Simultaneity on a moving train

[west-river]

Question from non-physicist. Hope there is a simple answer.
In Einstein's thought experiment with moving train and embankment: If there is an observer in the middle of one of the train cars and one light beam is flashed from inside each side of the car (same location on each sides) towards the observer in the middle between them, will the beams arrive to the observer (that is to the midpoint of the car) at the exact same time? That is, does the movement of the train effect whether the two beams arrive at the same time? If the two beams are on the embankment, one will arrive at the midpoint of the car earlier than the other? Is it the same if the two beams come from within the car?
Thanks

 

[.Scott]

Q1: Will light reach middle observer on both sides at same time?
A1: Yes
Q2: Does movement of train affect this?
A2: No. It doesn't matter what is happening outside of the train.
Q3: Will embankment beams reach observer at same time?
A3: It depends on exactly how the experiment is conducted. If each embankment beam starts at the instant that the corresponding train wall passes it, then yes. If the embankment beams are synchronized from a timer half way between them, No.
Q4: Is it the same if the beams come from within the car.
A4: See A3.

The core issue here is exactly as your title suggests. What constitutes "simultaneous" flashing of the two beams is dependent on the reference frame.

 

[west-river]

Thank you. I made a mistake in my second to last sentence. I accidentally added a question mark. I meant it as a statement.

In any case, this answers my question. Thank you.

 

[Ibix]

In Einstein's thought experiment with moving train and embankment: If there is an observer in the middle of one of the train cars and one light beam is flashed from inside each side of the car (same location on each sides) towards the observer in the middle between them, will the beams arrive to the observer (that is to the midpoint of the car) at the exact same time?

You aren't giving enough information to answer this. If the two lights flashed at what the train frame would call the same time, then yes. Otherwise no.

That is, does the movement of the train effect whether the two beams arrive at the same time?

Again, it depends what you mean. If the lights flash at what the train frame regards as the same time then the pulses will arrive in the middle at the same time. Otherwise they won't. This is independent of what speed the train is traveling at - it can always regard itself as at rest.

If the two beams are on the embankment, one will arrive at the midpoint of the car earlier than the other?

It doesn't matter where the lights are mounted, it matters whether they were triggered at the same time according to the train rest frame. Other frames will not agree that the lights flashed at the same time.

 

[Ibix]

Q1: Will light reach middle observer on both sides at same time?
A1: Yes

Careful - you are making assumptions about when the lights flash. You are probably making a reasonable assumption, but failing to state experimental conditions clearly (including which frame is being used for which bit of the description) is a very common cause of confusion with relativity.

 

[west-river]

OK. Thank you.
One more question please:
Say the two light are on the moving train and triggered at the same time. What about an observer on the embankment, at rest relative to the train? Will the beams arrive to him at the same time?

 

[Ibix]

same time

Same time according to who? The observer on the train or the observer on the embankment? They won't agree.

 

[west-river]

I should have said "Say the two lights are on the moving train and triggered at the same time relative to the observer on the train in the middle of the car"

 

[west-river]

To restate: If two lights on a moving train are triggered at the same time relative to an observer on the train exactly between them, will an observer on the bank who is exactly between them when they are triggered (relative to the the observer on the train) receive the beams at the same time relative to him?

 

[Ibix]

Right. Then they will arrive at the same time at the train observer. Whether or not they arrive at the same time at the embankment observer at the same time depends on where she is. If the train observer says the flashes were emitted at the same time as he passed the embankment observer then by the time the flashes reach the train observer he is no longer next to the embankment observer. So she does not see the flashes at the same time.

If the embankment and train observers are next to each other when the flashes arrive then of course they see them at the same time.

Edit: I see you specified the scenario a bit more while I was typing.

 

[west-river]

I think I have it now. I appreciate your help!

 

[PeroK]

To restate: If two lights on a moving train are triggered at the same time relative to an observer on the train exactly between them, will an observer on the bank who is exactly between them when they are triggered (relative to the the observer on the train) receive the beams at the same time relative to him?

Your question is self contradictory. If two events are simultaneous in one frame (e.g. the train frame), then they cannot be simultaneous in another frame (e.g. the platform frame).

In other words, if the lights are triggered at the same time on the train then they will not be triggered at the same time to someone on the platform, no matter where they are standing.

