Train Stopping Time: Different Observers, Different Times?

[adelmakram]

For the platform observer, he will calculate the train length before and after stop to be equal to 5000 feet. In his case, he does not have to wait for the echo to calculate the distance because he is not attached to any end of the train. He can only multiply the time of the light takes to travel from one end to another by the difference between the speed of the light and the speed of the train in case the light rays goes in direction of the train and by the sum of both speeds in case the ray goes in the opposite direction of the train.

So again the question is: as long as there is no change on the length of the train before and after stop as seen by the platform observer, then there is no compression after the train stop. The phenomena that is seen by the train observer after the train stop is not seen by the platform one after the train stop. In other words, for the platform observer (the contracted length before the train stop - the true length after the train stop)=0

 

[PeterDonis]

as long as there is no change on the length of the train before and after stop as seen by the platform observer, then there is no compression after the train stop.

Wrong. You are incorrectly equating "no change in actual length" with "no compression". That is not correct. It would be correct if reality were not relativistic; in non-relativistic, Newtonian mechanics, yes, for there to be "compression" (more precisely, release of tension), there has to be a change in actual length, because the unstressed length of an object in Newtonian mechanics does not change when the object changes its state of motion. But in relativity, that's no longer the case; an object's unstressed length *does* change when it changes its state of motion. So in relativity, there can be "compression" (or release of tension) when an object changes its state of motion, even if its actual length, with respect to a particular frame, does not change.

All this has been pointed out to you several times now, so there's no point in your continuing to contradict it and our continuing to point it out again. If you have no new arguments to present, then there's not much point in further discussion.

 

[ghwellsjr]

For the platform observer, he will calculate the train length before and after stop to be equal to 5000 feet. In his case, he does not have to wait for the echo to calculate the distance because he is not attached to any end of the train. He can only multiply the time of the light takes to travel from one end to another by the difference between the speed of the light and the speed of the train in case the light rays goes in direction of the train and by the sum of both speeds in case the ray goes in the opposite direction of the train.

There are, of course, numerous ways to make measurements, but that is a side issue to the main one having to do with compression. I don't understand your method for the platform observer to measure the length of the train and I would be interested in your further clarifying it. But since we all agree on what the two observers determine the length of the train to be before and after stopping according to the two reference frames, let's focus on the compression issue right now.

So again the question is: as long as there is no change on the length of the train before and after stop as seen by the platform observer, then there is no compression after the train stop. The phenomena that is seen by the train observer after the train stop is not seen by the platform one after the train stop. In other words, for the platform observer (the contracted length before the train stop - the true length after the train stop)=0

I think maybe you are now using a definition of "compression" to be "a shortening of length". If so, then you are correct, for the platform IRF, the train does not change length and so there is no compression.

However, throughout this thread, you have used the terms "tension" and "compression" to refer to a force. Let me quote what I said in post #8:

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.

After the train stops, the train observer can get out of the train and look at the track to see if there are any clamps firmly constraining the train to its length of 5000 feet. The platform observer can hop off the platform and do the same thing.

Do you agree that there must be some sort of restraint on the train to keep its length at 5000 feet after stopping or are you maintaining that if the brakes were applied at the front and rear of the train simultaneously in the platform frame, then after all the vibrations settle down, there will be no lateral force between the wheels and the rails?

 

[adelmakram]

So in relativity, there can be "compression" (or release of tension) when an object changes its state of motion, even if its actual length, with respect to a particular frame, does not change.

All this has been pointed out to you several times now, so there's no point in your continuing to contradict it and our continuing to point it out again. If you have no new arguments to present, then there's not much point in further discussion.

But frankly speaking, I don`t see any answer yet to my question in your comments so far. You mentioned many times that there would be a force even there is no change in the length but you didn't prove that. From where this force came? How did the change in the state of motion cause a force that is not length- dependent? How if the matter is not related to a compression but to a simple dynamical system where it depends on the distance between the 2 ends and that distance changes in one frame and not in another?

 

[adelmakram]

Do you agree that there must be some sort of restraint on the train to keep its length at 5000 feet after stopping or are you maintaining that if the brakes were applied at the front and rear of the train simultaneously in the platform frame, then after all the vibrations settle down, there will be no lateral force between the wheels and the rails?

