Einstein’s Train(and the lightning strikes)

Well what can we say about this; one of the most discussed thought experiments conceived by Einstein; yet one of the most misunderstood pieces in all of Relativity.

When I first read chapter IX of Einstein’s wonderful exposition:

Relativity: The Special and General Theory. 1920.

I found it made good sense being straightforward and logical and well reasoned making it relatively (groan) easy to understand.

Einstein’s train

The great man put much thought and effort into describing and explaining this thought experiment, repeatedly emphasising that he was treating it from the perspective of the observer on the Embankment; which is what we would expect as it is all about Relativity.

The experiment can only be described relative to some particular reference frame; in this case it is that of the Embankment.

Chapter IX.       The Relativity of Simultaneity              

“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. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every Event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises:

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.

 

Einstein's Train Fig 1

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] 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 (note1: As judged from the embankment.) of lightning occur, this point M’ naturally coincides with the point M, but it moves toward the right in the diagram with the velocity v of the train.[Fig. 2.] 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.[Fig. 3.] Now in reality (considered with reference to the railway embankment) [Fig. 2.] he is hastening toward the beam of light coming from B, [2] whilst he is riding on ahead of the beam of light coming from A.[3] Hence the observer will see the beam of light emitted from B earlier [4] than he will see that emitted from A.[5] 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. We thus arrive at the important result:

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an Event.”

The reference numbers inserted into Einstein’s writing match those in the diagram, Fig. 2. This is Einstein’s Fig. 1. expanded to a Space-Time diagram with the Frame of Reference of the Embankment shewn in blue with the train in red.

Einstein's Train Fig 2

Einstein asks the question “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?”

So what exactly is he asking here? It seems to be a straightforward query about how the two lightning strikes are related when considered relative to the train; yet we are dealing with two distinct reference frames here: that of the Embankment and that of the train and the answer will be different for each.

First of all Einstein introduces us to the perspective from the Embankment and only from the Embankment. That this is indeed the case is quite clear when considering his argument point-by-point:

  1. We are dealing, quite explicitly, with lightning strikes that are simultaneous with reference to  (as perceived from/as judged by) the observer on the railway embankment. We know this because it is specifically stated in the description.
  2. We are asked whether they will also be simultaneous  “… relatively to the train.”  It is important to note he says “relatively to the train” and not “relatively from the train”; which tells us it is when considered from the Embankment.
  3. He reminds us: “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, the only Frame of Reference in Einstein’s thought experiment in which the observer at M’ possesses any velocity, is the Frame of Reference of the embankment.
  4. “Now in reality (considered with reference to the railway embankment) he is hastening toward the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A.”   This immediately follows the last quote and quite explicitly confirms from which Frame of Reference that view is taken, for the observer in the train can only be moving toward point B and be “… riding on ahead of the beam of light coming from A.” as viewed from the embankment.
  5. “But the events A and B also correspond to positions A and B on the train.”  The observer on the train will be confident that he is stationary, at the point M’, mid-way between points A and B on the train and that from this perspective it is the embankment that is moving past him.
  6. “Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A.”   Again it can only be from the perspective of the embankment that the observer at M’ is travelling toward lightning flash B. While in the reference frame of that observer, she is stationary between the points A & B on the train.
  7. “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.”   It is only from the perspective of the embankment that this sentence makes any sense.  For it is only the embankment observer who could measure the train to be moving. To those who take the railway train as their reference-body it will be stationary and it will be the embankment that is moving. (Surely that is the whole point about Relativity?)
  8. “Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).”   How many times does the great man have to repeat:  “… are simultaneous with reference to the embankment …” ?    (that is simultaneous as measured from the embankment) and  “… are not simultaneous with respect to the train …”  (not simultaneous in relation to the train as measured from the Embankment).
  9. And finally: “… vice versa (relativity of simultaneity).”  Surely this can only be referring to the first part of the sentence, implying it could equally well be written that  “Events which are simultaneous with reference to the train are not simultaneous with respect to the embankment, …”

What else can vice versa mean but that the whole of what has been written above would apply in reverse if considered from the train observer’s point of view in position M’? – “The Relativity of Simultaneity” – then surely it means: “That simultaneity is only relative to the reference frame in which it is measured”.