 

[west-river]

Imagine train car moving right at v. M' is midpoint of train car and M is the point on Embankment when M and M' are aligned, say at time T' for M' and at T for M. At T' (for M', A and B beams of light are triggered which are simultaneous for M'. Imagine A and B are affixed to the outside of the car. Why isn't it true that AM = BM in terms of distance at T and T'? I am assuming that the emissions of A and B are instantaneous and that, at that instant, the train hasn't moved forward, so the point of emissions of A and B are the same whether the train moves or doesn't. -- Not sure what I'm getting wrong here. -- Thanks.

 

[jbriggs444]

View attachment 236753
Imagine train car moving right at v. M' is midpoint of train car and M is the point on Embankment when M and M' are aligned, say at time T' for M' and at T for M. At T' (for M', A and B beams of light are triggered which are simultaneous for M'. Imagine A and B are affixed to the outside of the car. Why isn't it true that AM = BM in terms of distance at T and T'?

There is no point in specifying time T and T'. Just set both M and M's to zero at the event where they pass one another.

With this simplification, both flashes are emitted at time 0 according to M'. They are simultaneous according to M'.

I am assuming that the emissions of A and B are instantaneous and that, at that instant, the train hasn't moved forward, so the point of emissions of A and B are the same whether the train moves or doesn't.

You seem to implicitly assume that "that instant" is a meaningful and universal specification. It is not.

According to M', "That instant" would include the set of events that all occur at time zero using clocks synchronized using the standards of M'.
According to M, "That instant" would include the set of events that all occur at time zero using clocks synchronized using the standards of M.

But the clocks of M and M' are systematically out of synch due to their relative velocities. Those two "that instant" sets only overlap at the midpoint where M and M' momentarily pass one another.

You've set up the problem so that "that instant" according to M' includes both flashes. But "that instant" according to M then includes neither.

 

[sdkfz]

View attachment 236753
Imagine train car moving right at v. M' is midpoint of train car and M is the point on Embankment when M and M' are aligned, say at time T' for M' and at T for M. At T' (for M', A and B beams of light are triggered which are simultaneous for M'. Imagine A and B are affixed to the outside of the car. Why isn't it true that AM = BM in terms of distance at T and T'? I am assuming that the emissions of A and B are instantaneous and that, at that instant, the train hasn't moved forward, so the point of emissions of A and B are the same whether the train moves or doesn't. -- Not sure what I'm getting wrong here. -- Thanks.

From the point of view of M', it is M that is moving -v.
Also, light speed is finite, not infinite.
So the flashes from A and B can not reach M at the same time ... because M is moving towards A and away from B.

 

[Ibix]

View attachment 236753
Imagine train car moving right at v. M' is midpoint of train car and M is the point on Embankment when M and M' are aligned, say at time T' for M' and at T for M. At T' (for M', A and B beams of light are triggered which are simultaneous for M'. Imagine A and B are affixed to the outside of the car. Why isn't it true that AM = BM in terms of distance at T and T'? I am assuming that the emissions of A and B are instantaneous and that, at that instant, the train hasn't moved forward, so the point of emissions of A and B are the same whether the train moves or doesn't. -- Not sure what I'm getting wrong here. -- Thanks.

If the flashes are simultaneous for M' they are not simultaneous for M. So the distances AM and BM are equal - but the flashes did not happen at the same time so M does not see them at the same time.

You've set this up so the flashes are simultaneous for M'. Ok.

From the perspective of M' the flashes are simultaneous and equidistant from M', so M' receives them simultaneously. The flashes are equidistant from where M was when the flashes were emitted, but M is moving towards the rear lamp so sees that flash first.

From the perspective of M, the rear flash happened before the front flash. They were equidistant from M, so M saw the rear flash first. M' was moving forward at just the right speed that the light traveling forwards and the light traveling backwards happened to meet where M' was at the time.

Note that both descriptions agree on direct observations: M' received the flashes simultaneously and M saw the rear flash first. They only disagree on the interpretation of those observations.

 

[FactChecker]

If the two observers are side-by-side, then the lights will reach both observers simultaneously. They will agree on that. There is no disagreement regarding the simultaneity of events that are not separated in the direction of relative motion. However, the two observers will disagree on whether the lights started from the ends of the train simultaneously. They can not agree on the simultaneity of events that are separated in the direction of relative motion.

