A relativity problem (train & platform)

[Ookke]

This is a "paradox-type" relativity problem that I cannot figure out. Hope it's OK to post this here. Sorry for English errors, it's not my native language. Here it goes:

Railway. Platform (1 km) by side of it. Train (100m).

Train approaches platform very high velocity, so that (in train's reference frame) the platform is only 10m long. The personnel in train can calculate the moment when the mid-point of train is exactly at the mid-point of the platform. Train brakes are set to operate at that moment. All cars of the train will start braking at the same time (all cars "live the same time", at least until the braking). No messages need to be passed at the time of the braking. Deceleration is very strong, but not "infinitely" strong (we can allow a small stopping distance for the train, few centimeters for example).

Now, what is happening from the viewpoint of different observers?

An observer at the platform will see very short train approaching, braking and stopping at the mid-point of the platform, and "stretching" to its original 100m length.

But an observer at the first car of the train will observe very different event. Short (10m) platform approaches and passes him by. No braking yet. Instead, the passenger sees terrain beyond the platform. He can even leave mark there (say, toss a candy paper out of the window). After a short but non-zero period of time, braking starts. Then he finds himself in a train, at halfway of 1km long platform.

How can it be? In platform's reference frame train never passed the platform, not even reached the end of the platform. In passenger's reference frame train went past the platform, and the passenger even left a mark there. The mark (candy paper) should be there. How can it be, if the train never passed platform? What went wrong?

 

[actionintegral]

You have no problem understanding the "stretching" of the train, but you have a problem with the "stretching" of the platform and terrain. Why?

 

[Ookke]

You have no problem understanding the "stretching" of the train, but you have a problem with the "stretching" of the platform and terrain. Why?

Stretching (when the length contraction ceases) of the platform or terrain is not itself a problem. However, I find it strange that the train passes by the platform in one reference frame, and not in another. The train even leaves a mark on the "other side" of the platform. From the platform's perspective, the train never was there.

 

[Farsight]

Ooke: Think in terms of a plane. When the plane is traveling fast, imagine it's flying high. The runway down below looks small, maybe 10m long, because of perspective. And the plane when viewed from the runway also looks small, maybe 1m long. When the plane is exactly halfway along the runway (not beyond the runway), it stops very quickly. This is like descending rapidly to the ground. Then the runway looks 1000m long and the plane looks 100m long. The perspective has gone, and now everything looks normal sized. Your relativistic train exhibits a different type of "perspective" to the plane, but it doesn't create a paradox.

 

[actionintegral]

I hope that I am not misunderstanding you. But I see a perfect symmetry between the platform and the train.

Paradoxed notwithstanding, I like the fact that you are exploring how frames "snap back" to a rest frame. But personally I don't feel comfortable starting out with such a complicated example.

 

[Ookke]

Think in terms of a plane.

The plane-analogy may work, if we think the length contraction just "seen" (more or less imaginary) phenomenon. But if I've understood correctly, the length contraction is real, as real as can be. The front-end of the train really goes past the platform (in train's frame). This is not the case with the plane.

 

[Farsight]

Sure the length contraction is real Ookke. But the front of the plane didn't go beyond the runway, and the front of the train didn't go beyond the platform. The train stops half way along the platform, like you said.

 

[Doc Al]

Train approaches platform very high velocity, so that (in train's reference frame) the platform is only 10m long. The personnel in train can calculate the moment when the mid-point of train is exactly at the mid-point of the platform. Train brakes are set to operate at that moment. All cars of the train will start braking at the same time (all cars "live the same time", at least until the braking).

"At the same time" only in the frame of the train.

No messages need to be passed at the time of the braking. Deceleration is very strong, but not "infinitely" strong (we can allow a small stopping distance for the train, few centimeters for example).

Now, what is happening from the viewpoint of different observers?

According to the platform observers, clocks in different parts of the train are wildly out of synch. Platform observers see the rear of the train brake long before the front brakes engage. (According to platform observers, when the rear brakes are applied, the front of the train is still moving fast as if nothing happened.)