In Fig. 3. the same events are drawn using the Train’s Frame of Reference. The Train is of course now stationary while the Embankment is moving with the velocity -v. The reflected lights meet at [1]. M’ meets the light from A [2] first at time tA [4], then the light from B [3] later at time tB [5].

(After some debate I have retained the convention of making the moving frame the primed frame, so M’ is mid-point between A and B on the Embankment as it is the Embankment that is moving in the Train’s Frame of Reference.

Einstein's Train Fig 3

For the observer, sitting in the middle of the train, his frame is at rest and point M, mid-way between A and B, is permanently located in his Frame of Reference; and the embankment would have mid-point M’, moving toward strike A. Leading to the inevitable conclusion that the observer at M in the train would see the strikes simultaneously, while the Embankment observer at M’ would see strike A before strike B.

The constant events that will be the same from whichever Frame they are observed are the two pulses of light meeting at the midpoint of the line AB. The question we must ask is: what are the coordinates of that location in the individual Frames of Reference?

Now within each Frame of Reference we have to consider what is at that mid-point of AB at the time the pulses of light meet.  And that this could be either M or M’ depending on which frame is stationary for the particular observer. M for the embankment and M’ for the train.

Summary

That is the Relativity of Simultaneity; each observer will see it within his own Frame of Reference but deny that it happens in any Frame of Reference that he observes to be moving.

Which is exactly what Einstein was saying when he wrote:

“People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every Event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment [my emphasis].”

 

How much clearer and more explicit can he possibly have stated it? Repeating the principle in his conclusion where he states:

“Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).”

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About TyroJack

For me the World is a logical place, in a Universe that works according to strict, straightforward and uncomplicated Laws. Physical Laws. Many of which we are aware of some of which we understand, some that we have working theories about, some we hypothesise about and some that are as yet pure guesswork. I believe in William of Ockham's principle, known as Occam's Razor: 'The fewer assumptions one makes, the more likely one is to be correct.' I like to apply this in life as well as science. I worked in various roles in IT for most of my working life which could be summed up as programming consultancy. Primarily I fixed problems from the most basic coding, to System design. I programmed systems that were bug-free. I am interested in everything and anything, but most of all in how things work.
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6 Responses to Einstein’s Train(and the lightning strikes)

  1. Bob says:

    You’re close but I don’t think your summary is quite right.

    When M determines that the flashes don’t arrive together at M’, that’s not something that can be contradicted by M’.

    Imagine M’ is holding a bomb that explodes if the flashes reach him or her at the same time from front and back. As M can determine (when the events happen to be simultaneous for M), the flashes don’t reach the bomb (M’) at the same time, and the bomb held by M’ won’t explode. You can’t then say “oh but from M’ perspective the flashes do arrive at the same time and the bomb does explode”. That’s a contradiction; the bomb either explodes or it doesn’t and both observers can say why: e.g. M says M’ was moving toward one flash and away from the other; M’ says the flashes were not simultaneous, both know the M’ bomb didn’t explode.

    You are correct that the roles can be reversed (vice versa). The train observer could consider two events to be simultaneous; but they’d be different events than any that the embankment observer considers simultaneous.

    They can’t both consider the same two events as simultaneous … which your summary seems to imply.

    • TyroJack says:

      Ah, I see where you are coming from and it is something that has beguiled the study of Relativity. The idea that one MUST be able to relate what happens to a single Frame of Reference; which is true, and valid and the only way we can determine what the observer in that frame will perceive and measure.
      BUT one cannot deduce from that one Frame of Reference what an observer in another Frame will perceive or measure!
      Doing so is automatically giving that first Frame, preference. It is making the view from that initial frame ‘reality’.
      We have three Spacetime Events. A and B where the lightning strikes and A and B on the train are colocated with A and B on the track and let us call it point C, dead centre between A and B where M and M’ are colocated.
      In any frame of reference Events are fixed, they cannot move as movement is a change of location over time and an Event only exist for a moment in time.
      Therefore the only location in Spacetime where those rays of light will meet is at point C.

      In any Frame Spacetime is fixed because that is how it is mapped by that frame of Reference; In the Embankment frame M is permanently located at point C and will measure simultaneity when the reflected lights meet there, while M’ is moving away – in the Embankment frame.
      In the train frame, the train is at rest in Spacetime and the train observer will remain at C, in that frame, and see the lights meet while M is moving away.

