1. Jul 25, 2010

### Nabeshin

A friend of mine posed this SR paradox to me a few weeks ago and I was unable to come up with a convincing answer (nor have I been convinced by any I've heard!). The problem is as follows:
Assume we have a train track which has certain gaps in it. At rest, the train which moves on the track is longer than the gaps (you can probably see where this is going already...). Now, when the train is in motion, from the train frame the track gaps are lorentz contracted. To the platform frame, the train is contracted and for a certain velocity, will actually be smaller than the gaps.

Now, it seems reasonable to me that the platform observer will see the train fall some distance. However, what troubles me is what the train observer will see. Owing to the fact that his train is very much larger than the gap in the tracks, intuition states he will notice no effect. In the (sparse) literature on this paradox, it seems that the train will indeed fall, but not much owing to it's high velocity. They somehow claim this solves the paradox, but I am more skeptical. The train cannot have some vertical motion in one frame and none in the other, for any discrepency will lead to a different outcome.

Feel free to posit the train as a rigid body, or perhaps use the more realistic assumption of a normal physical material. I feel the resolution to this lays somewhere in the mechanics of the train (in the trains frame) as it passes over the very slight gap, but cannot quite get things reconciled in my mind. Thoughts include some type of torquing motion where only part of the train is unsupported by the track.

Cheers

2. Jul 25, 2010

### starthaus

Rest assured that no matter how high the speed $v<c$, the train will not "fall through the cracks" .

Last edited: Jul 25, 2010
3. Jul 26, 2010

### AJ Bentley

The mistake is to assume that you can apply then Lorentz contraction in one dimension(x) whilst ignoring it in the perpendicular (y).
A proper analysis of the movement of the train in both directions eliminates the problem.

4. Jul 26, 2010

### JesseM

Even if the gap is smaller than the train, won't the front of the train start tipping downwards when it reaches the gap (and when the very front has not yet reached the position where the gap ends and the tracks start again)? A detailed analysis would presumably require modeling both the normal force upward from the tracks and the "gravitational" force downward (you can't actually have gravity in special relativity, but we could consider an electromagnetically charged train with an oppositely-charged planet below, or perhaps assume that the tracks were mounted in a rocket which was continuously accelerating upward)

5. Jul 26, 2010

### starthaus

Since the gap is always much smaller than the diameter of the wheels, this doesn't happen. The wheels always make contact with either one or with two adjoining tracks. One way to convince yourself is that , in the frame of the train, the wheels always ride on the tracks, even when they go across the gaps. (otherwise you would have a very rough ride :-)). So, in the frame of the train, the normal reaction of the track against the wheels is always non-zero.

6. Jul 26, 2010

### starthaus

There is no length contraction in the direction transverse to the direction of motion.

7. Jul 26, 2010

### JesseM

I don't understand, isn't a single wheel only in contact with one track? How can it be in contact with "adjoining tracks"?
Are there "gaps" in ordinary train tracks? And if so, are you claiming that when the wheels pass over them, they don't experience any downward falling motion, even if they only have time to fall a fraction of a millimeter in the time they pass over the gap?

And if you make the gap "much smaller than the diameter of the wheels" in the train frame, but make the train move so close to light speed that the gap is larger than the entire length of the train in the platform frame, then surely the train would be moving so fast that it wouldn't have time to fall very far before reaching the other end of the gap; I trust that a full calculation would show it only fell by the same fraction of a millimeter (or whatever) as each wheel briefly fell when crossing the gap in the train's own frame. Like I said, you would need to do a detailed calculation in which the train was a non-rigid object which could flex and bend, and where you'd take into account the normal force from every point on the tracks in contact with the wheels, along with whatever force was standing in for gravity...but as long as all the laws involved are Lorentz-invariant ones, it's guaranteed that both frames should agree on the maximum vertical displacement experienced by each wheel in crossing the gap.

8. Jul 26, 2010

### starthaus

When the wheel goes ove the gap between two adjoining tracks it makes contact with both tracks at the same time.

Not "in". "Between" adjoining sections. This is what the OP is about.

Yes, the wheels will go down by a fraction of mm. How is this relevant to your claim that the car will tip between the gaps? In the frame of the train, the wheels are in contact with the rails at all times

The point is that the train will not fall at all since the normals on the wheels is non-null.

You don't need that. You know that the normal reaction on the wheels is non-null in either the train frame (this is obvious), neither in the track frame (this is derived from analyzing the normal forces to the wheels). In other words: the wheels rest on the rails in both the train frame and in the track frame. The train doesn't fall thruogh the gaps in the train frame, neither does it fall in the track frame. This completely resolves the false "paradox"

Last edited: Jul 26, 2010
9. Jul 26, 2010

### JesseM

My claim was originally based on the assumption the gap was substantial in both frames, the OP didn't say anything about the gap being much smaller than the diameter of the wheel, that seems to be a new condition that you are introducing. In any case I think my statement is still correct, since if the frontmost wheels dips down by a fraction of a mm, then the front of the train will tip down slightly. "Tip" did not imply that the front of the train would fail to continue to ride the tracks after crossing the gap.
If the wheels dip down by a fraction of a mm, surely the train car connected to the wheels does as well? I didn't mean "fall" to imply free-fall, any small reduction in the normal force upwards resulting in net downward acceleration would count, even if brief and soon countered by an upward acceleration.

