# Meaning and logic of Einstein's train thought experiment

This seems to be a recurrent topic but I wasn't able to find any satisfactory discussion about this tought experiment.

I will quote the original story from Einstein's and put my doubts inside:

Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. [...]

Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A —> B of the embankment.

1. This definition of simultaneity seems indisputable for the embankment reference but what about the train reference? It seems to me that it would be suitable also for the train only if we assume invariance of the speed of light otherwise we should take into account the relative speed between the sources of light and the train. (It should be noted that the argument will not use invariance of speed of light later on so this seems to be the only possible place for this assumption).

2. So which is the physical theoretic model that we are using here? Classical mechanics? It wouldn't be consistent with the definition of simultaneity (as I said above). Maybe we are considering a hybrid model derived from classical-mechanics where we drop Galilean relativity and introduce the additional axiom of invariance of speed of light? In the latter case how can we be sure that this model does make sense? Isn't this hybrid model inherently inconsistent/incoherent? From inconsistent models we can derive any contradiction.

But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A —> B on the travelling train. Just when the flashes 1 of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.

1. If we are assuming invariance of speed of light then "$M'$ moves towards the right" only within the embankment reference. From the train reference $M'$ would not move ($M$ would in the opposite direction). Why do we choose this particular reference and not any other inertial reference?

If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A.

1. Wouldn't this description violate the assumption of invariance of the speed of light? It would amount to say - as far as I can see - that from the (inertial) reference of the train the light traveled equal distances with different times. Are we assuming invariance or not?

Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A.

1. It seems to me that from the reference of the train the situation is completely symmetric (unless we are assuming there is a special reference like aether, are we?): they are two inertial references with a relative speed and the light sources have $0$ initial velocity for both (i.e. it moves at $v=c$ in both references), so both observer should have the same experience. Putting it in another way: we could just think that we have a stationary train and a moving "rest of the world" and then we could riproduce the same reasoning we did before arriving to the conclusion that the events are simultaneous for the observer on the "static" train and not simultaneous for the observer on the moving embankment. What is there that breaks the symmetry?

Some remarks after understanding better

1. There is not a "model" for the dynamics but there are some principles or axioms that we can imagine will be part of the RR model and which have a small intersection with classical mechanics so it is reasonable to expect they will not be inconsistent/contradictory

2. For some reason I have been thinking that the effective actual simultaneity of the bolts was something that was stated or implied in advance with the description of the setting, but Einstein ask us to stop thinking in terms of absolute time references. This is why I have been thinking that there is a symmetry in the descriprion of the events from the two references.

3. I think there is something slightly misleading in the final part of the argument when Einsten says:

Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A.

In fact the only concept we are considering here is a purely techical definition of "simultaneity" and the observer just notice that this definition doesn't apply in his reference, but this is not an actual simultaneity (or actual non-simultaneity) in the sense we are familiar with (with a reference to absolute time): it is just a completely new definition and we just discovered that it defines a property that depends on the reference.

• The point of the experiment is that absolute time doesn't make sense, and that simultaneity in one frame of reference doesn't mean the same thing as simultaneity in another frame. – Jahan Claes Apr 15 '16 at 18:10
• Physics is not mathematics. The invariance of the speed of light has been experimentally confirmed with extremely high accuracy. You need to stop worrying about conceptual experiments that are essentially just teaching tools (that is really all this one is) and look at real world physics. – CuriousOne Apr 15 '16 at 18:28
• I'm not disputing special relativity, I just can't understand this specific thought experiment, it seems to me that it doesn't have a consistent logical foundation – Marco Disce Apr 15 '16 at 18:33
• Your assumption should be that it has a consistent logical foundation and to understand it. I think it will be difficult to try to explain it at a better level by written word here. – Peter R Apr 15 '16 at 19:40
• If you are not disputing special relativity, then would you accept an explanation of what is happening in this scenario in terms of the Lorentz transformation? – Chet Miller Apr 15 '16 at 23:42

Questions 1-2: The discussion leading to the paragraph in your link starts in Sec.VII of that ref. (see first 3 paragraphs therein), with the assumption of the postulate of the speed of light. Hence Einstein is endeavoring to show that accepting the light postulate necessarily implies relativity of simultaneity. He is discussing the propagation of the lightning bolts under the assumption that in both the embankment frame and in the train frame the bolts must be observed to propagate at the speed of light, regardless of the relative velocities of the sources that emitted them wrt the respective observers.

Question 3: The purpose is to show that 2 events that appear simultaneous in one inertial frame do not appear simultaneous in another inertial frame. In the present case frame 1 is that of the embankment, and frame 2 is that of the train. The train is introduced precisely to serve as frame 2, but you can substitute any other in its place.

