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Revision as of 13:41, 18 August 2006

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Rigid rods

I'm still of the opinion, that you are making a severe error in your presentation of the rigid rod and the BSP problem.

I'd prefer no to further fill the Talk:Bell's spaceship paradox page with this discussion but we can have a discussion here, starting with your picture and the theses you presented on it -- if you agree. In the (hopefully consensual) end, we can report our results, if relevant, at Talk:Bell's spaceship paradox.

Pjacobi 16:58, 3 May 2006 (UTC)

Not interested? --Pjacobi 20:12, 7 May 2006 (UTC)

I've only just found this item (I thought it was just some stuff about sourcing diagrams and hadn't scrolled down before). I slightly reworded it this morning, so what don't you agree with ? Rod Ball 13:58, 8 May 2006 (UTC)

Can we start with agreeing (or disagreeing) on some definitions (and direct consequences of them):

Proper time is time as measured by the clock for an observer who is traveling through spacetime. In SRT, if the path is giving in cartesian coordinates in some inertieal system, the proper time difference between start and end point of a path P is given by

${\displaystyle \Delta \tau =\int _{P}{\sqrt {dt^{2}-dx^{2}-dy^{2}-dz^{2}}}}$ (if we take c as 1)

And this gives the same result for any intertial system.

Proper acceleration is the acceleration measured in the comoving system, and in any inertial system given as

${\displaystyle a_{p}=a\,\gamma ^{3}={a \over (1-v^{2})^{3 \over 2}}}$

Do you agree?

Pjacobi 15:13, 8 May 2006 (UTC)

First item - I would say accompanying an observer rather than "for" (I'm not contradicting, just being as clear as possible). (I think we could drop the y and z coords also for this one dimensional prob.) But yes I would agree although bear in mind that for the ends of a rod or string ( I mean the ends of the proper length) the path length for the integral is different. Second item - I think one has to be very careful here but I'll save my reservations for later. The formula is acceptable for acceleration of individual points taken in isolation but "a" needs to be defined carefully when considering correlated points such as ends of proper length that move disproportionately in x-t space. That is, for a given increment of proper time, dx and dt are proportionately greater at one end than the other.

Just briefly, to help me see your point of view, am I right in thinking that we agree that the proper length (ie. as measured in its own frame) of the rod is constant and that it has the same velocity at each end at all proper times ? I hope we ageree this is so, in which case I wonder how you reconcile that, with the ends of the same proper length having different accelerations ? Rod Ball 19:51, 9 May 2006 (UTC)

Yes, a in the case if the rigid rod, the rod in comoving coordinates has constant lengths, that is exact the definition of rigidity. In SRT there is only a very limited set of rigid motions possible, but uniform acceleration is one of them, see Born rigidity.

I'm a bit confused about your reservations in the a case. The acceleration can be defined for each point separately.

Pjacobi 20:05, 9 May 2006 (UTC)

Yes, but an increment of proper time is an increment of angle about O that a) increments the velocity the same amount at each end and b) sweeps a greater path length on the hyperbola by exactly the inverse ratio of the separate "a" for each curve. There's no coincidence about this. The true acceleration of a curve depends on how fast you move along it. For concentric circles the accn. is proportional to r for a common radius but quite different if you just use cartesian coords on bits of the arc length.

I don't quite understand your first comment. If you have constant p.length and identical v at each end, does that alone not imply identical proper accn. at each end ? Rod Ball 07:59, 10 May 2006 (UTC)

If and only if both ends have equal proper time, and this notion is not equivalent to both ends being at the same time coordinate in the co-moving frame.

I assume, if we can agree on the definitions of proper time and proper acceleration (or you are at least willing to see what follows from these definitions, we should continue to worked example with concrete values to see what happens.

Taking still c=1, we can at first look at the two points travelling with x²=t²+3² and x²=t²+5² (in the lab frame), as pictured in your diagram above. OK to proceed with this?