 

[PeroK]

View attachment 236753
Imagine train car moving right at v. M' is midpoint of train car and M is the point on Embankment when M and M' are aligned, say at time T' for M' and at T for M. At T' (for M', A and B beams of light are triggered which are simultaneous for M'. Imagine A and B are affixed to the outside of the car. Why isn't it true that AM = BM in terms of distance at T and T'? I am assuming that the emissions of A and B are instantaneous and that, at that instant, the train hasn't moved forward, so the point of emissions of A and B are the same whether the train moves or doesn't. -- Not sure what I'm getting wrong here. -- Thanks.

Any diagram such as you have done can only be valid in one frame of reference. In this case that is the situation in the train frame of reference. To see the scenario in the platform frame, you would have to draw a different diagram. In particular, you would have to show the emission of flashes from A and B at different times in the platform frame.

It makes a big difference whether the train is moving relative to the platform of not. If the train is not moving, then the train and platform share a rest frame and your diagram is valid for any observer at rest in the train-platform frame. But, if the train is moving, then the train and platform do not share a rest frame; events that are simultaneous in one frame are not simultaneous in the other (unless they are also colocated, which essentially makes them the same event).

This is a fundamental difference between SR, as opposed to classical physics, where one diagram can show all events at a given (universal) point in time.

 

[west-river]

Thank you very much for all the help. I am going to take some time to absorb them. If I still don't understand, I may try one more time to formulate my question, but will not want to burden the forum anymore with my own inabilities.

Thanks again

 

[PeroK]

Thank you very much for all the help. I am going to take some time to absorb them. If I still don't understand, I may try one more time to formulate my question, but will not want to burden the forum anymore with my own inabilities.

Thanks again

It's not perfect, but this video deals with your scenario.

 

[west-river]

Thanks for the video. I hope you will bear with me a little longer. I want to get clear in my own mind, if it is possible.

1. Originally I started this thread with one question. My question was (I am now rephrasing it): "If there is a train car with an observer in the middle facing towards an embankment, and there is one light affixed to the wall on his left and one to the wall on his right wall, and the two lights are triggered to go off at the same time relative to the observer, and they do now go off, will the observer perceive them as simultaneous? I know this is true if the train is not moving relative to the embankment. Is it also true if the train is moving relative to the embankment?" -- If I am still phrasing it and seeing it incorrectly, I will give up (at least for now), as it will mean I've not understood anything of what you have all said.

2. I also asked a second question which, to me is more difficult. I think I can say why I am still confused even though I think I understand the answers given. From the frame of reference of the train and the observer in the middle of the car, there are two lights affixed to the far ends of the car. The observer is in the middle of the car (M') when it passes an observer on the embankment (at M) (though I am confused how this would be assessed -- but that is another problem). Now take the two sources of light at the two ends of the car and say they go off simultaneously from the point of view of the observer at M'. Now assume the beams of light (A and B) each are broken into the two parts, one that heads towards M' and the other that heads towards M. So there are, as it were, 4 beams AM', BM', AM, and BM. It seems to me that, once AM and BM are emitted they are no longer part of the train and are no longer moving in the direction of the train. Once they are emitted, they are, as it were, in the air and completely independent of the train and any of its relative movement. At that point the beams are in no way different from two beams emitted from points outside the train. And since AM and BM are = in distance, the two beams will arrive at M at the same time. True, if the train moved while the beams are being emitted, one will arrive at M before the other. But, if they are emitted exactly together and "instantaneously" (to use a word that seems to be suspect), then they are no longer attached to a moving train and are as if coming from a source independent of the train.

Again, thanks for your patience. I am good in other areas, but physics has never been one of them.

 

[PeroK]

1. Originally I started this thread with one question. My question was (I am now rephrasing it): "If there is a train car with an observer in the middle facing towards an embankment, and there is one light affixed to the wall on his left and one to the wall on his right wall, and the two lights are triggered to go off at the same time relative to the observer, and they do now go off, will the observer perceive them as simultaneous? I know this is true if the train is not moving relative to the embankment. Is it also true if the train is moving relative to the embankment?" -- If I am still phrasing it and seeing it incorrectly, I will give up (at least for now), as it will mean I've not understood anything of what you have all said.

Where are these walls? On the train or on the embankment?

The only difficulty in SR comes when you consider the relationship between events as measured in two reference frames. If you state any scenario and study it in one reference frame, then there are no issues. The speed of light is constant and that is all there is to it.