Yes I agree about this restraint. But I don`t get the meaning of lateral force. Can you please elaborate more about your reasoning?

 

[ghwellsjr]

Do you agree that there must be some sort of restraint on the train to keep its length at 5000 feet after stopping or are you maintaining that if the brakes were applied at the front and rear of the train simultaneously in the platform frame, then after all the vibrations settle down, there will be no lateral force between the wheels and the rails?

Yes I agree about this restraint. But I don`t get the meaning of lateral force. Can you please elaborate more about your reasoning?

Since you agree that there must be a restraint, then you agree that there is a lateral force (along the direction of the rails) between the rails and the wheels to keep the train from expanding back to about 6250 feet. Or, to put it another way, if there were no clamps restraining the length of the train, then even though the brakes are applied and the wheels stop turning, the wheels are ineffective in stopping the train simultaneously in the platform frame and they just slide on the rails until the train gets back near to its original Proper Length of 6250 feet (in the platform frame).

 

[adelmakram]

Since you agree that there must be a restraint, then you agree that there is a lateral force (along the direction of the rails) between the rails and the wheels to keep the train from expanding back to about 6250 feet.

This reply makes me even more confused: however there 1 urgent issue and one cold issue:

The first urgent issue is: Does that mean that every object moving with an apparent contracted length relative to an observer is under natural compression that to be relieved when it comes to a rest. If so , this means that the train or the coil in this example was under compression from the very beginning relative to the platform observer. For if this would be the case, a compression force should have started from the beginning of the motion and stayed all the time not only at the end. It also raises an important question, is the length contraction a mechanical phenomena?The less urgent issue: you illustrated before that for the train observer attached to the rear end, he will measures that the train length gets longer and then gets shorter before all vibrations dampen down. Now, during the lengthen phase which the train length increases to 7500 feet, there would be no compression but rather tension for the train observer, but for the platform observer, the force, if any, has to be applied all the time to keep the train length equals to 5000 feet.

 

[PeterDonis]

I think maybe you are now using a definition of "compression" to be "a shortening of length".

Which is *not* a correct definition, because, as I noted in a previous post, you can have "compression" (or release of tension, i.e., a change in the internal stress of the object) without any shortening of length in a particular frame. And it's clear that the OP really intends "compression" to mean "change in internal stress"; he is just incorrectly assuming that you can only have a change in internal stress if there is a change in length.

 

[PeterDonis]

You mentioned many times that there would be a force even there is no change in the length

More precisely, I said many times that there is a change in *unstressed length* even though there is no change in *actual length*. The change in unstressed length comes from a change in *internal* forces inside the object--i.e., the forces between its atoms--when its state of motion changes.

you didn't prove that.

I did not give a detailed proof because that would require constructing a detailed model of the object, atom by atom, including the internal forces between the atoms. However, I did give a sketch of how such a proof would go: internal forces between atoms are electromagnetic, and it's obvious from Maxwell's Equations that electromagnetic forces change when an object's state of motion changes.

From where this force came?

If you mean, where do the internal forces between the atoms come from, surely that's obvious: if they weren't there, the train would not hold itself together as a single object. It would just be a cloud of atoms flying around. So obviously there must be *some* force holding the atoms together. It turns out, when you look at the details, that that force is electromagnetic.

If you mean, where do the forces that change the train's state of motion come from, well, you're the one that specified the problem as involving a change in the train's state of motion, so if you don't know, how am I supposed to know? I was assuming, as was ghwellsjr, that brakes were applied at each end of the train.

How did the change in the state of motion cause a force that is not length-dependent?

The force doesn't have to be length-dependent; it just has to be motion-dependent, i.e., it has to change as the state of motion of the atoms in the train changes. Electromagnetic forces have that property; see above.

How if the matter is not related to a compression but to a simple dynamical system where it depends on the distance between the 2 ends and that distance changes in one frame and not in another?

What kind of dynamical system would have this property? Remember that in relativity, "distance" is frame-dependent, and the laws of physics can't be frame-dependent. So any correct relativistic force law can't have a force that depends on "distance".

Note carefully, however, that this does not mean forces themselves can't be frame-dependent; what can't be frame-dependent are the *laws* that the forces satisfy. So, for example, the law for electromagnetic forces are Maxwell's Equations and the Lorentz force law, and these are not frame-dependent. But particular forces that obey these laws *are* frame-dependent; the forces transform along with the fields that appear in the laws so that the same laws--the same equations--hold in every frame, with respect to quantities in that frame.