      It is all about how the events are measured within each frame. No preferences. Each measures Simultaneity in their frame, and that simultaneity can not be measured from the other frame.
      That is why it is the Relativity of Simultaneity.

      • Bob says:

        It is not making a preferred frame, to make measurements from one, about what the other frame will measure. Events occur in all frames! For example, if a camera flash pops on a train, you’ll know that the flash popped whether you were on that train sitting next to it, or on an embankment watching the train roll quickly past.

        It’s perfectly valid for one observer to figure what the other observer might observe. The point in the thought experiment by Einstein is that these observations can’t contradict; the flashes will either reach M’ at the same time or they won’t. M’ is a single location, so events occurring at M’ will be simultaneous or not, according to all observers. (e.g. the bomb held by M’ in my previous reply will either explode or it won’t). Since the flashes are stipulated (pre-defined for the experiment) to be simultaneous for M, it’s clear they can’t be simultaneous for M’. This isn’t an illusion for M, it’s the reality for both M and M’. This isn’t making the frame of M preferred, it just happens to be that A and B were simultaneous for M. Two – other – flashes might happen to be simultaneous for M’, and not for M.

        There’s nothing “special” about events being simultaneous. In the standard scenario, where we stipulate simultaneous flashes for M on the embankment; M’ will see the flashes at different times. But M’ will be able to figure that M would see the flashes at the same time – M’ does not disagree with M! M’ says “the flashes were not simultaneous, but M’ saw them at the same time because they were moving towards the later flash”. That’s not making the M’ frame any more “preferred” than the M frame is “preferred” for saying those particular flashes were simultaneous. All observers can observe events; that the events happen to be simultaneous in one frame is nothing to do with having a “preferred” frame.

        You really are 90% of the way there. It’s true M’ can consider themselves as at rest and M to be moving. And all the rest. But … any two events that M’ considers simultaneous can’t be simultaneous for M (and vice versa). Not just as some kind of illusion, but as the reality of their view on the Universe.

        What you are arguing for is the exact opposite of the Relativity of Simultaneity. You are arguing for the Absoluteness of Simultaneity.

  2. TyroJack says:

    “…the Absoluteness of Simultaneity.” – hmm.
    Consider Einstein’s test for simultaneity: the observer at the centre between two flashes of light that reach him simultaneously. This thought experiment uses exactly that scenario. So M is proving the simultaneity of the lightning strikes, because he is at rest equidistant from them.
    That is a cast iron guarantee; they are simultaneous. Scientifically tested!
    There is nothing relative about that! It is an absolute truth!
    Then why did he label it the Relativity of Simultaneity?

    In my original post I point out nine places in Einstein’s description where he insists that what he is describing is from the viewpoint of the embankment Observer.
    He is stationary and measures simultaneity while the moving observer does not. This has nothing to do with who is on the Embankment but everything to do with who is stationary and who is moving.

    The train observer can only be moving measured from another frame, because measured from his frame the roles are reversed and it is he who is at rest and the embankment that is moving.

    This is the Relativity of Simultaneity, it is relative to one who is at rest (that is from whose frame it is measured) but cannot be for a moving observer. It is this view as Einstein was at pains to explain that demonstrates relativity.

    The accepted interpretation, has nothing relative about it. The embankment observer measures simultaneity. Full stop.

    The biggest problem with the traditional accepted interpretation is in using what the embankment observer sees as the only reality. It is the view from one frame and is relative to that observer. It is not an absolute version of what is happening.

    M and M’ are not points fixed in Spacetime; they are points in frames of reference, which are relative to those individual maps of Spacetime.

    The point where the lights meet is the event. Events are fixed in Spacetime. Their coordinates are particular to the individual frames.
    In the Embankment Frame the train observer moves away from the lights meeting event; while from the Train Frame it is the embankment observer who is moving away – that is relativity – it is the essence of what relativity is about.

    The one thing I am not saying is that simultaneity is absolute; it is relative to the motion of each frame. It can only satisfy Einstein’s test for an observer at rest, for that is part of his test for simultaneity – the observer has to remain at the midpoint between A and B.