There is, however, in my humble opinion a really serious problem with learning SR from this lightning experiment. It over-emphasises the need for a single observer, whose observations are dependent on the travel time of light. Things are simultaneous in a frame of reference even if they emit no light and no one sees them. For example, two alarm clocks could go off at either end of a train. Each event has a time (in a given reference frame) and those times are either the same (events are simultaneous) or those times are different (events are not simultaneous). Simultaneity has nothing whatsover to do with the speed of light.

Unfortunately, almost all the available material online completely over-emphasises the role of light signals and light travel times in measuring the time an event takes place and whether two events are simultaneous.

2. I also asked a second question which, to me is more difficult. I think I can say why I am still confused even though I think I understand the answers given. From the frame of reference of the train and the observer in the middle of the car, there are two lights affixed to the far ends of the car. The observer is in the middle of the car (M') when it passes an observer on the embankment (at M) (though I am confused how this would be assessed -- but that is another problem). Now take the two sources of light at the two ends of the car and say they go off simultaneously from the point of view of the observer at M'. Now assume the beams of light (A and B) each are broken into the two parts, one that heads towards M' and the other that heads towards M. So there are, as it were, 4 beams AM', BM', AM, and BM. It seems to me that, once AM and BM are emitted they are no longer part of the train and are no longer moving in the direction of the train. Once they are emitted, they are, as it were, in the air and completely independent of the train and any of its relative movement. At that point the beams are in no way different from two beams emitted from points outside the train. And since AM and BM are = in distance, the two beams will arrive at M at the same time. True, if the train moved while the beams are being emitted, one will arrive at M before the other. But, if they are emitted exactly together and "instantaneously" (to use a word that seems to be suspect), then they are no longer attached to a moving train and are as if coming from a source independent of the train.

Light beams have no "knowledge" of being on a train or being emitted by a "moving" source or being received by a "moving" receiver. LIght beams move at the speed of light in all reference frames.

Also, there is no such thing as absolute motion. For example, in all your experiments you can (and in fact should) consider the train to be at rest and the platform to be moving when you are studying things from the reference frame of the train.

For example:

There is a train at rest with alarm clocks at either end. The alarm clocks go off at the same time in the train reference frame. Meanwhile, a platform is moving towards the train at high speed. In the reference frame of the platform, the alarm clocks do not go off at the same time. We don't immediately have to worry about how an observer or observers in the train or the platform frame actually measures the time the alarm clocks go off. That's a different problem.

In short, your attempt to understand SR has become tangled up with the nature of light signals from events. This is a common consequence of the train and lightning strikes experiment.

 

[west-river]

Very helpful for me. Thank you. I feel close to getting a handle on it.

I also like what you said about using the light experiment in learning SR. Of course, my appreciation for your point, has less meaning, since my understanding is so limited, but it makes sense to me.

Thanks again.

 

[PeroK]

Very helpful for me. Thank you. I feel close to getting a handle on it.

I also like what you said about using the light experiment in learning SR. Of course, my appreciation for your point, has less meaning, since my understanding is so limited, but it makes sense to me.

Thanks again.

Okay, here's an alternative approach to this. We have a train with a clock at the front. Call this clock T. We have a platform with a clock at either end. Call these clocks P1 and P2. By some process, it's not that important how, the clocks P1 and P2 are synchronised in the platform frame. This is actually closer to what we mean by a "frame of reference" than a single clock at a single location.

Now, the train approaches. Clock T reaches clock P1. You don't actually need any "observers" here, so I'm simply going to describe what the clocks read when they pass each other. This could be observed or recorded but it doesn't matter, The physics is the same.

This event has a certain time in the platform frame. We might as well assume that it's $t=0$. In the platform frame both clocks P1 and P2 read $0$ when this event takes place. And, we might as well assume that clock T reads $0$ at this point as well.

Now, the next important event is the clock T reaching clock P2. If we analyse things in the platform frame we know that clock T is time-dilated. Let's assume by a factor of 0.8. Let's assume also that the train takes 1 second (in the platform frame) to travel from P1 to P2. Then, when T passes P2 we know that:

Clock P2 reads $1s$ and clock $T$ reads $0.8s$. That is now a fact that everyone will agree on. The two clocks are at the same place and everyone who cares to observe or record this event will say: "Clock P1 showed $1s$ and clock $T$ showed $0.8s$ when they passed each other. If you want you could imagine that both clocks are stopped at that point, so their readings are frozen for all to see at their leisure!

