# Time to the train stop

## Main Question or Discussion Point

Lets have a train moving to the right side in the shown direction of the arrow.

From a point near the end A, 2 rays of light are emitted toward both ends A and B so as they appear to reach both ends at the same time for an external observer.

At the moment the 2 rays reaches the 2 ends at the same time relative to the external observer, the train stops moving.

For the train observer, both rays reach both ends at different times. More precisely, the light ray reaches A before B.

The question, will the 2 observers record 2 different times of the train stopping? This is given that they both know that their train is moving.

Alternatively, if they interpret that the platform is moving in the opposite direction to the indicated arrow in the figure, will they record 2 different times of platform stopping?

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Lets have a train moving to the right side in the shown direction of the arrow.

From a point near the end A, 2 rays of light are emitted toward both ends A and B so as they appear to reach both ends at the same time for an external observer.

At the moment the 2 rays reaches the 2 ends at the same time relative to the external observer, the train stops moving.

For the train observer, both rays reach both ends at different times. More precisely, the light ray reaches A before B.

The question, will the 2 observers record 2 different times of the train stopping? This is given that they both know that their train is moving.

Alternatively, if they interpret that the platform is moving in the opposite direction to the indicated arrow in the figure, will they record 2 different times of platform stopping?
What kind of work did you do to solve the problem? Can you post your equations?

ghwellsjr
Gold Member
Lets have a train moving to the right side in the shown direction of the arrow.

From a point near the end A, 2 rays of light are emitted toward both ends A and B so as they appear to reach both ends at the same time for an external observer.

At the moment the 2 rays reaches the 2 ends at the same time relative to the external observer, the train stops moving.

For the train observer, both rays reach both ends at different times. More precisely, the light ray reaches A before B.

The question, will the 2 observers record 2 different times of the train stopping? This is given that they both know that their train is moving.

Alternatively, if they interpret that the platform is moving in the opposite direction to the indicated arrow in the figure, will they record 2 different times of platform stopping?
I think you have already answered your own question. You said that the train stops when the light reaches both ends. And then you said for the external observer, this happens at one time and for the train observer, this happens at two different times, so how can it be that the train could stop at the same time for both observers?

More precisely, when you talk about the external observer, you mean according to the Inertial Reference Frame in which the external observer, or the platform, is at rest and when you talk about the train observer, you mean the IRF in which the train is moving prior to it stopping.

To make things clear, it helps to draw spacetime diagrams and use a specific scenario. Let's consider the train to be moving at 0.6c and to have a length in the platform IRF while it is moving of 5000 feet and we'll put the light source at 4000 feet from the rear of the train. In this spacetime diagram, I've shown the ends of the platform in green, the front of the train in black, the rear of the train in red, and the light source in blue:

As you can see, in the platform IRF, all parts of the train stop at the same time, at the Coordinate Time of 10 microseconds.

Now we can transform the above IRF to the train IRF by applying the Lorentz Transformation to all the Coordinates and we get this spacetime diagram:

Now you can see that the different parts of the train stop at different Coordinate Times ranging from 4.25 to 8 microseconds.

Does that clarify everything for you?

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As the different parts of the train stop at different times, then the part B is still moving while the part A comes to a stop.

Now, suppose, there is a coil stretched between A and B. So the B -end of the coil moves toward the A-end of the string when A comes to a stop. That creates an extra tension on the coil to exceed its breaking point and then it breaks down. However, for the platform observer both ends come to a stop at the same time and then no extra-tension or breaking down of the coil should occur.

I drew a diagram quoted from yours. I fainted out the line colours but leaving the black and the red to represent A and B time-lines, respectively. The first one shows the time when A stops but B is still moving. The second one, I rotate the diagram clock-wise to bring the A at the origin with its time-line is vertical.

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It is a mistake to think there is no extra tension from the point of view of the platform. In fact, due to Lorentz contraction the observer at the platform would expect the train to change length when it stops. That 's not happening which means the train must be under stress either before it stops or after it stops (or both).

ghwellsjr
Gold Member
You're welcome.

As the different parts of the train stop at different times, then the part B is still moving while the part A comes to a stop.
This isn't a good way to say what is happening. In the IRF in which the train is originally at rest as depicted in my second drawing above, part B is still at rest while part A suddenly moves towards it. So the train is getting compressed, starting from the front and extending toward the rear. This is kind of like a train parked on the track experiencing a head on collision from another train coming toward it and then dragging it down the track.