 

[PeterDonis]

I agree about this restraint.

I don't; at least, not with the version of the scenario we have been using, where there is zero stress in the train (and the coil) after it stops (relative to the platform). In that scenario, the restraint, which keeps the train's length at something different than its unstressed length, would have to be present *before* the train stops (relative to the platform).

If there is a restraint present after the train stops (relative to the platform), then the train (and the coil) must be under compression after it stops, and must have been under zero tension before it stopped (relative to the platform). I said many posts ago that that is also a valid scenario; but I didn't think it was the version we were discussing.

 

[PeterDonis]

Does that mean that every object moving with an apparent contracted length relative to an observer is under natural compression that to be relieved when it comes to a rest.

No, of course not. The only reason that happens in this scenario (and it's natural *tension*, not compression, if the stress is present before the train stops; I've pointed this out several times already) is because of the way you specified the scenario. You specified that, in the platform frame, the train's length does not change when it stops. *That* is what forces the change in stress in the train (either from tension to zero stress, or zero stress to compression).

is the length contraction a mechanical phenomena?

No. Changing an object's state of motion is a mechanical phenomenon. How the object's length changes, or doesn't change, relative to a particular frame, and how the internal stresses in the object change, or don't change, depends on how you change its state of motion.

The less urgent issue: you illustrated before that for the train observer attached to the rear end, he will measures that the train length gets longer and then gets shorter before all vibrations dampen down. Now, during the lengthen phase which the train length increases to 7500 feet, there would be no compression but rather tension for the train observer, but for the platform observer, the force, if any, has to be applied all the time to keep the train length equals to 5000 feet.

These are two different versions of the scenario again. If the train's total length is constrained, in the platform frame, to always be 5000 feet, then it can't lengthen and shorten in the train frame; it can only shorten. The vibrations ghwellsjr was describing cannot happen the way he described them in this case.

 

[ghwellsjr]

I think maybe you are now using a definition of "compression" to be "a shortening of length". If so, then you are correct, for the platform IRF, the train does not change length and so there is no compression.

Which is *not* a correct definition,

It certainly is one of the many correct definitions of compression according to dictionary.com.

because, as I noted in a previous post, you can have "compression" (or release of tension, i.e., a change in the internal stress of the object) without any shortening of length in a particular frame. And it's clear that the OP really intends "compression" to mean "change in internal stress"; he is just incorrectly assuming that you can only have a change in internal stress if there is a change in length.

And that is the point I made in post #38. But if you think I was unclear, then thanks for the added clarity.

 

[ghwellsjr]

I agree about this restraint.

I don't; at least, not with the version of the scenario we have been using, where there is zero stress in the train (and the coil) after it stops (relative to the platform). In that scenario, the restraint, which keeps the train's length at something different than its unstressed length, would have to be present *before* the train stops (relative to the platform).

If there is a restraint present after the train stops (relative to the platform), then the train (and the coil) must be under compression after it stops, and must have been under zero tension before it stopped (relative to the platform). I said many posts ago that that is also a valid scenario; but I didn't think it was the version we were discussing.

In the opening post, there was no tension, no compression, no coil. It was a question about timing as indicated by the title of this thread and for which I gave an answer in post #3. After that, the OP introduced the coil and an "extra tension" as a result of the train stopping that he thought would result in the coil breaking. I pointed out that this was wrong but that there was a compression and that SR was inadequate to determine if the coil would break.

I also pointed out that it was not enough to have just the two ends of the train stop simultaneously in the platform frame but that there needed to be clamps all along the length of the train programmed by timers to make all points of the train stop simultaneously.

So to avoid these problems, I suggested two separated trains with the coil stretched between them to avoid the issue of having any stress in a single train while it was moving. I still don't why you think that "version" of the scenario is the one that we were discussing. I also don't understand how you think such a tension could be enacted in a train moving on the tracks.

 

[PeterDonis]

In the opening post, there was no tension, no compression, no coil.

No coil, yes. No tension or compression, no. The initial specification of the scenario *requires* that there is either tension in the train before it stops, or compression after. There is *no* way to physically realize the scenario as the OP specified it without one of those two being present. So even though the OP did not say so, the OP scenario *does* have either tension or compression.

