Train Fall Paradox

[anuttarasammyak]

Train Fall Paradox

A train is running on a long bridge over a river. A series of bombs planted on the bridge by terrorists explode simultaneously, and the bridge collapses into dust in an instant. The train falls while keeping its cars in a horizontal line and hits the river. All the cars receive equal damage.

However, in the inertial frame of reference in which the train had been at rest, due to the relativity of simultaneity, the bombs at the front explode earlier. The first car falls before the second, and so on. The train tilts and falls. Therefore, the first car is damaged more severely upon the initial impact than the later ones.(*)

Isn't this a paradox?

Spoiler

No, it isn't. The last sentence (*) is wrong.

 

[Ibix]

I think the train actually distorts in its original rest frame, rather than tilting. Although it's a funny kind of stress-free distortion that's a relative of length contraction.

 

[PeterDonis]

A series of bombs planted on the bridge by terrorists explode simultaneously

Simultaneously in what frame? I assume you mean the rest frame of the bridge, but in any relativity scenario, you should always explicitly specify the frame for any frame-dependent things. This is particularly important with simultaneity since it is more often a source of confusion than anything else.

The train falls while keeping its cars in a horizontal line

I assume this means that, in the bridge rest frame, the bombs all go off at the instant when the train is entirely on the bridge.

The train tilts and falls.

We can assume this is true (although as @Ibix points out, "tilting" might not be the best way to describe what's going on). However...

Therefore, the first car is damaged more severely upon the initial impact than the later ones

...this does not follow. The fall is perpendicular to the direction of motion of the train, so each car of the train still falls the same distance and is damaged by the same amount. The only difference in the train rest frame is that the cars don't hit the bottom all at the same time, since they didn't start falling all at the same time.

 

[Nugatory]

Isn't this a paradox?

Most relativity "paradoxes" are better described as "apparent contradiction resulting from a hidden bogus assumption, often used as a pedagogical device to help students recognize bogus assumptions". By that definition, yes, it is a paradox. But....

The first car falls before the second, and so on. The train tilts and falls. Therefore, the first car is damaged more severely upon the initial impact than the later ones.

This assumes that when the train augurs in at an angle and the lead car hits first (note the hidden simultaneity assumption here) the cars behind are slowed. It has to be that way, right? They're in a train and the lead car suddenly stops? Of course the cars behind the lead car are slowed or stopped?
No.
We've just cleverly hidden the rigid rod fallacy and the well-understood bug-rivet paradox.

 

[Sagittarius A-Star]

A train is running on a long bridge over a river.

In the restframe of the train, the bridge-length maybe even smaller than the train-length due to length contraction, if the bridge moves fast enough in the train's rest-frame.

Your scenario has some similarities to the "length contraction paradox", published in 1961 by W. Rindler:
https://home.agh.edu.pl/~mariuszp/wfiis_stw/stw_rindler_lcp.pdf

 

[anuttarasammyak]

Thanks you all for good teachings. I add would add a rough hand drawing explaining the event.

 

[pervect]

Train Fall Paradox

A train is running on a long bridge over a river. A series of bombs planted on the bridge by terrorists explode simultaneously, and the bridge collapses into dust in an instant. The train falls while keeping its cars in a horizontal line and hits the river. All the cars receive equal damage.

However, in the inertial frame of reference in which the train had been at rest, due to the relativity of simultaneity, the bombs at the front explode earlier. The first car falls before the second, and so on. The train tilts and falls. Therefore, the first car is damaged more severely upon the initial impact than the later ones.(*)

Isn't this a paradox?

Spoiler

No, it isn't. The last sentence (*) is wrong.

Not a paradox, but a rather hard problem. To deal with the problem with special relativity, I'd suggest replacing gravity due to mass due to the "artificial" gravity crated by an accelerated platform.

The "paradox" then becomes the description of the accelerated platform with a different notion of simultaneity.

A very long time ago, we had a thread in which we derived two different (but, as nearly as I could tell at the time, both correct) coordinate systems for such a case. But

Train Fall Paradox

A train is running on a long bridge over a river. A series of bombs planted on the bridge by terrorists explode simultaneously, and the bridge collapses into dust in an instant. The train falls while keeping its cars in a horizontal line and hits the river. All the cars receive equal damage.

However, in the inertial frame of reference in which the train had been at rest, due to the relativity of simultaneity, the bombs at the front explode earlier. The first car falls before the second, and so on. The train tilts and falls. Therefore, the first car is damaged more severely upon the initial impact than the later ones.(*)

Isn't this a paradox?

Spoiler

No, it isn't. The last sentence (*) is wrong.

To avoid the complications of gravity, imagine we have the train mounted on Einstein's elevator to simulate gravity.

Then there is a different version of your "paradox". What is the shape of the floor of the elevator. We define it as being flat in some inertial frame, and it remains flat at different times using the clock synchronization of said inertial frame. But if we change our clock syn chronization, the "flat" floor becomes curved, as in the following image, taken from a paper on Thomas precession (an effect on a gyroscope mounted on said frame). Specifically it was taken from https://arxiv.org/abs/0708.2490v1

I once went so far as to derive a metric for the coordinates of this accelerating frame, one that turned into a long thread where other people developed a different metric. I believe both metrics were correct, in the end - If there's some interest I could try and find it, but it was a while ago and wound up being very long and not well organized. My metric had the property (by construction) that the spatial Christoffel symbols vanished, and the floor was curved. Others liked a metric where the floor was still "flat", but in their metric the spatial Christoffel symbols did not vanish.