    It is like time dilation and length contraction: how fast is the clock really running? it reads different times from within its frame where it is at rest from the time measured in any frame where it is moving, as a function of its speed. How does a clock read two different times at the same moment?
    Yet we know both times are correct and real! It is all relative.

  3. Bob says:

    TyroJack: “Consider Einstein’s test for simultaneity: the observer at the centre between two flashes of light that reach him simultaneously. This thought experiment uses exactly that scenario. So M is proving the simultaneity of the lightning strikes, because he is at rest equidistant from them.
    That is a cast iron guarantee; they are simultaneous. Scientifically tested!
    There is nothing relative about that! It is an absolute truth!”

    Yes, it’s true – for M.

    If the flashes happen to be simultaneous for M, then M (given this setup) will see the flashes simultaneously. And M’ will not (can’t) disagree! As I tried to help you see with the bomb in my first reply … M and M’ cannot disagree about what occurs at M or M’. In this case, M’ is agreeing with M, that the flashes reach M at the same time. (If M has the bomb, it will blow up. M’ will see the explosion!)

    But what you are doing here (with “cast iron” etc) is – assuming – that simultaneity is absolute; that events are either simultaneous or not. You are missing the point – that Einstein showed that simultaneity is in fact relative. It makes a difference who is looking.

    TyroJack: “Then why did he label it the Relativity of Simultaneity?”

    Because M’ will not agree that the events were simultaneous.

    As shown, from the point of view of M, the flashes reach M’ at different times. And as above, M and M’ can’t disagree about that! (e.g. If M’ is holding that bomb, neither M nor M’ will see it explode).

    In short:

    The observers must AGREE with each other about what they know each other saw; what they disagree about is whether that means the events were simultaneous.

    M says: “I am in the middle of the events and saw them at the same time, so they were simultaneous. But I know M’ is moving towards one event and away from the other, so M’ won’t see the events at the same time.”

    M’ says: “I am in the middle of the events and saw them at different times, so they were not simultaneous. But I know M is moving towards the later event and away from the earlier event, so have calculated that M will in fact see the events at the same time.”

    TyroJack: “The one thing I am not saying is that simultaneity is absolute; it is relative to the motion of each frame. It can only satisfy Einstein’s test for an observer at rest, for that is part of his test for simultaneity – the observer has to remain at the midpoint between A and B.”

    No, you ARE saying simultaneity is absolute. You started that reply with the insistence that if the events were simultaneous according to M, then the events ARE simultaneous, “cast iron guarantee”, “absolute truth”, etc. You think if M says the events are simultaneous, that that means every other observer (e.g. M’) must agree that the events are simultaneous. That’s absolute, not relative. If “relative” here simply amounts to an illusion about what each observer thinks the other sees, it could all have been written much more simply.

    Remember; M’ won’t disagree that the flashes reached M at the same time. But likewise, M’ won’t disagree that the flashes reach M’ at different times. This is consistent. You are creating an inconsistency – effectively claiming that M can see the flashes reach M’ at different times, but that M’ will somehow see the events at the same time. (The bomb goes off or it doesn’t!)

    (I’d add that you are taking the test for simultaenity much too literally; but don’t want to distract from the main point right now.)

    • TyroJack says:

      Thank you Bob, you argue your case well, and it is the accepted view of how this works in Relativity; I understand this very well and have been studying it for some years.

      Bob: “The observers must AGREE with each other about what they know each other saw; what they disagree about is whether that means the events were simultaneous.”

      Which implies that we need to examine what simultaneity means.

      Simultaneity requires two events that are equidistant in time from an observer.

      This must depend upon the way that Time is measured for that observer.

      The time axis of a frame of reference as described by Minkowski is drawn vertically and is seen as a fourth Cartesian axis. In such a diagram two events will be simultaneous if they lie on the same time coordinate, parallel to the x and y axes.

      While we know that for a moving frame time is dilated and length contracted and that this will effect such determination of simultaneity.

      There are two ways we measure/consider time:
      As ongoing background time e.g. the time of day or as an interval between two events.
      Background time is measured as proper time; time measured along the worldline of a clock such as the clock at the origin of a reference frame. This is the Cartesian time axis of the time dimension we are familiar with.
      The time interval from an event can be measured by the distance light would have travelled from that event – a sphere of light, if you will. This would best be measured using a Spherical Coordinate System.

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