It was a question about timing as indicated by the title of this thread and for which I gave an answer in post #3.

Yes, but as the subsequent discussion has made clear, the OP was really interested in more than just the timing. He was interested in the physical implications of "length contraction" (and part of the problem is that he's confused about the different possible meanings that term can have). Your spacetime diagrams implicitly contain the answers to those questions as well.

After that, the OP introduced the coil and an "extra tension" as a result of the train stopping that he thought would result in the coil breaking.

He thought he was introducing "extra tension"; but that's only because he didn't realize that his original specification of the problem already included tension (or compression), as above.

I pointed out that this was wrong but that there was a compression

That depends on whether we assume that the train is under zero stress after it stops (in which case it's under tension before it stops), or before it stops (in which case it's under compression after it stops).

SR was inadequate to determine if the coil would break.

Correct, you also need a material model of the coil.

I also pointed out that it was not enough to have just the two ends of the train stop simultaneously in the platform frame but that there needed to be clamps all along the length of the train programmed by timers to make all points of the train stop simultaneously.

This is true if you want every part of the train to stop simultaneously, yes, which is what the OP seems to have intended.

So to avoid these problems, I suggested two separated trains with the coil stretched between them to avoid the issue of having any stress in a single train while it was moving. I still don't why you think that "version" of the scenario is the one that we were discussing.

I don't. I think the one we were discussing is the OP's original scenario, with the assumption that the train is under zero stress after it stops. The OP never explicitly stated this, but I think that's what he originally had in mind (not realizing the full implications, of course). If the OP wants to specify a different assumption, that's fine. The real point is that the OP needs to understand *all* the implications of whatever specification he makes of the scenario.

I also don't understand how you think such a tension could be enacted in a train moving on the tracks.

It certainly wouldn't be easy to do; but I'm not the one that made up the scenario. One possible way would be to have an engine at each end of the train, with the front engine pulling forward and the rear engine pulling backward, and a frictionless track so that the center of mass of the train is moving inertially. Of course, in a real train, this wouldn't work at relativistic velocities because the train would be torn apart; we don't make trains out of materials that can sustain relativistic stresses. But in principle you could do it that way.

 

[PeterDonis]

It certainly is one of the many correct definitions of compression according to dictionary.com.

You're right, the statement I made was too strong. Your point in post #38 (with my clarification) is really what I was trying to say, and I should have just stuck to that.

 

[ghwellsjr]

Since you agree that there must be a restraint, then you agree that there is a lateral force (along the direction of the rails) between the rails and the wheels to keep the train from expanding back to about 6250 feet.

This reply makes me even more confused:

If you hadn't specified that both ends of the train stop at the same time in the platform frame but instead if you had just said that the train came to a stop the way all trains come to a stop, by applying their brakes over a long period of time, then the length of the train would end up near 6250 feet. We can't say exactly because Special Relativity cannot address that issue, as I said before.

however there 1 urgent issue and one cold issue:

The first urgent issue is: Does that mean that every object moving with an apparent contracted length relative to an observer is under natural compression that to be relieved when it comes to a rest. If so , this means that the train or the coil in this example was under compression from the very beginning relative to the platform observer. For if this would be the case, a compression force should have started from the beginning of the motion and stayed all the time not only at the end. It also raises an important question, is the length contraction a mechanical phenomena?

No, as I indicated in post #16 and gave you three examples, Length Contraction is a coordinate effect, it has nothing to do with anything mechanical happening. When you take an inertial object, or a non-inertial object during intervals when it is inertial, and you transform the coordinates from the frame in which the object is at rest where its length is its Proper Length, to a frame in which the object is moving, the object is Length Contracted to its Proper Length divided by gamma, exactly. No stresses or anything mechanical is involved when you simply change to a different frame with different coordinates. Don't get Length Contraction mixed up with the change in length as a result of acceleration. This change in length is not predictable by Special Relativity but we often like to approximate it as being equal to the Proper Length divided by gamma. Indeed, if we really accelerated any object instantaneously (like you did in your scenario) it would be instantly destroyed. If we used a realistic acceleration, the diagrams would be close to a million times larger and be impossible to comprehend, not to mention, boring.

