There is a relativistic paradox I was introduced by a friend in Ithaca (Courtenay Footman). It involves a vehicle travelling over a ravine relativistically, so it's in the general family of ladder paradoxes. There are many ways to phrase it, but here's a simple one.

A three-car train with each car of length $L$ is travelling over a bridge across a ravine. (Assume that the couplings between cars are stiff: this train does not bend.) Unfortunately, some evil genius has precisely removed a length $6L$ of the track. The train however is relativistic, travelling at a speed such that $\gamma \approx 12$.

In the reference frame of the ground, the train should appear to be 1/24th the size of the gap, which suggests that it will fall a little and should crash into the bridge opposite. But in the reference frame of the train, each car of the train is at least twice as long as the gap and aside from the very front wheel and the very back wheel, which need to be supported by rigidity, everything is constantly well-supported. This suggests that the train makes it. Who is right?

I vaguely remember that as an undergraduate knowing only a little relativity, I trusted the train's perspective because it's at rest in that perspective, and Courtenay said that this was ultimately correct after a lot of details got resolved. But I do not remember all of these details such that I could work it out for myself. So I'm looking for some discussion that's a little more well-sourced or authoritative.

What is this paradox called, and what literature has addressed the issue?

  • 1
    $\begingroup$ Is this just a variant of the pole in the barn or car in garage "paradox"? $\endgroup$
    – user81619
    Aug 2 '15 at 22:31
  • $\begingroup$ Falling Rod or Bug-Rivet ? These have no moving parts :) $\endgroup$
    – gox
    Aug 2 '15 at 22:44
  • $\begingroup$ @AcidJazz: that's the ladder paradox that I mentioned in the question. It has a crucial difference from them, which is that there are no clocks for an argument by simultaneity. $\endgroup$
    – CR Drost
    Aug 2 '15 at 23:04
  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ Aug 2 '15 at 23:34
  • $\begingroup$ @AlfredCentauri: I should have checked Wikipedia! "Rindler's Grid paradox" should be good enough, I guess; just as importantly, home.agh.edu.pl/~mariuszp/wfiis_stw/stw_rindler_lcp.pdf is where the idea was first published. If you'll flesh it into an answer I'd happily award you Best Answer. $\endgroup$
    – CR Drost
    Aug 3 '15 at 0:43

This is Wolfgang Rindler's "grid paradox", discussed on Wikipedia here and both proposed and resolved on this PDF by Rindler. Its resolution is the exact opposite of what you're saying; if you remember being "right" then you must have instead trusted the ground's perspective, not the train's.

The problem is essentially that our notion of "rigid" does not continue into the relativistic framework, since it requires instantaneous information transmission. Suppose we simply have two atoms which are attached together by some potential $U(|\vec r_2 - \vec r_1|)$ with a strong minimum at a distance $a,$ maybe we can say $U(L) = \frac12 k (L - a)^2.$

In Newtonian mechanics this would create rigidity for large $k$, but in special relativity, we have a time $\tau = a/c$ during which we can manipulate the one atom without any effect on the other atom. Let us just assume that both particles have mass $m$ and are at rest in the center-of-mass frame; then we give the "right" one an instantaneous impulse $\Delta p$. Then the particles attain a spacing of $\ell = \tau\Delta p/m$ before any signal reaches the "left" one informing it about our impulse. When this happens, a powerful restoring force should draw the "left" particle forward and the "right" particle backward, and to conserve momentum we're probably going to need a field in the middle to carry the momentum as it gets transferred back and forth. The mathematics here are complicated and I don't know that I've ever actually seen the full example worked out. However, the point would be that there is a certain "flexibility" mandated by relativistic information limits, and possibly an increase in the gradient of the gravitational field due to length contraction, which might both drive the leading edge of the train to bend sharply down into the gap, causing the inevitable carnage.


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