Bell spaceship paradox quantitatively

[bcrowell]

I only just this afternoon got around to catching up with this discussion, which has been great. I love PAllen's idea of using Herglotz-Noether here, which definitely clears up a logical hole in my previous treatment using the expansion scalar. Also, vanhees71 was absolutely right to point out that there was a lack of rigor in the more elementary treatment, where the length of the string was arbitrarily defined in the frame of one of the ships rather than the other.

I've rewritten the relevant portions of my SR book http://www.lightandmatter.com/sr/ , which are in sections 3.5.2 and 9.5.3-4 (currently near pp. 58 and 179, although those will eventually change). In the more elementary initial treatment, I give a rough argument that the error incurred by choosing one ship's frame rather than the other is of a lower order than the effect being discussed. However, I admit that this would be cumbersome to carry through in detail, and give the reader a pointer to the later and fancier treatment. In the later treatment, I use P. Allen's idea (and acknowledge him). If you've recently looked at the book and want to see the revisions, you may have to empty your browser's cache or reload the page or something.

Herglotz-Noether is a pretty sophisticated piece of machinery, and I didn't want to have to just invoke it cargo-cult style. I realized that the 1+1-dimensional version, which is all that's needed for our present purposes, is simple to prove and to state, so I wrote up a proof and included it in the book. BTW the 1+1 version is not a special case of 3+1; I have a brief discussion of this in the book.

My explanation and understanding of this paradox have benefited hugely from past and present discussions here on PF. Thanks! Any further comments would also be very welcome.

 

[vanhees71]

The treatment in your book, which btw. is simply great (also the GR book), is very clear now! Also for the more elementary treatment, I guess the use of Rindler coordinates for the front spaceship should be a great application for SRT in terms of accelerated frames.

My last quibble is also solved thanks to Peter Donis's posting #34: For the case, where $\alpha L>c^2$, you simply cannot connect a Born-rigid body of proper length $L_A$ at spaceship C. This comes very clearly out of my calculation for $\alpha L<c^2$: The rear end of the rigid rod must accelarate faster than spaceship C, and for $\alpha L \rightarrow c^2-0^+$ this acceleration tends to $\infty$, so that a rigid rod must immediately break in this limit even if it is not connected to spaceship B. This shows another aspect of the presence of the Rindler horizon: There cannot be a rigid body with too large extent. It's limited by the Rindler horizon, or stated in another way a Born-rigid body of given length cannot accelerate at arbitrarily large proper acceleration.

What I also realized is a lack in modern SRT books for physicists: All these old discussions are simply left out. That's why Pauli's review is so valuable, because it discusses all these issues (and it's written in 1921!). Now I've also ordered the two volumes about relativity by von Laue, which also contain all these issues in great detail. It's interesting, how (apparently outdated) topics simply disappear from modern textbooks although they are very valuable for strengthen one's understanding of the topic. Nowadays we only learn about relativistic hydrodynamics, which is of course great and very valuable in my field of relativistic heavy-ion collisions with a lot of also pretty recent new achievements like a systematic treatment of viscous hydrodynamics beyond the standard Israel-Stuart formalism, but that's another story. Of course relativistic hydro becomes also more an more important in GR.

Compared to this quite well understood issues (using relativistic transport and quantum transport approaches), I've the impression that a relativistic theory of elastic bodies is still not so much developed. In my Google serach about all these issues here discussed, I stumbled over a paper by Paria from 1965:

G. Paria, On relativistic elasticity, Acta Mechanica 3, 93 (1967)
http://dx.doi.org/10.1007/BF01453709

I guess, there should be something more recent, but it seems to be pretty nicely written. The only trouble is (also with these older sources, mentioned above) is the use of the $\mathrm{i} c t$ convention for the Minkowski metric, which I always hated ;-(.

Thanks again for this great discussion!

 

[vanhees71]

I've put my (now hopefully correct) more elementary analysis of the spaceship paradox into my SRT-FAQ draft. If you like, have a look:

http://fias.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[Vmedvil]

If they have the same mass, the distance is still L_A Different masses then that makes the problem much more difficult and are you taking in affect length contraction? If so then the space within the objects gets smaller by

http://www.4p8.com/eric.brasseur/erta1.gif
Sub

 

[PeterDonis]

If they have the same mass, the distance is still L_A

I don't understand what point you're trying to make.

 

[Vmedvil]

My Point is their distance won't change besides the Lorentz contraction effect with the same mass and acceleration which will make them slightly and I mean slightly more distant.

 

[PeterDonis]

My Point is their distance won't change besides the Lorentz contraction effect with the same mass and acceleration which will make them slightly and I mean slightly more distant.

I still don't understand. How are you defining "distance"? Have you read the previous posts in this thread, which have gone into this subject in detail?

 

[PeterDonis]

Vmdevil, you're still not making sense and you're bringing in a lot of stuff that' s not even relevant to this thread.