The less urgent issue: you illustrated before that for the train observer attached to the rear end, he will measures that the train length gets longer and then gets shorter before all vibrations dampen down. Now, during the lengthen phase which the train length increases to 7500 feet, there would be no compression but rather tension for the train observer, but for the platform observer, the force, if any, has to be applied all the time to keep the train length equals to 5000 feet.

Yes, I did mention vibrations but I shouldn't have because they have nothing to do with the length as determined by an observer. I should have said that the length determined by a non-inertial observer can fluctuate during the period of acceleration. I think this is what Peter was referring to at the end of post #46.

You should read again what Peter said in post #25:

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.

There is no standard definition for a non-inertial frame. I just used a particular definition that I like that produced the lengthening and then the shortening of the train before the non-inertial transients due to acceleration ended. And just to emphasize what Peter said, I will now show you what an observer at the front end of the train determines using exactly the same process as I used in post #35 so if you want to know the details of the process, read about them in that post.

Here is the diagram for the IRF in which the train starts off moving and ends up stopped:

Here is the log of the data the observer takes and the calculations he makes:

Sent	Rcvd	Avg	1/2 of
Time	Time	Time	diff

0.5	13	6.75	6.25
1.5	14	7.75	6.25
2.5	15	8.75	6.25
3.5	16	9.75	6.25
4.5	16.5	10.5	6
5.5	17	11.25	5.75
6.5	17.5	12	5.5
7.5	18	12.75	5.25
8.5	18.5	13.5	5
9.5	19	14.25	4.75
10.5	19.5	15	4.5
11.5	20	15.75	4.25
12.5	20.5	16.5	4
13.5	21	17.25	3.75
14	22	18	4
14.5	23	18.75	4.25
15	24	19.5	4.5
15.5	25	20.25	4.75
16	26	21	5
17	27	22	5
18	28	23	5

Here is the diagram of the non-inertial reference frame he constructs from the log.

Note that the train doesn't get longer, rather it shortens to less than 4000 feet and then lengthens back to 5000 feet.

And here is the diagram for the IRF in which the trains starts out at rest and ends up moving:

Once again, the frame has no bearing on the measurements and observations that are made, including those of stress.

 

[adelmakram]

What kind of dynamical system would have this property? Remember that in relativity, "distance" is frame-dependent, and the laws of physics can't be frame-dependent. So any correct relativistic force law can't have a force that depends on "distance".

Note carefully, however, that this does not mean forces themselves can't be frame-dependent; what can't be frame-dependent are the *laws* that the forces satisfy. So, for example, the law for electromagnetic forces are Maxwell's Equations and the Lorentz force law, and these are not frame-dependent. But particular forces that obey these laws *are* frame-dependent; the forces transform along with the fields that appear in the laws so that the same laws--the same equations--hold in every frame, with respect to quantities in that frame.

Here are 2 examples which I am just thinking about:

1) Consider a point charge located at the front end of the train and a magnetic field in the rear end where the magnetic field lines run perpendicular to the long axis of the train as shown in the figure. As the front end starts to move toward the rear end, a magnetic force will act on it and its direction will be determined by right hand rule. So, the point charge should move under the effect of this force in the upward direction, toward the roof of the train. Will that movement be observed in the platform observer?
2) How about gravitational force, how if there are 2 massive objects at both ends of the train. When the length shortens the force increase between them ( inverse square law), will this be the same for the platform observer?

 

[Dale]

1) Consider a point charge located at the front end of the train and a magnetic field in the rear end where the magnetic field lines run perpendicular to the long axis of the train as shown in the figure. As the front end starts to move toward the rear end, a magnetic force will act on it and its direction will be determined by right hand rule. So, the point charge should move under the effect of this force in the upward direction, toward the roof of the train. Will that movement be observed in the platform observer?

Transform the coordinates:
http://farside.ph.utexas.edu/teaching/em/lectures/node109.html

and transform the fields:
http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

Then apply the Lorentz force law.

2) How about gravitational force, how if there are 2 massive objects at both ends of the train. When the length shortens the force increase between them ( inverse square law), will this be the same for the platform observer?

That requires general relativity where the gravitational force does not strictly follow an inverse square law. We should stick with EM phenomena until you have learned SR.