Now, suppose, there is a coil stretched between A and B. So the B -end of the coil moves toward the A-end of the string when A comes to a stop. That creates an extra tension on the coil to exceed its breaking point and then it breaks down.
Again, this is backwards: the B-end remains stationary while the A-end suddenly moves toward it. This would create a compression, not a tension. Whether or not anything breaks is not an issue that Special Relativity can resolve. However, if it breaks in one frame, it breaks in all frames.

However, for the platform observer both ends come to a stop at the same time and then no extra-tension or breaking down of the coil should occur.
In the platform frame, the train, because it originally is moving, is Length Contracted. It's Proper Length is longer and can be determined by a platform observer. When all parts of the train are brought to a stop simultaneously, the Proper Length instantly changes from its original longer length to a shorter length. This will create compression but Special Relativity can't say whether or not anything breaks. That has to be answered by a materials or structural analysis.

I drew a diagram quoted from yours. I fainted out the line colours but leaving the black and the red to represent A and B time-lines, respectively. The first one shows the time when A stops but B is still moving. The second one, I rotate the diagram clock-wise to bring the A at the origin with its time-line is vertical.
I have no idea what your drawings are trying to depict which means you probably have no idea what mine are trying to depict.

The length of the train, which is equal to the length of the coil, as seen by the external observer is shorter when the train is moving than when the train comes to a stop. So when the train stops, the length contraction effect disappears and the coil length increases suddenly as seen by the external observer. This should cause tension on the coil not compression.

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ghwellsjr
Gold Member
The length of the train, which is equal to the length of the coil, as seen by the external observer is shorter when the train is moving than when the train comes to a stop. So when the train stops, the length contraction effect disappears and the coil length increases suddenly as seen by the external observer. This should cause tension on the coil not compression.
What you are describing could happen when a train stops in an unrestrained way which would result in different parts of the train stopping at different times in the platform IRF but you said the different parts of the train stop simultaneously in the platform IRF. You can look at my first drawing to see that the length of the train under that specification does not change in the platform IRF.

In order for all parts of a train (or any object) to stop simultaneously according to a frame, there must be something like clamps set up all along the track which simultaneously stop the train (by preprogrammed timers) that bring all the parts to a halt (in that frame) and keep the different parts from expanding back to their natural length.

I can't help it if you specified an unrealistic scenario but that's what you did and Special Relativity doesn't address the realisticness of a scenario. That's why we can specify instantaneous accelerations (or decelerations) and see what happens in different IRF's even if they can't actually happen in reality.

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ghwellsjr
Gold Member
Can I offer you a different scenario that may bring out the effects you want?

Consider two separate trains traveling down a track at the same speed with a coil stretched between them and then both trains suddenly stop simultaneously in the platform frame.

1 person
Can you tell the difference between the 2 scenario?

Here is my prepared reply on your previous post (No. 8) which I just about to post before you came with the new scenario...

"The set up aims to investigate the possible discrepancy between the observation of the coil length and the force exerted on it in the two spaces. Whether there is a mechanical constraint during stopping the train or not, this is not the issue. Because the same will be still held if we only use the coil alone as our set up and its two ends come to stop simultaneously relative to the external observer.

So the matter of force exerted on coil which appears to be tension for the external observer and compression for the train observer is still not clear for me."

ghwellsjr
Gold Member
Can you tell the difference between the 2 scenario?

Here is my prepared reply on your previous post (No. 8) which I just about to post before you came with the new scenario...

"The set up aims to investigate the possible discrepancy between the observation of the coil length and the force exerted on it in the two spaces. Whether there is a mechanical constraint during stopping the train or not, this is not the issue. Because the same will be still held if we only use the coil alone as our set up and its two ends come to stop simultaneously relative to the external observer.

So the matter of force exerted on coil which appears to be tension for the external observer and compression for the train observer is still not clear for me."
OK, then we'll go with my new scenario.