10. Jul 26, 2010

### starthaus

No, I am not introducing it, it is a fact of life.

Irrelevant, there is no way the train can pass through the gap since the normal reaction to the wheel is non-null in both frames at all times.

True but irrelevant. The train can't go throgh the gap. The non-null reaction of the track prevents it from doing that. This is not a kinematics problem, this is a static mechanics problem.

11. Jul 26, 2010

### yossell

Given our slow moving trains, yes. But I think the point of the paradox is that, if there is any gap at all then, from the stationary person's point of view, since there is no positive lower bound on the degree to which something can be lorentz contracted, if the train reaches a great enough speed its wheels (and even the train itself) will be (relative to the stationary observer) narrower than the gap - just by Lorentz contraction.

Accordingly, for a very short while, the train is unsupported in the observer's frame.

12. Jul 26, 2010

### starthaus

This is false. You are trying to solve this problem as a kinematics problem (which it isn't).

13. Jul 26, 2010

### curiousphoton

Guys,

Isn't this thought experiment just a version of Einstein's original though experiment that led him to SR? You know the one where he visualized someone dropping a rock out of a window of the train? To the person in the train, the rocks path is curved. To a person watching the train go by, the rocks path is straight down. Both make different observations / conclusions so who is right? According to Einstein, they BOTH are.

Back to the current thought experiment. The train will experience some downward displacement...the amount depending on your RF.

14. Jul 26, 2010

### JesseM

It's a thought-experiment, not life--in real life it's not practically possible for a train to travel at relativistic speeds! If one wants to consider a thought experiment where a large section of the track has been removed so the gap is half the size of the train, that's a perfectly legitimate thought-experiment which relativity should be able to deal with too.
Merely saying it's "non-null" is not sufficient to show both frames make precisely the same predictions, and it does not address the thought-experiment where the gap is large enough that a wheel can lose all contact with the tracks for some period of time, which may well have been what the OP was imagining.
Again "non-null" is too vague, if the normal force upward is less than the force downwards the wheel will accelerate downwards, if the wheel was sufficiently flexible it might squeeze through the gap like a glob of pudding, but more realistically once different points on the wheel are in contact with either side of the track at the front and back of the gap, the normal force will increase to balance out the downward force so the wheel can't go down any further.

15. Jul 26, 2010

### starthaus

In the frame of the train the train does not exhibit any "downward displacement" let alone fall through the gap between rails. Relativity forbids a different outcome of this experiment in the frame of the track (or any other inertial frame), i.e. the train does not fall through the gap no matter how large its speed wrt the tracks.

16. Jul 26, 2010

Staff Emeritus
This is a relativity of simultaneity problem:

Train frame order of events:
1. Front wheels enter gap
2. Front end of train begins to dip
3. Front wheels catch the tracks on the far side
4. Front end of train straightens out
5. Rear wheels enter gap
6. Rear end of train begins to dip
7. Rear wheels catch the tracks on the far side
8. Rear end of train straightens out

Station frame order of events:
1. Front wheels enter gap
2. Front end of train begins to dip
3. Rear wheels enter gap
4. Rear end of train begins to dip
5. Front wheels catch the tracks on the far side
6. Front end of train straightens out
7. Rear wheels catch the tracks on the far side
8. Rear end of train straightens out

17. Jul 26, 2010

### M Grandin

Maybe a simple answer is that SR is about constant velocities without accelerations.
But passing a gap, falling down somewhat due to gravity, implies acceleration - and therefore no longer SR applies. GR may solve the "paradox". :grumpy:

18. Jul 26, 2010

### Al68

LOL. It seems obvious that the intent of the OP was that the gap is large enough so that the scenario is meaningful as an SR thought experiment.

Why don't we just assume a train rest length of 100 meters, a gap of 80 meters rest length, and v=0.8c. I think that would meet the intent of the OP.
Yes, this is just a variation of the barn/pole paradox. The only thing the frames will disagree on is when each part of the train "dropped".

Last edited by a moderator: Jul 26, 2010
19. Jul 26, 2010

### yossell

I respectfully disagree on two counts. Firstly, I'm not trying to solve it, I'm just trying to present the problem.

Secondly, the formula for length contraction is: $$L' = L\sqrt{1-\frac{v^2}{c^2}}$$
It's clear that, by a high enough v, you can make L' as small as you like. So, if there's some space c between the tracks, then there's some v such that the train is contracted to a length smaller than c. So, for a period of time, it's unsupported.

Vanadium's points about relativity of simultaneity explain why you can't argue from the fact that there's always contact in train's frame to there always being contact in stationary frame.

edit: Beaten again. Jeez

20. Jul 26, 2010

### JesseM

Not true, SR can deal with accelerations fine, it's only gravity and curved spacetime that requires GR. You can analyze the motion of an accelerating object from the perspective of an inertial frame, and even if you choose to use an accelerating frame, the modern perspective is that this is still part of "SR" as long as there is no spacetime curvature. See this section of the Usenet Physics FAQ for more info.