 

[adelmakram]

Transform the coordinates:
http://farside.ph.utexas.edu/teaching/em/lectures/node109.html

and transform the fields:
http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

Then apply the Lorentz force law

Those transformation will not be applied in this case because the state of motion of 2 ends of the train are not the same for all reference frames. Please read the post from the beginning.

 

[Dale]

Those transformation will not be applied in this case because the state of motion of 2 ends of the train are not the same for all reference frames.

Yes, they do apply. They apply any time that gravitational effects are not significant. The motion of the 2 ends of the train is not relevant.

 

[adelmakram]

Yes, they do apply. They apply any time that gravitational effects are not significant. The motion of the 2 ends of the train is not relevant.

Fine, So would you please show how the platform observer could explain the upward motion of the point charge?

 

[Dale]

You should work it through on your own using the material I linked to above. It will be much more instructive for you that way.

Also, your brief description is not sufficiently clear for me to work it through without trying to guess what you had in mind.

 

[adelmakram]

You should work it through on your own using the material I linked to above. It will be much more instructive for you that way.

Also, your brief description is not sufficiently clear for me to work it through without trying to guess what you had in mind.

It is a difficult problem no matter what I have in my mind.

 

[adelmakram]

When both ends of the train stop at the same time relative to the platform observer, the point charge will only move when a magnetic field changes in intensity. But the magnitude of that change and the resultant direction of the force acting on the charge is not clear for me to calculate.

 

[Dale]

Are you familiar with matrices and linear algebra?

 

[adelmakram]

Are you familiar with matrices and linear algebra?

yes,

 

[Dale]

Then it is not that difficult. Using units where c=1 you simply set up the spacetime coordinate 4-vector: $r=(t,x,y,z)$ and the Lorentz transform matrix
$$\Lambda = \left(
\begin{array}{cccc}
\gamma & - \gamma v & 0 & 0 \\
-\gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) $$

And the electromagnetic field tensor:
$$F=\left(
\begin{array}{cccc}
0 & E_x & E_y & E_z \\
-E_x & 0 & B_z & -B_y \\
-E_y & -B_z & 0 & B_x \\
-E_z & B_y & -B_x & 0 \\
\end{array}
\right)$$

Then $r'=\Lambda r$ and $F'=\Lambda F \Lambda^T$

 

[adelmakram]

Then it is not that difficult.

For the train observer, the force acting on the point charge after the A-end starts to move with velocity v toward B-end is a magnetic force described by Lorentz.
F`= y qv x B`(the primed symbols indicate the field in the train frame of reference)
The direction is upward by right hand rule.

For the platform observer, there is no relative motion between the 2 ends, however, there is an expected change in the magnetic field after the train stops.

If we consider that the train moves along x-axis, and B` is directed along z-axis, then the transformation equation is:

B`z = γ [Bz – (v/c2) Ey ]

So the inverse transformation is:

Bz = γ [B`z + (v/c2) E`y ]

But E`y = 0

So Bz before = ^ [B`z ]
Where γ is Lorentz factor.

After the train stop,

Bz after = B`z

So Bz before > Bz after

So the point charge will experience a net electric force that depends on the rate of change of the Bz. The direction is governed by Lenz law. As there is reduction of the magnetic field, then the direction of the induced force on the charge will aim to oppose the change of the magnetic field. So the charge should move upward similar to what is seen by the train observer.

First, I am wondering if B is really reduced after the train comes to a stop? Once this is cleared the problem will be solved.

One more question, what will be the direction of B relative to the platform observer while the train is moving. I assumed in my calculation that the direction of B is the same as B` along the z direction.

 

[adelmakram]

Will the point electric charge really intersect the magnetic lines as it starts to move from A toward B?
I think the magnetic lines themselves will start moving too because they are confined within the space of the train, so as the magnetic lines moves toward B in the same velocity as the point charges does leading to no Lorentz force on it!.
On the other hand, the platform observer should notice the upward motion of the charge because it is an electric force induced on it from a changing magnetic field after the train comes to a stop.

Will that be a paradox?

 

[Dale]

Will the point electric charge really intersect the magnetic lines as it starts to move from A toward B?
I think the magnetic lines themselves will start moving too because they are confined within the space of the train, so as the magnetic lines moves toward B in the same velocity as the point charges does leading to no Lorentz force on it!.
On the other hand, the platform observer should notice the upward motion of the charge because it is an electric force induced on it from a changing magnetic field after the train comes to a stop.