Here is a spacetime diagram showing a coil stretched between two locomotives. We assume that the coil can both get shorter (because it has been previously stretched) or get longer (because it can stretch even more) without breaking. And because we have an unrealistic scenario (instantaneous deceleration), we have to assume that the forces propagate at the speed of light (otherwise, the coil will break as soon as one part of it decelerates instantly). Some of the midparts of the coil are shown in grey. They start out separated by a thousand feet in the platform frame:

As you can see, all the midpoints continue to travel at 0.6c when the two endpoints suddenly stop. The coil is compressed starting from the front (black end on the right) until the forces, propagating at the speed of light as shown by the thin black line, cause the midpoints to respond. Special Relativity cannot provide an answer to how they respond. We have to make that up in our unrealistic scenario. To make things reasonably simple, I have assumed that the various midpoints respond by changing direction until the forces from the trailing end of the coil (red end on the left) have a chance to further constrain the coil.

In a realistic scenario, a coil (or any other structure that is not destroyed) will oscillate or vibrate until the motions all dampen out and they all come to rest with respect to one another as constrained by their two endpoints. This would take a very long time, not just a few microseconds. But I just want to give you the flavor of what happens. Although I show a large tension or stretching on the left side, in a realistic scenario, this tension would be intertwined with compression resulting in the oscillations or vibrations.

But the final result will be a more or less equal compression (actually, less stretching) of all parts of the coil from one end to the other. It will not end up all in tension. Again, these are issues that Special Relativity cannot address, they have to be determined by materials analysis. But once we decide how the midpoints oscillate during the dampening process, we can transform to any other frame, including the frame in which the train was at rest in the beginning of the scenario:

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PeterDonis
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2019 Award
The length of the train, which is equal to the length of the coil, as seen by the external observer is shorter when the train is moving than when the train comes to a stop.
Not according to the way you specified the problem. The way you specified the problem, both ends of the train stop at the same time in the frame in which the train is originally moving (call this frame O). So in frame O, then length of the train does *not* change. In the frame in which the train starts out at rest (call this frame T), the two ends of the train change their state of motion (note that they *start* moving in this frame, so you mis-stated this part earlier) at different times, so the length of the train changes in frame T. (Btw, you should think carefully about *how* the train's length will change in frame T; you should conclude that it will get *shorter*.)

In this version of the problem, the coil will be under tension at the start of the experiment (while the train is moving in frame O), and the tension will be completely removed at the end of the experiment (when the train is at rest in frame O).

So when the train stops, the length contraction effect disappears and the coil length increases suddenly as seen by the external observer.
This is a *different* problem specification than your original one. In this version of the problem, the train's length changes in frame O, which means the two ends must stop moving (with respect to frame O) at different times, in just the right way to undo the length contraction effect of the train's motion, in frame O. So in this version of the problem, the coil will be slack (no tension or compression) the whole time; its apparent length will increase in frame O, but this is purely due to the change in relative motion and does not correspond to any change in the internal forces in the coil.

This explanation of how the coil does react when the train stops will still hold even if there is no effect of special relativity. In other words, the compression wave which starts from A end and the tension wave which starts from B end will still be observed for the ground observer as well as for the train observer because it is related to the inertia of the coil even if there no SR effect and both ends of the train stop at the same time relative to the train observer. But my query is another compression force which is initiated by the relative motion of A relative to B when A stops first for the train observer which is not paralleled relative to the ground one. Because for the ground observer, there is no relative motion of A toward B when the train comes to a stop.

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Not according to the way you specified the problem. The way you specified the problem, both ends of the train stop at the same time in the frame in which the train is originally moving (call this frame O). So in frame O, then length of the train does *not* change. In the frame in which the train starts out at rest (call this frame T), the two ends of the train change their state of motion (note that they *start* moving in this frame, so you mis-stated this part earlier) at different times, so the length of the train changes in frame T. (Btw, you should think carefully about *how* the train's length will change in frame T; you should conclude that it will get *shorter*.)

In this version of the problem, the coil will be under tension at the start of the experiment (while the train is moving in frame O), and the tension will be completely removed at the end of the experiment (when the train is at rest in frame O).