 

[Ookke]

Farsight: The front of the train never goes beyond the platform, in platform's reference frame. But how about the train's frame?

 

[JesseM]

Farsight: The front of the train never goes beyond the platform, in platform's reference frame. But how about the train's frame?

The front of the train certainly does pass the end of the platform--as you said, a rider at the front could even leave a mark at some point past the end, different frames cannot disagree about events which occur at a single place and time like the front of the train reaching a particular spot on the ground past the platform. Doc Al gave you the correct answer, it has to do with the fact that simultaneous braking in the train's frame doesn't look simultaneous in the platform's frame. (are you familiar with the relativity of simultaneity?) In the platform's frame, the back of the train starts braking before it even reaches the platform, the midpoint of the train starts breaking right as it passes the midpoint of the platform, and the front of the train starts breaking well after it has passed the platform.

 

[Ookke]

According to the platform observers, clocks in different parts of the train are wildly out of synch. Platform observers see the rear of the train brake long before the front brakes engage. (According to platform observers, when the rear brakes are applied, the front of the train is still moving fast as if nothing happened.)

No doubt there are out-of-sync issues, but I cannot see how the relativity of simultaneity is involved here.

Shouldn't the question "Does the front of the train go beyond the platform" be yes or no? Distances may be relative, but either it goes or not. Now it seems to do both.

 

[JesseM]

Shouldn't the question "Does the front of the train go beyond the platform" be yes or no?

Yes, it should be and it is. All frames agree that the front of the train goes past the platform, because in the platform's frame the different parts of the train begin braking at different times, and the front doesn't begin until long after the middle began, at which point the front has actually passed the end of the platform completely.

 

[Ookke]

All frames agree that the front of the train goes past the platform, because in the platform's frame the different parts of the train begin braking at different times, and the front doesn't begin until long after the middle began, at which point the front has actually passed the end of the platform completely.

Now it makes sense! Everything is not relative: the front of the train goes beyond the platform, in all frames. We might even say that the front takes some "reverse" during braking, of course a point-of-view issue.

I'd like to watch this kind of train arrival!

 

[actionintegral]

I I like the fact that you are exploring how frames "snap back" to a rest frame. But personally I don't feel comfortable starting out with such a complicated example.

I quote myself...

 

[jpr0]

there is a page with a nice explanation of something similar here:

http://galileoandeinstein.physics.virginia.edu/lectures/sreltwins.html

 

[actionintegral]

there is a page with a nice explanation of something similar here:

http://galileoandeinstein.physics.virginia.edu/lectures/sreltwins.html

Unless I skimmed it too quickly, that's not the same thing at all! Ookle's idea is waiting until the train is in the tunnel and then SLAMMING ON THE BRAKES! What a cool idea! The lorentz contraction stops suddenly and the train busts out of both ends of the tunnel! We could call it the Lorentz explosion!

 

[JesseM]

Unless I skimmed it too quickly, that's not the same thing at all! Ookle's idea is waiting until the train is in the tunnel and then SLAMMING ON THE BRAKES! What a cool idea! The lorentz contraction stops suddenly and the train busts out of both ends of the tunnel! We could call it the Lorentz explosion!

Well, in the frame of the tunnel, the train wouldn't bust out of both ends, instead the back end would begin to brake before it had entered the tunnel, the middle would begin to brake as it passed through the middle of the tunnel, and the front end wouldn't begin to brake until it had already passed completely through the tunnel and come out the other side. How exactly the train would stretch in this period would depend on the details of how it was accelerating--a "natural" way to do it might be to have the train accelerate in such a way that its length in the instantaneous inertial rest frame of its midpoint at any given moment was always 100m, and also to accelerate in such a way that the G-forces felt by an observer sitting in the middle of the train would feel constant throughout the acceleration. In this case I think the problem would be one of "Born rigid motion" discussed by pervect in post #22 of this thread and analyzed in more detail here:

http://www.mathpages.com/home/kmath422/kmath422.htm

 

[actionintegral]

A further simplification would involve two unconnected moving observers with synchronized clocks agree to 'slam on the brakes' at a predetermined time. After that phenomenon is analyzed, you could introduce some rod in the rest frame and play around with it.