Will that be a paradox?

I don't know why you keep rambling on about paradoxes. Maxwell's equations are invariant under the Lorentz transform so there will obviously never be any paradox. Using the usual tensor notation with c=1 and a (-+++) signature:

In the train frame:
$$F^{\mu\nu}=\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & B_z & 0 \\
0 & -B_z & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right)$$
$$U^{\mu}=(\gamma,\gamma v, 0,0)$$
so the four-force is
$$q U_{\mu}F^{\mu\nu} = (0,0,q\gamma B_z v,0)$$

In the particle's frame:
$$F'^{\mu\nu}=\left(
\begin{array}{cccc}
0 & 0 & -\gamma B_z v & 0 \\
0 & 0 & \gamma B_z & 0 \\
\gamma B_z v & -\gamma B_z & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right)$$
$$U'^{\mu}=(1,0, 0,0)$$
so the four-force is once again
$$q U'_{\mu}F'^{\mu\nu} = (0,0,q\gamma B_z v,0)$$

No paradox. Do the math.

 

[adelmakram]

No paradox. Do the math.

Thank you for your formulation. But that was irrelevant to my concern. I did not ask about the invariance of Lorentz force relative to the point charge frame.

I asked whether the point charge will still intersect the magnetic field lines as it starts moving from A to B. And I wondered whether the magnetic field lines themselves will move too as long as they are confined in the same space. For if that would be the case, there will be no force on the charge. Then I questioned whether there would be a paradox if the charge would not experience a Lorentz force because it does not intersect the B`-lines in the frame of the train but it experience that force in the platform frame because of the change of the amplitude of B due to sudden stop of the train.

 

[Dale]

I wondered whether the magnetic field lines themselves will move too as long as they are confined in the same space. For if that would be the case, there will be no force on the charge.

Where in the Lorentz force law is the velocity of the magnetic field? Write down the equation and ask about the meaning of any term that you don't recognize.

Thank you for your formulation. But that was irrelevant to my concern. I did not ask about the invariance of Lorentz force relative to the point charge frame.
... I questioned whether there would be a paradox if the charge would not experience a Lorentz force because it does not intersect the B`-lines in the frame of the train but it experience that force in the platform frame because of the change of the amplitude of B due to sudden stop of the train.

It is completely relevant. It disproves your suggestion of a paradox by explicitly calculating the supposedly paradoxical quantity. And even if the fields are not what you had in mind (your description is unclear as I mentioned earlier), the method is instructive and relevant.

Regarding the dynamic EM effects due to the stopping of the train. Whatever the field and forces are in one frame, regardless of how complicated, as long as they satisfy Maxwell's equations and the Lorentz force law in one frame they will satisfy them in all frames. The complicating details are not important, but you are welcome to calculate them following my example.

 

[adelmakram]

I don't know why you keep rambling on about paradoxes.

I can not promise that I will stop doing that, but I might call them potential paradoxes.

Consider the same scenario but this time there are 2 points charges, one of them is a positive charge put at A-end and the other is a negative charge put at B-end and I am interested to examine the electric force between them relative to different observers.

For the platform observer, the force value should not change after the train comes to a complete stop because the electric field in the direction of the motion does not change; Ex =E`x

For the train observer, the length of the train changes and so does the electric force between the 2 points charges according to the Coulomb law which creates a potential paradox.

 

[Dale]

The correct expression for the fields from an arbitrarily moving point charge is called the Lienard Wiechert potential. Coulomb's law only applies to electrostatic situations.

You will need to do the math on your own. If you get a paradox then go back and check your work since you made an error.

 

[adelmakram]

Since you agree that there must be a restraint, then you agree that there is a lateral force (along the direction of the rails) between the rails and the wheels to keep the train from expanding back to about 6250 feet.

But how do we know that the original length at the start of the motion was 6250 feet? Is that possible that the train would start moving with both ends accelerating simultaneously? If so, the train length would remain unaltered before the motion, during the motion and after the stop as long as both ends have the same states of acceleration all times. So the existence of the lateral force that hinders the train from getting back to its original length depends on the prior information at the start of the motion which makes it non-deterministic !