This is a *different* problem specification than your original one. In this version of the problem, the train's length changes in frame O, which means the two ends must stop moving (with respect to frame O) at different times, in just the right way to undo the length contraction effect of the train's motion, in frame O. So in this version of the problem, the coil will be slack (no tension or compression) the whole time; its apparent length will increase in frame O, but this is purely due to the change in relative motion and does not correspond to any change in the internal forces in the coil.
what are the 2 versions of the problem? In my version, I did not speak about the time the train starts moving. I started from an originally moving train relative to a ground observer. Would you please correct me here: The length of a moving train relative to a ground observer is shorter compared with the length when the train comes to a stop relative to the same observer! Was this the meaning of length contraction?

ghwellsjr
Gold Member
This explanation of how the coil does react when the train stops will still hold even if there is no effect of special relativity. In other words, the compression wave which starts from A end and the tension wave which starts from B end will still be observed for the ground observer as well as for the train observer because it is related to the inertia of the coil even if there no SR effect and both ends of the train stop at the same time relative to the train observer. But my query is another compression force which is initiated by the relative motion of A relative to B when A stops first for the train observer which is not paralleled relative to the ground one. Because for the ground observer, there is no relative motion of A toward B when the train comes to a stop.
You're making progress because in post #4 you said there was extra tension in the final state of the coil:
Now, suppose, there is a coil stretched between A and B. So the B -end of the coil moves toward the A-end of the string when A comes to a stop. That creates an extra tension on the coil to exceed its breaking point and then it breaks down. However, for the platform observer both ends come to a stop at the same time and then no extra-tension or breaking down of the coil should occur.
...and now you are realizing that the coil ends up in compression.

So if there was no compression or tension in the coil at the start of the scenario, in other words the two locomotives cannot measure any force on their attachments to the coil, then they will both measure a compression or a force trying to push the locomotives apart at the end of the scenario. It doesn't matter what frame we desire to describe the scenario in, they all will come to the same conclusion, although the "explanations" may be different.

So in the "train" frame, that is, the frame in which the train is originally at rest, we might want to "explain" by saying that the A end starts moving toward the B end before the B end starts moving and so the two ends come closer together and in the "platform" frame, we might "explain" by saying that the Proper Length of the coil is 6250 feet originally and 5000 feet finally so it experiences compression.

ghwellsjr
Gold Member
Would you please correct me here: The length of a moving train relative to a ground observer is shorter compared with the length when the train comes to a stop relative to the same observer! Was this the meaning of length contraction?
No, that is not the meaning of Length Contraction. Length Contraction is the ratio of the length of an object in a frame in which the object is moving compared (or divided by) the length of the object in a frame in which the object is not moving. It has nothing to do with the length of an object before and after it experiences an acceleration. As I said before, Special Relativity cannot address that issue, that is a materials or structural issue. It's no different than asking the question of how the length of a nail changes when you hit it with a hammer. Special Relativity cannot answer that question.

If you look at my diagrams, there are several examples of Length Contraction:

The length of the platform (depicted by the two green lines) in its rest frame is 7000 feet. In the frame in which it is moving at 0.6c, its length is 5600 feet.

The original length of the train in its rest frame is 6250 feet. In the frame in which it is orginally moving, its length is 5000 feet.

The final length of the train in the frame in which it is not moving is 5000 feet. In the frame in which it ends up moving, its length is 4000 feet.

In all these cases, we can calculate the ratio of the contracted length to the Proper Length (the length in its rest frame) as the inverse of gamma. But this only works for an object during periods of time when it is inertial, that is, not experiencing any acceleration or vibration anywhere along its length. In other words, in the frame in which the train is originally at rest, the length contraction formula does not work during the time that one end of the train starts moving until the other end starts moving.

PeterDonis
Mentor
2019 Award
what are the 2 versions of the problem?
Version #1 is the one you explicitly described; the key thing about it, as I note below, is that you explicitly specified that both ends of the train stop moving at the same time, relative to the ground observer.

Version #2 is the one you implicitly switched to when you said that when the train stops, "length contraction disappears", i.e., the length of the train, as measured by the ground observer, changes when it stops. For this to happen, the two ends of the train *cannot* stop moving at the same time, relative to the ground observer; if they do, as I note below, the length of the train, relative to the ground observer, cannot change.

In my version, I did not speak about the time the train starts moving.
Yes, you did, although you didn't realize you were doing so. In the frame of the ground observer, the train is moving, and then stops moving. But in the frame of the train observer (the one who starts out moving along with the train), the train is at rest, and then starts moving.

Would you please correct me here: The length of a moving train relative to a ground observer is shorter compared with the length when the train comes to a stop relative to the same observer!
It depends on how the train stops moving. In your scenario, you explicitly *specified* that both ends of the train stop moving at the same time, relative to the ground observer. That means the length of the train, as measured by the ground observer, *cannot change*.

