a great variety of claims; some of the more thoughtful include these:
 * the paradoxes arise from failure to distinguish between competing notions of states of acceleration/rotation,
 * the paradoxes arise from failure to specify suitable operational procedures,
 * specifically, from a failure to distinguish between competing notions of distance which can be employed by accelerating observers,
 * specifically, from failure to distinguish between e.g. radar distance "in the large" and from integrating the Landau line element over the path of a null geodesic,
 * the paradoxes arise from failure to consider desychronization phenomena, and is essentially topological,
 * the paradoxes arise from failure to recognize hidden assumptions about accelerated ideal clocks,
 * the paradoxes arise from failure to use reasonable idealizations,
 * the paradoxes can be resolved without appeal to any material model,
 * specifically, the paradoxes arise from failure to correctly apply Lorentz transformations,
 * resolving the paradoxes requires more general transformations,
 * the paradoxes arise from failure to distinguish between charts, frame fields, and "frames" as the latter word is used in elementary discussion of special relativity,
 * questions concerning the geometry (physical behavior) of accelerated rods (spun up disks) cannot be resolved without introducing a mechanical model of the rod/disk,
 * the paradoxes arise from failure to specify reasonable initial conditions (e.g. in a material model),
 * the answers depend upon how the rod is acclerated (how the disk is spun up),
 * Each author typically seizes upon just one simple idea, analyzes this single consideration, and concludes that the resolution is perfectly simple if you just think about it the right way! But of course, none of them agree on just what is "the right way to think about the problem", and not surprisingly their conclusions often disagree.  Even worse, authors typically are very careless in summarizing their conclusions in natural language, thus adding another layer of entirely unneccessary confusion.  Some of the papers are in fact so incomprehensible from start to finish as to be essentially useless.


 * So which of the above considerations are correct? The answer is that (almost) all of these observations have a kernel of truth, but clearly none of them can resolve the various paradoxes to everyone's satisfaction.  In particular, regarding the last claim, the one common to almost every paper in this literature, "every thing you need to consider" is indeed simple, but there are many simple things to consider, depending upon how careful/imaginative/thoughtful you are (and how willing you are to broaden your perspective if you wind up with a conclusion which contradicts the mainstream).


 * At times, the reader is apt to grow so annoyed with typical authors in this genre, who tend to have made little if any effort to acquaint themselves in advance with the appropriate mathematical techniques, who rarely bother to employ suitable notation, and who rarely seem to have attempted to express themselves clearly, that one becomes tempted to conclude that this literature concerns a manufactured controversy which was invented, not to make students think, but to give really bad physicists something to write about.


 * A particularly objectionable feature of this literature is insistence upon using archaic and misleading terminology, as seen in questions like "do accelerated rods exhibit Lorentz contraction?". Sheesh, asking a question like that almost guarantees failure to say anything worthwhile!


 * There are some individual exceptions to the general rule of inarticulate awfulness, of course, but it is striking that apart from J. S. Bell and the early pioneers, very few contributors to this literature are known--- at least not to me--- from other contributions to physics. I mostly read papers on relativistic physics, so this suggests that if these authors are in fact well known in other subfields, their contributions to the paradoxical literature must constitute mere dabbling.  That might explain--- but not justify--- the generally poor quality of these contributions.---CH 03:40, 16 May 2006 (UTC)


 * Hi Hillman. I'd made minor edits before, but hadn't signed up until recently. I'm the GJM who added the title of one of the papers cited.


 * I was completely unfamiliar with Europhysics Letters. I usually have web access to journals through my physics department account, but that journal was not among them. According to its online catalog, my university (LSU) only carried it from 1986 to 1987. I meant to check the stacks there and in the physics dept. library one weekend. I found that article via google just recently and it was news to me since I last looked at the problem in 1998. I was not aware of the reputation of Nuovo Cimento; I've rarely seen it mentioned. I've just completed a protracted undergraduate career and have not read many articles outside of Physical Review and the American Journal of Physics except those of historical interest.


 * Calculating the tension on the string due to its contraction (or the ships' separation) is beyond my efforts if not my current abilities. Nonetheless, even if there could be no string so weak as to break under the tension, it doesn't change the fact that there would be tension on the string owing to its contraction from the unaccelerated observer's perspective or the separation of the ships from their perspectives.


 * Some years ago, I had the displeasure of presenting this paradox to my professors who got it wrong and being the object of jokes and sneers from my classmates for it. It seems odd to have suffered for advancing the mainstream. Eventually my professors did persuade themselves that the string breaks, though my classmates did not to my knowledge learn of that. I could go on.


 * I hope that clarifies the nature of my worry about the weak string. Having thought about it some more, I suppose it is silly to worry about an airtight presentation of the problem given the nature of the errors commonly made. -- Gregory Merchan 22:11, 16 May 2006 (UTC)


 * One tiny quibble: tension in the cable is due to differential accelerations or to elongation as measured by an observer riding somewhere on the cable itself; the so-called "Lorentz contraction" is concerns coordinate descriptions and is analogous to the fact that |sin| <= 1; coordinate descriptions cannot cause physioal effects. But otherwise I agree.  Wow, sorry you were subjected to sneers!  Oh well, I think you are enjoying the last laugh :-/  ---CH 01:39, 19 May 2006 (UTC)