Now, we analyse things from the train frame. In this frame, clock P1 reads $t=0$ as it passes clock T. Same as before. And, clock T reads $0.8s$ when clock P2 passes it, reading $1s$. There's no alternative. This is a fact that could be verified by the clocks being stopped at that point if need be.

But,wait in minute. In the train frame it is clock P2 that is time-dilated. And by the same factor $0.8$. Therefore, in the time it takes clock T to advance by $0.8s$, clock P2 advances by only $0.64s$.

But, clock P2 reads $1s$ when it passes clock T. Therefore, in the train frame, it must have read $0.36s$ when clock P1 passed clock T.

Therefore, using time dilation and inescapable logic we can deduce that:

In the train frame, when clock P1 passed clock T, clock P1 read $0s$ and clock P2 read $0.36s$. And, therefore, despite being synchronised in the platform frame, the clocks P1 and P2 are not synchronised in the train frame. Therefore, simultaneity is reference frame dependent.

 

[Janus]

Thank you very much for all the help. I am going to take some time to absorb them. If I still don't understand, I may try one more time to formulate my question, but will not want to burden the forum anymore with my own inabilities.

Thanks again

Consider this set pair of animations showing flashes arriving at both a train and embankment observer at the same time:
First according to the embankment observer:

He remains at a fixed point between the two sources, halfway between them for the entire time between emission and reception, thus seeing the flashes at the same time means that the flashes were emitted at the same time for him.
Now consider the same flashes of light according to the train observer:

He is, like the embankment observer, halfway between the sources of the flash when he detects them. However, this means he could not have been an equal distance from them when either flash was emitted, as at any moment before detecting the flashes he would be closer to the left source than he was to the right source. Since he must measure the speed at which the each flash travels as being c relative to himself, this means in order for the light of each flash to both reach him at the same moment he is halfway between them, the right flash has to leave its source before the left flash leaves its source, and the initial emissions of each flash cannot be simultaneous.

This particular animation doesn't take length contraction into account, ( something that would have to be accounted for if you wanted to extend the train so that parts of it passed the sources at the moments of emission) so it only demonstrates the principle behind the Relativity of Simultaneity and is not an exact representation.

 

[sdkfz]

... But, if they are emitted exactly together and "instantaneously" ...

What you are doing here is assuming that simultaneity is absolute, and the point of this kind of thought experiment is to question whether that's true. Your scenario stipulates that the flashes are simultaneous for the train observer; the question is then whether they are also simultaneous for the embankment observer.

... True, if the train moved while the beams are being emitted, one will arrive at M before the other. ...

The embankment observer is entitled to consider themselves as at rest. Also, given the setup of the experiment, they were exactly between the two lights at the time they flashed (according to the train observer, who considers them simultaneous).

If M is exactly between the sources of two flashes, and since light speed is constant, then if the flashes were simultaneous, they would arrive at the same time. As you note, they don't, so for M they were not simultaneous.(The original has lightning strikes hit across the tracks and the ends of the train, to remove quibbles about where the source of the light is.)

 

[west-river]

"Now, the next important event is the clock T reaching clock P2. If we analyse things in the platform frame we know that clock T is time-dilated. Let's assume by a factor of 0.8. Let's assume also that the train takes 1 second (in the platform frame) to travel from P1 to P2. Then, when T passes P2 we know that:
Clock P2 reads $1s$ and clock $T$ reads $0.8s$."

Does the fact that platform analysis shows clock T will read 0.8s and not 1s show that clock T will actually read 0.8s? -- I am thinking as a philosopher who has just trying to understand physics, so please take my comments and questions with a grain of salt.

"And, therefore, despite being synchronised in the platform frame, the clocks P1 and P2 are not synchronised in the train frame. Therefore, simultaneity is reference frame dependent."

Again, I wonder at my own thinking, but isn't an alternative reading of your argument be SR is incorrect in part?

I have another diagram for what may possibly be a distantly related thought. The image below was distorted in making a photo, but the idea should be clear.

It seems to me there are at least two ways of measuring simultaneity for a person on the platform and a person on the train. Start with the train observer: He could wait for the light to hit him at M' and make a judgment which would lead to him saying that A and B are not simultaneous (Einstein's point, I think). And the person on the bank at M can also judge in at least two ways. He can wait for the the light from A and B to hit him and then judge, in which case A and B will seem simultaneous.