No. As ghwellsjr pointed out, length contraction means the length of a moving train as measured by a ground observer while it is moving relative to the ground observer, is shorter than the length of the same train as measured by an observer at rest on the train, *while it is moving* relative to the ground observer. In other words, the two measurements must measure the train while it is in the same state of motion. Stopping the train changes its state of motion.

and in the "platform" frame, we might "explain" by saying that the Proper Length of the coil is 6250 feet originally and 5000 feet finally so it experiences compression.
You said that for the ground observer, he will not see any change in the train length of 5000 feet before and after stop according to your first diagram. So the compression wave which starts at A for the train observer should not be seen by the ground observer! at the time the train stops. If the ground observer has to see any compression, he should observe this force when the train starts moving. In this case the proper length of 6250 drops down to 5000 feet during moving.

In this version of the problem, the coil will be under tension at the start of the experiment (while the train is moving in frame O), and the tension will be completely removed at the end of the experiment (when the train is at rest in frame O).
Why will the coil be under tension at the start?

This is a *different* problem specification than your original one. In this version of the problem, the train's length changes in frame O, which means the two ends must stop moving (with respect to frame O) at different times, in just the right way to undo the length contraction effect of the train's motion, in frame O. So in this version of the problem, the coil will be slack (no tension or compression) the whole time; its apparent length will increase in frame O, but this is purely due to the change in relative motion and does not correspond to any change in the internal forces in the coil.
Why does not the change in the relative motion cause any change in the internal force of the coil?

In this version of the problem, the coil will be under tension at the start of the experiment (while the train is moving in frame O), and the tension will be completely removed at the end of the experiment (when the train is at rest in frame O).
Why will be a tension at the start?

the coil will be slack (no tension or compression) the whole time; its apparent length will increase in frame O, but this is purely due to the change in relative motion and does not correspond to any change in the internal forces in the coil.
How the change of the relative motion between the 2 ends of the coil will not affect the internal force of the coil?

In other words, the two measurements must measure the train while it is in the same state of motion. Stopping the train changes its state of motion.
So how does the length contraction work from the beginning of the motion as long as both train ends can not be in the same state of motion relative to all observers?

In other words, the two measurements must measure the train while it is in the same state of motion. Stopping the train changes its state of motion.
So how does the length contraction work in the beginning of train motion as long as both train ends can not be in the same state of motion relative to all observers?

And one important question: How does the length contraction work from the beginning of the motion as long as both train ends can not be in the same states of motion relative to all observers?

PeterDonis
Mentor
2019 Award
Why will the coil be under tension at the start?
I should have said, it will be under tension at the start, if we *assume* that it is under zero tension at the end, when the train is at rest in the observer's frame. But you could alternately assume that the coil was under zero tension at the start, which would lead to it being under *compression* at the end, when the train is at rest in the observer's frame. The key point is that the internal stress has to go from more to less tension (or less to greater compression) in this version of the scenario.

To see why, consider what is happening in this scenario when the train stops. As the scenario is specified in this version, both ends of the train stop at the same time in the observer frame. Suppose that after the ends stop, the train is under zero internal stress. The stopping operation does not change the length of the train (in this version) as measured in the observer's frame; call this length L. But that means that the "natural", or unstressed, length of the train before it stopped must have been smaller than L, with respect to the observer, because of length contraction. So in order for the actual length of the train to be L when the train is moving, relative to the observer, the train must be under tension, because its actual length is greater than its unstressed length.

Why does not the change in the relative motion cause any change in the internal force of the coil?
Because in this version, the two ends of the train stop at *different* times in the observer frame, and the times are related in just the right way to keep the train at zero internal stress, by changing the length of the train, as measured in the observer frame, so that it is always equal to its unstressed length, in that frame, taking length contraction into account.

PeterDonis
Mentor
2019 Award
So how does the length contraction work from the beginning of the motion as long as both train ends can not be in the same state of motion relative to all observers?
Length contraction is not well-defined relative to a given observer if the two ends of the train are in relative motion, with respect to that observer. That's because the train's "length" itself is not well-defined, relative to that observer, if the two ends of the train are in relative motion.

Bear in mind that length contraction, like "length" itself, is a derived phenomenon in relativity; it's not fundamental. The fundamental objects are the worldlines of the parts of the train, which are invariant curves in spacetime and can be described without even choosing a reference frame. Length contraction, time dilation, relativity of simultaneity, etc., are not necessary to describe the physics; the only reason we talk about them is that our minds are evolved to perceive things in these terms.