Pjacobi 09:10, 10 May 2006 (UTC)

Sure. Rod Ball 11:38, 10 May 2006 (UTC)

I'm in a hurry and offline tomorrow, so let my throw in some conclusions from x²=t²+3² (for positive t and x):

• ${\displaystyle x={\sqrt {t^{2}+3^{2}}}}$
• ${\displaystyle {dx \over dt}={t \over {\sqrt {t^{2}+3^{2}}}}}$
• ${\displaystyle t={\sqrt {x^{2}-3^{2}}}}$
• ${\displaystyle {dt \over dx}={x \over {\sqrt {x^{2}-3^{2}}}}}$
• ${\displaystyle {d\tau \over dt}={\sqrt {({dt \over dt})^{2}-({dx \over dt})^{2}}}={\sqrt {1-{t^{2} \over t^{2}+3^{2}}}}={\sqrt {3^{2} \over t^{2}+3^{2}}}}$
• Integrating gives ${\displaystyle \tau (T)-\tau (0)=\int _{0}^{T}{\sqrt {3^{2} \over t^{2}+3^{2}}}dt=3\,arcsinh({T \over 3})}$
• If we choose τ (0) = 0, reverting to t as variable, and omitting the argument of τ this reads ${\displaystyle \tau =3\,arcsinh({t \over 3})}$
• This now gives ${\displaystyle t=3\,sinh({\tau \over 3})}$
• And ${\displaystyle x={\sqrt {3^{2}sinh^{2}({\tau \over 3})+3^{2}}}={\sqrt {3^{2}cosh^{2}({\tau \over 3})}}=3\,cosh({\tau \over 3})}$

So, that should give a solid base for some numerical examples and pictures. Please check my calculations, I'm notorous for having typos in formulas. Also point out, whether any of these mathematical steps look physically unsound to you.

Pjacobi 21:27, 10 May 2006 (UTC)

This seems reasonable, and in general... (x/y)=cosh(tau/y), (t/y)=sinh(tau/y), where y=x(o)

(I corrected a few minor typo's) Rod Ball 10:36, 12 May 2006 (UTC)

Thanks for the corrections.

Before dusting off my Python and Matplotlib skills to make some more diagrams, I'd like to interrupt the mathematics for a short break to discuss physics, or even philosophy. The rationale for this is the notion of "proper length", you mentioned above. "Proper time" and "proper acceleration" are observer local effects, but proper length, for example between both ends of the rods, when the observer is at one end, is not local. It requires a notion simultanity. Which point of the world line of the other end is simultaneous to a given point on the worldline of the observer?

It's at the core of relativity theory, that there is no absolute simultanity, the only invariant notions are, whether two spacetime points are space-like or time-like separated. Now, for inertial observers, there is a natural choice, the Einstein synchronisation. But even in that case it is considered a convention only - at least by significant faction of philosophers and physicists, see the literature and links of that article, especially .

For accelarated observes no single choice of simultanity has all the nice properties of the intertial case. The most common convention is, to use at each point of the wordline the simultanity hyperplane of an inertial observe with the same position and velocity. This is the concept underlying Born rigidity. The Rindler coordinates are another example. But another possibility are Märzke-Wheeler coordinates a.k.a. Radar time, see .

Fortunately, it is my opinion that neither for the rigid rod nor for BSP these complications doesn't make a significant difference (and I'd consider those authors, who base their arguments on either of these coordinate choices, to be confused or confusing). Two arguments:

• If in doubt, we can always switch off drive for some time and make measurements resp. calculatuinb in inertial frames.
• The difference between different simultanity conventions gradually disappear when going near the observers world line: As we more and more "zoom in" the world line more and more looks like a straight one. And whereas we tend to use "large distances" (length times acceleration comparable with c²), to make the diagrams better showing the effects, we can always use a small l·a for calculations.

Pjacobi 21:28, 12 May 2006 (UTC)

You say "no absolute simutaneity". Agree but this means between relatively moving reference frames. You seem to be taking it too far and possibly denying a notion of simultaneity within a single inertial frame. There is no problem establishing simultaneity among different points in the same inertial RF, but it breaks down and becomes "shifted" as between RF's moving relative to one another. Rod Ball 18:36, 13 May 2006 (UTC)

As I've said, in inertial frames there is natural (but argueably still conventional) notion of simultanity. But regarding general observations in non-inertual frames, there is not. But sorry to be confusing myself. I just wanted to point out, that the problem of non-unique definitions of simultanity will not change the results here. --Pjacobi 20:08, 13 May 2006 (UTC)

Rubicon reached?