Question 4: No, the assumption does not violate the invariance of the speed of light - and this is the entire point. In the embankment frame the bolts are seen as emitted simultaneously at equal distances away and moving at the speed of light in opposite directions, while the train observer is moving at lower velocity toward one bolt and away from the other. The embankment observer sees the bolts reaching him simultaneously after traveling equal times, but he also observes them reaching the train observer separately, at different moments. From the point of view of the train observer, the bolts are also emitted equal distances away, but at different times, so for him the bolts travel equal distances, but arrive at his location at different moments. At the same time, this train observer sees the embankment observer moving in the opposite direction toward one bolt and away from the other, in such a way that the two bolts reach him (the embankment observer) simultaneously.

Question 5: Technically the light sources cannot be at rest wrt both frames. But given the postulate of the speed of light it really does not matter in what frame they are actually at rest, and this is what the thought experiment emphasizes. What is important is that although the emission events take place symmetrically in space for both observers, they occur simultaneously for one observer and not simultaneously for the other. This is what breaks the symmetry. The symmetrical situation would have instead the bolts striking simultaneously for the train observer, but not simultaneously for the embankment observer. And you are right about the symmetrical conclusion: the latter situation would serve to show that events simultaneous in the train frame are not simultaneous in the embankment frame.

I think that, rather than trying to pick apart the logic of the thought experiment which may or may not be well-worded (yes, I realise I just criticised Einstein), it is easier to just work one out of your own and see what is going on.

## Experimental setup

To that end, consider two frames of reference moving relative to one another: one is going to be his embankment frame, and one will be the train frame. They are both essentially one-dimensional (in space, there is time too). The train frame is moving to the right with respect to the embankment frame. In the embankment frame there are two devices which can be made either to emit clicks (sound) or to emit flashes of light.

So, OK, there are three measurements you are allowed to do (in either frame):

• you can find the position where the flashes of light or clicks of sound from each of the devices reach you simultaneously, and mark that.
• you can mark the position in the train frame where the devices were when they flashed/clicked: perhaps you do this by having them fire a bullet at the train which leaves a mark, which is very close to them so you don't have to worry about the time of travel of the bullet;
• you can measure the speed of light or the speed of sound from each device (ie you can measure the left-going and the right-going light/sound seperately).

## Sound

So then you run the train past and the clickers go off, and the above three measurements are done (well: two of them in the embankment frame), and you find the following things:

• in the embankment frame, the position where the clicks are heard simultaneously is just half-way between the devices, and the speed of sound in each direction is the same;
• in the train frame, the position where the clicks are heard simultaneously is not half-way between the marks left by the devices but is shifted towards the left of the train, but the speeds of sound in each direction are also not the same -- the left-going sound is moving faster than the right-going sound.

So you do the maths and both people conclude the same thing: the clicks were made simultaneously, because the difference in the speeds of sound as measured in the train frame are just sufficient to account for the difference in the place where they are heard simultaneously. All is well with the world.

## Light

Now you do the same experiment with flashes of light.

• In the embankment frame all is well: the flashes are seen simultaneously half-way between the two devices, and the speed of light is measured as the same in each direction. Once again you conclude that the flashes were simultaneous.
• In the train frame all is not well: the flashes of light are seen simultaneously at a position shifted left, as before, but the speed of light is measured as the same for each direction.

So the only conclusion from this is that the flashes, in the train frame, were not simultaneous: the left-hand flash must have happened after the right-hand flash. The critical thing that makes this be true is that the speed of light is always the same, in either direction, in either frame (this doesn't show that both frames measure the same speed of light as each other, just that they always measure it the same in either direction).

In other words observers in the two frames can disagree about whether events which do not occur at the same place are simultaneous.

## A variation

What I have described is not quite the same as the Einstein experiment, because the two frames measure the position-of-simultaneity independently from each other. A variation is to arrange life so that each frame makes a mark in the other frame (using the bullet trick again) at the point of simultaneity. And it is reasonably obvious both that these marks always agree with where the person in the other frame thought was the point of simultaneity, and that the point is shifted left from the centre of the train in both cases.

• I can understand that the situation is asymmetric between train and embankment when we consider sound waves because air is at rest only in one of the two references but what would break the symmetry in the case of light? What is inherently different between train and embankment references? – Marco Disce Apr 16 '16 at 22:48
• @MarcoDisce Nothing is, and that's the point. For sound you can always pick a special privileged frame based on measuring the speed of sound, while for light there is no such frame. So for light something else has to give, and that is that there is no well-defined notion of simultaneity. – tfb Apr 17 '16 at 16:13