Bell spaceship paradox quantitatively

[loislane]

There seems not to be a clear solution of the paradox in elementary terms, and one must argue with the expansion scalar as in bcrowell's book, but as I said in a previous posting in this thread, it's not clear to me, why choosing this particular world-line congruence to fill the world-line tube between the ships. So this is also arbitrary.

This might be because the expansion scalar in the book's example is restricted to one spatial dimension. As is well known a 3-surface has vanishing expansion in a static spacetime.

 

[PeterDonis]

By expanding I meant: not everywhere non-expanding.

But this is a much weaker statement than my "on average positive expansion".

What I would say is there is no invariant global distance, nor is there any 'rest length'. However, there are perfectly consistent, frame dependent, global distances.

Agreed, this is a better way of saying it.

 

[vanhees71]

Just to make my problem in #32 clear again, here's the explicit calculation if one considers a Born-rigid rod with one end fixed at the leading spaceship. With the same notation as in #32 the other end of the rigid rod is given by the world line (written in coordinates in from A)
$$x_{\text{rod}}=x_C(\tau)-\hat{\Lambda}^{-1}(\tau)=\begin{pmatrix}
(1-\alpha L_A)/\alpha \sinh(\alpha \tau) \\ (1-\alpha L_A)/\alpha \cosh(\alpha \tau)
\end{pmatrix}.$$
As long as $\alpha L_A<1$ there's no problem. Plotting the world lines one sees that at any instant of time $t>0$ in A the rod is too short to reach to B, i.e., tighing the end of the rod to B would need to stretch it more and more, and thus a "real" (non-rigid) rod would eventually break.

But now, if $1-\alpha L_A<0$, a problem occurs. Then for $t>0$ in A one has $x_{\text{rod}}^1(\tau)<0$ (note that $\tau<0$ to have $x^0(\tau)=c t(\tau)>0$). This doesn't make sense to me. That would mean in this case the rigid rod at any instant $t>0$ is longer (sic!) than $L_A$. So something must be wrong in this argument either, but I still cannot figure out what that might be.

Another solution to the problem is, when one slightly changes the conditions by assuming that after some finite time $t_{\text{end}}$ the two spaceships stop accelerating (e.g., because their fuel is over ;-)) and then drift both with the same constant velocity. Then you have a common rest frame of the spaceships, and the distance in A is still $L_A$ but this is the Lorentz contracted proper distance (which now is well defined as the measured distance in the common restframe) which is thus longer by the $\gamma$ factor $L_{\text{proper}}=\gamma L_A$.

So, maybe the entire trouble is the somewhat artificial example of constant proper acceleration (also known as hyperbolic motion for obvious reasons). This example seems to be a notorious trouble maker as there is the (for me also quite paradoxical) calculation that the Lienard Wiechert retarded solutions of the Maxwell equations for a point charge in hyperbolic motion seem to indicate that there is no radiation field at all, while if stopping the acceleration after some finite time, one finds radiation as expected (see, e.g., Pauli's famous monography as a reference).

 

[PeterDonis]

if $1-\alpha L_A < 0$, a problem occurs

Yes, this is because it's physically impossible to have a rigid rod trailing the leading spaceship with a length longer than $1 / \alpha$ (or $c^2 / \alpha$ in conventional units), since that would put the trailing edge of the rod behind the leading spaceship's Rindler horizon. If you try to base an analysis on physically impossible assumptions, naturally you're going to get incorrect results.

maybe the entire trouble is the somewhat artificial example of constant proper acceleration

No, the trouble is that you are trying to construct a model that's physically impossible; see above. Having a rigid rod "pushed" ahead of the trailing spaceship doesn't have this problem because the Rindler horizon is always behind an accelerating object.

 

[PAllen]

Yes, this is because it's physically impossible to have a rigid rod trailing the leading spaceship with a length longer than $1 / \alpha$ (or $c^2 / \alpha$ in conventional units), since that would put the trailing edge of the rod behind the leading spaceship's Rindler horizon. If you try to base an analysis on physically impossible assumptions, naturally you're going to get incorrect results.

The way this would play out is that if you tried to specify the rigid congruence, you wouldn't be able to extend it as a timelike congruence beyond this limit. Physically, as I explained in an earlier post, this means that an unconstrained object longer than this must break if it is pulled with the stated proper acceleration, irrespective of any theory of matter. In practice, it would break at a much shorter length assuming you are able to generate the stated proper acceleration of the leading edge.

 

[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.