 

[Ookke]

If there is a wall at the end of the platform, and the brakes are set as described (operate when mid-points meet), what will happen?

Front of the train never makes it to braking (collides, instead), the rest of the train brakes. After full-stop, the train is at midway of the platform, with broken nose. Right?

 

[actionintegral]

If you look at my simplification, you will see that you are assuming the
train is rigid. That is an invalid assumption.

 

[JesseM]

If you look at my simplification, you will see that you are assuming the
train is rigid. That is an invalid assumption.

In what way is Ookke assuming the train is rigid? Anyway, you can make the assumption of "Born rigidity" which I mentioned above, even if you are using your simplification of unconnected clocks traveling alongside each other and accelerating according to some preset schedule (you can time each clock's acceleration such that their distance in their instantaneous inertial rest frame at each moment remains constant).

 

[Farsight]

Ooke: the front of the train doesn't really pass the end of the platform. If it did, and if all parts of it stopped instantly, you end up with a train 10 kilometres long. The two are not in the same reference frame, they each have a different perspective on space and each other. Talking about the front of the train "passing" the front of the platform assumes they're in the same reference frame, and the result is you have your cake and eat it and then cry paradox.

 

[MeJennifer]

Ooke: the front of the train doesn't really pass the end of the platform.

Huh?

Ooke writes:

But an observer at the first car of the train will observe very different event. Short (10m) platform approaches and passes him by. No braking yet. Instead, the passenger sees terrain beyond the platform. He can even leave mark there (say, toss a candy paper out of the window). After a short but non-zero period of time, braking starts.

That implies that the front of the train factually passed the platform!

If it did, and if all parts of it stopped instantly, you end up with a train 10 kilometres long.

Sorry but that does not make any sense. How do you come to that conclusion?

Talking about the front of the train "passing" the front of the platform assumes they're in the same reference frame, and the result is you have your cake and eat it and then cry paradox.

Not true.

 

[JesseM]

Ooke: the front of the train doesn't really pass the end of the platform. If it did, and if all parts of it stopped instantly

"Instantly" in what frame? Ookke specified that all parts of the train brake at the same moment in the inertial rest frame of the train before it begins to accelerate, which necessarily means that in the platform's frame, the back brakes before it reaches the platform, the middle brakes at the moment it passes the middle of the platform, and the front brakes after it has already completely passed the platform.

Talking about the front of the train "passing" the front of the platform assumes they're in the same reference frame, and the result is you have your cake and eat it and then cry paradox.

If the train passes the front of the platform in one frame, it must pass it in all frames. Different reference frames cannot disagree about local events which occur at a particular time and spatial location, like one object passing arbitrarily close to another.

 

[Farsight]

MeJennifer:

Factual? Ooke's passenger can't admire the countryside. For one it's going by a bit fast, it's foreshortened, and the only light he'd see would be a blue shifted blob straight ahead. If he tossed a sweet wrapper out of the window it would just keep going until it hit something like a nuclear bomb.

The train is 100m long passing a platform apparently 10m long. If it stops instantly, every part of it stays where it was. So it would have to stay ten times as long as the platform, which is really 1km long. I'm saying this won't happen, because trains traveling at c stopping instantly isn't factual either.

It's the mixture of relativity and the genteel thought experiment that's causing the problems here.

 

[Farsight]

If the train passes the front of the platform in one frame, it must pass it in all frames. Different reference frames cannot disagree about local events which occur at a particular time and spatial location, like one object passing arbitrarily close to another.

No problem. The point is that if the train passed the front of the platform it passed the front of the platform. It doesn't go backwards when Ooke slams on the brakes to end up not passing the front of the platform.