Despite patient efforts to explain matters to you, you are still pushing your strange POV at Bell's spaceship paradox. Your article space edits can easily be reverted but the the endless discussions on the talk page are even worse. Making a bilance, your current Misplaced Pages edit behaviour just drains resources (edit time of contributing physicists). I concede that you seem have read quite a lot articles on the subjects, but your failure of understanding basic SR facts, leave me puzzled, how you can become an asset and not an obligation for Misplaced Pages.

Perhaps you just should try editing a completely different subject (and perhaps and after a significant pause only re-read the BSP arguments and a textbook)?

Pjacobi 22:26, 16 August 2006 (UTC)

You are hardly in a position to explain things to me, rather it's the other way around. I have deeply researched this topic and know and understand it better than the four of you. This is quite evident from the comments that are made in the talk page. Your contribution has been little else than to keep restating the equations of hyperbolic motion which have no bearing on the problem at all since it is only stipulated that the acceleration regimes be the same, not that they take any particlar profile. Remember also that ChrisH had never heard of the problem before and I had great difficulty persuading him to even consult the literature, such was his arrogance. He didn't understand the problem and thought the 2nd s'ship unnecessary because he wanted to model the string elasticity with concatenated springs !! I particularly didn't like his dishonesty in blustering and not acknowledging some elementary errors when they were pointed out to him. He even denied not having read the topic before, when he eventually got round to doing so. His analysis is incorrect due partly to errors and partly to omitting to include relativity.

Against incessant repetition of unconnected statements and unsupported assertions of the correctness of D&B and Bell, I have managed to pursue a variety of arguments and approaches on the talk page that have considerably clarified the issues and shown that the divergence of opinion on string-breaking is due to the unconscious use of different conceptual versions of relativity. As I describe in my most recent entries, because of the near-identity of Lorentz's theory and Einstein's SR for constant motion, it has become not uncommon for some physicists to think of SR in terms of the more common-sense notions of Lorentz's theory. This is, of course, openly stated at length by Bell and has been adopted more widely, although not always knowingly.

However what has not been generally recognised is that for accelerated motion, Lorentz's theory and SR differ markedly in their predictions. In particlar in Bell's problem where Lorentz's theory has the string contracting and the empty space between s'ships unable to, both being accurately observable from launch, in SR the string does not actually contract but is only measured as doing so from the launch frame and this corresponds exactly to the s'ship distance and measurements of it from launch. In other words I show that the statement that the s'ships "cannot get closer from the launch frame unless they have different accelerations" (which seems to be the main plank of your argument) is demonstrably false in the context of special relativity, where one does not always expect things to be as simple as in Newtonian physics.

You complain that I am draining resources of contributing physicists. What arrogance !! What makes you think that you (and Harald & Ed) are any better qualified to edit Misplaced Pages than I ?Any contributor, whether professional scientist, or degree level (like myself), or hobbyist has at best one to a few areas of specialisation in which they may have expert knowledge, whilst perhaps having a "rough and ready" acquaintance with a broader range. This can also change somewhat as a result of intense research or prolonged disengagement. I sense that SR for many years, or decades even, has been neglected due to understandable eagerness to get stuck into the hot topics of GR & cosmology, and is all too often treated in a sloppy and superficial way because it is thought to have been completely worked out and done and dusted prior to WWII.Far from failing to understand basic SR facts, as you put it, I am clarifying what are SR facts by distinguishing contaminating ideas from Lorentz's theory that have infected much SR thinking. A muddled mixture of the two theories is worse than either on its own. If you do not have sufficient interest or detailed knowledge in BSP & SR, perhaps it is you who should think of moving to a different topic where you can contribute more usefully. Rod Ball 11:22, 18 August 2006 (UTC)