Now whatever criteria Einstein used that he thought could establish for an observer on the train and one on the embankment that M and M' are at the same place at the same time, the same criteria could be used to establish that A and A' and B and B' are at the same place at the same time. So A', M', and B' align with A and M and B at the same time.

Now we might imagine 4 clocks all set at 0 and made to trigger when they are hit by a light coming from A and B and to mark that moment. So a second way for an observer at M to tell if A and B were simultaneous would be to walk over to A and B, after the fact and see if they were triggered at the same time. And the same for an observer on the train at M'. He could decide not to trust his eyes but to go and examine the two clocks and use them to establish simultaneity.

In this case it seems to me, both observers will pronounce that A and B were simultaneous, even if they will think the start and finish times of A and B are different.

 

[PeroK]

Again, I wonder at my own thinking, but isn't an alternative reading of your argument be SR is incorrect in part?

Well, yes, of course. But, then in physics you do things called experiments that decide the issue. Experiments show SR's analysis of space and time to be correct and absolute Newtonian space and time to be only an approximation.

That's the difference between physics and philosophy. In physics nature decides and if your theory does not agree with experiment it's wrong. As Feyman put it: "It doesn't matter what your name is or how clever you are, it's wrong."

What we've presented to you here is the theory. But, nothing we have presented compels SR to be a true reflection of nature. That it is (a true reflection of nature) hinges on the outcome of real experiments.

PS as has been said many times on this forum, you are about 110 years too late to debate whether SR might be wrong.

 

[Ibix]

He could decide not to trust his eyes but to go and examine the two clocks and use them to establish simultaneity.

How were the clocks synchronised? (Hint: any process you specify will lead to the clocks being synchronised in one frame but not the other).

 

[west-river]

OK. Back to the drawing board. I can only imagine what my thoughts might sound like to the physicist, but I am genuinely trying to figure things out and needed to put them out somewhere for a reality test. I don't know anywhere else to go with my questions. Thanks.

 

[sdkfz]

... Now whatever criteria Einstein used that he thought could establish for an observer on the train and one on the embankment that M and M' are at the same place at the same time, the same criteria could be used to establish that A and A' and B and B' are at the same place at the same time. So A', M', and B' align with A and M and B at the same time. ...

Don't forget the relative movement between M and M', and the finite speed of light. By the time the light reaches M and M' from A/A' and B/B', M and M' will no longer be adjacent. The situation at the start is not the situation a moment later.

...
Now we might imagine 4 clocks all set at 0 and made to trigger when they are hit by a light coming from A and B and to mark that moment. So a second way for an observer at M to tell if A and B were simultaneous would be to walk over to A and B, after the fact and see if they were triggered at the same time. And the same for an observer on the train at M'. He could decide not to trust his eyes but to go and examine the two clocks and use them to establish simultaneity. ...

Your setup here isn't clear to me, but I'd note that the issue has never been that M disagrees with what M' thinks, or vice versa. That is, if M' considers the events to be simultaneous, M may say "no they were not", but can agree that "well, they were for you".

A common way to visualise this is with M and M' wearing bombs that will explode if the light from the events reaches them at the same time. If the flashes are simultaneous for M', he or she explodes. M can see that, they can't claim the explosion did not occur! (Note that M did not explode in this case.)

 

[west-river]

Interesting! Thanks again. All this is very useful to me in my search for understanding and is much appreciated.

 

[Janus]

"Now, the next important event is the clock T reaching clock P2. If we analyse things in the platform frame we know that clock T is time-dilated. Let's assume by a factor of 0.8. Let's assume also that the train takes 1 second (in the platform frame) to travel from P1 to P2. Then, when T passes P2 we know that:
Clock P2 reads $1s$ and clock $T$ reads $0.8s$."

Does the fact that platform analysis shows clock T will read 0.8s and not 1s show that clock T will actually read 0.8s? -- I am thinking as a philosopher who has just trying to understand physics, so please take my comments and questions with a grain of salt.

"And, therefore, despite being synchronised in the platform frame, the clocks P1 and P2 are not synchronised in the train frame. Therefore, simultaneity is reference frame dependent."

Again, I wonder at my own thinking, but isn't an alternative reading of your argument be SR is incorrect in part?

I have another diagram for what may possibly be a distantly related thought. The image below was distorted in making a photo, but the idea should be clear.