Edit: sorry, I get what you mean now. I was thinking whole trains and whole platforms. Apologies. Perhaps we should resurrect the collision to guarantee local events. A platform ending with buffers perhaps?

 

[JesseM]

The train is 100m long passing a platform apparently 10m long. If it stops instantly, every part of it stays where it was.

Ookke didn't say it stopped instantly. If it did, though, it would indeed have to stretch. In these sorts of thought-experiments it's always easier to think about a set of separate objects representing the front, back and middle of the train which accelerate according to some predetermined schedule, with no rigid structure connecting them. So if they each stopped instantly in the frame where they were originally at rest before accelerating, that would mean the object representing the back stopped before reaching the platform, the object representing the middle stopped at the center of the platform, and the object representing the end stopped after passing completely by the platform.

 

[MeJennifer]

Factual? Ooke's passenger can't admire the countryside. For one it's going by a bit fast, it's foreshortened, and the only light he'd see would be a blue shifted blob straight ahead. If he tossed a sweet wrapper out of the window it would just keep going until it hit something like a nuclear bomb.

Yes and?
The point is that he passed the end of the platform. And I fully agree with JesseM that there can not be any ambiguity about this from the perspective of another frame.

If you claim that it could not be determined then how do you claim it could be determined that the train stops in the middle?

The train is 100m long passing a platform apparently 10m long. If it stops instantly, every part of it stays where it was. So it would have to stay ten times as long as the platform, which is really 1km long.

Sorry Farshight but you are completely wrong about this. In no single scenario would the train become 1km long.

Ookke didn't say it stopped instantly. If it did, though, it would indeed have to stretch.

It does, but not to 1km.

 

[JesseM]

No problem. The point is that if the train passed the front of the platform it passed the front of the platform.

Only if you assume the train stays rigid in its own instantaneous rest frame at each moment, the "Born rigidity" condition I mentioned above. If you don't impose this condition, then the train need not stay 100 m long after it comes to a stop. If you do impose this condition, I don't think you'd see any backwards motion, you'd just see the back, middle and front of the train decelerate over three different finite periods of time, with the back end taking longer to come to rest than the middle, and the middle taking longer than the front, all coming to rest at the same moment in the platform's frame. I believe all three ends would come to rest after passing the far end of the platform, since I think the front end is already more than 100 m past the end of the platform when it begins to decelerate in the platform's frame, but I'd have to check the math to make sure of that.

 

[Farsight]

Whoa. I'm NOT saying the train ends up 10km long. I'm saying that's not factual, but is an illustration of where the looseness of the thought experiment takes us. And I'm saying that if the train stopped in the middle of the platform the front of the train DID NOT pass the front of the platform.

Don't be rude, MeJennifer.

 

[MeJennifer]

Whoa. I'm NOT saying the train ends up 10km long. I'm saying that's not factual, but is an illustration of where the looseness of the thought experiment takes us. And I'm saying that if the train stopped in the middle of the platform the front of the train DID NOT pass the front of the platform.

Don't be rude, MeJennifer.

I am not sure anymore what you are saying

Does the train pass the end of the platform: YES
Does the train get longer when it decellerates from the perspective of the frame of the platform: YES
Does the train get longer when it is stopped than the length as measured in the train's restframe: NO

Do you agree?

 

[JesseM]

Whoa. I'm NOT saying the train ends up 10km long. I'm saying that's not factual, but is an illustration of where the looseness of the thought experiment takes us. And I'm saying that if the train stopped in the middle of the platform the front of the train DID NOT pass the front of the platform.

If the train does not remain rigid, then the train can elongate, and it definitely will if each part accelerates instantaneously. And if you add something like the Born rigidity condition, the train must accelerate over an extended period, so even if the middle of the train begins to accelerate at the moment it passes the middle of the platform, it may not stop until it has passed the end of the platform.

 

[Farsight]

Yes of course I agree MeJennifer. Sorry if I wasn't very clear but please reread my posts. Thank you for that Jesse.