It seems to me there are at least two ways of measuring simultaneity for a person on the platform and a person on the train. Start with the train observer: He could wait for the light to hit him at M' and make a judgment which would lead to him saying that A and B are not simultaneous (Einstein's point, I think). And the person on the bank at M can also judge in at least two ways. He can wait for the the light from A and B to hit him and then judge, in which case A and B will seem simultaneous.

View attachment 236784

Now whatever criteria Einstein used that he thought could establish for an observer on the train and one on the embankment that M and M' are at the same place at the same time, the same criteria could be used to establish that A and A' and B and B' are at the same place at the same time. So A', M', and B' align with A and M and B at the same time.

Now we might imagine 4 clocks all set at 0 and made to trigger when they are hit by a light coming from A and B and to mark that moment. So a second way for an observer at M to tell if A and B were simultaneous would be to walk over to A and B, after the fact and see if they were triggered at the same time. And the same for an observer on the train at M'. He could decide not to trust his eyes but to go and examine the two clocks and use them to establish simultaneity.

In this case it seems to me, both observers will pronounce that A and B were simultaneous, even if they will think the start and finish times of A and B are different.

Now you are going to have to deal with the length contraction issue I mentioned earlier.

Here's the original train experiment as it occurs according to the embankment.

Flashes are emitted from the red dots as the ends of the train reaches them and the trian observer is next to the embankment observer. Train observer runs into right flash before left flash catches up to him. One thing to keep in mind is that since the train is in motion with respect to the embankment, the train is length contracted according to the embankment and this length contraction is what allows the train to fit exactly between the red dots.

However, according to anyone at rest with respect to the train, the train is not length contracted, but instead, it is the embankment that it in motion and length contracted. Thus the train cannot fit between the red dots, the front of the train hit the right red dot before the back of the train reaches the left red dot. Since both the train and embankment have to agree that the flashes are emitted when the ends of the trains pass the red dots, the flashes cannot be emitted at the same time according to the train, like this:

Note that according to the train, the flashes still meet a the the embankment observer. Also note that the midpoint of the train is next to the same point of the tracks when each the light flashes reaches him in both animations.
A and A' being next to each other is simultaneous with B and B' being next to each other in the embankment frame, but this is not true for the Train frame. when A and A' are next to each other B and B' are apart,and B and B' are together A nd A' are apart.

 

[FactChecker]

Now whatever criteria Einstein used that he thought could establish for an observer on the train and one on the embankment that M and M' are at the same place at the same time, the same criteria could be used to establish that A and A' and B and B' are at the same place at the same time. So A', M', and B' align with A and M and B at the same time. (emphasis mine)

This is wrong. "at the same time" is dependent on the reference frame. Observer M will say that the clocks at A' and B' have been set out of synchronization. And M' will say the same thing about A and B. They can not agree on what "the same time" means.

Suppose M and M' both synchronize their respective clocks at the ends by flashing a light and saying that time 0 is when the light reaches the ends. Then M would say that M' has synchronized the A' and B' clocks wrong with clock B' time 0 being too late compared to clock A' time 0 (clock B' ran away from the light source and clock A' ran toward it). Similarly, M' would say that M has synchronized the A and B clocks wrong with clock A time 0 being too late compared to clock B time 0 (clock A ran away from the light source and clock B ran toward it). This is the root cause of the problem with agreeing on simultaneity.

You might object that there are better ways for M and M' to synchronize their respective clocks at the ends of the train. But the experiments measuring the speed of light contradict that. No one was ever able to detect any different speed of light no matter what the velocity of motion was. That means that any physical method of synchronizing clocks kept coming up with the same synchronization that they would have had if they had used light. It shows that synchronizing clocks using light travel time is deeply compatible all physical processes.
And you might say, as your diagram implies, that "at the same time" means when all the points are side-by-side. But don't forget that observer M sees that the length of the train has gotten shorter. So they no longer line up as you show it. Some points are side-by-side at different times from other points -- another example of "simultaneous" not being well-defined.
You can see how closely intertwined all this is. The mathematics and logic of SR all works together and is consistent. When thinking about it, one can not ignore any of its effects on simultaneity, time, or length.

 

[PeroK]

Interesting! Thanks again. All this is very useful to me in my search for understanding and is much appreciated.

If you really want to learn SR, then there is no substitute for a good-quality textbook. My recommendation would be Helliwell:

https://www.goodreads.com/book/show/6453378-special-relativity

You do need some high-school maths, of course, but the main conceptual issues require very little in the way of prerequisites.